/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern append(a,a,a) w.r.t. the given Prolog program could not be shown: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2) PiTRS (3) DependencyPairsProof [EQUIVALENT, 0 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) PiDP (7) UsableRulesProof [EQUIVALENT, 0 ms] (8) PiDP (9) PiDPToQDPProof [SOUND, 0 ms] (10) QDP (11) PrologToPiTRSProof [SOUND, 0 ms] (12) PiTRS (13) DependencyPairsProof [EQUIVALENT, 0 ms] (14) PiDP (15) DependencyGraphProof [EQUIVALENT, 0 ms] (16) PiDP (17) UsableRulesProof [EQUIVALENT, 0 ms] (18) PiDP (19) PiDPToQDPProof [SOUND, 0 ms] (20) QDP (21) PrologToTRSTransformerProof [SOUND, 0 ms] (22) QTRS (23) QTRSRRRProof [EQUIVALENT, 0 ms] (24) QTRS (25) QTRSRRRProof [EQUIVALENT, 4 ms] (26) QTRS (27) Overlay + Local Confluence [EQUIVALENT, 0 ms] (28) QTRS (29) DependencyPairsProof [EQUIVALENT, 0 ms] (30) QDP (31) UsableRulesProof [EQUIVALENT, 0 ms] (32) QDP (33) QReductionProof [EQUIVALENT, 0 ms] (34) QDP (35) PrologToIRSwTTransformerProof [SOUND, 0 ms] (36) IRSwT (37) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (38) IRSwT (39) IntTRSCompressionProof [EQUIVALENT, 21 ms] (40) IRSwT (41) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (42) IRSwT (43) IRSwTTerminationDigraphProof [EQUIVALENT, 6 ms] (44) IRSwT (45) FilterProof [EQUIVALENT, 0 ms] (46) IntTRS (47) IntTRSPeriodicNontermProof [COMPLETE, 7 ms] (48) NO (49) PrologToDTProblemTransformerProof [SOUND, 0 ms] (50) TRIPLES (51) TriplesToPiDPProof [SOUND, 0 ms] (52) PiDP (53) DependencyGraphProof [EQUIVALENT, 0 ms] (54) PiDP (55) PiDPToQDPProof [SOUND, 0 ms] (56) QDP ---------------------------------------- (0) Obligation: Clauses: append([], L, L). append(.(H, L1), L2, .(H, L3)) :- append(L1, L2, L3). append1([], L, L). append1(.(H, L1), L2, .(H, L3)) :- append1(L1, L2, L3). Query: append(a,a,a) ---------------------------------------- (1) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 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: append_in_aaa([], L, L) -> append_out_aaa([], L, L) append_in_aaa(.(H, L1), L2, .(H, L3)) -> U1_aaa(H, L1, L2, L3, append_in_aaa(L1, L2, L3)) U1_aaa(H, L1, L2, L3, append_out_aaa(L1, L2, L3)) -> append_out_aaa(.(H, L1), L2, .(H, L3)) The argument filtering Pi contains the following mapping: append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa(x1) U1_aaa(x1, x2, x3, x4, x5) = U1_aaa(x5) .(x1, x2) = .(x2) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: append_in_aaa([], L, L) -> append_out_aaa([], L, L) append_in_aaa(.(H, L1), L2, .(H, L3)) -> U1_aaa(H, L1, L2, L3, append_in_aaa(L1, L2, L3)) U1_aaa(H, L1, L2, L3, append_out_aaa(L1, L2, L3)) -> append_out_aaa(.(H, L1), L2, .(H, L3)) The argument filtering Pi contains the following mapping: append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa(x1) U1_aaa(x1, x2, x3, x4, x5) = U1_aaa(x5) .(x1, x2) = .(x2) ---------------------------------------- (3) 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: APPEND_IN_AAA(.(H, L1), L2, .(H, L3)) -> U1_AAA(H, L1, L2, L3, append_in_aaa(L1, L2, L3)) APPEND_IN_AAA(.(H, L1), L2, .(H, L3)) -> APPEND_IN_AAA(L1, L2, L3) The TRS R consists of the following rules: append_in_aaa([], L, L) -> append_out_aaa([], L, L) append_in_aaa(.(H, L1), L2, .(H, L3)) -> U1_aaa(H, L1, L2, L3, append_in_aaa(L1, L2, L3)) U1_aaa(H, L1, L2, L3, append_out_aaa(L1, L2, L3)) -> append_out_aaa(.(H, L1), L2, .(H, L3)) The argument filtering Pi contains the following mapping: append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa(x1) U1_aaa(x1, x2, x3, x4, x5) = U1_aaa(x5) .(x1, x2) = .(x2) APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA U1_AAA(x1, x2, x3, x4, x5) = U1_AAA(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND_IN_AAA(.(H, L1), L2, .(H, L3)) -> U1_AAA(H, L1, L2, L3, append_in_aaa(L1, L2, L3)) APPEND_IN_AAA(.(H, L1), L2, .(H, L3)) -> APPEND_IN_AAA(L1, L2, L3) The TRS R consists of the following rules: append_in_aaa([], L, L) -> append_out_aaa([], L, L) append_in_aaa(.(H, L1), L2, .(H, L3)) -> U1_aaa(H, L1, L2, L3, append_in_aaa(L1, L2, L3)) U1_aaa(H, L1, L2, L3, append_out_aaa(L1, L2, L3)) -> append_out_aaa(.(H, L1), L2, .(H, L3)) The argument filtering Pi contains the following mapping: append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa(x1) U1_aaa(x1, x2, x3, x4, x5) = U1_aaa(x5) .(x1, x2) = .(x2) APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA U1_AAA(x1, x2, x3, x4, x5) = U1_AAA(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node. ---------------------------------------- (6) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND_IN_AAA(.(H, L1), L2, .(H, L3)) -> APPEND_IN_AAA(L1, L2, L3) The TRS R consists of the following rules: append_in_aaa([], L, L) -> append_out_aaa([], L, L) append_in_aaa(.(H, L1), L2, .(H, L3)) -> U1_aaa(H, L1, L2, L3, append_in_aaa(L1, L2, L3)) U1_aaa(H, L1, L2, L3, append_out_aaa(L1, L2, L3)) -> append_out_aaa(.(H, L1), L2, .(H, L3)) The argument filtering Pi contains the following mapping: append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa(x1) U1_aaa(x1, x2, x3, x4, x5) = U1_aaa(x5) .(x1, x2) = .(x2) APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (7) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (8) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND_IN_AAA(.(H, L1), L2, .(H, L3)) -> APPEND_IN_AAA(L1, L2, L3) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (9) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (10) 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. ---------------------------------------- (11) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 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: append_in_aaa([], L, L) -> append_out_aaa([], L, L) append_in_aaa(.(H, L1), L2, .(H, L3)) -> U1_aaa(H, L1, L2, L3, append_in_aaa(L1, L2, L3)) U1_aaa(H, L1, L2, L3, append_out_aaa(L1, L2, L3)) -> append_out_aaa(.(H, L1), L2, .(H, L3)) The argument filtering Pi contains the following mapping: append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa(x1) U1_aaa(x1, x2, x3, x4, x5) = U1_aaa(x5) .(x1, x2) = .(x2) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (12) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: append_in_aaa([], L, L) -> append_out_aaa([], L, L) append_in_aaa(.(H, L1), L2, .(H, L3)) -> U1_aaa(H, L1, L2, L3, append_in_aaa(L1, L2, L3)) U1_aaa(H, L1, L2, L3, append_out_aaa(L1, L2, L3)) -> append_out_aaa(.(H, L1), L2, .(H, L3)) The argument filtering Pi contains the following mapping: append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa(x1) U1_aaa(x1, x2, x3, x4, x5) = U1_aaa(x5) .(x1, x2) = .(x2) ---------------------------------------- (13) 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: APPEND_IN_AAA(.(H, L1), L2, .(H, L3)) -> U1_AAA(H, L1, L2, L3, append_in_aaa(L1, L2, L3)) APPEND_IN_AAA(.(H, L1), L2, .(H, L3)) -> APPEND_IN_AAA(L1, L2, L3) The TRS R consists of the following rules: append_in_aaa([], L, L) -> append_out_aaa([], L, L) append_in_aaa(.(H, L1), L2, .(H, L3)) -> U1_aaa(H, L1, L2, L3, append_in_aaa(L1, L2, L3)) U1_aaa(H, L1, L2, L3, append_out_aaa(L1, L2, L3)) -> append_out_aaa(.(H, L1), L2, .(H, L3)) The argument filtering Pi contains the following mapping: append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa(x1) U1_aaa(x1, x2, x3, x4, x5) = U1_aaa(x5) .(x1, x2) = .(x2) APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA U1_AAA(x1, x2, x3, x4, x5) = U1_AAA(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND_IN_AAA(.(H, L1), L2, .(H, L3)) -> U1_AAA(H, L1, L2, L3, append_in_aaa(L1, L2, L3)) APPEND_IN_AAA(.(H, L1), L2, .(H, L3)) -> APPEND_IN_AAA(L1, L2, L3) The TRS R consists of the following rules: append_in_aaa([], L, L) -> append_out_aaa([], L, L) append_in_aaa(.(H, L1), L2, .(H, L3)) -> U1_aaa(H, L1, L2, L3, append_in_aaa(L1, L2, L3)) U1_aaa(H, L1, L2, L3, append_out_aaa(L1, L2, L3)) -> append_out_aaa(.(H, L1), L2, .(H, L3)) The argument filtering Pi contains the following mapping: append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa(x1) U1_aaa(x1, x2, x3, x4, x5) = U1_aaa(x5) .(x1, x2) = .(x2) APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA U1_AAA(x1, x2, x3, x4, x5) = U1_AAA(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (15) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node. ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND_IN_AAA(.(H, L1), L2, .(H, L3)) -> APPEND_IN_AAA(L1, L2, L3) The TRS R consists of the following rules: append_in_aaa([], L, L) -> append_out_aaa([], L, L) append_in_aaa(.(H, L1), L2, .(H, L3)) -> U1_aaa(H, L1, L2, L3, append_in_aaa(L1, L2, L3)) U1_aaa(H, L1, L2, L3, append_out_aaa(L1, L2, L3)) -> append_out_aaa(.(H, L1), L2, .(H, L3)) The argument filtering Pi contains the following mapping: append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa(x1) U1_aaa(x1, x2, x3, x4, x5) = U1_aaa(x5) .(x1, x2) = .(x2) APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (17) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (18) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND_IN_AAA(.(H, L1), L2, .(H, L3)) -> APPEND_IN_AAA(L1, L2, L3) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (19) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (20) 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. ---------------------------------------- (21) PrologToTRSTransformerProof (SOUND) Transformed Prolog program to TRS. { "root": 1, "program": { "directives": [], "clauses": [ [ "(append ([]) L L)", null ], [ "(append (. H L1) L2 (. H L3))", "(append L1 L2 L3)" ], [ "(append1 ([]) L L)", null ], [ "(append1 (. H L1) L2 (. H L3))", "(append1 L1 L2 L3)" ] ] }, "graph": { "nodes": { "22": { "goal": [ { "clause": 0, "scope": 1, "term": "(append T1 T2 T3)" }, { "clause": 1, "scope": 1, "term": "(append T1 T2 T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1": { "goal": [{ "clause": -1, "scope": -1, "term": "(append T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "23": { "goal": [{ "clause": 0, "scope": 1, "term": "(append T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "24": { "goal": [{ "clause": 1, "scope": 1, "term": "(append T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "135": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "136": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "137": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "138": { "goal": [{ "clause": -1, "scope": -1, "term": "(append T21 T22 T23)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "139": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes" }, "edges": [ { "from": 1, "to": 22, "label": "CASE" }, { "from": 22, "to": 23, "label": "PARALLEL" }, { "from": 22, "to": 24, "label": "PARALLEL" }, { "from": 23, "to": 135, "label": "EVAL with clause\nappend([], X5, X5).\nand substitutionT1 -> [],\nT2 -> T8,\nX5 -> T8,\nT3 -> T8" }, { "from": 23, "to": 136, "label": "EVAL-BACKTRACK" }, { "from": 24, "to": 138, "label": "EVAL with clause\nappend(.(X14, X15), X16, .(X14, X17)) :- append(X15, X16, X17).\nand substitutionX14 -> T17,\nX15 -> T21,\nT1 -> .(T17, T21),\nT2 -> T22,\nX16 -> T22,\nX17 -> T23,\nT3 -> .(T17, T23),\nT18 -> T21,\nT19 -> T22,\nT20 -> T23" }, { "from": 24, "to": 139, "label": "EVAL-BACKTRACK" }, { "from": 135, "to": 137, "label": "SUCCESS" }, { "from": 138, "to": 1, "label": "INSTANCE with matching:\nT1 -> T21\nT2 -> T22\nT3 -> T23" } ], "type": "Graph" } } ---------------------------------------- (22) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f1_in -> f1_out1 f1_in -> U1(f1_in) U1(f1_out1) -> f1_out1 Q is empty. ---------------------------------------- (23) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U1(x_1)) = x_1 POL(f1_in) = 2 POL(f1_out1) = 0 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: f1_in -> f1_out1 ---------------------------------------- (24) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f1_in -> U1(f1_in) U1(f1_out1) -> f1_out1 Q is empty. ---------------------------------------- (25) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U1(x_1)) = 2*x_1 POL(f1_in) = 0 POL(f1_out1) = 1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U1(f1_out1) -> f1_out1 ---------------------------------------- (26) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f1_in -> U1(f1_in) Q is empty. ---------------------------------------- (27) Overlay + Local Confluence (EQUIVALENT) The TRS is overlay and locally confluent. By [NOC] we can switch to innermost. ---------------------------------------- (28) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f1_in -> U1(f1_in) The set Q consists of the following terms: f1_in ---------------------------------------- (29) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (30) Obligation: Q DP problem: The TRS P consists of the following rules: F1_IN -> F1_IN The TRS R consists of the following rules: f1_in -> U1(f1_in) The set Q consists of the following terms: f1_in We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (31) 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. ---------------------------------------- (32) Obligation: Q DP problem: The TRS P consists of the following rules: F1_IN -> F1_IN R is empty. The set Q consists of the following terms: f1_in We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (33) 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]. f1_in ---------------------------------------- (34) Obligation: Q DP problem: The TRS P consists of the following rules: F1_IN -> F1_IN R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (35) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 2, "program": { "directives": [], "clauses": [ [ "(append ([]) L L)", null ], [ "(append (. H L1) L2 (. H L3))", "(append L1 L2 L3)" ], [ "(append1 ([]) L L)", null ], [ "(append1 (. H L1) L2 (. H L3))", "(append1 L1 L2 L3)" ] ] }, "graph": { "nodes": { "78": { "goal": [{ "clause": -1, "scope": -1, "term": "(append T21 T22 T23)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "2": { "goal": [{ "clause": -1, "scope": -1, "term": "(append T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "79": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "6": { "goal": [ { "clause": 0, "scope": 1, "term": "(append T1 T2 T3)" }, { "clause": 1, "scope": 1, "term": "(append T1 T2 T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "73": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "74": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "20": { "goal": [{ "clause": 0, "scope": 1, "term": "(append T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "75": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "21": { "goal": [{ "clause": 1, "scope": 1, "term": "(append T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 2, "to": 6, "label": "CASE" }, { "from": 6, "to": 20, "label": "PARALLEL" }, { "from": 6, "to": 21, "label": "PARALLEL" }, { "from": 20, "to": 73, "label": "EVAL with clause\nappend([], X5, X5).\nand substitutionT1 -> [],\nT2 -> T8,\nX5 -> T8,\nT3 -> T8" }, { "from": 20, "to": 74, "label": "EVAL-BACKTRACK" }, { "from": 21, "to": 78, "label": "EVAL with clause\nappend(.(X14, X15), X16, .(X14, X17)) :- append(X15, X16, X17).\nand substitutionX14 -> T17,\nX15 -> T21,\nT1 -> .(T17, T21),\nT2 -> T22,\nX16 -> T22,\nX17 -> T23,\nT3 -> .(T17, T23),\nT18 -> T21,\nT19 -> T22,\nT20 -> T23" }, { "from": 21, "to": 79, "label": "EVAL-BACKTRACK" }, { "from": 73, "to": 75, "label": "SUCCESS" }, { "from": 78, "to": 2, "label": "INSTANCE with matching:\nT1 -> T21\nT2 -> T22\nT3 -> T23" } ], "type": "Graph" } } ---------------------------------------- (36) Obligation: Rules: f6_out -> f2_out :|: TRUE f2_in -> f6_in :|: TRUE f78_in -> f2_in :|: TRUE f2_out -> f78_out :|: TRUE f21_in -> f79_in :|: TRUE f78_out -> f21_out :|: TRUE f21_in -> f78_in :|: TRUE f79_out -> f21_out :|: TRUE f6_in -> f20_in :|: TRUE f21_out -> f6_out :|: TRUE f20_out -> f6_out :|: TRUE f6_in -> f21_in :|: TRUE Start term: f2_in ---------------------------------------- (37) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f2_in -> f6_in :|: TRUE f78_in -> f2_in :|: TRUE f21_in -> f78_in :|: TRUE f6_in -> f21_in :|: TRUE ---------------------------------------- (38) Obligation: Rules: f2_in -> f6_in :|: TRUE f78_in -> f2_in :|: TRUE f21_in -> f78_in :|: TRUE f6_in -> f21_in :|: TRUE ---------------------------------------- (39) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (40) Obligation: Rules: f21_in -> f21_in :|: TRUE ---------------------------------------- (41) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (42) Obligation: Rules: f21_in -> f21_in :|: TRUE ---------------------------------------- (43) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f21_in -> f21_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (44) Obligation: Termination digraph: Nodes: (1) f21_in -> f21_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (45) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f21_in() Replaced non-predefined constructor symbols by 0. ---------------------------------------- (46) Obligation: Rules: f21_in -> f21_in :|: TRUE ---------------------------------------- (47) IntTRSPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc) -> f(1) :|: pc = 1 && TRUE Witness term starting non-terminating reduction: f(1) ---------------------------------------- (48) NO ---------------------------------------- (49) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 3, "program": { "directives": [], "clauses": [ [ "(append ([]) L L)", null ], [ "(append (. H L1) L2 (. H L3))", "(append L1 L2 L3)" ], [ "(append1 ([]) L L)", null ], [ "(append1 (. H L1) L2 (. H L3))", "(append1 L1 L2 L3)" ] ] }, "graph": { "nodes": { "type": "Nodes", "150": { "goal": [{ "clause": -1, "scope": -1, "term": "(append T47 T48 T49)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "151": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "152": { "goal": [ { "clause": 0, "scope": 3, "term": "(append T47 T48 T49)" }, { "clause": 1, "scope": 3, "term": "(append T47 T48 T49)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "153": { "goal": [{ "clause": 0, "scope": 3, "term": "(append T47 T48 T49)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "154": { "goal": [{ "clause": 1, "scope": 3, "term": "(append T47 T48 T49)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "155": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "156": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "157": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "158": { "goal": [{ "clause": -1, "scope": -1, "term": "(append T67 T68 T69)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "159": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "12": { "goal": [ { "clause": 0, "scope": 1, "term": "(append T1 T2 T3)" }, { "clause": 1, "scope": 1, "term": "(append T1 T2 T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "140": { "goal": [{ "clause": -1, "scope": -1, "term": "(append T14 T15 T16)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "141": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "142": { "goal": [ { "clause": 0, "scope": 2, "term": "(append T14 T15 T16)" }, { "clause": 1, "scope": 2, "term": "(append T14 T15 T16)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "143": { "goal": [{ "clause": 0, "scope": 2, "term": "(append T14 T15 T16)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "144": { "goal": [{ "clause": 1, "scope": 2, "term": "(append T14 T15 T16)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "145": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "3": { "goal": [{ "clause": -1, "scope": -1, "term": "(append T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "146": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "147": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "148": { "goal": [{ "clause": -1, "scope": -1, "term": "(append T34 T35 T36)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "149": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "81": { "goal": [ { "clause": -1, "scope": -1, "term": "(true)" }, { "clause": 1, "scope": 1, "term": "(append T1 T2 T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "82": { "goal": [{ "clause": 1, "scope": 1, "term": "(append T1 T2 T3)" }], "kb": { "nonunifying": [[ "(append T1 T2 T3)", "(append ([]) X2 X2)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X2"], "exprvars": [] } }, "83": { "goal": [{ "clause": 1, "scope": 1, "term": "(append T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 3, "to": 12, "label": "CASE" }, { "from": 12, "to": 81, "label": "EVAL with clause\nappend([], X2, X2).\nand substitutionT1 -> [],\nT2 -> T5,\nX2 -> T5,\nT3 -> T5" }, { "from": 12, "to": 82, "label": "EVAL-BACKTRACK" }, { "from": 81, "to": 83, "label": "SUCCESS" }, { "from": 82, "to": 150, "label": "EVAL with clause\nappend(.(X34, X35), X36, .(X34, X37)) :- append(X35, X36, X37).\nand substitutionX34 -> T43,\nX35 -> T47,\nT1 -> .(T43, T47),\nT2 -> T48,\nX36 -> T48,\nX37 -> T49,\nT3 -> .(T43, T49),\nT44 -> T47,\nT45 -> T48,\nT46 -> T49" }, { "from": 82, "to": 151, "label": "EVAL-BACKTRACK" }, { "from": 83, "to": 140, "label": "EVAL with clause\nappend(.(X7, X8), X9, .(X7, X10)) :- append(X8, X9, X10).\nand substitutionX7 -> T10,\nX8 -> T14,\nT1 -> .(T10, T14),\nT2 -> T15,\nX9 -> T15,\nX10 -> T16,\nT3 -> .(T10, T16),\nT11 -> T14,\nT12 -> T15,\nT13 -> T16" }, { "from": 83, "to": 141, "label": "EVAL-BACKTRACK" }, { "from": 140, "to": 142, "label": "CASE" }, { "from": 142, "to": 143, "label": "PARALLEL" }, { "from": 142, "to": 144, "label": "PARALLEL" }, { "from": 143, "to": 145, "label": "EVAL with clause\nappend([], X15, X15).\nand substitutionT14 -> [],\nT15 -> T21,\nX15 -> T21,\nT16 -> T21" }, { "from": 143, "to": 146, "label": "EVAL-BACKTRACK" }, { "from": 144, "to": 148, "label": "EVAL with clause\nappend(.(X24, X25), X26, .(X24, X27)) :- append(X25, X26, X27).\nand substitutionX24 -> T30,\nX25 -> T34,\nT14 -> .(T30, T34),\nT15 -> T35,\nX26 -> T35,\nX27 -> T36,\nT16 -> .(T30, T36),\nT31 -> T34,\nT32 -> T35,\nT33 -> T36" }, { "from": 144, "to": 149, "label": "EVAL-BACKTRACK" }, { "from": 145, "to": 147, "label": "SUCCESS" }, { "from": 148, "to": 3, "label": "INSTANCE with matching:\nT1 -> T34\nT2 -> T35\nT3 -> T36" }, { "from": 150, "to": 152, "label": "CASE" }, { "from": 152, "to": 153, "label": "PARALLEL" }, { "from": 152, "to": 154, "label": "PARALLEL" }, { "from": 153, "to": 155, "label": "EVAL with clause\nappend([], X42, X42).\nand substitutionT47 -> [],\nT48 -> T54,\nX42 -> T54,\nT49 -> T54" }, { "from": 153, "to": 156, "label": "EVAL-BACKTRACK" }, { "from": 154, "to": 158, "label": "EVAL with clause\nappend(.(X51, X52), X53, .(X51, X54)) :- append(X52, X53, X54).\nand substitutionX51 -> T63,\nX52 -> T67,\nT47 -> .(T63, T67),\nT48 -> T68,\nX53 -> T68,\nX54 -> T69,\nT49 -> .(T63, T69),\nT64 -> T67,\nT65 -> T68,\nT66 -> T69" }, { "from": 154, "to": 159, "label": "EVAL-BACKTRACK" }, { "from": 155, "to": 157, "label": "SUCCESS" }, { "from": 158, "to": 3, "label": "INSTANCE with matching:\nT1 -> T67\nT2 -> T68\nT3 -> T69" } ], "type": "Graph" } } ---------------------------------------- (50) Obligation: Triples: appendA(.(X1, .(X2, X3)), X4, .(X1, .(X2, X5))) :- appendA(X3, X4, X5). appendA(.(X1, .(X2, X3)), X4, .(X1, .(X2, X5))) :- appendA(X3, X4, X5). Clauses: appendcA([], X1, X1). appendcA(.(X1, []), X2, .(X1, X2)). appendcA(.(X1, .(X2, X3)), X4, .(X1, .(X2, X5))) :- appendcA(X3, X4, X5). appendcA(.(X1, []), X2, .(X1, X2)). appendcA(.(X1, .(X2, X3)), X4, .(X1, .(X2, X5))) :- appendcA(X3, X4, X5). Afs: appendA(x1, x2, x3) = appendA ---------------------------------------- (51) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: appendA_in_3: (f,f,f) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: APPENDA_IN_AAA(.(X1, .(X2, X3)), X4, .(X1, .(X2, X5))) -> U1_AAA(X1, X2, X3, X4, X5, appendA_in_aaa(X3, X4, X5)) APPENDA_IN_AAA(.(X1, .(X2, X3)), X4, .(X1, .(X2, X5))) -> APPENDA_IN_AAA(X3, X4, X5) R is empty. The argument filtering Pi contains the following mapping: appendA_in_aaa(x1, x2, x3) = appendA_in_aaa .(x1, x2) = .(x2) APPENDA_IN_AAA(x1, x2, x3) = APPENDA_IN_AAA U1_AAA(x1, x2, x3, x4, x5, x6) = U1_AAA(x6) We have to consider all (P,R,Pi)-chains Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES ---------------------------------------- (52) Obligation: Pi DP problem: The TRS P consists of the following rules: APPENDA_IN_AAA(.(X1, .(X2, X3)), X4, .(X1, .(X2, X5))) -> U1_AAA(X1, X2, X3, X4, X5, appendA_in_aaa(X3, X4, X5)) APPENDA_IN_AAA(.(X1, .(X2, X3)), X4, .(X1, .(X2, X5))) -> APPENDA_IN_AAA(X3, X4, X5) R is empty. The argument filtering Pi contains the following mapping: appendA_in_aaa(x1, x2, x3) = appendA_in_aaa .(x1, x2) = .(x2) APPENDA_IN_AAA(x1, x2, x3) = APPENDA_IN_AAA U1_AAA(x1, x2, x3, x4, x5, x6) = U1_AAA(x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (53) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node. ---------------------------------------- (54) Obligation: Pi DP problem: The TRS P consists of the following rules: APPENDA_IN_AAA(.(X1, .(X2, X3)), X4, .(X1, .(X2, X5))) -> APPENDA_IN_AAA(X3, X4, X5) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) APPENDA_IN_AAA(x1, x2, x3) = APPENDA_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (55) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (56) Obligation: Q DP problem: The TRS P consists of the following rules: APPENDA_IN_AAA -> APPENDA_IN_AAA R is empty. Q is empty. We have to consider all (P,Q,R)-chains.