YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Left Termination of the query pattern f(g,g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToDTProblemTransformerProof [SOUND, 0 ms] (2) TRIPLES (3) TriplesToPiDPProof [SOUND, 0 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: f(RES, [], RES). f([], .(Head, Tail), RES) :- f(.(Head, Tail), Tail, RES). f(.(Head, Tail), Y, RES) :- f(Y, Tail, RES). Query: f(g,g,a) ---------------------------------------- (1) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 9, "program": { "directives": [], "clauses": [ [ "(f RES ([]) RES)", null ], [ "(f ([]) (. Head Tail) RES)", "(f (. Head Tail) Tail RES)" ], [ "(f (. Head Tail) Y RES)", "(f Y Tail RES)" ] ] }, "graph": { "nodes": { "66": { "goal": [ { "clause": 1, "scope": 1, "term": "(f T5 ([]) T3)" }, { "clause": 2, "scope": 1, "term": "(f T5 ([]) T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T5"], "free": [], "exprvars": [] } }, "67": { "goal": [{ "clause": 2, "scope": 1, "term": "(f T5 ([]) T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T5"], "free": [], "exprvars": [] } }, "170": { "goal": [{ "clause": 0, "scope": 4, "term": "(f T80 T79 T82)" }], "kb": { "nonunifying": [[ "(f (. T78 T79) T80 T3)", "(f X2 ([]) X2)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T78", "T79", "T80" ], "free": ["X2"], "exprvars": [] } }, "171": { "goal": [ { "clause": 1, "scope": 4, "term": "(f T80 T79 T82)" }, { "clause": 2, "scope": 4, "term": "(f T80 T79 T82)" } ], "kb": { "nonunifying": [[ "(f (. T78 T79) T80 T3)", "(f X2 ([]) X2)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T78", "T79", "T80" ], "free": ["X2"], "exprvars": [] } }, "type": "Nodes", "172": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "173": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "174": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "175": { "goal": [{ "clause": 1, "scope": 4, "term": "(f T80 T79 T82)" }], "kb": { "nonunifying": [[ "(f (. T78 T79) T80 T3)", "(f X2 ([]) X2)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T78", "T79", "T80" ], "free": ["X2"], "exprvars": [] } }, "110": { "goal": [ { "clause": 2, "scope": 3, "term": "(f (. T36 T37) T37 T39)" }, { "clause": -1, "scope": 3, "term": null }, { "clause": 2, "scope": 1, "term": "(f ([]) (. T36 T37) T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T36", "T37" ], "free": [], "exprvars": [] } }, "176": { "goal": [{ "clause": 2, "scope": 4, "term": "(f T80 T79 T82)" }], "kb": { "nonunifying": [[ "(f (. T78 T79) T80 T3)", "(f X2 ([]) X2)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T78", "T79", "T80" ], "free": ["X2"], "exprvars": [] } }, "111": { "goal": [{ "clause": 2, "scope": 3, "term": "(f (. T36 T37) T37 T39)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T36", "T37" ], "free": [], "exprvars": [] } }, "177": { "goal": [{ "clause": -1, "scope": -1, "term": "(f (. T100 T101) T101 T103)" }], "kb": { "nonunifying": [[ "(f (. T78 (. T100 T101)) ([]) T3)", "(f X2 ([]) X2)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T78", "T100", "T101" ], "free": ["X2"], "exprvars": [] } }, "112": { "goal": [ { "clause": -1, "scope": 3, "term": null }, { "clause": 2, "scope": 1, "term": "(f ([]) (. T36 T37) T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T36", "T37" ], "free": [], "exprvars": [] } }, "178": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "113": { "goal": [{ "clause": -1, "scope": -1, "term": "(f T69 T69 T71)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T69"], "free": [], "exprvars": [] } }, "157": { "goal": [{ "clause": 2, "scope": 1, "term": "(f ([]) (. T36 T37) T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T36", "T37" ], "free": [], "exprvars": [] } }, "179": { "goal": [{ "clause": -1, "scope": -1, "term": "(f T118 T117 T120)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T117", "T118" ], "free": [], "exprvars": [] } }, "158": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "159": { "goal": [{ "clause": -1, "scope": -1, "term": "(f T80 T79 T82)" }], "kb": { "nonunifying": [[ "(f (. T78 T79) T80 T3)", "(f X2 ([]) X2)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T78", "T79", "T80" ], "free": ["X2"], "exprvars": [] } }, "90": { "goal": [{ "clause": -1, "scope": -1, "term": "(f (. T25 T26) T26 T28)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T25", "T26" ], "free": [], "exprvars": [] } }, "91": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "53": { "goal": [ { "clause": 0, "scope": 1, "term": "(f T1 T2 T3)" }, { "clause": 1, "scope": 1, "term": "(f T1 T2 T3)" }, { "clause": 2, "scope": 1, "term": "(f T1 T2 T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T2" ], "free": [], "exprvars": [] } }, "77": { "goal": [{ "clause": -1, "scope": -1, "term": "(f ([]) T10 T12)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": [], "exprvars": [] } }, "99": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "78": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "79": { "goal": [ { "clause": 0, "scope": 2, "term": "(f ([]) T10 T12)" }, { "clause": 1, "scope": 2, "term": "(f ([]) T10 T12)" }, { "clause": 2, "scope": 2, "term": "(f ([]) T10 T12)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": [], "exprvars": [] } }, "180": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "100": { "goal": [ { "clause": -1, "scope": -1, "term": "(f (. T36 T37) T37 T39)" }, { "clause": 2, "scope": 1, "term": "(f ([]) (. T36 T37) T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T36", "T37" ], "free": [], "exprvars": [] } }, "168": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "103": { "goal": [{ "clause": 2, "scope": 1, "term": "(f T1 T2 T3)" }], "kb": { "nonunifying": [ [ "(f T1 T2 T3)", "(f X2 ([]) X2)" ], [ "(f T1 T2 T3)", "(f ([]) (. X43 X44) X45)" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T2" ], "free": [ "X2", "X43", "X44", "X45" ], "exprvars": [] } }, "169": { "goal": [ { "clause": 0, "scope": 4, "term": "(f T80 T79 T82)" }, { "clause": 1, "scope": 4, "term": "(f T80 T79 T82)" }, { "clause": 2, "scope": 4, "term": "(f T80 T79 T82)" } ], "kb": { "nonunifying": [[ "(f (. T78 T79) T80 T3)", "(f X2 ([]) X2)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T78", "T79", "T80" ], "free": ["X2"], "exprvars": [] } }, "104": { "goal": [ { "clause": 0, "scope": 3, "term": "(f (. T36 T37) T37 T39)" }, { "clause": 1, "scope": 3, "term": "(f (. T36 T37) T37 T39)" }, { "clause": 2, "scope": 3, "term": "(f (. T36 T37) T37 T39)" }, { "clause": -1, "scope": 3, "term": null }, { "clause": 2, "scope": 1, "term": "(f ([]) (. T36 T37) T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T36", "T37" ], "free": [], "exprvars": [] } }, "105": { "goal": [{ "clause": 0, "scope": 3, "term": "(f (. T36 T37) T37 T39)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T36", "T37" ], "free": [], "exprvars": [] } }, "106": { "goal": [ { "clause": 1, "scope": 3, "term": "(f (. T36 T37) T37 T39)" }, { "clause": 2, "scope": 3, "term": "(f (. T36 T37) T37 T39)" }, { "clause": -1, "scope": 3, "term": null }, { "clause": 2, "scope": 1, "term": "(f ([]) (. T36 T37) T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T36", "T37" ], "free": [], "exprvars": [] } }, "80": { "goal": [{ "clause": 0, "scope": 2, "term": "(f ([]) T10 T12)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": [], "exprvars": [] } }, "107": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "9": { "goal": [{ "clause": -1, "scope": -1, "term": "(f T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T2" ], "free": [], "exprvars": [] } }, "81": { "goal": [ { "clause": 1, "scope": 2, "term": "(f ([]) T10 T12)" }, { "clause": 2, "scope": 2, "term": "(f ([]) T10 T12)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": [], "exprvars": [] } }, "108": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "82": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "109": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "83": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "84": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "85": { "goal": [{ "clause": 1, "scope": 2, "term": "(f ([]) T10 T12)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": [], "exprvars": [] } }, "64": { "goal": [ { "clause": -1, "scope": -1, "term": "(true)" }, { "clause": 1, "scope": 1, "term": "(f T5 ([]) T3)" }, { "clause": 2, "scope": 1, "term": "(f T5 ([]) T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T5"], "free": [], "exprvars": [] } }, "86": { "goal": [{ "clause": 2, "scope": 2, "term": "(f ([]) T10 T12)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": [], "exprvars": [] } }, "65": { "goal": [ { "clause": 1, "scope": 1, "term": "(f T1 T2 T3)" }, { "clause": 2, "scope": 1, "term": "(f T1 T2 T3)" } ], "kb": { "nonunifying": [[ "(f T1 T2 T3)", "(f X2 ([]) X2)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T2" ], "free": ["X2"], "exprvars": [] } } }, "edges": [ { "from": 9, "to": 53, "label": "CASE" }, { "from": 53, "to": 64, "label": "EVAL with clause\nf(X2, [], X2).\nand substitutionT1 -> T5,\nX2 -> T5,\nT2 -> [],\nT3 -> T5" }, { "from": 53, "to": 65, "label": "EVAL-BACKTRACK" }, { "from": 64, "to": 66, "label": "SUCCESS" }, { "from": 65, "to": 100, "label": "EVAL with clause\nf([], .(X43, X44), X45) :- f(.(X43, X44), X44, X45).\nand substitutionT1 -> [],\nX43 -> T36,\nX44 -> T37,\nT2 -> .(T36, T37),\nT3 -> T39,\nX45 -> T39,\nT38 -> T39" }, { "from": 65, "to": 103, "label": "EVAL-BACKTRACK" }, { "from": 66, "to": 67, "label": "BACKTRACK\nfor clause: f([], .(Head, Tail), RES) :- f(.(Head, Tail), Tail, RES)because of non-unification" }, { "from": 67, "to": 77, "label": "EVAL with clause\nf(.(X10, X11), X12, X13) :- f(X12, X11, X13).\nand substitutionX10 -> T9,\nX11 -> T10,\nT5 -> .(T9, T10),\nX12 -> [],\nT3 -> T12,\nX13 -> T12,\nT11 -> T12" }, { "from": 67, "to": 78, "label": "EVAL-BACKTRACK" }, { "from": 77, "to": 79, "label": "CASE" }, { "from": 79, "to": 80, "label": "PARALLEL" }, { "from": 79, "to": 81, "label": "PARALLEL" }, { "from": 80, "to": 82, "label": "EVAL with clause\nf(X18, [], X18).\nand substitutionX18 -> [],\nT10 -> [],\nT12 -> []" }, { "from": 80, "to": 83, "label": "EVAL-BACKTRACK" }, { "from": 81, "to": 85, "label": "PARALLEL" }, { "from": 81, "to": 86, "label": "PARALLEL" }, { "from": 82, "to": 84, "label": "SUCCESS" }, { "from": 85, "to": 90, "label": "EVAL with clause\nf([], .(X31, X32), X33) :- f(.(X31, X32), X32, X33).\nand substitutionX31 -> T25,\nX32 -> T26,\nT10 -> .(T25, T26),\nT12 -> T28,\nX33 -> T28,\nT27 -> T28" }, { "from": 85, "to": 91, "label": "EVAL-BACKTRACK" }, { "from": 86, "to": 99, "label": "BACKTRACK\nfor clause: f(.(Head, Tail), Y, RES) :- f(Y, Tail, RES)because of non-unification" }, { "from": 90, "to": 9, "label": "INSTANCE with matching:\nT1 -> .(T25, T26)\nT2 -> T26\nT3 -> T28" }, { "from": 100, "to": 104, "label": "CASE" }, { "from": 103, "to": 159, "label": "EVAL with clause\nf(.(X88, X89), X90, X91) :- f(X90, X89, X91).\nand substitutionX88 -> T78,\nX89 -> T79,\nT1 -> .(T78, T79),\nT2 -> T80,\nX90 -> T80,\nT3 -> T82,\nX91 -> T82,\nT81 -> T82" }, { "from": 103, "to": 168, "label": "EVAL-BACKTRACK" }, { "from": 104, "to": 105, "label": "PARALLEL" }, { "from": 104, "to": 106, "label": "PARALLEL" }, { "from": 105, "to": 107, "label": "EVAL with clause\nf(X50, [], X50).\nand substitutionT36 -> T48,\nT37 -> [],\nX50 -> .(T48, []),\nT49 -> [],\nT39 -> .(T48, [])" }, { "from": 105, "to": 108, "label": "EVAL-BACKTRACK" }, { "from": 106, "to": 110, "label": "BACKTRACK\nfor clause: f([], .(Head, Tail), RES) :- f(.(Head, Tail), Tail, RES)because of non-unification" }, { "from": 107, "to": 109, "label": "SUCCESS" }, { "from": 110, "to": 111, "label": "PARALLEL" }, { "from": 110, "to": 112, "label": "PARALLEL" }, { "from": 111, "to": 113, "label": "ONLY EVAL with clause\nf(.(X74, X75), X76, X77) :- f(X76, X75, X77).\nand substitutionT36 -> T68,\nX74 -> T68,\nT37 -> T69,\nX75 -> T69,\nX76 -> T69,\nT39 -> T71,\nX77 -> T71,\nT70 -> T71" }, { "from": 112, "to": 157, "label": "FAILURE" }, { "from": 113, "to": 9, "label": "INSTANCE with matching:\nT1 -> T69\nT2 -> T69\nT3 -> T71" }, { "from": 157, "to": 158, "label": "BACKTRACK\nfor clause: f(.(Head, Tail), Y, RES) :- f(Y, Tail, RES)because of non-unification" }, { "from": 159, "to": 169, "label": "CASE" }, { "from": 169, "to": 170, "label": "PARALLEL" }, { "from": 169, "to": 171, "label": "PARALLEL" }, { "from": 170, "to": 172, "label": "EVAL with clause\nf(X96, [], X96).\nand substitutionT80 -> T87,\nX96 -> T87,\nT79 -> [],\nT82 -> T87" }, { "from": 170, "to": 173, "label": "EVAL-BACKTRACK" }, { "from": 171, "to": 175, "label": "PARALLEL" }, { "from": 171, "to": 176, "label": "PARALLEL" }, { "from": 172, "to": 174, "label": "SUCCESS" }, { "from": 175, "to": 177, "label": "EVAL with clause\nf([], .(X109, X110), X111) :- f(.(X109, X110), X110, X111).\nand substitutionT80 -> [],\nX109 -> T100,\nX110 -> T101,\nT79 -> .(T100, T101),\nT82 -> T103,\nX111 -> T103,\nT102 -> T103" }, { "from": 175, "to": 178, "label": "EVAL-BACKTRACK" }, { "from": 176, "to": 179, "label": "EVAL with clause\nf(.(X122, X123), X124, X125) :- f(X124, X123, X125).\nand substitutionX122 -> T116,\nX123 -> T117,\nT80 -> .(T116, T117),\nT79 -> T118,\nX124 -> T118,\nT82 -> T120,\nX125 -> T120,\nT119 -> T120" }, { "from": 176, "to": 180, "label": "EVAL-BACKTRACK" }, { "from": 177, "to": 9, "label": "INSTANCE with matching:\nT1 -> .(T100, T101)\nT2 -> T101\nT3 -> T103" }, { "from": 179, "to": 9, "label": "INSTANCE with matching:\nT1 -> T118\nT2 -> T117\nT3 -> T120" } ], "type": "Graph" } } ---------------------------------------- (2) Obligation: Triples: fA(.(X1, .(X2, X3)), [], X4) :- fA(.(X2, X3), X3, X4). fA([], .(X1, X2), X3) :- fA(X2, X2, X3). fA(.(X1, .(X2, X3)), [], X4) :- fA(.(X2, X3), X3, X4). fA(.(X1, X2), .(X3, X4), X5) :- fA(X2, X4, X5). Clauses: fcA(X1, [], X1). fcA(.(X1, []), [], []). fcA(.(X1, .(X2, X3)), [], X4) :- fcA(.(X2, X3), X3, X4). fcA([], .(X1, []), .(X1, [])). fcA([], .(X1, X2), X3) :- fcA(X2, X2, X3). fcA(.(X1, []), X2, X2). fcA(.(X1, .(X2, X3)), [], X4) :- fcA(.(X2, X3), X3, X4). fcA(.(X1, X2), .(X3, X4), X5) :- fcA(X2, X4, X5). Afs: fA(x1, x2, x3) = fA(x1, x2) ---------------------------------------- (3) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: fA_in_3: (b,b,f) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: FA_IN_GGA(.(X1, .(X2, X3)), [], X4) -> U1_GGA(X1, X2, X3, X4, fA_in_gga(.(X2, X3), X3, X4)) FA_IN_GGA(.(X1, .(X2, X3)), [], X4) -> FA_IN_GGA(.(X2, X3), X3, X4) FA_IN_GGA([], .(X1, X2), X3) -> U2_GGA(X1, X2, X3, fA_in_gga(X2, X2, X3)) FA_IN_GGA([], .(X1, X2), X3) -> FA_IN_GGA(X2, X2, X3) FA_IN_GGA(.(X1, X2), .(X3, X4), X5) -> U3_GGA(X1, X2, X3, X4, X5, fA_in_gga(X2, X4, X5)) FA_IN_GGA(.(X1, X2), .(X3, X4), X5) -> FA_IN_GGA(X2, X4, X5) R is empty. The argument filtering Pi contains the following mapping: fA_in_gga(x1, x2, x3) = fA_in_gga(x1, x2) .(x1, x2) = .(x1, x2) [] = [] FA_IN_GGA(x1, x2, x3) = FA_IN_GGA(x1, x2) U1_GGA(x1, x2, x3, x4, x5) = U1_GGA(x1, x2, x3, x5) U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x2, x4) U3_GGA(x1, x2, x3, x4, x5, x6) = U3_GGA(x1, x2, x3, x4, x6) 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: FA_IN_GGA(.(X1, .(X2, X3)), [], X4) -> U1_GGA(X1, X2, X3, X4, fA_in_gga(.(X2, X3), X3, X4)) FA_IN_GGA(.(X1, .(X2, X3)), [], X4) -> FA_IN_GGA(.(X2, X3), X3, X4) FA_IN_GGA([], .(X1, X2), X3) -> U2_GGA(X1, X2, X3, fA_in_gga(X2, X2, X3)) FA_IN_GGA([], .(X1, X2), X3) -> FA_IN_GGA(X2, X2, X3) FA_IN_GGA(.(X1, X2), .(X3, X4), X5) -> U3_GGA(X1, X2, X3, X4, X5, fA_in_gga(X2, X4, X5)) FA_IN_GGA(.(X1, X2), .(X3, X4), X5) -> FA_IN_GGA(X2, X4, X5) R is empty. The argument filtering Pi contains the following mapping: fA_in_gga(x1, x2, x3) = fA_in_gga(x1, x2) .(x1, x2) = .(x1, x2) [] = [] FA_IN_GGA(x1, x2, x3) = FA_IN_GGA(x1, x2) U1_GGA(x1, x2, x3, x4, x5) = U1_GGA(x1, x2, x3, x5) U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x2, x4) U3_GGA(x1, x2, x3, x4, x5, x6) = U3_GGA(x1, x2, x3, x4, x6) 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: FA_IN_GGA(.(X1, X2), .(X3, X4), X5) -> FA_IN_GGA(X2, X4, X5) FA_IN_GGA(.(X1, .(X2, X3)), [], X4) -> FA_IN_GGA(.(X2, X3), X3, X4) FA_IN_GGA([], .(X1, X2), X3) -> FA_IN_GGA(X2, X2, X3) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) [] = [] FA_IN_GGA(x1, x2, x3) = FA_IN_GGA(x1, x2) 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: FA_IN_GGA(.(X1, X2), .(X3, X4)) -> FA_IN_GGA(X2, X4) FA_IN_GGA(.(X1, .(X2, X3)), []) -> FA_IN_GGA(.(X2, X3), X3) FA_IN_GGA([], .(X1, X2)) -> FA_IN_GGA(X2, X2) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (9) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *FA_IN_GGA(.(X1, X2), .(X3, X4)) -> FA_IN_GGA(X2, X4) The graph contains the following edges 1 > 1, 2 > 2 *FA_IN_GGA(.(X1, .(X2, X3)), []) -> FA_IN_GGA(.(X2, X3), X3) The graph contains the following edges 1 > 1, 1 > 2 *FA_IN_GGA([], .(X1, X2)) -> FA_IN_GGA(X2, X2) The graph contains the following edges 2 > 1, 2 > 2 ---------------------------------------- (10) YES