MAYBE proof of /export/starexec/sandbox/benchmark/theBenchmark.pl # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Left Termination of the query pattern select(g,a,a) w.r.t. the given Prolog program could not be shown: (0) Prolog (1) PrologToTRSTransformerProof [SOUND, 0 ms] (2) QTRS (3) DependencyPairsProof [EQUIVALENT, 3 ms] (4) QDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) QDP (7) MNOCProof [EQUIVALENT, 0 ms] (8) QDP (9) UsableRulesProof [EQUIVALENT, 0 ms] (10) QDP (11) QReductionProof [EQUIVALENT, 0 ms] (12) QDP (13) PrologToPiTRSProof [SOUND, 0 ms] (14) PiTRS (15) DependencyPairsProof [EQUIVALENT, 0 ms] (16) PiDP (17) DependencyGraphProof [EQUIVALENT, 0 ms] (18) PiDP (19) UsableRulesProof [EQUIVALENT, 0 ms] (20) PiDP (21) PiDPToQDPProof [SOUND, 0 ms] (22) QDP (23) PrologToPiTRSProof [SOUND, 0 ms] (24) PiTRS (25) DependencyPairsProof [EQUIVALENT, 0 ms] (26) PiDP (27) DependencyGraphProof [EQUIVALENT, 0 ms] (28) PiDP (29) UsableRulesProof [EQUIVALENT, 0 ms] (30) PiDP (31) PiDPToQDPProof [SOUND, 0 ms] (32) QDP (33) PrologToIRSwTTransformerProof [SOUND, 0 ms] (34) IRSwT (35) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (36) IRSwT (37) IntTRSCompressionProof [EQUIVALENT, 0 ms] (38) IRSwT (39) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (40) IRSwT (41) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms] (42) IRSwT (43) FilterProof [EQUIVALENT, 0 ms] (44) IntTRS (45) IntTRSPeriodicNontermProof [COMPLETE, 5 ms] (46) NO (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: select(X, .(X, Xs), Xs). select(X, .(Y, Xs), .(Y, Zs)) :- select(X, Xs, Zs). Query: select(g,a,a) ---------------------------------------- (1) PrologToTRSTransformerProof (SOUND) Transformed Prolog program to TRS. { "root": 3, "program": { "directives": [], "clauses": [ [ "(select X (. X Xs) Xs)", null ], [ "(select X (. Y Xs) (. Y Zs))", "(select X Xs Zs)" ] ] }, "graph": { "nodes": { "13": { "goal": [{ "clause": 0, "scope": 1, "term": "(select T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "3": { "goal": [{ "clause": -1, "scope": -1, "term": "(select T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "14": { "goal": [{ "clause": 1, "scope": 1, "term": "(select T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "16": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "6": { "goal": [ { "clause": 0, "scope": 1, "term": "(select T1 T2 T3)" }, { "clause": 1, "scope": 1, "term": "(select T1 T2 T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "17": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "18": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "40": { "goal": [{ "clause": -1, "scope": -1, "term": "(select T22 T26 T27)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T22"], "free": [], "exprvars": [] } }, "41": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 3, "to": 6, "label": "CASE" }, { "from": 6, "to": 13, "label": "PARALLEL" }, { "from": 6, "to": 14, "label": "PARALLEL" }, { "from": 13, "to": 16, "label": "EVAL with clause\nselect(X9, .(X9, X10), X10).\nand substitutionT1 -> T12,\nX9 -> T12,\nX10 -> T13,\nT2 -> .(T12, T13),\nT3 -> T13" }, { "from": 13, "to": 17, "label": "EVAL-BACKTRACK" }, { "from": 14, "to": 40, "label": "EVAL with clause\nselect(X19, .(X20, X21), .(X20, X22)) :- select(X19, X21, X22).\nand substitutionT1 -> T22,\nX19 -> T22,\nX20 -> T23,\nX21 -> T26,\nT2 -> .(T23, T26),\nX22 -> T27,\nT3 -> .(T23, T27),\nT24 -> T26,\nT25 -> T27" }, { "from": 14, "to": 41, "label": "EVAL-BACKTRACK" }, { "from": 16, "to": 18, "label": "SUCCESS" }, { "from": 40, "to": 3, "label": "INSTANCE with matching:\nT1 -> T22\nT2 -> T26\nT3 -> T27" } ], "type": "Graph" } } ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f3_in(T12) -> f3_out1 f3_in(T22) -> U1(f3_in(T22), T22) U1(f3_out1, T22) -> f3_out1 Q is empty. ---------------------------------------- (3) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (4) Obligation: Q DP problem: The TRS P consists of the following rules: F3_IN(T22) -> U1^1(f3_in(T22), T22) F3_IN(T22) -> F3_IN(T22) The TRS R consists of the following rules: f3_in(T12) -> f3_out1 f3_in(T22) -> U1(f3_in(T22), T22) U1(f3_out1, T22) -> f3_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (6) Obligation: Q DP problem: The TRS P consists of the following rules: F3_IN(T22) -> F3_IN(T22) The TRS R consists of the following rules: f3_in(T12) -> f3_out1 f3_in(T22) -> U1(f3_in(T22), T22) U1(f3_out1, T22) -> f3_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (7) MNOCProof (EQUIVALENT) We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R. ---------------------------------------- (8) Obligation: Q DP problem: The TRS P consists of the following rules: F3_IN(T22) -> F3_IN(T22) The TRS R consists of the following rules: f3_in(T12) -> f3_out1 f3_in(T22) -> U1(f3_in(T22), T22) U1(f3_out1, T22) -> f3_out1 The set Q consists of the following terms: f3_in(x0) U1(f3_out1, x0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (9) 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. ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: F3_IN(T22) -> F3_IN(T22) R is empty. The set Q consists of the following terms: f3_in(x0) U1(f3_out1, x0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (11) 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]. f3_in(x0) U1(f3_out1, x0) ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: F3_IN(T22) -> F3_IN(T22) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (13) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: select_in_3: (b,f,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: select_in_gaa(X, .(X, Xs), Xs) -> select_out_gaa(X, .(X, Xs), Xs) select_in_gaa(X, .(Y, Xs), .(Y, Zs)) -> U1_gaa(X, Y, Xs, Zs, select_in_gaa(X, Xs, Zs)) U1_gaa(X, Y, Xs, Zs, select_out_gaa(X, Xs, Zs)) -> select_out_gaa(X, .(Y, Xs), .(Y, Zs)) The argument filtering Pi contains the following mapping: select_in_gaa(x1, x2, x3) = select_in_gaa(x1) select_out_gaa(x1, x2, x3) = select_out_gaa U1_gaa(x1, x2, x3, x4, x5) = U1_gaa(x5) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (14) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: select_in_gaa(X, .(X, Xs), Xs) -> select_out_gaa(X, .(X, Xs), Xs) select_in_gaa(X, .(Y, Xs), .(Y, Zs)) -> U1_gaa(X, Y, Xs, Zs, select_in_gaa(X, Xs, Zs)) U1_gaa(X, Y, Xs, Zs, select_out_gaa(X, Xs, Zs)) -> select_out_gaa(X, .(Y, Xs), .(Y, Zs)) The argument filtering Pi contains the following mapping: select_in_gaa(x1, x2, x3) = select_in_gaa(x1) select_out_gaa(x1, x2, x3) = select_out_gaa U1_gaa(x1, x2, x3, x4, x5) = U1_gaa(x5) ---------------------------------------- (15) 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: SELECT_IN_GAA(X, .(Y, Xs), .(Y, Zs)) -> U1_GAA(X, Y, Xs, Zs, select_in_gaa(X, Xs, Zs)) SELECT_IN_GAA(X, .(Y, Xs), .(Y, Zs)) -> SELECT_IN_GAA(X, Xs, Zs) The TRS R consists of the following rules: select_in_gaa(X, .(X, Xs), Xs) -> select_out_gaa(X, .(X, Xs), Xs) select_in_gaa(X, .(Y, Xs), .(Y, Zs)) -> U1_gaa(X, Y, Xs, Zs, select_in_gaa(X, Xs, Zs)) U1_gaa(X, Y, Xs, Zs, select_out_gaa(X, Xs, Zs)) -> select_out_gaa(X, .(Y, Xs), .(Y, Zs)) The argument filtering Pi contains the following mapping: select_in_gaa(x1, x2, x3) = select_in_gaa(x1) select_out_gaa(x1, x2, x3) = select_out_gaa U1_gaa(x1, x2, x3, x4, x5) = U1_gaa(x5) SELECT_IN_GAA(x1, x2, x3) = SELECT_IN_GAA(x1) U1_GAA(x1, x2, x3, x4, x5) = U1_GAA(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: SELECT_IN_GAA(X, .(Y, Xs), .(Y, Zs)) -> U1_GAA(X, Y, Xs, Zs, select_in_gaa(X, Xs, Zs)) SELECT_IN_GAA(X, .(Y, Xs), .(Y, Zs)) -> SELECT_IN_GAA(X, Xs, Zs) The TRS R consists of the following rules: select_in_gaa(X, .(X, Xs), Xs) -> select_out_gaa(X, .(X, Xs), Xs) select_in_gaa(X, .(Y, Xs), .(Y, Zs)) -> U1_gaa(X, Y, Xs, Zs, select_in_gaa(X, Xs, Zs)) U1_gaa(X, Y, Xs, Zs, select_out_gaa(X, Xs, Zs)) -> select_out_gaa(X, .(Y, Xs), .(Y, Zs)) The argument filtering Pi contains the following mapping: select_in_gaa(x1, x2, x3) = select_in_gaa(x1) select_out_gaa(x1, x2, x3) = select_out_gaa U1_gaa(x1, x2, x3, x4, x5) = U1_gaa(x5) SELECT_IN_GAA(x1, x2, x3) = SELECT_IN_GAA(x1) U1_GAA(x1, x2, x3, x4, x5) = U1_GAA(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (17) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node. ---------------------------------------- (18) Obligation: Pi DP problem: The TRS P consists of the following rules: SELECT_IN_GAA(X, .(Y, Xs), .(Y, Zs)) -> SELECT_IN_GAA(X, Xs, Zs) The TRS R consists of the following rules: select_in_gaa(X, .(X, Xs), Xs) -> select_out_gaa(X, .(X, Xs), Xs) select_in_gaa(X, .(Y, Xs), .(Y, Zs)) -> U1_gaa(X, Y, Xs, Zs, select_in_gaa(X, Xs, Zs)) U1_gaa(X, Y, Xs, Zs, select_out_gaa(X, Xs, Zs)) -> select_out_gaa(X, .(Y, Xs), .(Y, Zs)) The argument filtering Pi contains the following mapping: select_in_gaa(x1, x2, x3) = select_in_gaa(x1) select_out_gaa(x1, x2, x3) = select_out_gaa U1_gaa(x1, x2, x3, x4, x5) = U1_gaa(x5) SELECT_IN_GAA(x1, x2, x3) = SELECT_IN_GAA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (19) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (20) Obligation: Pi DP problem: The TRS P consists of the following rules: SELECT_IN_GAA(X, .(Y, Xs), .(Y, Zs)) -> SELECT_IN_GAA(X, Xs, Zs) R is empty. The argument filtering Pi contains the following mapping: SELECT_IN_GAA(x1, x2, x3) = SELECT_IN_GAA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (21) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (22) Obligation: Q DP problem: The TRS P consists of the following rules: SELECT_IN_GAA(X) -> SELECT_IN_GAA(X) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (23) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: select_in_3: (b,f,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: select_in_gaa(X, .(X, Xs), Xs) -> select_out_gaa(X, .(X, Xs), Xs) select_in_gaa(X, .(Y, Xs), .(Y, Zs)) -> U1_gaa(X, Y, Xs, Zs, select_in_gaa(X, Xs, Zs)) U1_gaa(X, Y, Xs, Zs, select_out_gaa(X, Xs, Zs)) -> select_out_gaa(X, .(Y, Xs), .(Y, Zs)) The argument filtering Pi contains the following mapping: select_in_gaa(x1, x2, x3) = select_in_gaa(x1) select_out_gaa(x1, x2, x3) = select_out_gaa(x1) U1_gaa(x1, x2, x3, x4, x5) = U1_gaa(x1, x5) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (24) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: select_in_gaa(X, .(X, Xs), Xs) -> select_out_gaa(X, .(X, Xs), Xs) select_in_gaa(X, .(Y, Xs), .(Y, Zs)) -> U1_gaa(X, Y, Xs, Zs, select_in_gaa(X, Xs, Zs)) U1_gaa(X, Y, Xs, Zs, select_out_gaa(X, Xs, Zs)) -> select_out_gaa(X, .(Y, Xs), .(Y, Zs)) The argument filtering Pi contains the following mapping: select_in_gaa(x1, x2, x3) = select_in_gaa(x1) select_out_gaa(x1, x2, x3) = select_out_gaa(x1) U1_gaa(x1, x2, x3, x4, x5) = U1_gaa(x1, x5) ---------------------------------------- (25) 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: SELECT_IN_GAA(X, .(Y, Xs), .(Y, Zs)) -> U1_GAA(X, Y, Xs, Zs, select_in_gaa(X, Xs, Zs)) SELECT_IN_GAA(X, .(Y, Xs), .(Y, Zs)) -> SELECT_IN_GAA(X, Xs, Zs) The TRS R consists of the following rules: select_in_gaa(X, .(X, Xs), Xs) -> select_out_gaa(X, .(X, Xs), Xs) select_in_gaa(X, .(Y, Xs), .(Y, Zs)) -> U1_gaa(X, Y, Xs, Zs, select_in_gaa(X, Xs, Zs)) U1_gaa(X, Y, Xs, Zs, select_out_gaa(X, Xs, Zs)) -> select_out_gaa(X, .(Y, Xs), .(Y, Zs)) The argument filtering Pi contains the following mapping: select_in_gaa(x1, x2, x3) = select_in_gaa(x1) select_out_gaa(x1, x2, x3) = select_out_gaa(x1) U1_gaa(x1, x2, x3, x4, x5) = U1_gaa(x1, x5) SELECT_IN_GAA(x1, x2, x3) = SELECT_IN_GAA(x1) U1_GAA(x1, x2, x3, x4, x5) = U1_GAA(x1, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (26) Obligation: Pi DP problem: The TRS P consists of the following rules: SELECT_IN_GAA(X, .(Y, Xs), .(Y, Zs)) -> U1_GAA(X, Y, Xs, Zs, select_in_gaa(X, Xs, Zs)) SELECT_IN_GAA(X, .(Y, Xs), .(Y, Zs)) -> SELECT_IN_GAA(X, Xs, Zs) The TRS R consists of the following rules: select_in_gaa(X, .(X, Xs), Xs) -> select_out_gaa(X, .(X, Xs), Xs) select_in_gaa(X, .(Y, Xs), .(Y, Zs)) -> U1_gaa(X, Y, Xs, Zs, select_in_gaa(X, Xs, Zs)) U1_gaa(X, Y, Xs, Zs, select_out_gaa(X, Xs, Zs)) -> select_out_gaa(X, .(Y, Xs), .(Y, Zs)) The argument filtering Pi contains the following mapping: select_in_gaa(x1, x2, x3) = select_in_gaa(x1) select_out_gaa(x1, x2, x3) = select_out_gaa(x1) U1_gaa(x1, x2, x3, x4, x5) = U1_gaa(x1, x5) SELECT_IN_GAA(x1, x2, x3) = SELECT_IN_GAA(x1) U1_GAA(x1, x2, x3, x4, x5) = U1_GAA(x1, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (27) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node. ---------------------------------------- (28) Obligation: Pi DP problem: The TRS P consists of the following rules: SELECT_IN_GAA(X, .(Y, Xs), .(Y, Zs)) -> SELECT_IN_GAA(X, Xs, Zs) The TRS R consists of the following rules: select_in_gaa(X, .(X, Xs), Xs) -> select_out_gaa(X, .(X, Xs), Xs) select_in_gaa(X, .(Y, Xs), .(Y, Zs)) -> U1_gaa(X, Y, Xs, Zs, select_in_gaa(X, Xs, Zs)) U1_gaa(X, Y, Xs, Zs, select_out_gaa(X, Xs, Zs)) -> select_out_gaa(X, .(Y, Xs), .(Y, Zs)) The argument filtering Pi contains the following mapping: select_in_gaa(x1, x2, x3) = select_in_gaa(x1) select_out_gaa(x1, x2, x3) = select_out_gaa(x1) U1_gaa(x1, x2, x3, x4, x5) = U1_gaa(x1, x5) SELECT_IN_GAA(x1, x2, x3) = SELECT_IN_GAA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (29) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (30) Obligation: Pi DP problem: The TRS P consists of the following rules: SELECT_IN_GAA(X, .(Y, Xs), .(Y, Zs)) -> SELECT_IN_GAA(X, Xs, Zs) R is empty. The argument filtering Pi contains the following mapping: SELECT_IN_GAA(x1, x2, x3) = SELECT_IN_GAA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (31) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (32) Obligation: Q DP problem: The TRS P consists of the following rules: SELECT_IN_GAA(X) -> SELECT_IN_GAA(X) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (33) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 2, "program": { "directives": [], "clauses": [ [ "(select X (. X Xs) Xs)", null ], [ "(select X (. Y Xs) (. Y Zs))", "(select X Xs Zs)" ] ] }, "graph": { "nodes": { "11": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "23": { "goal": [{ "clause": -1, "scope": -1, "term": "(select T22 T26 T27)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T22"], "free": [], "exprvars": [] } }, "2": { "goal": [{ "clause": -1, "scope": -1, "term": "(select T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "24": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "15": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "5": { "goal": [ { "clause": 0, "scope": 1, "term": "(select T1 T2 T3)" }, { "clause": 1, "scope": 1, "term": "(select T1 T2 T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "7": { "goal": [{ "clause": 0, "scope": 1, "term": "(select T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "8": { "goal": [{ "clause": 1, "scope": 1, "term": "(select T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "9": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes" }, "edges": [ { "from": 2, "to": 5, "label": "CASE" }, { "from": 5, "to": 7, "label": "PARALLEL" }, { "from": 5, "to": 8, "label": "PARALLEL" }, { "from": 7, "to": 9, "label": "EVAL with clause\nselect(X9, .(X9, X10), X10).\nand substitutionT1 -> T12,\nX9 -> T12,\nX10 -> T13,\nT2 -> .(T12, T13),\nT3 -> T13" }, { "from": 7, "to": 11, "label": "EVAL-BACKTRACK" }, { "from": 8, "to": 23, "label": "EVAL with clause\nselect(X19, .(X20, X21), .(X20, X22)) :- select(X19, X21, X22).\nand substitutionT1 -> T22,\nX19 -> T22,\nX20 -> T23,\nX21 -> T26,\nT2 -> .(T23, T26),\nX22 -> T27,\nT3 -> .(T23, T27),\nT24 -> T26,\nT25 -> T27" }, { "from": 8, "to": 24, "label": "EVAL-BACKTRACK" }, { "from": 9, "to": 15, "label": "SUCCESS" }, { "from": 23, "to": 2, "label": "INSTANCE with matching:\nT1 -> T22\nT2 -> T26\nT3 -> T27" } ], "type": "Graph" } } ---------------------------------------- (34) Obligation: Rules: f5_out(T1) -> f2_out(T1) :|: TRUE f2_in(x) -> f5_in(x) :|: TRUE f7_out(x1) -> f5_out(x1) :|: TRUE f5_in(x2) -> f7_in(x2) :|: TRUE f5_in(x3) -> f8_in(x3) :|: TRUE f8_out(x4) -> f5_out(x4) :|: TRUE f23_in(T22) -> f2_in(T22) :|: TRUE f2_out(x5) -> f23_out(x5) :|: TRUE f8_in(x6) -> f24_in :|: TRUE f8_in(x7) -> f23_in(x7) :|: TRUE f23_out(x8) -> f8_out(x8) :|: TRUE f24_out -> f8_out(x9) :|: TRUE Start term: f2_in(T1) ---------------------------------------- (35) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f2_in(x) -> f5_in(x) :|: TRUE f5_in(x3) -> f8_in(x3) :|: TRUE f23_in(T22) -> f2_in(T22) :|: TRUE f8_in(x7) -> f23_in(x7) :|: TRUE ---------------------------------------- (36) Obligation: Rules: f2_in(x) -> f5_in(x) :|: TRUE f5_in(x3) -> f8_in(x3) :|: TRUE f23_in(T22) -> f2_in(T22) :|: TRUE f8_in(x7) -> f23_in(x7) :|: TRUE ---------------------------------------- (37) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (38) Obligation: Rules: f23_in(T22:0) -> f23_in(T22:0) :|: TRUE ---------------------------------------- (39) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (40) Obligation: Rules: f23_in(T22:0) -> f23_in(T22:0) :|: TRUE ---------------------------------------- (41) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f23_in(T22:0) -> f23_in(T22:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (42) Obligation: Termination digraph: Nodes: (1) f23_in(T22:0) -> f23_in(T22:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (43) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f23_in(VARIABLE) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (44) Obligation: Rules: f23_in(T22:0) -> f23_in(T22:0) :|: TRUE ---------------------------------------- (45) IntTRSPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc, T22:0) -> f(1, T22:0) :|: pc = 1 && TRUE Witness term starting non-terminating reduction: f(1, -8) ---------------------------------------- (46) NO ---------------------------------------- (47) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 1, "program": { "directives": [], "clauses": [ [ "(select X (. X Xs) Xs)", null ], [ "(select X (. Y Xs) (. Y Zs))", "(select X Xs Zs)" ] ] }, "graph": { "nodes": { "22": { "goal": [ { "clause": 0, "scope": 2, "term": "(select T12 T16 T17)" }, { "clause": 1, "scope": 2, "term": "(select T12 T16 T17)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T12"], "free": [], "exprvars": [] } }, "25": { "goal": [{ "clause": 0, "scope": 2, "term": "(select T12 T16 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T12"], "free": [], "exprvars": [] } }, "26": { "goal": [{ "clause": 1, "scope": 2, "term": "(select T12 T16 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T12"], "free": [], "exprvars": [] } }, "27": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "28": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "29": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "111": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "94": { "goal": [{ "clause": -1, "scope": -1, "term": "(select T50 T54 T55)" }], "kb": { "nonunifying": [[ "(select T50 T2 T3)", "(select X3 (. X3 X4) X4)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T50"], "free": [ "X3", "X4" ], "exprvars": [] } }, "95": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "30": { "goal": [{ "clause": -1, "scope": -1, "term": "(select T36 T40 T41)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T36"], "free": [], "exprvars": [] } }, "96": { "goal": [ { "clause": 0, "scope": 3, "term": "(select T50 T54 T55)" }, { "clause": 1, "scope": 3, "term": "(select T50 T54 T55)" } ], "kb": { "nonunifying": [[ "(select T50 T2 T3)", "(select X3 (. X3 X4) X4)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T50"], "free": [ "X3", "X4" ], "exprvars": [] } }, "31": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "10": { "goal": [ { "clause": -1, "scope": -1, "term": "(true)" }, { "clause": 1, "scope": 1, "term": "(select T6 T2 T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T6"], "free": [], "exprvars": [] } }, "12": { "goal": [{ "clause": 1, "scope": 1, "term": "(select T1 T2 T3)" }], "kb": { "nonunifying": [[ "(select T1 T2 T3)", "(select X3 (. X3 X4) X4)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [ "X3", "X4" ], "exprvars": [] } }, "19": { "goal": [{ "clause": 1, "scope": 1, "term": "(select T6 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T6"], "free": [], "exprvars": [] } }, "1": { "goal": [{ "clause": -1, "scope": -1, "term": "(select T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "4": { "goal": [ { "clause": 0, "scope": 1, "term": "(select T1 T2 T3)" }, { "clause": 1, "scope": 1, "term": "(select T1 T2 T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "103": { "goal": [{ "clause": 0, "scope": 3, "term": "(select T50 T54 T55)" }], "kb": { "nonunifying": [[ "(select T50 T2 T3)", "(select X3 (. X3 X4) X4)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T50"], "free": [ "X3", "X4" ], "exprvars": [] } }, "104": { "goal": [{ "clause": 1, "scope": 3, "term": "(select T50 T54 T55)" }], "kb": { "nonunifying": [[ "(select T50 T2 T3)", "(select X3 (. X3 X4) X4)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T50"], "free": [ "X3", "X4" ], "exprvars": [] } }, "105": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "106": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "107": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "109": { "goal": [{ "clause": -1, "scope": -1, "term": "(select T74 T78 T79)" }], "kb": { "nonunifying": [[ "(select T74 T2 T3)", "(select X3 (. X3 X4) X4)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T74"], "free": [ "X3", "X4" ], "exprvars": [] } }, "20": { "goal": [{ "clause": -1, "scope": -1, "term": "(select T12 T16 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T12"], "free": [], "exprvars": [] } }, "21": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 4, "label": "CASE" }, { "from": 4, "to": 10, "label": "EVAL with clause\nselect(X3, .(X3, X4), X4).\nand substitutionT1 -> T6,\nX3 -> T6,\nX4 -> T7,\nT2 -> .(T6, T7),\nT3 -> T7" }, { "from": 4, "to": 12, "label": "EVAL-BACKTRACK" }, { "from": 10, "to": 19, "label": "SUCCESS" }, { "from": 12, "to": 94, "label": "EVAL with clause\nselect(X43, .(X44, X45), .(X44, X46)) :- select(X43, X45, X46).\nand substitutionT1 -> T50,\nX43 -> T50,\nX44 -> T51,\nX45 -> T54,\nT2 -> .(T51, T54),\nX46 -> T55,\nT3 -> .(T51, T55),\nT52 -> T54,\nT53 -> T55" }, { "from": 12, "to": 95, "label": "EVAL-BACKTRACK" }, { "from": 19, "to": 20, "label": "EVAL with clause\nselect(X9, .(X10, X11), .(X10, X12)) :- select(X9, X11, X12).\nand substitutionT6 -> T12,\nX9 -> T12,\nX10 -> T13,\nX11 -> T16,\nT2 -> .(T13, T16),\nX12 -> T17,\nT3 -> .(T13, T17),\nT14 -> T16,\nT15 -> T17" }, { "from": 19, "to": 21, "label": "EVAL-BACKTRACK" }, { "from": 20, "to": 22, "label": "CASE" }, { "from": 22, "to": 25, "label": "PARALLEL" }, { "from": 22, "to": 26, "label": "PARALLEL" }, { "from": 25, "to": 27, "label": "EVAL with clause\nselect(X21, .(X21, X22), X22).\nand substitutionT12 -> T26,\nX21 -> T26,\nX22 -> T27,\nT16 -> .(T26, T27),\nT17 -> T27" }, { "from": 25, "to": 28, "label": "EVAL-BACKTRACK" }, { "from": 26, "to": 30, "label": "EVAL with clause\nselect(X31, .(X32, X33), .(X32, X34)) :- select(X31, X33, X34).\nand substitutionT12 -> T36,\nX31 -> T36,\nX32 -> T37,\nX33 -> T40,\nT16 -> .(T37, T40),\nX34 -> T41,\nT17 -> .(T37, T41),\nT38 -> T40,\nT39 -> T41" }, { "from": 26, "to": 31, "label": "EVAL-BACKTRACK" }, { "from": 27, "to": 29, "label": "SUCCESS" }, { "from": 30, "to": 1, "label": "INSTANCE with matching:\nT1 -> T36\nT2 -> T40\nT3 -> T41" }, { "from": 94, "to": 96, "label": "CASE" }, { "from": 96, "to": 103, "label": "PARALLEL" }, { "from": 96, "to": 104, "label": "PARALLEL" }, { "from": 103, "to": 105, "label": "EVAL with clause\nselect(X55, .(X55, X56), X56).\nand substitutionT50 -> T64,\nX55 -> T64,\nX56 -> T65,\nT54 -> .(T64, T65),\nT55 -> T65" }, { "from": 103, "to": 106, "label": "EVAL-BACKTRACK" }, { "from": 104, "to": 109, "label": "EVAL with clause\nselect(X65, .(X66, X67), .(X66, X68)) :- select(X65, X67, X68).\nand substitutionT50 -> T74,\nX65 -> T74,\nX66 -> T75,\nX67 -> T78,\nT54 -> .(T75, T78),\nX68 -> T79,\nT55 -> .(T75, T79),\nT76 -> T78,\nT77 -> T79" }, { "from": 104, "to": 111, "label": "EVAL-BACKTRACK" }, { "from": 105, "to": 107, "label": "SUCCESS" }, { "from": 109, "to": 1, "label": "INSTANCE with matching:\nT1 -> T74\nT2 -> T78\nT3 -> T79" } ], "type": "Graph" } } ---------------------------------------- (48) Obligation: Triples: selectA(X1, .(X2, .(X3, X4)), .(X2, .(X3, X5))) :- selectA(X1, X4, X5). selectA(X1, .(X2, .(X3, X4)), .(X2, .(X3, X5))) :- selectA(X1, X4, X5). Clauses: selectcA(X1, .(X1, X2), X2). selectcA(X1, .(X2, .(X1, X3)), .(X2, X3)). selectcA(X1, .(X2, .(X3, X4)), .(X2, .(X3, X5))) :- selectcA(X1, X4, X5). selectcA(X1, .(X2, .(X1, X3)), .(X2, X3)). selectcA(X1, .(X2, .(X3, X4)), .(X2, .(X3, X5))) :- selectcA(X1, X4, X5). Afs: selectA(x1, x2, x3) = selectA(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: selectA_in_3: (b,f,f) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: SELECTA_IN_GAA(X1, .(X2, .(X3, X4)), .(X2, .(X3, X5))) -> U1_GAA(X1, X2, X3, X4, X5, selectA_in_gaa(X1, X4, X5)) SELECTA_IN_GAA(X1, .(X2, .(X3, X4)), .(X2, .(X3, X5))) -> SELECTA_IN_GAA(X1, X4, X5) R is empty. The argument filtering Pi contains the following mapping: selectA_in_gaa(x1, x2, x3) = selectA_in_gaa(x1) .(x1, x2) = .(x2) SELECTA_IN_GAA(x1, x2, x3) = SELECTA_IN_GAA(x1) U1_GAA(x1, x2, x3, x4, x5, x6) = U1_GAA(x1, x6) 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: SELECTA_IN_GAA(X1, .(X2, .(X3, X4)), .(X2, .(X3, X5))) -> U1_GAA(X1, X2, X3, X4, X5, selectA_in_gaa(X1, X4, X5)) SELECTA_IN_GAA(X1, .(X2, .(X3, X4)), .(X2, .(X3, X5))) -> SELECTA_IN_GAA(X1, X4, X5) R is empty. The argument filtering Pi contains the following mapping: selectA_in_gaa(x1, x2, x3) = selectA_in_gaa(x1) .(x1, x2) = .(x2) SELECTA_IN_GAA(x1, x2, x3) = SELECTA_IN_GAA(x1) U1_GAA(x1, x2, x3, x4, x5, x6) = U1_GAA(x1, x6) 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: SELECTA_IN_GAA(X1, .(X2, .(X3, X4)), .(X2, .(X3, X5))) -> SELECTA_IN_GAA(X1, X4, X5) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) SELECTA_IN_GAA(x1, x2, x3) = SELECTA_IN_GAA(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: SELECTA_IN_GAA(X1) -> SELECTA_IN_GAA(X1) R is empty. Q is empty. We have to consider all (P,Q,R)-chains.