/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.pl /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- MAYBE proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern member(g,a) w.r.t. the given Prolog program could not be shown: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2)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) PrologToTRSTransformerProof [SOUND, 0 ms] (12) QTRS (13) DependencyPairsProof [EQUIVALENT, 0 ms] (14) QDP (15) DependencyGraphProof [EQUIVALENT, 0 ms] (16) QDP (17) MNOCProof [EQUIVALENT, 0 ms] (18) QDP (19) UsableRulesProof [EQUIVALENT, 0 ms] (20) QDP (21) QReductionProof [EQUIVALENT, 0 ms] (22) QDP (23) PrologToIRSwTTransformerProof [SOUND, 0 ms] (24) IRSwT (25) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (26) IRSwT (27) IntTRSCompressionProof [EQUIVALENT, 20 ms] (28) IRSwT (29) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (30) IRSwT (31) IRSwTTerminationDigraphProof [EQUIVALENT, 5 ms] (32) IRSwT (33) FilterProof [EQUIVALENT, 0 ms] (34) IntTRS (35) IntTRSNonPeriodicNontermProof [COMPLETE, 7 ms] (36) NO (37) PrologToPiTRSProof [SOUND, 0 ms] (38) PiTRS (39) DependencyPairsProof [EQUIVALENT, 0 ms] (40) PiDP (41) DependencyGraphProof [EQUIVALENT, 0 ms] (42) PiDP (43) UsableRulesProof [EQUIVALENT, 0 ms] (44) PiDP (45) PiDPToQDPProof [SOUND, 0 ms] (46) QDP (47) PrologToDTProblemTransformerProof [SOUND, 0 ms] (48) TRIPLES (49) TriplesToPiDPProof [SOUND, 0 ms] (50) PiDP (51) DependencyGraphProof [EQUIVALENT, 0 ms] (52) PiDP (53) PiDPToQDPProof [SOUND, 0 ms] (54) QDP ---------------------------------------- (0) Obligation: Clauses: member(X, .(X, X1)). member(X, .(X2, Xs)) :- member(X, Xs). Query: member(g,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: member_in_2: (b,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: member_in_ga(X, .(X, X1)) -> member_out_ga(X, .(X, X1)) member_in_ga(X, .(X2, Xs)) -> U1_ga(X, X2, Xs, member_in_ga(X, Xs)) U1_ga(X, X2, Xs, member_out_ga(X, Xs)) -> member_out_ga(X, .(X2, Xs)) The argument filtering Pi contains the following mapping: member_in_ga(x1, x2) = member_in_ga(x1) member_out_ga(x1, x2) = member_out_ga(x1) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: member_in_ga(X, .(X, X1)) -> member_out_ga(X, .(X, X1)) member_in_ga(X, .(X2, Xs)) -> U1_ga(X, X2, Xs, member_in_ga(X, Xs)) U1_ga(X, X2, Xs, member_out_ga(X, Xs)) -> member_out_ga(X, .(X2, Xs)) The argument filtering Pi contains the following mapping: member_in_ga(x1, x2) = member_in_ga(x1) member_out_ga(x1, x2) = member_out_ga(x1) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) ---------------------------------------- (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: MEMBER_IN_GA(X, .(X2, Xs)) -> U1_GA(X, X2, Xs, member_in_ga(X, Xs)) MEMBER_IN_GA(X, .(X2, Xs)) -> MEMBER_IN_GA(X, Xs) The TRS R consists of the following rules: member_in_ga(X, .(X, X1)) -> member_out_ga(X, .(X, X1)) member_in_ga(X, .(X2, Xs)) -> U1_ga(X, X2, Xs, member_in_ga(X, Xs)) U1_ga(X, X2, Xs, member_out_ga(X, Xs)) -> member_out_ga(X, .(X2, Xs)) The argument filtering Pi contains the following mapping: member_in_ga(x1, x2) = member_in_ga(x1) member_out_ga(x1, x2) = member_out_ga(x1) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) MEMBER_IN_GA(x1, x2) = MEMBER_IN_GA(x1) U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBER_IN_GA(X, .(X2, Xs)) -> U1_GA(X, X2, Xs, member_in_ga(X, Xs)) MEMBER_IN_GA(X, .(X2, Xs)) -> MEMBER_IN_GA(X, Xs) The TRS R consists of the following rules: member_in_ga(X, .(X, X1)) -> member_out_ga(X, .(X, X1)) member_in_ga(X, .(X2, Xs)) -> U1_ga(X, X2, Xs, member_in_ga(X, Xs)) U1_ga(X, X2, Xs, member_out_ga(X, Xs)) -> member_out_ga(X, .(X2, Xs)) The argument filtering Pi contains the following mapping: member_in_ga(x1, x2) = member_in_ga(x1) member_out_ga(x1, x2) = member_out_ga(x1) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) MEMBER_IN_GA(x1, x2) = MEMBER_IN_GA(x1) U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4) 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: MEMBER_IN_GA(X, .(X2, Xs)) -> MEMBER_IN_GA(X, Xs) The TRS R consists of the following rules: member_in_ga(X, .(X, X1)) -> member_out_ga(X, .(X, X1)) member_in_ga(X, .(X2, Xs)) -> U1_ga(X, X2, Xs, member_in_ga(X, Xs)) U1_ga(X, X2, Xs, member_out_ga(X, Xs)) -> member_out_ga(X, .(X2, Xs)) The argument filtering Pi contains the following mapping: member_in_ga(x1, x2) = member_in_ga(x1) member_out_ga(x1, x2) = member_out_ga(x1) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) MEMBER_IN_GA(x1, x2) = MEMBER_IN_GA(x1) 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: MEMBER_IN_GA(X, .(X2, Xs)) -> MEMBER_IN_GA(X, Xs) R is empty. The argument filtering Pi contains the following mapping: MEMBER_IN_GA(x1, x2) = MEMBER_IN_GA(x1) 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: MEMBER_IN_GA(X) -> MEMBER_IN_GA(X) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (11) PrologToTRSTransformerProof (SOUND) Transformed Prolog program to TRS. { "root": 1, "program": { "directives": [], "clauses": [ [ "(member X (. X X1))", null ], [ "(member X (. X2 Xs))", "(member X Xs)" ] ] }, "graph": { "nodes": { "33": { "goal": [ { "clause": 0, "scope": 1, "term": "(member T1 T2)" }, { "clause": 1, "scope": 1, "term": "(member T1 T2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "1": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "67": { "goal": [{ "clause": 0, "scope": 1, "term": "(member T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "68": { "goal": [{ "clause": 1, "scope": 1, "term": "(member T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "69": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "70": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "71": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "72": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T19 T22)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T19"], "free": [], "exprvars": [] } }, "type": "Nodes", "73": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 33, "label": "CASE" }, { "from": 33, "to": 67, "label": "PARALLEL" }, { "from": 33, "to": 68, "label": "PARALLEL" }, { "from": 67, "to": 69, "label": "EVAL with clause\nmember(X11, .(X11, X12)).\nand substitutionT1 -> T11,\nX11 -> T11,\nX12 -> T12,\nT2 -> .(T11, T12)" }, { "from": 67, "to": 70, "label": "EVAL-BACKTRACK" }, { "from": 68, "to": 72, "label": "EVAL with clause\nmember(X19, .(X20, X21)) :- member(X19, X21).\nand substitutionT1 -> T19,\nX19 -> T19,\nX20 -> T20,\nX21 -> T22,\nT2 -> .(T20, T22),\nT21 -> T22" }, { "from": 68, "to": 73, "label": "EVAL-BACKTRACK" }, { "from": 69, "to": 71, "label": "SUCCESS" }, { "from": 72, "to": 1, "label": "INSTANCE with matching:\nT1 -> T19\nT2 -> T22" } ], "type": "Graph" } } ---------------------------------------- (12) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f1_in(T11) -> f1_out1 f1_in(T19) -> U1(f1_in(T19), T19) U1(f1_out1, T19) -> f1_out1 Q is empty. ---------------------------------------- (13) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (14) Obligation: Q DP problem: The TRS P consists of the following rules: F1_IN(T19) -> U1^1(f1_in(T19), T19) F1_IN(T19) -> F1_IN(T19) The TRS R consists of the following rules: f1_in(T11) -> f1_out1 f1_in(T19) -> U1(f1_in(T19), T19) U1(f1_out1, T19) -> f1_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (15) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (16) Obligation: Q DP problem: The TRS P consists of the following rules: F1_IN(T19) -> F1_IN(T19) The TRS R consists of the following rules: f1_in(T11) -> f1_out1 f1_in(T19) -> U1(f1_in(T19), T19) U1(f1_out1, T19) -> f1_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (17) MNOCProof (EQUIVALENT) We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R. ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: F1_IN(T19) -> F1_IN(T19) The TRS R consists of the following rules: f1_in(T11) -> f1_out1 f1_in(T19) -> U1(f1_in(T19), T19) U1(f1_out1, T19) -> f1_out1 The set Q consists of the following terms: f1_in(x0) U1(f1_out1, x0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (19) 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. ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: F1_IN(T19) -> F1_IN(T19) R is empty. The set Q consists of the following terms: f1_in(x0) U1(f1_out1, x0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (21) 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(x0) U1(f1_out1, x0) ---------------------------------------- (22) Obligation: Q DP problem: The TRS P consists of the following rules: F1_IN(T19) -> F1_IN(T19) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (23) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 3, "program": { "directives": [], "clauses": [ [ "(member X (. X X1))", null ], [ "(member X (. X2 Xs))", "(member X Xs)" ] ] }, "graph": { "nodes": { "66": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "3": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "6": { "goal": [ { "clause": 0, "scope": 1, "term": "(member T1 T2)" }, { "clause": 1, "scope": 1, "term": "(member T1 T2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "9": { "goal": [{ "clause": 0, "scope": 1, "term": "(member T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "type": "Nodes", "62": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "63": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "64": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "10": { "goal": [{ "clause": 1, "scope": 1, "term": "(member T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "65": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T19 T22)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T19"], "free": [], "exprvars": [] } } }, "edges": [ { "from": 3, "to": 6, "label": "CASE" }, { "from": 6, "to": 9, "label": "PARALLEL" }, { "from": 6, "to": 10, "label": "PARALLEL" }, { "from": 9, "to": 62, "label": "EVAL with clause\nmember(X11, .(X11, X12)).\nand substitutionT1 -> T11,\nX11 -> T11,\nX12 -> T12,\nT2 -> .(T11, T12)" }, { "from": 9, "to": 63, "label": "EVAL-BACKTRACK" }, { "from": 10, "to": 65, "label": "EVAL with clause\nmember(X19, .(X20, X21)) :- member(X19, X21).\nand substitutionT1 -> T19,\nX19 -> T19,\nX20 -> T20,\nX21 -> T22,\nT2 -> .(T20, T22),\nT21 -> T22" }, { "from": 10, "to": 66, "label": "EVAL-BACKTRACK" }, { "from": 62, "to": 64, "label": "SUCCESS" }, { "from": 65, "to": 3, "label": "INSTANCE with matching:\nT1 -> T19\nT2 -> T22" } ], "type": "Graph" } } ---------------------------------------- (24) Obligation: Rules: f3_out(T19) -> f65_out(T19) :|: TRUE f65_in(x) -> f3_in(x) :|: TRUE f6_in(T1) -> f10_in(T1) :|: TRUE f6_in(x1) -> f9_in(x1) :|: TRUE f9_out(x2) -> f6_out(x2) :|: TRUE f10_out(x3) -> f6_out(x3) :|: TRUE f10_in(x4) -> f66_in :|: TRUE f65_out(x5) -> f10_out(x5) :|: TRUE f10_in(x6) -> f65_in(x6) :|: TRUE f66_out -> f10_out(x7) :|: TRUE f6_out(x8) -> f3_out(x8) :|: TRUE f3_in(x9) -> f6_in(x9) :|: TRUE Start term: f3_in(T1) ---------------------------------------- (25) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f65_in(x) -> f3_in(x) :|: TRUE f6_in(T1) -> f10_in(T1) :|: TRUE f10_in(x6) -> f65_in(x6) :|: TRUE f3_in(x9) -> f6_in(x9) :|: TRUE ---------------------------------------- (26) Obligation: Rules: f65_in(x) -> f3_in(x) :|: TRUE f6_in(T1) -> f10_in(T1) :|: TRUE f10_in(x6) -> f65_in(x6) :|: TRUE f3_in(x9) -> f6_in(x9) :|: TRUE ---------------------------------------- (27) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (28) Obligation: Rules: f6_in(T1:0) -> f6_in(T1:0) :|: TRUE ---------------------------------------- (29) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (30) Obligation: Rules: f6_in(T1:0) -> f6_in(T1:0) :|: TRUE ---------------------------------------- (31) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f6_in(T1:0) -> f6_in(T1:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (32) Obligation: Termination digraph: Nodes: (1) f6_in(T1:0) -> f6_in(T1:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (33) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f6_in(VARIABLE) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (34) Obligation: Rules: f6_in(T1:0) -> f6_in(T1:0) :|: TRUE ---------------------------------------- (35) IntTRSNonPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc, T1:0) -> f(1, T1:0) :|: pc = 1 && TRUE Proved unsatisfiability of the following formula, indicating that the system is never left after entering: (((run2_0 = ((1 * 1)) and run2_1 = ((run1_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 run2_1 = ((run1_1 * 1))) and (((run1_0 * 1)) = ((1 * 1)) and T)) ---------------------------------------- (36) NO ---------------------------------------- (37) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: member_in_2: (b,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: member_in_ga(X, .(X, X1)) -> member_out_ga(X, .(X, X1)) member_in_ga(X, .(X2, Xs)) -> U1_ga(X, X2, Xs, member_in_ga(X, Xs)) U1_ga(X, X2, Xs, member_out_ga(X, Xs)) -> member_out_ga(X, .(X2, Xs)) The argument filtering Pi contains the following mapping: member_in_ga(x1, x2) = member_in_ga(x1) member_out_ga(x1, x2) = member_out_ga U1_ga(x1, x2, x3, x4) = U1_ga(x4) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (38) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: member_in_ga(X, .(X, X1)) -> member_out_ga(X, .(X, X1)) member_in_ga(X, .(X2, Xs)) -> U1_ga(X, X2, Xs, member_in_ga(X, Xs)) U1_ga(X, X2, Xs, member_out_ga(X, Xs)) -> member_out_ga(X, .(X2, Xs)) The argument filtering Pi contains the following mapping: member_in_ga(x1, x2) = member_in_ga(x1) member_out_ga(x1, x2) = member_out_ga U1_ga(x1, x2, x3, x4) = U1_ga(x4) ---------------------------------------- (39) 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: MEMBER_IN_GA(X, .(X2, Xs)) -> U1_GA(X, X2, Xs, member_in_ga(X, Xs)) MEMBER_IN_GA(X, .(X2, Xs)) -> MEMBER_IN_GA(X, Xs) The TRS R consists of the following rules: member_in_ga(X, .(X, X1)) -> member_out_ga(X, .(X, X1)) member_in_ga(X, .(X2, Xs)) -> U1_ga(X, X2, Xs, member_in_ga(X, Xs)) U1_ga(X, X2, Xs, member_out_ga(X, Xs)) -> member_out_ga(X, .(X2, Xs)) The argument filtering Pi contains the following mapping: member_in_ga(x1, x2) = member_in_ga(x1) member_out_ga(x1, x2) = member_out_ga U1_ga(x1, x2, x3, x4) = U1_ga(x4) MEMBER_IN_GA(x1, x2) = MEMBER_IN_GA(x1) U1_GA(x1, x2, x3, x4) = U1_GA(x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (40) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBER_IN_GA(X, .(X2, Xs)) -> U1_GA(X, X2, Xs, member_in_ga(X, Xs)) MEMBER_IN_GA(X, .(X2, Xs)) -> MEMBER_IN_GA(X, Xs) The TRS R consists of the following rules: member_in_ga(X, .(X, X1)) -> member_out_ga(X, .(X, X1)) member_in_ga(X, .(X2, Xs)) -> U1_ga(X, X2, Xs, member_in_ga(X, Xs)) U1_ga(X, X2, Xs, member_out_ga(X, Xs)) -> member_out_ga(X, .(X2, Xs)) The argument filtering Pi contains the following mapping: member_in_ga(x1, x2) = member_in_ga(x1) member_out_ga(x1, x2) = member_out_ga U1_ga(x1, x2, x3, x4) = U1_ga(x4) MEMBER_IN_GA(x1, x2) = MEMBER_IN_GA(x1) U1_GA(x1, x2, x3, x4) = U1_GA(x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (41) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node. ---------------------------------------- (42) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBER_IN_GA(X, .(X2, Xs)) -> MEMBER_IN_GA(X, Xs) The TRS R consists of the following rules: member_in_ga(X, .(X, X1)) -> member_out_ga(X, .(X, X1)) member_in_ga(X, .(X2, Xs)) -> U1_ga(X, X2, Xs, member_in_ga(X, Xs)) U1_ga(X, X2, Xs, member_out_ga(X, Xs)) -> member_out_ga(X, .(X2, Xs)) The argument filtering Pi contains the following mapping: member_in_ga(x1, x2) = member_in_ga(x1) member_out_ga(x1, x2) = member_out_ga U1_ga(x1, x2, x3, x4) = U1_ga(x4) MEMBER_IN_GA(x1, x2) = MEMBER_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (43) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (44) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBER_IN_GA(X, .(X2, Xs)) -> MEMBER_IN_GA(X, Xs) R is empty. The argument filtering Pi contains the following mapping: MEMBER_IN_GA(x1, x2) = MEMBER_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (45) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (46) Obligation: Q DP problem: The TRS P consists of the following rules: MEMBER_IN_GA(X) -> MEMBER_IN_GA(X) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (47) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 2, "program": { "directives": [], "clauses": [ [ "(member X (. X X1))", null ], [ "(member X (. X2 Xs))", "(member X Xs)" ] ] }, "graph": { "nodes": { "88": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "89": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "90": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T30 T33)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T30"], "free": [], "exprvars": [] } }, "91": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "92": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T41 T44)" }], "kb": { "nonunifying": [[ "(member T41 T2)", "(member X5 (. X5 X6))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T41"], "free": [ "X5", "X6" ], "exprvars": [] } }, "93": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "94": { "goal": [ { "clause": 0, "scope": 3, "term": "(member T41 T44)" }, { "clause": 1, "scope": 3, "term": "(member T41 T44)" } ], "kb": { "nonunifying": [[ "(member T41 T2)", "(member X5 (. X5 X6))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T41"], "free": [ "X5", "X6" ], "exprvars": [] } }, "95": { "goal": [{ "clause": 0, "scope": 3, "term": "(member T41 T44)" }], "kb": { "nonunifying": [[ "(member T41 T2)", "(member X5 (. X5 X6))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T41"], "free": [ "X5", "X6" ], "exprvars": [] } }, "96": { "goal": [{ "clause": 1, "scope": 3, "term": "(member T41 T44)" }], "kb": { "nonunifying": [[ "(member T41 T2)", "(member X5 (. X5 X6))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T41"], "free": [ "X5", "X6" ], "exprvars": [] } }, "97": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "98": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "99": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "78": { "goal": [ { "clause": 0, "scope": 1, "term": "(member T1 T2)" }, { "clause": 1, "scope": 1, "term": "(member T1 T2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "79": { "goal": [ { "clause": -1, "scope": -1, "term": "(true)" }, { "clause": 1, "scope": 1, "term": "(member T5 T2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T5"], "free": [], "exprvars": [] } }, "100": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T61 T64)" }], "kb": { "nonunifying": [[ "(member T61 T2)", "(member X5 (. X5 X6))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T61"], "free": [ "X5", "X6" ], "exprvars": [] } }, "2": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "101": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "80": { "goal": [{ "clause": 1, "scope": 1, "term": "(member T1 T2)" }], "kb": { "nonunifying": [[ "(member T1 T2)", "(member X5 (. X5 X6))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [ "X5", "X6" ], "exprvars": [] } }, "81": { "goal": [{ "clause": 1, "scope": 1, "term": "(member T5 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T5"], "free": [], "exprvars": [] } }, "82": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T10 T13)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": [], "exprvars": [] } }, "83": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "84": { "goal": [ { "clause": 0, "scope": 2, "term": "(member T10 T13)" }, { "clause": 1, "scope": 2, "term": "(member T10 T13)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": [], "exprvars": [] } }, "85": { "goal": [{ "clause": 0, "scope": 2, "term": "(member T10 T13)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": [], "exprvars": [] } }, "86": { "goal": [{ "clause": 1, "scope": 2, "term": "(member T10 T13)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": [], "exprvars": [] } }, "87": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 2, "to": 78, "label": "CASE" }, { "from": 78, "to": 79, "label": "EVAL with clause\nmember(X5, .(X5, X6)).\nand substitutionT1 -> T5,\nX5 -> T5,\nX6 -> T6,\nT2 -> .(T5, T6)" }, { "from": 78, "to": 80, "label": "EVAL-BACKTRACK" }, { "from": 79, "to": 81, "label": "SUCCESS" }, { "from": 80, "to": 92, "label": "EVAL with clause\nmember(X39, .(X40, X41)) :- member(X39, X41).\nand substitutionT1 -> T41,\nX39 -> T41,\nX40 -> T42,\nX41 -> T44,\nT2 -> .(T42, T44),\nT43 -> T44" }, { "from": 80, "to": 93, "label": "EVAL-BACKTRACK" }, { "from": 81, "to": 82, "label": "EVAL with clause\nmember(X10, .(X11, X12)) :- member(X10, X12).\nand substitutionT5 -> T10,\nX10 -> T10,\nX11 -> T11,\nX12 -> T13,\nT2 -> .(T11, T13),\nT12 -> T13" }, { "from": 81, "to": 83, "label": "EVAL-BACKTRACK" }, { "from": 82, "to": 84, "label": "CASE" }, { "from": 84, "to": 85, "label": "PARALLEL" }, { "from": 84, "to": 86, "label": "PARALLEL" }, { "from": 85, "to": 87, "label": "EVAL with clause\nmember(X21, .(X21, X22)).\nand substitutionT10 -> T22,\nX21 -> T22,\nX22 -> T23,\nT13 -> .(T22, T23)" }, { "from": 85, "to": 88, "label": "EVAL-BACKTRACK" }, { "from": 86, "to": 90, "label": "EVAL with clause\nmember(X29, .(X30, X31)) :- member(X29, X31).\nand substitutionT10 -> T30,\nX29 -> T30,\nX30 -> T31,\nX31 -> T33,\nT13 -> .(T31, T33),\nT32 -> T33" }, { "from": 86, "to": 91, "label": "EVAL-BACKTRACK" }, { "from": 87, "to": 89, "label": "SUCCESS" }, { "from": 90, "to": 2, "label": "INSTANCE with matching:\nT1 -> T30\nT2 -> T33" }, { "from": 92, "to": 94, "label": "CASE" }, { "from": 94, "to": 95, "label": "PARALLEL" }, { "from": 94, "to": 96, "label": "PARALLEL" }, { "from": 95, "to": 97, "label": "EVAL with clause\nmember(X50, .(X50, X51)).\nand substitutionT41 -> T53,\nX50 -> T53,\nX51 -> T54,\nT44 -> .(T53, T54)" }, { "from": 95, "to": 98, "label": "EVAL-BACKTRACK" }, { "from": 96, "to": 100, "label": "EVAL with clause\nmember(X58, .(X59, X60)) :- member(X58, X60).\nand substitutionT41 -> T61,\nX58 -> T61,\nX59 -> T62,\nX60 -> T64,\nT44 -> .(T62, T64),\nT63 -> T64" }, { "from": 96, "to": 101, "label": "EVAL-BACKTRACK" }, { "from": 97, "to": 99, "label": "SUCCESS" }, { "from": 100, "to": 2, "label": "INSTANCE with matching:\nT1 -> T61\nT2 -> T64" } ], "type": "Graph" } } ---------------------------------------- (48) Obligation: Triples: memberA(X1, .(X2, .(X3, X4))) :- memberA(X1, X4). memberA(X1, .(X2, .(X3, X4))) :- memberA(X1, X4). Clauses: membercA(X1, .(X1, X2)). membercA(X1, .(X2, .(X1, X3))). membercA(X1, .(X2, .(X3, X4))) :- membercA(X1, X4). membercA(X1, .(X2, .(X1, X3))). membercA(X1, .(X2, .(X3, X4))) :- membercA(X1, X4). Afs: memberA(x1, x2) = memberA(x1) ---------------------------------------- (49) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: memberA_in_2: (b,f) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: MEMBERA_IN_GA(X1, .(X2, .(X3, X4))) -> U1_GA(X1, X2, X3, X4, memberA_in_ga(X1, X4)) MEMBERA_IN_GA(X1, .(X2, .(X3, X4))) -> MEMBERA_IN_GA(X1, X4) R is empty. The argument filtering Pi contains the following mapping: memberA_in_ga(x1, x2) = memberA_in_ga(x1) .(x1, x2) = .(x2) MEMBERA_IN_GA(x1, x2) = MEMBERA_IN_GA(x1) U1_GA(x1, x2, x3, x4, x5) = U1_GA(x1, x5) We have to consider all (P,R,Pi)-chains Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES ---------------------------------------- (50) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBERA_IN_GA(X1, .(X2, .(X3, X4))) -> U1_GA(X1, X2, X3, X4, memberA_in_ga(X1, X4)) MEMBERA_IN_GA(X1, .(X2, .(X3, X4))) -> MEMBERA_IN_GA(X1, X4) R is empty. The argument filtering Pi contains the following mapping: memberA_in_ga(x1, x2) = memberA_in_ga(x1) .(x1, x2) = .(x2) MEMBERA_IN_GA(x1, x2) = MEMBERA_IN_GA(x1) U1_GA(x1, x2, x3, x4, x5) = U1_GA(x1, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (51) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node. ---------------------------------------- (52) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBERA_IN_GA(X1, .(X2, .(X3, X4))) -> MEMBERA_IN_GA(X1, X4) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) MEMBERA_IN_GA(x1, x2) = MEMBERA_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (53) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (54) Obligation: Q DP problem: The TRS P consists of the following rules: MEMBERA_IN_GA(X1) -> MEMBERA_IN_GA(X1) R is empty. Q is empty. We have to consider all (P,Q,R)-chains.