/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.pl /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- MAYBE proof of /export/starexec/sandbox/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern p(a) w.r.t. the given Prolog program could not be shown: (0) Prolog (1) PrologToTRSTransformerProof [SOUND, 0 ms] (2) QTRS (3) QTRSRRRProof [EQUIVALENT, 61 ms] (4) QTRS (5) QTRSRRRProof [EQUIVALENT, 12 ms] (6) QTRS (7) QTRSRRRProof [EQUIVALENT, 2 ms] (8) QTRS (9) QTRSRRRProof [EQUIVALENT, 0 ms] (10) QTRS (11) PrologToDTProblemTransformerProof [SOUND, 0 ms] (12) TRIPLES (13) TriplesToPiDPProof [SOUND, 0 ms] (14) PiDP (15) DependencyGraphProof [EQUIVALENT, 0 ms] (16) PiDP (17) PiDPToQDPProof [SOUND, 0 ms] (18) QDP (19) PrologToIRSwTTransformerProof [SOUND, 0 ms] (20) IRSwT (21) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (22) IRSwT (23) IntTRSCompressionProof [EQUIVALENT, 0 ms] (24) IRSwT (25) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (26) IRSwT (27) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms] (28) IRSwT (29) FilterProof [EQUIVALENT, 0 ms] (30) IntTRS (31) IntTRSNonPeriodicNontermProof [COMPLETE, 7 ms] (32) NO (33) PrologToPiTRSProof [SOUND, 0 ms] (34) PiTRS (35) DependencyPairsProof [EQUIVALENT, 0 ms] (36) PiDP (37) DependencyGraphProof [EQUIVALENT, 0 ms] (38) PiDP (39) UsableRulesProof [EQUIVALENT, 0 ms] (40) PiDP (41) PiDPToQDPProof [SOUND, 0 ms] (42) QDP (43) PrologToPiTRSProof [SOUND, 0 ms] (44) PiTRS (45) DependencyPairsProof [EQUIVALENT, 0 ms] (46) PiDP (47) DependencyGraphProof [EQUIVALENT, 0 ms] (48) PiDP (49) UsableRulesProof [EQUIVALENT, 0 ms] (50) PiDP (51) PiDPToQDPProof [SOUND, 0 ms] (52) QDP ---------------------------------------- (0) Obligation: Clauses: p(X) :- ','(l(X), q(X)). q(.(A, [])). r(1). l([]). l(.(H, T)) :- ','(r(H), l(T)). Query: p(a) ---------------------------------------- (1) PrologToTRSTransformerProof (SOUND) Transformed Prolog program to TRS. { "root": 3, "program": { "directives": [], "clauses": [ [ "(p X)", "(',' (l X) (q X))" ], [ "(q (. A ([])))", null ], [ "(r (1))", null ], [ "(l ([]))", null ], [ "(l (. H T))", "(',' (r H) (l T))" ] ] }, "graph": { "nodes": { "56": { "goal": [{ "clause": 1, "scope": 4, "term": "(q T7)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T7"], "free": [], "exprvars": [] } }, "24": { "goal": [ { "clause": 3, "scope": 2, "term": "(l T6)" }, { "clause": 4, "scope": 2, "term": "(l T6)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "46": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (r T14) (l T15))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "25": { "goal": [{ "clause": 3, "scope": 2, "term": "(l T6)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "47": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "58": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "26": { "goal": [{ "clause": 4, "scope": 2, "term": "(l T6)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "48": { "goal": [{ "clause": 2, "scope": 3, "term": "(',' (r T14) (l T15))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "27": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "49": { "goal": [{ "clause": -1, "scope": -1, "term": "(l T16)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "29": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "19": { "goal": [{ "clause": -1, "scope": -1, "term": "(l T6)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "3": { "goal": [{ "clause": -1, "scope": -1, "term": "(p T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "4": { "goal": [{ "clause": 0, "scope": 1, "term": "(p T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "60": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "50": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "62": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "31": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "10": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (l T6) (q T6))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "21": { "goal": [{ "clause": -1, "scope": -1, "term": "(q T7)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T7"], "free": [], "exprvars": [] } } }, "edges": [ { "from": 3, "to": 4, "label": "CASE" }, { "from": 4, "to": 10, "label": "ONLY EVAL with clause\np(X3) :- ','(l(X3), q(X3)).\nand substitutionT1 -> T6,\nX3 -> T6,\nT5 -> T6" }, { "from": 10, "to": 19, "label": "SPLIT 1" }, { "from": 10, "to": 21, "label": "SPLIT 2\nnew knowledge:\nT7 is ground\nreplacements:T6 -> T7" }, { "from": 19, "to": 24, "label": "CASE" }, { "from": 21, "to": 56, "label": "CASE" }, { "from": 24, "to": 25, "label": "PARALLEL" }, { "from": 24, "to": 26, "label": "PARALLEL" }, { "from": 25, "to": 27, "label": "EVAL with clause\nl([]).\nand substitutionT6 -> []" }, { "from": 25, "to": 29, "label": "EVAL-BACKTRACK" }, { "from": 26, "to": 46, "label": "EVAL with clause\nl(.(X8, X9)) :- ','(r(X8), l(X9)).\nand substitutionX8 -> T14,\nX9 -> T15,\nT6 -> .(T14, T15),\nT12 -> T14,\nT13 -> T15" }, { "from": 26, "to": 47, "label": "EVAL-BACKTRACK" }, { "from": 27, "to": 31, "label": "SUCCESS" }, { "from": 46, "to": 48, "label": "CASE" }, { "from": 48, "to": 49, "label": "EVAL with clause\nr(1).\nand substitutionT14 -> 1,\nT15 -> T16" }, { "from": 48, "to": 50, "label": "EVAL-BACKTRACK" }, { "from": 49, "to": 19, "label": "INSTANCE with matching:\nT6 -> T16" }, { "from": 56, "to": 58, "label": "EVAL with clause\nq(.(X12, [])).\nand substitutionX12 -> T19,\nT7 -> .(T19, [])" }, { "from": 56, "to": 60, "label": "EVAL-BACKTRACK" }, { "from": 58, "to": 62, "label": "SUCCESS" } ], "type": "Graph" } } ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f3_in -> U1(f10_in) U1(f10_out1(T6)) -> f3_out1(T6) f19_in -> f19_out1([]) f19_in -> U2(f19_in) U2(f19_out1(T16)) -> f19_out1(.(1, T16)) f21_in(.(T19, [])) -> f21_out1 f10_in -> U3(f19_in) U3(f19_out1(T7)) -> U4(f21_in(T7), T7) U4(f21_out1, T7) -> f10_out1(T7) Q is empty. ---------------------------------------- (3) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(.(x_1, x_2)) = x_1 + 2*x_2 POL(1) = 0 POL(U1(x_1)) = 2*x_1 POL(U2(x_1)) = 2*x_1 POL(U3(x_1)) = 2*x_1 POL(U4(x_1, x_2)) = x_1 + x_2 POL([]) = 0 POL(f10_in) = 0 POL(f10_out1(x_1)) = x_1 POL(f19_in) = 0 POL(f19_out1(x_1)) = x_1 POL(f21_in(x_1)) = x_1 POL(f21_out1) = 0 POL(f3_in) = 1 POL(f3_out1(x_1)) = 2*x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: f3_in -> U1(f10_in) ---------------------------------------- (4) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: U1(f10_out1(T6)) -> f3_out1(T6) f19_in -> f19_out1([]) f19_in -> U2(f19_in) U2(f19_out1(T16)) -> f19_out1(.(1, T16)) f21_in(.(T19, [])) -> f21_out1 f10_in -> U3(f19_in) U3(f19_out1(T7)) -> U4(f21_in(T7), T7) U4(f21_out1, T7) -> f10_out1(T7) Q is empty. ---------------------------------------- (5) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(.(x_1, x_2)) = 2*x_1 + x_2 POL(1) = 0 POL(U1(x_1)) = 2*x_1 POL(U2(x_1)) = x_1 POL(U3(x_1)) = 2*x_1 POL(U4(x_1, x_2)) = x_1 + 2*x_2 POL([]) = 0 POL(f10_in) = 2 POL(f10_out1(x_1)) = x_1 POL(f19_in) = 0 POL(f19_out1(x_1)) = 2*x_1 POL(f21_in(x_1)) = x_1 POL(f21_out1) = 0 POL(f3_out1(x_1)) = 2*x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: f10_in -> U3(f19_in) ---------------------------------------- (6) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: U1(f10_out1(T6)) -> f3_out1(T6) f19_in -> f19_out1([]) f19_in -> U2(f19_in) U2(f19_out1(T16)) -> f19_out1(.(1, T16)) f21_in(.(T19, [])) -> f21_out1 U3(f19_out1(T7)) -> U4(f21_in(T7), T7) U4(f21_out1, T7) -> f10_out1(T7) Q is empty. ---------------------------------------- (7) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(.(x_1, x_2)) = 2*x_1 + x_2 POL(1) = 0 POL(U1(x_1)) = 2*x_1 POL(U2(x_1)) = x_1 POL(U3(x_1)) = 1 + 2*x_1 POL(U4(x_1, x_2)) = 2 + 2*x_1 + x_2 POL([]) = 0 POL(f10_out1(x_1)) = 1 + x_1 POL(f19_in) = 2 POL(f19_out1(x_1)) = 2 + 2*x_1 POL(f21_in(x_1)) = x_1 POL(f21_out1) = 0 POL(f3_out1(x_1)) = 2 + 2*x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U3(f19_out1(T7)) -> U4(f21_in(T7), T7) U4(f21_out1, T7) -> f10_out1(T7) ---------------------------------------- (8) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: U1(f10_out1(T6)) -> f3_out1(T6) f19_in -> f19_out1([]) f19_in -> U2(f19_in) U2(f19_out1(T16)) -> f19_out1(.(1, T16)) f21_in(.(T19, [])) -> f21_out1 Q is empty. ---------------------------------------- (9) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(.(x_1, x_2)) = 2*x_1 + x_2 POL(1) = 0 POL(U1(x_1)) = 2 + x_1 POL(U2(x_1)) = 2*x_1 POL([]) = 0 POL(f10_out1(x_1)) = 2 + 2*x_1 POL(f19_in) = 0 POL(f19_out1(x_1)) = x_1 POL(f21_in(x_1)) = 1 + x_1 POL(f21_out1) = 1 POL(f3_out1(x_1)) = 2*x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U1(f10_out1(T6)) -> f3_out1(T6) ---------------------------------------- (10) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f19_in -> f19_out1([]) f19_in -> U2(f19_in) U2(f19_out1(T16)) -> f19_out1(.(1, T16)) f21_in(.(T19, [])) -> f21_out1 Q is empty. ---------------------------------------- (11) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 1, "program": { "directives": [], "clauses": [ [ "(p X)", "(',' (l X) (q X))" ], [ "(q (. A ([])))", null ], [ "(r (1))", null ], [ "(l ([]))", null ], [ "(l (. H T))", "(',' (r H) (l T))" ] ] }, "graph": { "nodes": { "22": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (l T13) (q (. (1) T13)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "23": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "28": { "goal": [{ "clause": -1, "scope": -1, "term": "(l T13)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "51": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (r T21) (l T22))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "30": { "goal": [{ "clause": -1, "scope": -1, "term": "(q (. (1) T14))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T14"], "free": [], "exprvars": [] } }, "52": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "53": { "goal": [{ "clause": 2, "scope": 6, "term": "(',' (r T21) (l T22))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "32": { "goal": [ { "clause": 3, "scope": 5, "term": "(l T13)" }, { "clause": 4, "scope": 5, "term": "(l T13)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "54": { "goal": [{ "clause": -1, "scope": -1, "term": "(l T23)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "11": { "goal": [{ "clause": 3, "scope": 2, "term": "(',' (l T4) (q T4))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "33": { "goal": [{ "clause": 3, "scope": 5, "term": "(l T13)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "55": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "12": { "goal": [{ "clause": 4, "scope": 2, "term": "(',' (l T4) (q T4))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "34": { "goal": [{ "clause": 4, "scope": 5, "term": "(l T13)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "13": { "goal": [{ "clause": -1, "scope": -1, "term": "(q ([]))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "57": { "goal": [{ "clause": 1, "scope": 7, "term": "(q (. (1) T14))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T14"], "free": [], "exprvars": [] } }, "14": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "15": { "goal": [{ "clause": 1, "scope": 3, "term": "(q ([]))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "37": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "59": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "16": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "38": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "17": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (',' (r T11) (l T12)) (q (. T11 T12)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "39": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "18": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1": { "goal": [{ "clause": -1, "scope": -1, "term": "(p T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "5": { "goal": [{ "clause": 0, "scope": 1, "term": "(p T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "7": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (l T4) (q T4))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "9": { "goal": [ { "clause": 3, "scope": 2, "term": "(',' (l T4) (q T4))" }, { "clause": 4, "scope": 2, "term": "(',' (l T4) (q T4))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "61": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "63": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "20": { "goal": [{ "clause": 2, "scope": 4, "term": "(',' (',' (r T11) (l T12)) (q (. T11 T12)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 5, "label": "CASE" }, { "from": 5, "to": 7, "label": "ONLY EVAL with clause\np(X2) :- ','(l(X2), q(X2)).\nand substitutionT1 -> T4,\nX2 -> T4,\nT3 -> T4" }, { "from": 7, "to": 9, "label": "CASE" }, { "from": 9, "to": 11, "label": "PARALLEL" }, { "from": 9, "to": 12, "label": "PARALLEL" }, { "from": 11, "to": 13, "label": "EVAL with clause\nl([]).\nand substitutionT4 -> []" }, { "from": 11, "to": 14, "label": "EVAL-BACKTRACK" }, { "from": 12, "to": 17, "label": "EVAL with clause\nl(.(X8, X9)) :- ','(r(X8), l(X9)).\nand substitutionX8 -> T11,\nX9 -> T12,\nT4 -> .(T11, T12),\nT9 -> T11,\nT10 -> T12" }, { "from": 12, "to": 18, "label": "EVAL-BACKTRACK" }, { "from": 13, "to": 15, "label": "CASE" }, { "from": 15, "to": 16, "label": "BACKTRACK\nfor clause: q(.(A, []))because of non-unification" }, { "from": 17, "to": 20, "label": "CASE" }, { "from": 20, "to": 22, "label": "EVAL with clause\nr(1).\nand substitutionT11 -> 1,\nT12 -> T13" }, { "from": 20, "to": 23, "label": "EVAL-BACKTRACK" }, { "from": 22, "to": 28, "label": "SPLIT 1" }, { "from": 22, "to": 30, "label": "SPLIT 2\nnew knowledge:\nT14 is ground\nreplacements:T13 -> T14" }, { "from": 28, "to": 32, "label": "CASE" }, { "from": 30, "to": 57, "label": "CASE" }, { "from": 32, "to": 33, "label": "PARALLEL" }, { "from": 32, "to": 34, "label": "PARALLEL" }, { "from": 33, "to": 37, "label": "EVAL with clause\nl([]).\nand substitutionT13 -> []" }, { "from": 33, "to": 38, "label": "EVAL-BACKTRACK" }, { "from": 34, "to": 51, "label": "EVAL with clause\nl(.(X14, X15)) :- ','(r(X14), l(X15)).\nand substitutionX14 -> T21,\nX15 -> T22,\nT13 -> .(T21, T22),\nT19 -> T21,\nT20 -> T22" }, { "from": 34, "to": 52, "label": "EVAL-BACKTRACK" }, { "from": 37, "to": 39, "label": "SUCCESS" }, { "from": 51, "to": 53, "label": "CASE" }, { "from": 53, "to": 54, "label": "EVAL with clause\nr(1).\nand substitutionT21 -> 1,\nT22 -> T23" }, { "from": 53, "to": 55, "label": "EVAL-BACKTRACK" }, { "from": 54, "to": 28, "label": "INSTANCE with matching:\nT13 -> T23" }, { "from": 57, "to": 59, "label": "EVAL with clause\nq(.(X18, [])).\nand substitutionX18 -> 1,\nT14 -> []" }, { "from": 57, "to": 61, "label": "EVAL-BACKTRACK" }, { "from": 59, "to": 63, "label": "SUCCESS" } ], "type": "Graph" } } ---------------------------------------- (12) Obligation: Triples: lA(.(1, X1)) :- lA(X1). pB(.(1, X1)) :- lA(X1). Clauses: lcA([]). lcA(.(1, X1)) :- lcA(X1). Afs: pB(x1) = pB ---------------------------------------- (13) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: pB_in_1: (f) lA_in_1: (f) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: PB_IN_A(.(1, X1)) -> U2_A(X1, lA_in_a(X1)) PB_IN_A(.(1, X1)) -> LA_IN_A(X1) LA_IN_A(.(1, X1)) -> U1_A(X1, lA_in_a(X1)) LA_IN_A(.(1, X1)) -> LA_IN_A(X1) R is empty. The argument filtering Pi contains the following mapping: lA_in_a(x1) = lA_in_a .(x1, x2) = .(x1, x2) 1 = 1 PB_IN_A(x1) = PB_IN_A U2_A(x1, x2) = U2_A(x2) LA_IN_A(x1) = LA_IN_A U1_A(x1, x2) = U1_A(x2) We have to consider all (P,R,Pi)-chains Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: PB_IN_A(.(1, X1)) -> U2_A(X1, lA_in_a(X1)) PB_IN_A(.(1, X1)) -> LA_IN_A(X1) LA_IN_A(.(1, X1)) -> U1_A(X1, lA_in_a(X1)) LA_IN_A(.(1, X1)) -> LA_IN_A(X1) R is empty. The argument filtering Pi contains the following mapping: lA_in_a(x1) = lA_in_a .(x1, x2) = .(x1, x2) 1 = 1 PB_IN_A(x1) = PB_IN_A U2_A(x1, x2) = U2_A(x2) LA_IN_A(x1) = LA_IN_A U1_A(x1, x2) = U1_A(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (15) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes. ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: LA_IN_A(.(1, X1)) -> LA_IN_A(X1) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) 1 = 1 LA_IN_A(x1) = LA_IN_A 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: LA_IN_A -> LA_IN_A R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (19) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 2, "program": { "directives": [], "clauses": [ [ "(p X)", "(',' (l X) (q X))" ], [ "(q (. A ([])))", null ], [ "(r (1))", null ], [ "(l ([]))", null ], [ "(l (. H T))", "(',' (r H) (l T))" ] ] }, "graph": { "nodes": { "44": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "66": { "goal": [{ "clause": 2, "scope": 3, "term": "(',' (r T14) (l T15))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "45": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "67": { "goal": [{ "clause": -1, "scope": -1, "term": "(l T16)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "35": { "goal": [{ "clause": -1, "scope": -1, "term": "(l T6)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "68": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "36": { "goal": [{ "clause": -1, "scope": -1, "term": "(q T7)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T7"], "free": [], "exprvars": [] } }, "69": { "goal": [{ "clause": 1, "scope": 4, "term": "(q T7)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T7"], "free": [], "exprvars": [] } }, "type": "Nodes", "2": { "goal": [{ "clause": -1, "scope": -1, "term": "(p T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "6": { "goal": [{ "clause": 0, "scope": 1, "term": "(p T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "8": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (l T6) (q T6))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "70": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "71": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "72": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "40": { "goal": [ { "clause": 3, "scope": 2, "term": "(l T6)" }, { "clause": 4, "scope": 2, "term": "(l T6)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "41": { "goal": [{ "clause": 3, "scope": 2, "term": "(l T6)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "42": { "goal": [{ "clause": 4, "scope": 2, "term": "(l T6)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "64": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (r T14) (l T15))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "43": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "65": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 2, "to": 6, "label": "CASE" }, { "from": 6, "to": 8, "label": "ONLY EVAL with clause\np(X3) :- ','(l(X3), q(X3)).\nand substitutionT1 -> T6,\nX3 -> T6,\nT5 -> T6" }, { "from": 8, "to": 35, "label": "SPLIT 1" }, { "from": 8, "to": 36, "label": "SPLIT 2\nnew knowledge:\nT7 is ground\nreplacements:T6 -> T7" }, { "from": 35, "to": 40, "label": "CASE" }, { "from": 36, "to": 69, "label": "CASE" }, { "from": 40, "to": 41, "label": "PARALLEL" }, { "from": 40, "to": 42, "label": "PARALLEL" }, { "from": 41, "to": 43, "label": "EVAL with clause\nl([]).\nand substitutionT6 -> []" }, { "from": 41, "to": 44, "label": "EVAL-BACKTRACK" }, { "from": 42, "to": 64, "label": "EVAL with clause\nl(.(X8, X9)) :- ','(r(X8), l(X9)).\nand substitutionX8 -> T14,\nX9 -> T15,\nT6 -> .(T14, T15),\nT12 -> T14,\nT13 -> T15" }, { "from": 42, "to": 65, "label": "EVAL-BACKTRACK" }, { "from": 43, "to": 45, "label": "SUCCESS" }, { "from": 64, "to": 66, "label": "CASE" }, { "from": 66, "to": 67, "label": "EVAL with clause\nr(1).\nand substitutionT14 -> 1,\nT15 -> T16" }, { "from": 66, "to": 68, "label": "EVAL-BACKTRACK" }, { "from": 67, "to": 35, "label": "INSTANCE with matching:\nT6 -> T16" }, { "from": 69, "to": 70, "label": "EVAL with clause\nq(.(X12, [])).\nand substitutionX12 -> T19,\nT7 -> .(T19, [])" }, { "from": 69, "to": 71, "label": "EVAL-BACKTRACK" }, { "from": 70, "to": 72, "label": "SUCCESS" } ], "type": "Graph" } } ---------------------------------------- (20) Obligation: Rules: f67_out -> f66_out :|: TRUE f66_in -> f68_in :|: TRUE f68_out -> f66_out :|: TRUE f66_in -> f67_in :|: TRUE f67_in -> f35_in :|: TRUE f35_out -> f67_out :|: TRUE f35_in -> f40_in :|: TRUE f40_out -> f35_out :|: TRUE f64_out -> f42_out :|: TRUE f65_out -> f42_out :|: TRUE f42_in -> f65_in :|: TRUE f42_in -> f64_in :|: TRUE f64_in -> f66_in :|: TRUE f66_out -> f64_out :|: TRUE f40_in -> f41_in :|: TRUE f41_out -> f40_out :|: TRUE f42_out -> f40_out :|: TRUE f40_in -> f42_in :|: TRUE f6_out -> f2_out :|: TRUE f2_in -> f6_in :|: TRUE f8_out -> f6_out :|: TRUE f6_in -> f8_in :|: TRUE f36_out(T7) -> f8_out :|: TRUE f8_in -> f35_in :|: TRUE f35_out -> f36_in(x) :|: TRUE Start term: f2_in ---------------------------------------- (21) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f66_in -> f67_in :|: TRUE f67_in -> f35_in :|: TRUE f35_in -> f40_in :|: TRUE f42_in -> f64_in :|: TRUE f64_in -> f66_in :|: TRUE f40_in -> f42_in :|: TRUE ---------------------------------------- (22) Obligation: Rules: f66_in -> f67_in :|: TRUE f67_in -> f35_in :|: TRUE f35_in -> f40_in :|: TRUE f42_in -> f64_in :|: TRUE f64_in -> f66_in :|: TRUE f40_in -> f42_in :|: TRUE ---------------------------------------- (23) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (24) Obligation: Rules: f42_in -> f42_in :|: TRUE ---------------------------------------- (25) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (26) Obligation: Rules: f42_in -> f42_in :|: TRUE ---------------------------------------- (27) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f42_in -> f42_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (28) Obligation: Termination digraph: Nodes: (1) f42_in -> f42_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (29) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f42_in() Replaced non-predefined constructor symbols by 0. ---------------------------------------- (30) Obligation: Rules: f42_in -> f42_in :|: TRUE ---------------------------------------- (31) IntTRSNonPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc) -> f(1) :|: pc = 1 && TRUE Proved unsatisfiability of the following formula, indicating that the system is never left after entering: ((run2_0 = ((1 * 1)) and (((run1_0 * 1)) = ((1 * 1)) and T)) and !(((run2_0 * 1)) = ((1 * 1)) and T)) Proved satisfiability of the following formula, indicating that the system is entered at least once: (run2_0 = ((1 * 1)) and (((run1_0 * 1)) = ((1 * 1)) and T)) ---------------------------------------- (32) NO ---------------------------------------- (33) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: p_in_1: (f) l_in_1: (f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: p_in_a(X) -> U1_a(X, l_in_a(X)) l_in_a([]) -> l_out_a([]) l_in_a(.(H, T)) -> U3_a(H, T, r_in_a(H)) r_in_a(1) -> r_out_a(1) U3_a(H, T, r_out_a(H)) -> U4_a(H, T, l_in_a(T)) U4_a(H, T, l_out_a(T)) -> l_out_a(.(H, T)) U1_a(X, l_out_a(X)) -> U2_a(X, q_in_g(X)) q_in_g(.(A, [])) -> q_out_g(.(A, [])) U2_a(X, q_out_g(X)) -> p_out_a(X) The argument filtering Pi contains the following mapping: p_in_a(x1) = p_in_a U1_a(x1, x2) = U1_a(x2) l_in_a(x1) = l_in_a l_out_a(x1) = l_out_a(x1) U3_a(x1, x2, x3) = U3_a(x3) r_in_a(x1) = r_in_a r_out_a(x1) = r_out_a(x1) U4_a(x1, x2, x3) = U4_a(x1, x3) U2_a(x1, x2) = U2_a(x1, x2) q_in_g(x1) = q_in_g(x1) .(x1, x2) = .(x1, x2) [] = [] q_out_g(x1) = q_out_g p_out_a(x1) = p_out_a(x1) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (34) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: p_in_a(X) -> U1_a(X, l_in_a(X)) l_in_a([]) -> l_out_a([]) l_in_a(.(H, T)) -> U3_a(H, T, r_in_a(H)) r_in_a(1) -> r_out_a(1) U3_a(H, T, r_out_a(H)) -> U4_a(H, T, l_in_a(T)) U4_a(H, T, l_out_a(T)) -> l_out_a(.(H, T)) U1_a(X, l_out_a(X)) -> U2_a(X, q_in_g(X)) q_in_g(.(A, [])) -> q_out_g(.(A, [])) U2_a(X, q_out_g(X)) -> p_out_a(X) The argument filtering Pi contains the following mapping: p_in_a(x1) = p_in_a U1_a(x1, x2) = U1_a(x2) l_in_a(x1) = l_in_a l_out_a(x1) = l_out_a(x1) U3_a(x1, x2, x3) = U3_a(x3) r_in_a(x1) = r_in_a r_out_a(x1) = r_out_a(x1) U4_a(x1, x2, x3) = U4_a(x1, x3) U2_a(x1, x2) = U2_a(x1, x2) q_in_g(x1) = q_in_g(x1) .(x1, x2) = .(x1, x2) [] = [] q_out_g(x1) = q_out_g p_out_a(x1) = p_out_a(x1) ---------------------------------------- (35) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: Pi DP problem: The TRS P consists of the following rules: P_IN_A(X) -> U1_A(X, l_in_a(X)) P_IN_A(X) -> L_IN_A(X) L_IN_A(.(H, T)) -> U3_A(H, T, r_in_a(H)) L_IN_A(.(H, T)) -> R_IN_A(H) U3_A(H, T, r_out_a(H)) -> U4_A(H, T, l_in_a(T)) U3_A(H, T, r_out_a(H)) -> L_IN_A(T) U1_A(X, l_out_a(X)) -> U2_A(X, q_in_g(X)) U1_A(X, l_out_a(X)) -> Q_IN_G(X) The TRS R consists of the following rules: p_in_a(X) -> U1_a(X, l_in_a(X)) l_in_a([]) -> l_out_a([]) l_in_a(.(H, T)) -> U3_a(H, T, r_in_a(H)) r_in_a(1) -> r_out_a(1) U3_a(H, T, r_out_a(H)) -> U4_a(H, T, l_in_a(T)) U4_a(H, T, l_out_a(T)) -> l_out_a(.(H, T)) U1_a(X, l_out_a(X)) -> U2_a(X, q_in_g(X)) q_in_g(.(A, [])) -> q_out_g(.(A, [])) U2_a(X, q_out_g(X)) -> p_out_a(X) The argument filtering Pi contains the following mapping: p_in_a(x1) = p_in_a U1_a(x1, x2) = U1_a(x2) l_in_a(x1) = l_in_a l_out_a(x1) = l_out_a(x1) U3_a(x1, x2, x3) = U3_a(x3) r_in_a(x1) = r_in_a r_out_a(x1) = r_out_a(x1) U4_a(x1, x2, x3) = U4_a(x1, x3) U2_a(x1, x2) = U2_a(x1, x2) q_in_g(x1) = q_in_g(x1) .(x1, x2) = .(x1, x2) [] = [] q_out_g(x1) = q_out_g p_out_a(x1) = p_out_a(x1) P_IN_A(x1) = P_IN_A U1_A(x1, x2) = U1_A(x2) L_IN_A(x1) = L_IN_A U3_A(x1, x2, x3) = U3_A(x3) R_IN_A(x1) = R_IN_A U4_A(x1, x2, x3) = U4_A(x1, x3) U2_A(x1, x2) = U2_A(x1, x2) Q_IN_G(x1) = Q_IN_G(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (36) Obligation: Pi DP problem: The TRS P consists of the following rules: P_IN_A(X) -> U1_A(X, l_in_a(X)) P_IN_A(X) -> L_IN_A(X) L_IN_A(.(H, T)) -> U3_A(H, T, r_in_a(H)) L_IN_A(.(H, T)) -> R_IN_A(H) U3_A(H, T, r_out_a(H)) -> U4_A(H, T, l_in_a(T)) U3_A(H, T, r_out_a(H)) -> L_IN_A(T) U1_A(X, l_out_a(X)) -> U2_A(X, q_in_g(X)) U1_A(X, l_out_a(X)) -> Q_IN_G(X) The TRS R consists of the following rules: p_in_a(X) -> U1_a(X, l_in_a(X)) l_in_a([]) -> l_out_a([]) l_in_a(.(H, T)) -> U3_a(H, T, r_in_a(H)) r_in_a(1) -> r_out_a(1) U3_a(H, T, r_out_a(H)) -> U4_a(H, T, l_in_a(T)) U4_a(H, T, l_out_a(T)) -> l_out_a(.(H, T)) U1_a(X, l_out_a(X)) -> U2_a(X, q_in_g(X)) q_in_g(.(A, [])) -> q_out_g(.(A, [])) U2_a(X, q_out_g(X)) -> p_out_a(X) The argument filtering Pi contains the following mapping: p_in_a(x1) = p_in_a U1_a(x1, x2) = U1_a(x2) l_in_a(x1) = l_in_a l_out_a(x1) = l_out_a(x1) U3_a(x1, x2, x3) = U3_a(x3) r_in_a(x1) = r_in_a r_out_a(x1) = r_out_a(x1) U4_a(x1, x2, x3) = U4_a(x1, x3) U2_a(x1, x2) = U2_a(x1, x2) q_in_g(x1) = q_in_g(x1) .(x1, x2) = .(x1, x2) [] = [] q_out_g(x1) = q_out_g p_out_a(x1) = p_out_a(x1) P_IN_A(x1) = P_IN_A U1_A(x1, x2) = U1_A(x2) L_IN_A(x1) = L_IN_A U3_A(x1, x2, x3) = U3_A(x3) R_IN_A(x1) = R_IN_A U4_A(x1, x2, x3) = U4_A(x1, x3) U2_A(x1, x2) = U2_A(x1, x2) Q_IN_G(x1) = Q_IN_G(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (37) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 6 less nodes. ---------------------------------------- (38) Obligation: Pi DP problem: The TRS P consists of the following rules: U3_A(H, T, r_out_a(H)) -> L_IN_A(T) L_IN_A(.(H, T)) -> U3_A(H, T, r_in_a(H)) The TRS R consists of the following rules: p_in_a(X) -> U1_a(X, l_in_a(X)) l_in_a([]) -> l_out_a([]) l_in_a(.(H, T)) -> U3_a(H, T, r_in_a(H)) r_in_a(1) -> r_out_a(1) U3_a(H, T, r_out_a(H)) -> U4_a(H, T, l_in_a(T)) U4_a(H, T, l_out_a(T)) -> l_out_a(.(H, T)) U1_a(X, l_out_a(X)) -> U2_a(X, q_in_g(X)) q_in_g(.(A, [])) -> q_out_g(.(A, [])) U2_a(X, q_out_g(X)) -> p_out_a(X) The argument filtering Pi contains the following mapping: p_in_a(x1) = p_in_a U1_a(x1, x2) = U1_a(x2) l_in_a(x1) = l_in_a l_out_a(x1) = l_out_a(x1) U3_a(x1, x2, x3) = U3_a(x3) r_in_a(x1) = r_in_a r_out_a(x1) = r_out_a(x1) U4_a(x1, x2, x3) = U4_a(x1, x3) U2_a(x1, x2) = U2_a(x1, x2) q_in_g(x1) = q_in_g(x1) .(x1, x2) = .(x1, x2) [] = [] q_out_g(x1) = q_out_g p_out_a(x1) = p_out_a(x1) L_IN_A(x1) = L_IN_A U3_A(x1, x2, x3) = U3_A(x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (39) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (40) Obligation: Pi DP problem: The TRS P consists of the following rules: U3_A(H, T, r_out_a(H)) -> L_IN_A(T) L_IN_A(.(H, T)) -> U3_A(H, T, r_in_a(H)) The TRS R consists of the following rules: r_in_a(1) -> r_out_a(1) The argument filtering Pi contains the following mapping: r_in_a(x1) = r_in_a r_out_a(x1) = r_out_a(x1) .(x1, x2) = .(x1, x2) L_IN_A(x1) = L_IN_A U3_A(x1, x2, x3) = U3_A(x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (41) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (42) Obligation: Q DP problem: The TRS P consists of the following rules: U3_A(r_out_a(H)) -> L_IN_A L_IN_A -> U3_A(r_in_a) The TRS R consists of the following rules: r_in_a -> r_out_a(1) The set Q consists of the following terms: r_in_a We have to consider all (P,Q,R)-chains. ---------------------------------------- (43) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: p_in_1: (f) l_in_1: (f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: p_in_a(X) -> U1_a(X, l_in_a(X)) l_in_a([]) -> l_out_a([]) l_in_a(.(H, T)) -> U3_a(H, T, r_in_a(H)) r_in_a(1) -> r_out_a(1) U3_a(H, T, r_out_a(H)) -> U4_a(H, T, l_in_a(T)) U4_a(H, T, l_out_a(T)) -> l_out_a(.(H, T)) U1_a(X, l_out_a(X)) -> U2_a(X, q_in_g(X)) q_in_g(.(A, [])) -> q_out_g(.(A, [])) U2_a(X, q_out_g(X)) -> p_out_a(X) The argument filtering Pi contains the following mapping: p_in_a(x1) = p_in_a U1_a(x1, x2) = U1_a(x2) l_in_a(x1) = l_in_a l_out_a(x1) = l_out_a(x1) U3_a(x1, x2, x3) = U3_a(x3) r_in_a(x1) = r_in_a r_out_a(x1) = r_out_a(x1) U4_a(x1, x2, x3) = U4_a(x1, x3) U2_a(x1, x2) = U2_a(x1, x2) q_in_g(x1) = q_in_g(x1) .(x1, x2) = .(x1, x2) [] = [] q_out_g(x1) = q_out_g(x1) p_out_a(x1) = p_out_a(x1) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (44) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: p_in_a(X) -> U1_a(X, l_in_a(X)) l_in_a([]) -> l_out_a([]) l_in_a(.(H, T)) -> U3_a(H, T, r_in_a(H)) r_in_a(1) -> r_out_a(1) U3_a(H, T, r_out_a(H)) -> U4_a(H, T, l_in_a(T)) U4_a(H, T, l_out_a(T)) -> l_out_a(.(H, T)) U1_a(X, l_out_a(X)) -> U2_a(X, q_in_g(X)) q_in_g(.(A, [])) -> q_out_g(.(A, [])) U2_a(X, q_out_g(X)) -> p_out_a(X) The argument filtering Pi contains the following mapping: p_in_a(x1) = p_in_a U1_a(x1, x2) = U1_a(x2) l_in_a(x1) = l_in_a l_out_a(x1) = l_out_a(x1) U3_a(x1, x2, x3) = U3_a(x3) r_in_a(x1) = r_in_a r_out_a(x1) = r_out_a(x1) U4_a(x1, x2, x3) = U4_a(x1, x3) U2_a(x1, x2) = U2_a(x1, x2) q_in_g(x1) = q_in_g(x1) .(x1, x2) = .(x1, x2) [] = [] q_out_g(x1) = q_out_g(x1) p_out_a(x1) = p_out_a(x1) ---------------------------------------- (45) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: Pi DP problem: The TRS P consists of the following rules: P_IN_A(X) -> U1_A(X, l_in_a(X)) P_IN_A(X) -> L_IN_A(X) L_IN_A(.(H, T)) -> U3_A(H, T, r_in_a(H)) L_IN_A(.(H, T)) -> R_IN_A(H) U3_A(H, T, r_out_a(H)) -> U4_A(H, T, l_in_a(T)) U3_A(H, T, r_out_a(H)) -> L_IN_A(T) U1_A(X, l_out_a(X)) -> U2_A(X, q_in_g(X)) U1_A(X, l_out_a(X)) -> Q_IN_G(X) The TRS R consists of the following rules: p_in_a(X) -> U1_a(X, l_in_a(X)) l_in_a([]) -> l_out_a([]) l_in_a(.(H, T)) -> U3_a(H, T, r_in_a(H)) r_in_a(1) -> r_out_a(1) U3_a(H, T, r_out_a(H)) -> U4_a(H, T, l_in_a(T)) U4_a(H, T, l_out_a(T)) -> l_out_a(.(H, T)) U1_a(X, l_out_a(X)) -> U2_a(X, q_in_g(X)) q_in_g(.(A, [])) -> q_out_g(.(A, [])) U2_a(X, q_out_g(X)) -> p_out_a(X) The argument filtering Pi contains the following mapping: p_in_a(x1) = p_in_a U1_a(x1, x2) = U1_a(x2) l_in_a(x1) = l_in_a l_out_a(x1) = l_out_a(x1) U3_a(x1, x2, x3) = U3_a(x3) r_in_a(x1) = r_in_a r_out_a(x1) = r_out_a(x1) U4_a(x1, x2, x3) = U4_a(x1, x3) U2_a(x1, x2) = U2_a(x1, x2) q_in_g(x1) = q_in_g(x1) .(x1, x2) = .(x1, x2) [] = [] q_out_g(x1) = q_out_g(x1) p_out_a(x1) = p_out_a(x1) P_IN_A(x1) = P_IN_A U1_A(x1, x2) = U1_A(x2) L_IN_A(x1) = L_IN_A U3_A(x1, x2, x3) = U3_A(x3) R_IN_A(x1) = R_IN_A U4_A(x1, x2, x3) = U4_A(x1, x3) U2_A(x1, x2) = U2_A(x1, x2) Q_IN_G(x1) = Q_IN_G(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (46) Obligation: Pi DP problem: The TRS P consists of the following rules: P_IN_A(X) -> U1_A(X, l_in_a(X)) P_IN_A(X) -> L_IN_A(X) L_IN_A(.(H, T)) -> U3_A(H, T, r_in_a(H)) L_IN_A(.(H, T)) -> R_IN_A(H) U3_A(H, T, r_out_a(H)) -> U4_A(H, T, l_in_a(T)) U3_A(H, T, r_out_a(H)) -> L_IN_A(T) U1_A(X, l_out_a(X)) -> U2_A(X, q_in_g(X)) U1_A(X, l_out_a(X)) -> Q_IN_G(X) The TRS R consists of the following rules: p_in_a(X) -> U1_a(X, l_in_a(X)) l_in_a([]) -> l_out_a([]) l_in_a(.(H, T)) -> U3_a(H, T, r_in_a(H)) r_in_a(1) -> r_out_a(1) U3_a(H, T, r_out_a(H)) -> U4_a(H, T, l_in_a(T)) U4_a(H, T, l_out_a(T)) -> l_out_a(.(H, T)) U1_a(X, l_out_a(X)) -> U2_a(X, q_in_g(X)) q_in_g(.(A, [])) -> q_out_g(.(A, [])) U2_a(X, q_out_g(X)) -> p_out_a(X) The argument filtering Pi contains the following mapping: p_in_a(x1) = p_in_a U1_a(x1, x2) = U1_a(x2) l_in_a(x1) = l_in_a l_out_a(x1) = l_out_a(x1) U3_a(x1, x2, x3) = U3_a(x3) r_in_a(x1) = r_in_a r_out_a(x1) = r_out_a(x1) U4_a(x1, x2, x3) = U4_a(x1, x3) U2_a(x1, x2) = U2_a(x1, x2) q_in_g(x1) = q_in_g(x1) .(x1, x2) = .(x1, x2) [] = [] q_out_g(x1) = q_out_g(x1) p_out_a(x1) = p_out_a(x1) P_IN_A(x1) = P_IN_A U1_A(x1, x2) = U1_A(x2) L_IN_A(x1) = L_IN_A U3_A(x1, x2, x3) = U3_A(x3) R_IN_A(x1) = R_IN_A U4_A(x1, x2, x3) = U4_A(x1, x3) U2_A(x1, x2) = U2_A(x1, x2) Q_IN_G(x1) = Q_IN_G(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (47) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 6 less nodes. ---------------------------------------- (48) Obligation: Pi DP problem: The TRS P consists of the following rules: U3_A(H, T, r_out_a(H)) -> L_IN_A(T) L_IN_A(.(H, T)) -> U3_A(H, T, r_in_a(H)) The TRS R consists of the following rules: p_in_a(X) -> U1_a(X, l_in_a(X)) l_in_a([]) -> l_out_a([]) l_in_a(.(H, T)) -> U3_a(H, T, r_in_a(H)) r_in_a(1) -> r_out_a(1) U3_a(H, T, r_out_a(H)) -> U4_a(H, T, l_in_a(T)) U4_a(H, T, l_out_a(T)) -> l_out_a(.(H, T)) U1_a(X, l_out_a(X)) -> U2_a(X, q_in_g(X)) q_in_g(.(A, [])) -> q_out_g(.(A, [])) U2_a(X, q_out_g(X)) -> p_out_a(X) The argument filtering Pi contains the following mapping: p_in_a(x1) = p_in_a U1_a(x1, x2) = U1_a(x2) l_in_a(x1) = l_in_a l_out_a(x1) = l_out_a(x1) U3_a(x1, x2, x3) = U3_a(x3) r_in_a(x1) = r_in_a r_out_a(x1) = r_out_a(x1) U4_a(x1, x2, x3) = U4_a(x1, x3) U2_a(x1, x2) = U2_a(x1, x2) q_in_g(x1) = q_in_g(x1) .(x1, x2) = .(x1, x2) [] = [] q_out_g(x1) = q_out_g(x1) p_out_a(x1) = p_out_a(x1) L_IN_A(x1) = L_IN_A U3_A(x1, x2, x3) = U3_A(x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (49) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (50) Obligation: Pi DP problem: The TRS P consists of the following rules: U3_A(H, T, r_out_a(H)) -> L_IN_A(T) L_IN_A(.(H, T)) -> U3_A(H, T, r_in_a(H)) The TRS R consists of the following rules: r_in_a(1) -> r_out_a(1) The argument filtering Pi contains the following mapping: r_in_a(x1) = r_in_a r_out_a(x1) = r_out_a(x1) .(x1, x2) = .(x1, x2) L_IN_A(x1) = L_IN_A U3_A(x1, x2, x3) = U3_A(x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (51) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (52) Obligation: Q DP problem: The TRS P consists of the following rules: U3_A(r_out_a(H)) -> L_IN_A L_IN_A -> U3_A(r_in_a) The TRS R consists of the following rules: r_in_a -> r_out_a(1) The set Q consists of the following terms: r_in_a We have to consider all (P,Q,R)-chains.