MAYBE proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Left Termination of the query pattern fl(a,a,g) w.r.t. the given Prolog program could not be shown: (0) Prolog (1) PrologToTRSTransformerProof [SOUND, 0 ms] (2) QTRS (3) QTRSRRRProof [EQUIVALENT, 63 ms] (4) QTRS (5) QTRSRRRProof [EQUIVALENT, 0 ms] (6) QTRS (7) QTRSRRRProof [EQUIVALENT, 0 ms] (8) QTRS (9) Overlay + Local Confluence [EQUIVALENT, 0 ms] (10) QTRS (11) DependencyPairsProof [EQUIVALENT, 0 ms] (12) QDP (13) UsableRulesProof [EQUIVALENT, 0 ms] (14) QDP (15) QReductionProof [EQUIVALENT, 0 ms] (16) QDP (17) NonTerminationLoopProof [COMPLETE, 0 ms] (18) NO (19) PrologToIRSwTTransformerProof [SOUND, 0 ms] (20) AND (21) IRSwT (22) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (23) IRSwT (24) IntTRSCompressionProof [EQUIVALENT, 41 ms] (25) IRSwT (26) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (27) IRSwT (28) IRSwTTerminationDigraphProof [EQUIVALENT, 6 ms] (29) IRSwT (30) FilterProof [EQUIVALENT, 0 ms] (31) IntTRS (32) IntTRSNonPeriodicNontermProof [COMPLETE, 6 ms] (33) NO (34) IRSwT (35) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (36) IRSwT (37) IntTRSCompressionProof [EQUIVALENT, 8 ms] (38) IRSwT (39) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (40) IRSwT (41) IRSwTTerminationDigraphProof [EQUIVALENT, 11 ms] (42) IRSwT (43) FilterProof [EQUIVALENT, 0 ms] (44) IntTRS (45) PrologToPiTRSProof [SOUND, 0 ms] (46) PiTRS (47) DependencyPairsProof [EQUIVALENT, 5 ms] (48) PiDP (49) DependencyGraphProof [EQUIVALENT, 0 ms] (50) AND (51) PiDP (52) UsableRulesProof [EQUIVALENT, 0 ms] (53) PiDP (54) PiDPToQDPProof [SOUND, 0 ms] (55) QDP (56) NonTerminationLoopProof [COMPLETE, 0 ms] (57) NO (58) PiDP (59) UsableRulesProof [EQUIVALENT, 0 ms] (60) PiDP (61) PiDPToQDPProof [SOUND, 0 ms] (62) QDP (63) QDPSizeChangeProof [EQUIVALENT, 0 ms] (64) YES (65) PrologToPiTRSProof [SOUND, 0 ms] (66) PiTRS (67) DependencyPairsProof [EQUIVALENT, 7 ms] (68) PiDP (69) DependencyGraphProof [EQUIVALENT, 0 ms] (70) AND (71) PiDP (72) UsableRulesProof [EQUIVALENT, 0 ms] (73) PiDP (74) PiDPToQDPProof [SOUND, 0 ms] (75) QDP (76) NonTerminationLoopProof [COMPLETE, 0 ms] (77) NO (78) PiDP (79) UsableRulesProof [EQUIVALENT, 0 ms] (80) PiDP (81) PiDPToQDPProof [SOUND, 0 ms] (82) QDP (83) QDPSizeChangeProof [EQUIVALENT, 0 ms] (84) YES (85) PrologToDTProblemTransformerProof [SOUND, 0 ms] (86) TRIPLES (87) TriplesToPiDPProof [SOUND, 0 ms] (88) PiDP (89) DependencyGraphProof [EQUIVALENT, 0 ms] (90) PiDP (91) PiDPToQDPProof [SOUND, 0 ms] (92) QDP (93) UsableRulesReductionPairsProof [EQUIVALENT, 4 ms] (94) QDP (95) DependencyGraphProof [EQUIVALENT, 0 ms] (96) QDP (97) NonTerminationLoopProof [COMPLETE, 0 ms] (98) NO ---------------------------------------- (0) Obligation: Clauses: fl([], [], 0). fl(.(E, X), R, s(Z)) :- ','(append(E, Y, R), fl(X, Y, Z)). append([], X, X). append(.(X, Xs), Ys, .(X, Zs)) :- append(Xs, Ys, Zs). Query: fl(a,a,g) ---------------------------------------- (1) PrologToTRSTransformerProof (SOUND) Transformed Prolog program to TRS. { "root": 2, "program": { "directives": [], "clauses": [ [ "(fl ([]) ([]) (0))", null ], [ "(fl (. E X) R (s Z))", "(',' (append E Y R) (fl X Y Z))" ], [ "(append ([]) X X)", null ], [ "(append (. X Xs) Ys (. X Zs))", "(append Xs Ys Zs)" ] ] }, "graph": { "nodes": { "11": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "66": { "goal": [{ "clause": -1, "scope": -1, "term": "(append T39 X47 T40)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X47"], "exprvars": [] } }, "12": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "67": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "13": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "48": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (append T16 X14 T17) (fl T18 X14 T15))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T15"], "free": ["X14"], "exprvars": [] } }, "49": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "2": { "goal": [{ "clause": -1, "scope": -1, "term": "(fl T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } }, "5": { "goal": [ { "clause": 0, "scope": 1, "term": "(fl T1 T2 T3)" }, { "clause": 1, "scope": 1, "term": "(fl T1 T2 T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } }, "7": { "goal": [{ "clause": 0, "scope": 1, "term": "(fl T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } }, "8": { "goal": [{ "clause": 1, "scope": 1, "term": "(fl T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } }, "60": { "goal": [ { "clause": 2, "scope": 2, "term": "(append T16 X14 T17)" }, { "clause": 3, "scope": 2, "term": "(append T16 X14 T17)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X14"], "exprvars": [] } }, "50": { "goal": [{ "clause": -1, "scope": -1, "term": "(append T16 X14 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X14"], "exprvars": [] } }, "61": { "goal": [{ "clause": 2, "scope": 2, "term": "(append T16 X14 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X14"], "exprvars": [] } }, "51": { "goal": [{ "clause": -1, "scope": -1, "term": "(fl T22 T21 T15)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T15"], "free": [], "exprvars": [] } }, "62": { "goal": [{ "clause": 3, "scope": 2, "term": "(append T16 X14 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X14"], "exprvars": [] } }, "63": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "64": { "goal": [], "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": 5, "label": "CASE" }, { "from": 5, "to": 7, "label": "PARALLEL" }, { "from": 5, "to": 8, "label": "PARALLEL" }, { "from": 7, "to": 11, "label": "EVAL with clause\nfl([], [], 0).\nand substitutionT1 -> [],\nT2 -> [],\nT3 -> 0" }, { "from": 7, "to": 12, "label": "EVAL-BACKTRACK" }, { "from": 8, "to": 48, "label": "EVAL with clause\nfl(.(X10, X11), X12, s(X13)) :- ','(append(X10, X14, X12), fl(X11, X14, X13)).\nand substitutionX10 -> T16,\nX11 -> T18,\nT1 -> .(T16, T18),\nT2 -> T17,\nX12 -> T17,\nX13 -> T15,\nT3 -> s(T15),\nT12 -> T16,\nT14 -> T17,\nT13 -> T18" }, { "from": 8, "to": 49, "label": "EVAL-BACKTRACK" }, { "from": 11, "to": 13, "label": "SUCCESS" }, { "from": 48, "to": 50, "label": "SPLIT 1" }, { "from": 48, "to": 51, "label": "SPLIT 2\nreplacements:X14 -> T21,\nT18 -> T22" }, { "from": 50, "to": 60, "label": "CASE" }, { "from": 51, "to": 2, "label": "INSTANCE with matching:\nT1 -> T22\nT2 -> T21\nT3 -> T15" }, { "from": 60, "to": 61, "label": "PARALLEL" }, { "from": 60, "to": 62, "label": "PARALLEL" }, { "from": 61, "to": 63, "label": "EVAL with clause\nappend([], X31, X31).\nand substitutionT16 -> [],\nX14 -> T29,\nX31 -> T29,\nT17 -> T29,\nX32 -> T29" }, { "from": 61, "to": 64, "label": "EVAL-BACKTRACK" }, { "from": 62, "to": 66, "label": "EVAL with clause\nappend(.(X43, X44), X45, .(X43, X46)) :- append(X44, X45, X46).\nand substitutionX43 -> T36,\nX44 -> T39,\nT16 -> .(T36, T39),\nX14 -> X47,\nX45 -> X47,\nX46 -> T40,\nT17 -> .(T36, T40),\nT37 -> T39,\nT38 -> T40" }, { "from": 62, "to": 67, "label": "EVAL-BACKTRACK" }, { "from": 63, "to": 65, "label": "SUCCESS" }, { "from": 66, "to": 50, "label": "INSTANCE with matching:\nT16 -> T39\nX14 -> X47\nT17 -> T40" } ], "type": "Graph" } } ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f2_in(0) -> f2_out1 f2_in(s(T15)) -> U1(f48_in(T15), s(T15)) U1(f48_out1, s(T15)) -> f2_out1 f50_in -> f50_out1 f50_in -> U2(f50_in) U2(f50_out1) -> f50_out1 f48_in(T15) -> U3(f50_in, T15) U3(f50_out1, T15) -> U4(f2_in(T15), T15) U4(f2_out1, T15) -> f48_out1 Q is empty. ---------------------------------------- (3) QTRSRRRProof (EQUIVALENT) Used ordering: f2_in/1(YES) 0/0) f2_out1/0) s/1(YES) U1/2(YES,YES) f48_in/1(YES) f48_out1/0) f50_in/0) f50_out1/0) U2/1)YES( U3/2(YES,YES) U4/2(YES,YES) Quasi precedence: 0 > f2_out1 > f48_out1 > [U1_2, U4_2] s_1 > f2_out1 > f48_out1 > [U1_2, U4_2] s_1 > f48_in_1 > [f50_in, f50_out1] > [f2_in_1, U3_2] > [U1_2, U4_2] Status: f2_in_1: multiset status 0: multiset status f2_out1: multiset status s_1: [1] U1_2: multiset status f48_in_1: multiset status f48_out1: multiset status f50_in: multiset status f50_out1: multiset status U3_2: multiset status U4_2: [2,1] With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: f2_in(0) -> f2_out1 f2_in(s(T15)) -> U1(f48_in(T15), s(T15)) U1(f48_out1, s(T15)) -> f2_out1 f48_in(T15) -> U3(f50_in, T15) U3(f50_out1, T15) -> U4(f2_in(T15), T15) U4(f2_out1, T15) -> f48_out1 ---------------------------------------- (4) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f50_in -> f50_out1 f50_in -> U2(f50_in) U2(f50_out1) -> f50_out1 Q is empty. ---------------------------------------- (5) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U2(x_1)) = x_1 POL(f50_in) = 2 POL(f50_out1) = 0 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: f50_in -> f50_out1 ---------------------------------------- (6) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f50_in -> U2(f50_in) U2(f50_out1) -> f50_out1 Q is empty. ---------------------------------------- (7) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U2(x_1)) = 2*x_1 POL(f50_in) = 0 POL(f50_out1) = 1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U2(f50_out1) -> f50_out1 ---------------------------------------- (8) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f50_in -> U2(f50_in) Q is empty. ---------------------------------------- (9) Overlay + Local Confluence (EQUIVALENT) The TRS is overlay and locally confluent. By [NOC] we can switch to innermost. ---------------------------------------- (10) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f50_in -> U2(f50_in) The set Q consists of the following terms: f50_in ---------------------------------------- (11) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: F50_IN -> F50_IN The TRS R consists of the following rules: f50_in -> U2(f50_in) The set Q consists of the following terms: f50_in We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (13) 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. ---------------------------------------- (14) Obligation: Q DP problem: The TRS P consists of the following rules: F50_IN -> F50_IN R is empty. The set Q consists of the following terms: f50_in We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (15) 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]. f50_in ---------------------------------------- (16) Obligation: Q DP problem: The TRS P consists of the following rules: F50_IN -> F50_IN R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (17) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by semiunifying a rule from P directly. s = F50_IN evaluates to t =F50_IN Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence The DP semiunifies directly so there is only one rewrite step from F50_IN to F50_IN. ---------------------------------------- (18) NO ---------------------------------------- (19) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 1, "program": { "directives": [], "clauses": [ [ "(fl ([]) ([]) (0))", null ], [ "(fl (. E X) R (s Z))", "(',' (append E Y R) (fl X Y Z))" ], [ "(append ([]) X X)", null ], [ "(append (. X Xs) Ys (. X Zs))", "(append Xs Ys Zs)" ] ] }, "graph": { "nodes": { "22": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "55": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "56": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "57": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "14": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "58": { "goal": [{ "clause": -1, "scope": -1, "term": "(append T39 X47 T40)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X47"], "exprvars": [] } }, "59": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "16": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "17": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "1": { "goal": [{ "clause": -1, "scope": -1, "term": "(fl T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } }, "6": { "goal": [ { "clause": 0, "scope": 1, "term": "(fl T1 T2 T3)" }, { "clause": 1, "scope": 1, "term": "(fl T1 T2 T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } }, "9": { "goal": [{ "clause": 0, "scope": 1, "term": "(fl T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } }, "30": { "goal": [{ "clause": -1, "scope": -1, "term": "(append T16 X14 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X14"], "exprvars": [] } }, "52": { "goal": [ { "clause": 2, "scope": 2, "term": "(append T16 X14 T17)" }, { "clause": 3, "scope": 2, "term": "(append T16 X14 T17)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X14"], "exprvars": [] } }, "31": { "goal": [{ "clause": -1, "scope": -1, "term": "(fl T22 T21 T15)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T15"], "free": [], "exprvars": [] } }, "53": { "goal": [{ "clause": 2, "scope": 2, "term": "(append T16 X14 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X14"], "exprvars": [] } }, "10": { "goal": [{ "clause": 1, "scope": 1, "term": "(fl T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } }, "21": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (append T16 X14 T17) (fl T18 X14 T15))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T15"], "free": ["X14"], "exprvars": [] } }, "54": { "goal": [{ "clause": 3, "scope": 2, "term": "(append T16 X14 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X14"], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 6, "label": "CASE" }, { "from": 6, "to": 9, "label": "PARALLEL" }, { "from": 6, "to": 10, "label": "PARALLEL" }, { "from": 9, "to": 14, "label": "EVAL with clause\nfl([], [], 0).\nand substitutionT1 -> [],\nT2 -> [],\nT3 -> 0" }, { "from": 9, "to": 16, "label": "EVAL-BACKTRACK" }, { "from": 10, "to": 21, "label": "EVAL with clause\nfl(.(X10, X11), X12, s(X13)) :- ','(append(X10, X14, X12), fl(X11, X14, X13)).\nand substitutionX10 -> T16,\nX11 -> T18,\nT1 -> .(T16, T18),\nT2 -> T17,\nX12 -> T17,\nX13 -> T15,\nT3 -> s(T15),\nT12 -> T16,\nT14 -> T17,\nT13 -> T18" }, { "from": 10, "to": 22, "label": "EVAL-BACKTRACK" }, { "from": 14, "to": 17, "label": "SUCCESS" }, { "from": 21, "to": 30, "label": "SPLIT 1" }, { "from": 21, "to": 31, "label": "SPLIT 2\nreplacements:X14 -> T21,\nT18 -> T22" }, { "from": 30, "to": 52, "label": "CASE" }, { "from": 31, "to": 1, "label": "INSTANCE with matching:\nT1 -> T22\nT2 -> T21\nT3 -> T15" }, { "from": 52, "to": 53, "label": "PARALLEL" }, { "from": 52, "to": 54, "label": "PARALLEL" }, { "from": 53, "to": 55, "label": "EVAL with clause\nappend([], X31, X31).\nand substitutionT16 -> [],\nX14 -> T29,\nX31 -> T29,\nT17 -> T29,\nX32 -> T29" }, { "from": 53, "to": 56, "label": "EVAL-BACKTRACK" }, { "from": 54, "to": 58, "label": "EVAL with clause\nappend(.(X43, X44), X45, .(X43, X46)) :- append(X44, X45, X46).\nand substitutionX43 -> T36,\nX44 -> T39,\nT16 -> .(T36, T39),\nX14 -> X47,\nX45 -> X47,\nX46 -> T40,\nT17 -> .(T36, T40),\nT37 -> T39,\nT38 -> T40" }, { "from": 54, "to": 59, "label": "EVAL-BACKTRACK" }, { "from": 55, "to": 57, "label": "SUCCESS" }, { "from": 58, "to": 30, "label": "INSTANCE with matching:\nT16 -> T39\nX14 -> X47\nT17 -> T40" } ], "type": "Graph" } } ---------------------------------------- (20) Complex Obligation (AND) ---------------------------------------- (21) Obligation: Rules: f52_in -> f53_in :|: TRUE f54_out -> f52_out :|: TRUE f52_in -> f54_in :|: TRUE f53_out -> f52_out :|: TRUE f52_out -> f30_out :|: TRUE f30_in -> f52_in :|: TRUE f58_out -> f54_out :|: TRUE f54_in -> f59_in :|: TRUE f59_out -> f54_out :|: TRUE f54_in -> f58_in :|: TRUE f58_in -> f30_in :|: TRUE f30_out -> f58_out :|: TRUE f1_in(T3) -> f6_in(T3) :|: TRUE f6_out(x) -> f1_out(x) :|: TRUE f6_in(x1) -> f10_in(x1) :|: TRUE f6_in(x2) -> f9_in(x2) :|: TRUE f9_out(x3) -> f6_out(x3) :|: TRUE f10_out(x4) -> f6_out(x4) :|: TRUE f21_out(T15) -> f10_out(s(T15)) :|: TRUE f10_in(x5) -> f22_in :|: TRUE f22_out -> f10_out(x6) :|: TRUE f10_in(s(x7)) -> f21_in(x7) :|: TRUE f31_out(x8) -> f21_out(x8) :|: TRUE f21_in(x9) -> f30_in :|: TRUE f30_out -> f31_in(x10) :|: TRUE Start term: f1_in(T3) ---------------------------------------- (22) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f52_in -> f54_in :|: TRUE f30_in -> f52_in :|: TRUE f54_in -> f58_in :|: TRUE f58_in -> f30_in :|: TRUE ---------------------------------------- (23) Obligation: Rules: f52_in -> f54_in :|: TRUE f30_in -> f52_in :|: TRUE f54_in -> f58_in :|: TRUE f58_in -> f30_in :|: TRUE ---------------------------------------- (24) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (25) Obligation: Rules: f30_in -> f30_in :|: TRUE ---------------------------------------- (26) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (27) Obligation: Rules: f30_in -> f30_in :|: TRUE ---------------------------------------- (28) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f30_in -> f30_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (29) Obligation: Termination digraph: Nodes: (1) f30_in -> f30_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (30) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f30_in() Replaced non-predefined constructor symbols by 0. ---------------------------------------- (31) Obligation: Rules: f30_in -> f30_in :|: TRUE ---------------------------------------- (32) 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)) ---------------------------------------- (33) NO ---------------------------------------- (34) Obligation: Rules: f55_in -> f55_out :|: TRUE f52_out -> f30_out :|: TRUE f30_in -> f52_in :|: TRUE f6_in(T3) -> f10_in(T3) :|: TRUE f6_in(x) -> f9_in(x) :|: TRUE f9_out(x1) -> f6_out(x1) :|: TRUE f10_out(x2) -> f6_out(x2) :|: TRUE f53_in -> f55_in :|: TRUE f55_out -> f53_out :|: TRUE f53_in -> f56_in :|: TRUE f56_out -> f53_out :|: TRUE f31_in(T15) -> f1_in(T15) :|: TRUE f1_out(x3) -> f31_out(x3) :|: TRUE f58_in -> f30_in :|: TRUE f30_out -> f58_out :|: TRUE f31_out(x4) -> f21_out(x4) :|: TRUE f21_in(x5) -> f30_in :|: TRUE f30_out -> f31_in(x6) :|: TRUE f52_in -> f53_in :|: TRUE f54_out -> f52_out :|: TRUE f52_in -> f54_in :|: TRUE f53_out -> f52_out :|: TRUE f21_out(x7) -> f10_out(s(x7)) :|: TRUE f10_in(x8) -> f22_in :|: TRUE f22_out -> f10_out(x9) :|: TRUE f10_in(s(x10)) -> f21_in(x10) :|: TRUE f1_in(x11) -> f6_in(x11) :|: TRUE f6_out(x12) -> f1_out(x12) :|: TRUE f58_out -> f54_out :|: TRUE f54_in -> f59_in :|: TRUE f59_out -> f54_out :|: TRUE f54_in -> f58_in :|: TRUE Start term: f1_in(T3) ---------------------------------------- (35) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f55_in -> f55_out :|: TRUE f52_out -> f30_out :|: TRUE f30_in -> f52_in :|: TRUE f6_in(T3) -> f10_in(T3) :|: TRUE f53_in -> f55_in :|: TRUE f55_out -> f53_out :|: TRUE f31_in(T15) -> f1_in(T15) :|: TRUE f58_in -> f30_in :|: TRUE f30_out -> f58_out :|: TRUE f21_in(x5) -> f30_in :|: TRUE f30_out -> f31_in(x6) :|: TRUE f52_in -> f53_in :|: TRUE f54_out -> f52_out :|: TRUE f52_in -> f54_in :|: TRUE f53_out -> f52_out :|: TRUE f10_in(s(x10)) -> f21_in(x10) :|: TRUE f1_in(x11) -> f6_in(x11) :|: TRUE f58_out -> f54_out :|: TRUE f54_in -> f58_in :|: TRUE ---------------------------------------- (36) Obligation: Rules: f55_in -> f55_out :|: TRUE f52_out -> f30_out :|: TRUE f30_in -> f52_in :|: TRUE f6_in(T3) -> f10_in(T3) :|: TRUE f53_in -> f55_in :|: TRUE f55_out -> f53_out :|: TRUE f31_in(T15) -> f1_in(T15) :|: TRUE f58_in -> f30_in :|: TRUE f30_out -> f58_out :|: TRUE f21_in(x5) -> f30_in :|: TRUE f30_out -> f31_in(x6) :|: TRUE f52_in -> f53_in :|: TRUE f54_out -> f52_out :|: TRUE f52_in -> f54_in :|: TRUE f53_out -> f52_out :|: TRUE f10_in(s(x10)) -> f21_in(x10) :|: TRUE f1_in(x11) -> f6_in(x11) :|: TRUE f58_out -> f54_out :|: TRUE f54_in -> f58_in :|: TRUE ---------------------------------------- (37) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (38) Obligation: Rules: f30_in -> f52_out :|: TRUE f52_out -> f30_in :|: TRUE f52_out -> f52_out :|: TRUE f30_in -> f30_in :|: TRUE ---------------------------------------- (39) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (40) Obligation: Rules: f30_in -> f52_out :|: TRUE f52_out -> f30_in :|: TRUE f52_out -> f52_out :|: TRUE f30_in -> f30_in :|: TRUE ---------------------------------------- (41) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f30_in -> f52_out :|: TRUE (2) f52_out -> f30_in :|: TRUE (3) f52_out -> f52_out :|: TRUE (4) f30_in -> f30_in :|: TRUE Arcs: (1) -> (2), (3) (2) -> (1), (4) (3) -> (2), (3) (4) -> (1), (4) This digraph is fully evaluated! ---------------------------------------- (42) Obligation: Termination digraph: Nodes: (1) f30_in -> f52_out :|: TRUE (2) f30_in -> f30_in :|: TRUE (3) f52_out -> f30_in :|: TRUE (4) f52_out -> f52_out :|: TRUE Arcs: (1) -> (3), (4) (2) -> (1), (2) (3) -> (1), (2) (4) -> (3), (4) This digraph is fully evaluated! ---------------------------------------- (43) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f30_in() f52_out() Replaced non-predefined constructor symbols by 0. ---------------------------------------- (44) Obligation: Rules: f30_in -> f52_out :|: TRUE f30_in -> f30_in :|: TRUE f52_out -> f30_in :|: TRUE f52_out -> f52_out :|: TRUE ---------------------------------------- (45) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: fl_in_3: (f,f,b) append_in_3: (f,f,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: fl_in_aag([], [], 0) -> fl_out_aag([], [], 0) fl_in_aag(.(E, X), R, s(Z)) -> U1_aag(E, X, R, Z, append_in_aaa(E, Y, R)) append_in_aaa([], X, X) -> append_out_aaa([], X, X) append_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U3_aaa(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) U3_aaa(X, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(X, Xs), Ys, .(X, Zs)) U1_aag(E, X, R, Z, append_out_aaa(E, Y, R)) -> U2_aag(E, X, R, Z, fl_in_aag(X, Y, Z)) U2_aag(E, X, R, Z, fl_out_aag(X, Y, Z)) -> fl_out_aag(.(E, X), R, s(Z)) The argument filtering Pi contains the following mapping: fl_in_aag(x1, x2, x3) = fl_in_aag(x3) 0 = 0 fl_out_aag(x1, x2, x3) = fl_out_aag s(x1) = s(x1) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x4, x5) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa U3_aaa(x1, x2, x3, x4, x5) = U3_aaa(x5) U2_aag(x1, x2, x3, x4, x5) = U2_aag(x5) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (46) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: fl_in_aag([], [], 0) -> fl_out_aag([], [], 0) fl_in_aag(.(E, X), R, s(Z)) -> U1_aag(E, X, R, Z, append_in_aaa(E, Y, R)) append_in_aaa([], X, X) -> append_out_aaa([], X, X) append_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U3_aaa(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) U3_aaa(X, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(X, Xs), Ys, .(X, Zs)) U1_aag(E, X, R, Z, append_out_aaa(E, Y, R)) -> U2_aag(E, X, R, Z, fl_in_aag(X, Y, Z)) U2_aag(E, X, R, Z, fl_out_aag(X, Y, Z)) -> fl_out_aag(.(E, X), R, s(Z)) The argument filtering Pi contains the following mapping: fl_in_aag(x1, x2, x3) = fl_in_aag(x3) 0 = 0 fl_out_aag(x1, x2, x3) = fl_out_aag s(x1) = s(x1) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x4, x5) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa U3_aaa(x1, x2, x3, x4, x5) = U3_aaa(x5) U2_aag(x1, x2, x3, x4, x5) = U2_aag(x5) ---------------------------------------- (47) 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: FL_IN_AAG(.(E, X), R, s(Z)) -> U1_AAG(E, X, R, Z, append_in_aaa(E, Y, R)) FL_IN_AAG(.(E, X), R, s(Z)) -> APPEND_IN_AAA(E, Y, R) APPEND_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> U3_AAA(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) APPEND_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AAA(Xs, Ys, Zs) U1_AAG(E, X, R, Z, append_out_aaa(E, Y, R)) -> U2_AAG(E, X, R, Z, fl_in_aag(X, Y, Z)) U1_AAG(E, X, R, Z, append_out_aaa(E, Y, R)) -> FL_IN_AAG(X, Y, Z) The TRS R consists of the following rules: fl_in_aag([], [], 0) -> fl_out_aag([], [], 0) fl_in_aag(.(E, X), R, s(Z)) -> U1_aag(E, X, R, Z, append_in_aaa(E, Y, R)) append_in_aaa([], X, X) -> append_out_aaa([], X, X) append_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U3_aaa(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) U3_aaa(X, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(X, Xs), Ys, .(X, Zs)) U1_aag(E, X, R, Z, append_out_aaa(E, Y, R)) -> U2_aag(E, X, R, Z, fl_in_aag(X, Y, Z)) U2_aag(E, X, R, Z, fl_out_aag(X, Y, Z)) -> fl_out_aag(.(E, X), R, s(Z)) The argument filtering Pi contains the following mapping: fl_in_aag(x1, x2, x3) = fl_in_aag(x3) 0 = 0 fl_out_aag(x1, x2, x3) = fl_out_aag s(x1) = s(x1) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x4, x5) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa U3_aaa(x1, x2, x3, x4, x5) = U3_aaa(x5) U2_aag(x1, x2, x3, x4, x5) = U2_aag(x5) FL_IN_AAG(x1, x2, x3) = FL_IN_AAG(x3) U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x4, x5) APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA U3_AAA(x1, x2, x3, x4, x5) = U3_AAA(x5) U2_AAG(x1, x2, x3, x4, x5) = U2_AAG(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (48) Obligation: Pi DP problem: The TRS P consists of the following rules: FL_IN_AAG(.(E, X), R, s(Z)) -> U1_AAG(E, X, R, Z, append_in_aaa(E, Y, R)) FL_IN_AAG(.(E, X), R, s(Z)) -> APPEND_IN_AAA(E, Y, R) APPEND_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> U3_AAA(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) APPEND_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AAA(Xs, Ys, Zs) U1_AAG(E, X, R, Z, append_out_aaa(E, Y, R)) -> U2_AAG(E, X, R, Z, fl_in_aag(X, Y, Z)) U1_AAG(E, X, R, Z, append_out_aaa(E, Y, R)) -> FL_IN_AAG(X, Y, Z) The TRS R consists of the following rules: fl_in_aag([], [], 0) -> fl_out_aag([], [], 0) fl_in_aag(.(E, X), R, s(Z)) -> U1_aag(E, X, R, Z, append_in_aaa(E, Y, R)) append_in_aaa([], X, X) -> append_out_aaa([], X, X) append_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U3_aaa(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) U3_aaa(X, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(X, Xs), Ys, .(X, Zs)) U1_aag(E, X, R, Z, append_out_aaa(E, Y, R)) -> U2_aag(E, X, R, Z, fl_in_aag(X, Y, Z)) U2_aag(E, X, R, Z, fl_out_aag(X, Y, Z)) -> fl_out_aag(.(E, X), R, s(Z)) The argument filtering Pi contains the following mapping: fl_in_aag(x1, x2, x3) = fl_in_aag(x3) 0 = 0 fl_out_aag(x1, x2, x3) = fl_out_aag s(x1) = s(x1) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x4, x5) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa U3_aaa(x1, x2, x3, x4, x5) = U3_aaa(x5) U2_aag(x1, x2, x3, x4, x5) = U2_aag(x5) FL_IN_AAG(x1, x2, x3) = FL_IN_AAG(x3) U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x4, x5) APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA U3_AAA(x1, x2, x3, x4, x5) = U3_AAA(x5) U2_AAG(x1, x2, x3, x4, x5) = U2_AAG(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (49) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 3 less nodes. ---------------------------------------- (50) Complex Obligation (AND) ---------------------------------------- (51) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AAA(Xs, Ys, Zs) The TRS R consists of the following rules: fl_in_aag([], [], 0) -> fl_out_aag([], [], 0) fl_in_aag(.(E, X), R, s(Z)) -> U1_aag(E, X, R, Z, append_in_aaa(E, Y, R)) append_in_aaa([], X, X) -> append_out_aaa([], X, X) append_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U3_aaa(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) U3_aaa(X, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(X, Xs), Ys, .(X, Zs)) U1_aag(E, X, R, Z, append_out_aaa(E, Y, R)) -> U2_aag(E, X, R, Z, fl_in_aag(X, Y, Z)) U2_aag(E, X, R, Z, fl_out_aag(X, Y, Z)) -> fl_out_aag(.(E, X), R, s(Z)) The argument filtering Pi contains the following mapping: fl_in_aag(x1, x2, x3) = fl_in_aag(x3) 0 = 0 fl_out_aag(x1, x2, x3) = fl_out_aag s(x1) = s(x1) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x4, x5) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa U3_aaa(x1, x2, x3, x4, x5) = U3_aaa(x5) U2_aag(x1, x2, x3, x4, x5) = U2_aag(x5) APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (52) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (53) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AAA(Xs, Ys, Zs) R is empty. The argument filtering Pi contains the following mapping: APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (54) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (55) Obligation: Q DP problem: The TRS P consists of the following rules: APPEND_IN_AAA -> APPEND_IN_AAA R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (56) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by semiunifying a rule from P directly. s = APPEND_IN_AAA evaluates to t =APPEND_IN_AAA Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence The DP semiunifies directly so there is only one rewrite step from APPEND_IN_AAA to APPEND_IN_AAA. ---------------------------------------- (57) NO ---------------------------------------- (58) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_AAG(E, X, R, Z, append_out_aaa(E, Y, R)) -> FL_IN_AAG(X, Y, Z) FL_IN_AAG(.(E, X), R, s(Z)) -> U1_AAG(E, X, R, Z, append_in_aaa(E, Y, R)) The TRS R consists of the following rules: fl_in_aag([], [], 0) -> fl_out_aag([], [], 0) fl_in_aag(.(E, X), R, s(Z)) -> U1_aag(E, X, R, Z, append_in_aaa(E, Y, R)) append_in_aaa([], X, X) -> append_out_aaa([], X, X) append_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U3_aaa(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) U3_aaa(X, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(X, Xs), Ys, .(X, Zs)) U1_aag(E, X, R, Z, append_out_aaa(E, Y, R)) -> U2_aag(E, X, R, Z, fl_in_aag(X, Y, Z)) U2_aag(E, X, R, Z, fl_out_aag(X, Y, Z)) -> fl_out_aag(.(E, X), R, s(Z)) The argument filtering Pi contains the following mapping: fl_in_aag(x1, x2, x3) = fl_in_aag(x3) 0 = 0 fl_out_aag(x1, x2, x3) = fl_out_aag s(x1) = s(x1) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x4, x5) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa U3_aaa(x1, x2, x3, x4, x5) = U3_aaa(x5) U2_aag(x1, x2, x3, x4, x5) = U2_aag(x5) FL_IN_AAG(x1, x2, x3) = FL_IN_AAG(x3) U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (59) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (60) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_AAG(E, X, R, Z, append_out_aaa(E, Y, R)) -> FL_IN_AAG(X, Y, Z) FL_IN_AAG(.(E, X), R, s(Z)) -> U1_AAG(E, X, R, Z, append_in_aaa(E, Y, R)) The TRS R consists of the following rules: append_in_aaa([], X, X) -> append_out_aaa([], X, X) append_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U3_aaa(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) U3_aaa(X, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(X, Xs), Ys, .(X, Zs)) The argument filtering Pi contains the following mapping: s(x1) = s(x1) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa U3_aaa(x1, x2, x3, x4, x5) = U3_aaa(x5) FL_IN_AAG(x1, x2, x3) = FL_IN_AAG(x3) U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (61) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (62) Obligation: Q DP problem: The TRS P consists of the following rules: U1_AAG(Z, append_out_aaa) -> FL_IN_AAG(Z) FL_IN_AAG(s(Z)) -> U1_AAG(Z, append_in_aaa) The TRS R consists of the following rules: append_in_aaa -> append_out_aaa append_in_aaa -> U3_aaa(append_in_aaa) U3_aaa(append_out_aaa) -> append_out_aaa The set Q consists of the following terms: append_in_aaa U3_aaa(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (63) 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: *FL_IN_AAG(s(Z)) -> U1_AAG(Z, append_in_aaa) The graph contains the following edges 1 > 1 *U1_AAG(Z, append_out_aaa) -> FL_IN_AAG(Z) The graph contains the following edges 1 >= 1 ---------------------------------------- (64) YES ---------------------------------------- (65) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: fl_in_3: (f,f,b) append_in_3: (f,f,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: fl_in_aag([], [], 0) -> fl_out_aag([], [], 0) fl_in_aag(.(E, X), R, s(Z)) -> U1_aag(E, X, R, Z, append_in_aaa(E, Y, R)) append_in_aaa([], X, X) -> append_out_aaa([], X, X) append_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U3_aaa(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) U3_aaa(X, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(X, Xs), Ys, .(X, Zs)) U1_aag(E, X, R, Z, append_out_aaa(E, Y, R)) -> U2_aag(E, X, R, Z, fl_in_aag(X, Y, Z)) U2_aag(E, X, R, Z, fl_out_aag(X, Y, Z)) -> fl_out_aag(.(E, X), R, s(Z)) The argument filtering Pi contains the following mapping: fl_in_aag(x1, x2, x3) = fl_in_aag(x3) 0 = 0 fl_out_aag(x1, x2, x3) = fl_out_aag(x3) s(x1) = s(x1) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x4, x5) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa U3_aaa(x1, x2, x3, x4, x5) = U3_aaa(x5) U2_aag(x1, x2, x3, x4, x5) = U2_aag(x4, x5) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (66) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: fl_in_aag([], [], 0) -> fl_out_aag([], [], 0) fl_in_aag(.(E, X), R, s(Z)) -> U1_aag(E, X, R, Z, append_in_aaa(E, Y, R)) append_in_aaa([], X, X) -> append_out_aaa([], X, X) append_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U3_aaa(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) U3_aaa(X, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(X, Xs), Ys, .(X, Zs)) U1_aag(E, X, R, Z, append_out_aaa(E, Y, R)) -> U2_aag(E, X, R, Z, fl_in_aag(X, Y, Z)) U2_aag(E, X, R, Z, fl_out_aag(X, Y, Z)) -> fl_out_aag(.(E, X), R, s(Z)) The argument filtering Pi contains the following mapping: fl_in_aag(x1, x2, x3) = fl_in_aag(x3) 0 = 0 fl_out_aag(x1, x2, x3) = fl_out_aag(x3) s(x1) = s(x1) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x4, x5) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa U3_aaa(x1, x2, x3, x4, x5) = U3_aaa(x5) U2_aag(x1, x2, x3, x4, x5) = U2_aag(x4, x5) ---------------------------------------- (67) 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: FL_IN_AAG(.(E, X), R, s(Z)) -> U1_AAG(E, X, R, Z, append_in_aaa(E, Y, R)) FL_IN_AAG(.(E, X), R, s(Z)) -> APPEND_IN_AAA(E, Y, R) APPEND_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> U3_AAA(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) APPEND_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AAA(Xs, Ys, Zs) U1_AAG(E, X, R, Z, append_out_aaa(E, Y, R)) -> U2_AAG(E, X, R, Z, fl_in_aag(X, Y, Z)) U1_AAG(E, X, R, Z, append_out_aaa(E, Y, R)) -> FL_IN_AAG(X, Y, Z) The TRS R consists of the following rules: fl_in_aag([], [], 0) -> fl_out_aag([], [], 0) fl_in_aag(.(E, X), R, s(Z)) -> U1_aag(E, X, R, Z, append_in_aaa(E, Y, R)) append_in_aaa([], X, X) -> append_out_aaa([], X, X) append_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U3_aaa(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) U3_aaa(X, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(X, Xs), Ys, .(X, Zs)) U1_aag(E, X, R, Z, append_out_aaa(E, Y, R)) -> U2_aag(E, X, R, Z, fl_in_aag(X, Y, Z)) U2_aag(E, X, R, Z, fl_out_aag(X, Y, Z)) -> fl_out_aag(.(E, X), R, s(Z)) The argument filtering Pi contains the following mapping: fl_in_aag(x1, x2, x3) = fl_in_aag(x3) 0 = 0 fl_out_aag(x1, x2, x3) = fl_out_aag(x3) s(x1) = s(x1) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x4, x5) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa U3_aaa(x1, x2, x3, x4, x5) = U3_aaa(x5) U2_aag(x1, x2, x3, x4, x5) = U2_aag(x4, x5) FL_IN_AAG(x1, x2, x3) = FL_IN_AAG(x3) U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x4, x5) APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA U3_AAA(x1, x2, x3, x4, x5) = U3_AAA(x5) U2_AAG(x1, x2, x3, x4, x5) = U2_AAG(x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (68) Obligation: Pi DP problem: The TRS P consists of the following rules: FL_IN_AAG(.(E, X), R, s(Z)) -> U1_AAG(E, X, R, Z, append_in_aaa(E, Y, R)) FL_IN_AAG(.(E, X), R, s(Z)) -> APPEND_IN_AAA(E, Y, R) APPEND_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> U3_AAA(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) APPEND_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AAA(Xs, Ys, Zs) U1_AAG(E, X, R, Z, append_out_aaa(E, Y, R)) -> U2_AAG(E, X, R, Z, fl_in_aag(X, Y, Z)) U1_AAG(E, X, R, Z, append_out_aaa(E, Y, R)) -> FL_IN_AAG(X, Y, Z) The TRS R consists of the following rules: fl_in_aag([], [], 0) -> fl_out_aag([], [], 0) fl_in_aag(.(E, X), R, s(Z)) -> U1_aag(E, X, R, Z, append_in_aaa(E, Y, R)) append_in_aaa([], X, X) -> append_out_aaa([], X, X) append_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U3_aaa(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) U3_aaa(X, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(X, Xs), Ys, .(X, Zs)) U1_aag(E, X, R, Z, append_out_aaa(E, Y, R)) -> U2_aag(E, X, R, Z, fl_in_aag(X, Y, Z)) U2_aag(E, X, R, Z, fl_out_aag(X, Y, Z)) -> fl_out_aag(.(E, X), R, s(Z)) The argument filtering Pi contains the following mapping: fl_in_aag(x1, x2, x3) = fl_in_aag(x3) 0 = 0 fl_out_aag(x1, x2, x3) = fl_out_aag(x3) s(x1) = s(x1) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x4, x5) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa U3_aaa(x1, x2, x3, x4, x5) = U3_aaa(x5) U2_aag(x1, x2, x3, x4, x5) = U2_aag(x4, x5) FL_IN_AAG(x1, x2, x3) = FL_IN_AAG(x3) U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x4, x5) APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA U3_AAA(x1, x2, x3, x4, x5) = U3_AAA(x5) U2_AAG(x1, x2, x3, x4, x5) = U2_AAG(x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (69) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 3 less nodes. ---------------------------------------- (70) Complex Obligation (AND) ---------------------------------------- (71) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AAA(Xs, Ys, Zs) The TRS R consists of the following rules: fl_in_aag([], [], 0) -> fl_out_aag([], [], 0) fl_in_aag(.(E, X), R, s(Z)) -> U1_aag(E, X, R, Z, append_in_aaa(E, Y, R)) append_in_aaa([], X, X) -> append_out_aaa([], X, X) append_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U3_aaa(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) U3_aaa(X, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(X, Xs), Ys, .(X, Zs)) U1_aag(E, X, R, Z, append_out_aaa(E, Y, R)) -> U2_aag(E, X, R, Z, fl_in_aag(X, Y, Z)) U2_aag(E, X, R, Z, fl_out_aag(X, Y, Z)) -> fl_out_aag(.(E, X), R, s(Z)) The argument filtering Pi contains the following mapping: fl_in_aag(x1, x2, x3) = fl_in_aag(x3) 0 = 0 fl_out_aag(x1, x2, x3) = fl_out_aag(x3) s(x1) = s(x1) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x4, x5) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa U3_aaa(x1, x2, x3, x4, x5) = U3_aaa(x5) U2_aag(x1, x2, x3, x4, x5) = U2_aag(x4, x5) APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (72) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (73) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AAA(Xs, Ys, Zs) R is empty. The argument filtering Pi contains the following mapping: APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (74) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (75) Obligation: Q DP problem: The TRS P consists of the following rules: APPEND_IN_AAA -> APPEND_IN_AAA R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (76) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by semiunifying a rule from P directly. s = APPEND_IN_AAA evaluates to t =APPEND_IN_AAA Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence The DP semiunifies directly so there is only one rewrite step from APPEND_IN_AAA to APPEND_IN_AAA. ---------------------------------------- (77) NO ---------------------------------------- (78) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_AAG(E, X, R, Z, append_out_aaa(E, Y, R)) -> FL_IN_AAG(X, Y, Z) FL_IN_AAG(.(E, X), R, s(Z)) -> U1_AAG(E, X, R, Z, append_in_aaa(E, Y, R)) The TRS R consists of the following rules: fl_in_aag([], [], 0) -> fl_out_aag([], [], 0) fl_in_aag(.(E, X), R, s(Z)) -> U1_aag(E, X, R, Z, append_in_aaa(E, Y, R)) append_in_aaa([], X, X) -> append_out_aaa([], X, X) append_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U3_aaa(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) U3_aaa(X, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(X, Xs), Ys, .(X, Zs)) U1_aag(E, X, R, Z, append_out_aaa(E, Y, R)) -> U2_aag(E, X, R, Z, fl_in_aag(X, Y, Z)) U2_aag(E, X, R, Z, fl_out_aag(X, Y, Z)) -> fl_out_aag(.(E, X), R, s(Z)) The argument filtering Pi contains the following mapping: fl_in_aag(x1, x2, x3) = fl_in_aag(x3) 0 = 0 fl_out_aag(x1, x2, x3) = fl_out_aag(x3) s(x1) = s(x1) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x4, x5) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa U3_aaa(x1, x2, x3, x4, x5) = U3_aaa(x5) U2_aag(x1, x2, x3, x4, x5) = U2_aag(x4, x5) FL_IN_AAG(x1, x2, x3) = FL_IN_AAG(x3) U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (79) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (80) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_AAG(E, X, R, Z, append_out_aaa(E, Y, R)) -> FL_IN_AAG(X, Y, Z) FL_IN_AAG(.(E, X), R, s(Z)) -> U1_AAG(E, X, R, Z, append_in_aaa(E, Y, R)) The TRS R consists of the following rules: append_in_aaa([], X, X) -> append_out_aaa([], X, X) append_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U3_aaa(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) U3_aaa(X, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(X, Xs), Ys, .(X, Zs)) The argument filtering Pi contains the following mapping: s(x1) = s(x1) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa U3_aaa(x1, x2, x3, x4, x5) = U3_aaa(x5) FL_IN_AAG(x1, x2, x3) = FL_IN_AAG(x3) U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (81) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (82) Obligation: Q DP problem: The TRS P consists of the following rules: U1_AAG(Z, append_out_aaa) -> FL_IN_AAG(Z) FL_IN_AAG(s(Z)) -> U1_AAG(Z, append_in_aaa) The TRS R consists of the following rules: append_in_aaa -> append_out_aaa append_in_aaa -> U3_aaa(append_in_aaa) U3_aaa(append_out_aaa) -> append_out_aaa The set Q consists of the following terms: append_in_aaa U3_aaa(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (83) 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: *FL_IN_AAG(s(Z)) -> U1_AAG(Z, append_in_aaa) The graph contains the following edges 1 > 1 *U1_AAG(Z, append_out_aaa) -> FL_IN_AAG(Z) The graph contains the following edges 1 >= 1 ---------------------------------------- (84) YES ---------------------------------------- (85) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 3, "program": { "directives": [], "clauses": [ [ "(fl ([]) ([]) (0))", null ], [ "(fl (. E X) R (s Z))", "(',' (append E Y R) (fl X Y Z))" ], [ "(append ([]) X X)", null ], [ "(append (. X Xs) Ys (. X Zs))", "(append Xs Ys Zs)" ] ] }, "graph": { "nodes": { "23": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (append T15 X13 T16) (fl T17 X13 T14))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T14"], "free": ["X13"], "exprvars": [] } }, "24": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "68": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (append T34 X38 T35) (fl T36 X38 T14))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T14"], "free": ["X38"], "exprvars": [] } }, "25": { "goal": [ { "clause": 2, "scope": 2, "term": "(',' (append T15 X13 T16) (fl T17 X13 T14))" }, { "clause": 3, "scope": 2, "term": "(',' (append T15 X13 T16) (fl T17 X13 T14))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T14"], "free": ["X13"], "exprvars": [] } }, "69": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "15": { "goal": [ { "clause": -1, "scope": -1, "term": "(true)" }, { "clause": 1, "scope": 1, "term": "(fl T1 T2 (0))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "26": { "goal": [{ "clause": 2, "scope": 2, "term": "(',' (append T15 X13 T16) (fl T17 X13 T14))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T14"], "free": ["X13"], "exprvars": [] } }, "27": { "goal": [{ "clause": 3, "scope": 2, "term": "(',' (append T15 X13 T16) (fl T17 X13 T14))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T14"], "free": ["X13"], "exprvars": [] } }, "28": { "goal": [{ "clause": -1, "scope": -1, "term": "(fl T23 T24 T14)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T14"], "free": [], "exprvars": [] } }, "18": { "goal": [{ "clause": 1, "scope": 1, "term": "(fl T1 T2 T3)" }], "kb": { "nonunifying": [[ "(fl T1 T2 T3)", "(fl ([]) ([]) (0))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } }, "29": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "19": { "goal": [{ "clause": 1, "scope": 1, "term": "(fl T1 T2 (0))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "3": { "goal": [{ "clause": -1, "scope": -1, "term": "(fl T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } }, "4": { "goal": [ { "clause": 0, "scope": 1, "term": "(fl T1 T2 T3)" }, { "clause": 1, "scope": 1, "term": "(fl T1 T2 T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } }, "20": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 3, "to": 4, "label": "CASE" }, { "from": 4, "to": 15, "label": "EVAL with clause\nfl([], [], 0).\nand substitutionT1 -> [],\nT2 -> [],\nT3 -> 0" }, { "from": 4, "to": 18, "label": "EVAL-BACKTRACK" }, { "from": 15, "to": 19, "label": "SUCCESS" }, { "from": 18, "to": 23, "label": "EVAL with clause\nfl(.(X9, X10), X11, s(X12)) :- ','(append(X9, X13, X11), fl(X10, X13, X12)).\nand substitutionX9 -> T15,\nX10 -> T17,\nT1 -> .(T15, T17),\nT2 -> T16,\nX11 -> T16,\nX12 -> T14,\nT3 -> s(T14),\nT11 -> T15,\nT13 -> T16,\nT12 -> T17" }, { "from": 18, "to": 24, "label": "EVAL-BACKTRACK" }, { "from": 19, "to": 20, "label": "BACKTRACK\nfor clause: fl(.(E, X), R, s(Z)) :- ','(append(E, Y, R), fl(X, Y, Z))because of non-unification" }, { "from": 23, "to": 25, "label": "CASE" }, { "from": 25, "to": 26, "label": "PARALLEL" }, { "from": 25, "to": 27, "label": "PARALLEL" }, { "from": 26, "to": 28, "label": "EVAL with clause\nappend([], X22, X22).\nand substitutionT15 -> [],\nX13 -> T24,\nX22 -> T24,\nT16 -> T24,\nX23 -> T24,\nT17 -> T23,\nT22 -> T24" }, { "from": 26, "to": 29, "label": "EVAL-BACKTRACK" }, { "from": 27, "to": 68, "label": "EVAL with clause\nappend(.(X34, X35), X36, .(X34, X37)) :- append(X35, X36, X37).\nand substitutionX34 -> T31,\nX35 -> T34,\nT15 -> .(T31, T34),\nX13 -> X38,\nX36 -> X38,\nX37 -> T35,\nT16 -> .(T31, T35),\nT32 -> T34,\nT33 -> T35,\nT17 -> T36" }, { "from": 27, "to": 69, "label": "EVAL-BACKTRACK" }, { "from": 28, "to": 3, "label": "INSTANCE with matching:\nT1 -> T23\nT2 -> T24\nT3 -> T14" }, { "from": 68, "to": 23, "label": "INSTANCE with matching:\nT15 -> T34\nX13 -> X38\nT16 -> T35\nT17 -> T36" } ], "type": "Graph" } } ---------------------------------------- (86) Obligation: Triples: pB([], X1, X1, X2, X3) :- flA(X2, X1, X3). pB(.(X1, X2), X3, .(X1, X4), X5, X6) :- pB(X2, X3, X4, X5, X6). flA(.(X1, X2), X3, s(X4)) :- pB(X1, X5, X3, X2, X4). Clauses: flcA([], [], 0). flcA(.(X1, X2), X3, s(X4)) :- qcB(X1, X5, X3, X2, X4). qcB([], X1, X1, X2, X3) :- flcA(X2, X1, X3). qcB(.(X1, X2), X3, .(X1, X4), X5, X6) :- qcB(X2, X3, X4, X5, X6). Afs: flA(x1, x2, x3) = flA(x3) ---------------------------------------- (87) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: flA_in_3: (f,f,b) pB_in_5: (f,f,f,f,b) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: FLA_IN_AAG(.(X1, X2), X3, s(X4)) -> U3_AAG(X1, X2, X3, X4, pB_in_aaaag(X1, X5, X3, X2, X4)) FLA_IN_AAG(.(X1, X2), X3, s(X4)) -> PB_IN_AAAAG(X1, X5, X3, X2, X4) PB_IN_AAAAG([], X1, X1, X2, X3) -> U1_AAAAG(X1, X2, X3, flA_in_aag(X2, X1, X3)) PB_IN_AAAAG([], X1, X1, X2, X3) -> FLA_IN_AAG(X2, X1, X3) PB_IN_AAAAG(.(X1, X2), X3, .(X1, X4), X5, X6) -> U2_AAAAG(X1, X2, X3, X4, X5, X6, pB_in_aaaag(X2, X3, X4, X5, X6)) PB_IN_AAAAG(.(X1, X2), X3, .(X1, X4), X5, X6) -> PB_IN_AAAAG(X2, X3, X4, X5, X6) R is empty. The argument filtering Pi contains the following mapping: flA_in_aag(x1, x2, x3) = flA_in_aag(x3) s(x1) = s(x1) pB_in_aaaag(x1, x2, x3, x4, x5) = pB_in_aaaag(x5) FLA_IN_AAG(x1, x2, x3) = FLA_IN_AAG(x3) U3_AAG(x1, x2, x3, x4, x5) = U3_AAG(x4, x5) PB_IN_AAAAG(x1, x2, x3, x4, x5) = PB_IN_AAAAG(x5) U1_AAAAG(x1, x2, x3, x4) = U1_AAAAG(x3, x4) U2_AAAAG(x1, x2, x3, x4, x5, x6, x7) = U2_AAAAG(x6, x7) We have to consider all (P,R,Pi)-chains Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES ---------------------------------------- (88) Obligation: Pi DP problem: The TRS P consists of the following rules: FLA_IN_AAG(.(X1, X2), X3, s(X4)) -> U3_AAG(X1, X2, X3, X4, pB_in_aaaag(X1, X5, X3, X2, X4)) FLA_IN_AAG(.(X1, X2), X3, s(X4)) -> PB_IN_AAAAG(X1, X5, X3, X2, X4) PB_IN_AAAAG([], X1, X1, X2, X3) -> U1_AAAAG(X1, X2, X3, flA_in_aag(X2, X1, X3)) PB_IN_AAAAG([], X1, X1, X2, X3) -> FLA_IN_AAG(X2, X1, X3) PB_IN_AAAAG(.(X1, X2), X3, .(X1, X4), X5, X6) -> U2_AAAAG(X1, X2, X3, X4, X5, X6, pB_in_aaaag(X2, X3, X4, X5, X6)) PB_IN_AAAAG(.(X1, X2), X3, .(X1, X4), X5, X6) -> PB_IN_AAAAG(X2, X3, X4, X5, X6) R is empty. The argument filtering Pi contains the following mapping: flA_in_aag(x1, x2, x3) = flA_in_aag(x3) s(x1) = s(x1) pB_in_aaaag(x1, x2, x3, x4, x5) = pB_in_aaaag(x5) FLA_IN_AAG(x1, x2, x3) = FLA_IN_AAG(x3) U3_AAG(x1, x2, x3, x4, x5) = U3_AAG(x4, x5) PB_IN_AAAAG(x1, x2, x3, x4, x5) = PB_IN_AAAAG(x5) U1_AAAAG(x1, x2, x3, x4) = U1_AAAAG(x3, x4) U2_AAAAG(x1, x2, x3, x4, x5, x6, x7) = U2_AAAAG(x6, x7) We have to consider all (P,R,Pi)-chains ---------------------------------------- (89) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes. ---------------------------------------- (90) Obligation: Pi DP problem: The TRS P consists of the following rules: FLA_IN_AAG(.(X1, X2), X3, s(X4)) -> PB_IN_AAAAG(X1, X5, X3, X2, X4) PB_IN_AAAAG([], X1, X1, X2, X3) -> FLA_IN_AAG(X2, X1, X3) PB_IN_AAAAG(.(X1, X2), X3, .(X1, X4), X5, X6) -> PB_IN_AAAAG(X2, X3, X4, X5, X6) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) FLA_IN_AAG(x1, x2, x3) = FLA_IN_AAG(x3) PB_IN_AAAAG(x1, x2, x3, x4, x5) = PB_IN_AAAAG(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (91) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (92) Obligation: Q DP problem: The TRS P consists of the following rules: FLA_IN_AAG(s(X4)) -> PB_IN_AAAAG(X4) PB_IN_AAAAG(X3) -> FLA_IN_AAG(X3) PB_IN_AAAAG(X6) -> PB_IN_AAAAG(X6) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (93) UsableRulesReductionPairsProof (EQUIVALENT) By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well. The following dependency pairs can be deleted: FLA_IN_AAG(s(X4)) -> PB_IN_AAAAG(X4) No rules are removed from R. Used ordering: POLO with Polynomial interpretation [POLO]: POL(FLA_IN_AAG(x_1)) = 2 + x_1 POL(PB_IN_AAAAG(x_1)) = 2 + x_1 POL(s(x_1)) = 2 + 2*x_1 ---------------------------------------- (94) Obligation: Q DP problem: The TRS P consists of the following rules: PB_IN_AAAAG(X3) -> FLA_IN_AAG(X3) PB_IN_AAAAG(X6) -> PB_IN_AAAAG(X6) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (95) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (96) Obligation: Q DP problem: The TRS P consists of the following rules: PB_IN_AAAAG(X6) -> PB_IN_AAAAG(X6) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (97) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by semiunifying a rule from P directly. s = PB_IN_AAAAG(X6) evaluates to t =PB_IN_AAAAG(X6) Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence The DP semiunifies directly so there is only one rewrite step from PB_IN_AAAAG(X6) to PB_IN_AAAAG(X6). ---------------------------------------- (98) NO