/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern q() w.r.t. the given Prolog program could not be shown: (0) Prolog (1) PrologToTRSTransformerProof [SOUND, 0 ms] (2) QTRS (3) QTRSRRRProof [EQUIVALENT, 35 ms] (4) QTRS (5) Overlay + Local Confluence [EQUIVALENT, 0 ms] (6) QTRS (7) DependencyPairsProof [EQUIVALENT, 0 ms] (8) QDP (9) UsableRulesProof [EQUIVALENT, 0 ms] (10) QDP (11) QReductionProof [EQUIVALENT, 0 ms] (12) QDP (13) PrologToDTProblemTransformerProof [SOUND, 0 ms] (14) TRIPLES (15) TriplesToPiDPProof [SOUND, 0 ms] (16) PiDP (17) DependencyGraphProof [EQUIVALENT, 0 ms] (18) PiDP (19) PiDPToQDPProof [EQUIVALENT, 3 ms] (20) QDP (21) PrologToIRSwTTransformerProof [SOUND, 0 ms] (22) IRSwT (23) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (24) IRSwT (25) IntTRSCompressionProof [EQUIVALENT, 21 ms] (26) IRSwT (27) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (28) IRSwT (29) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms] (30) IRSwT (31) FilterProof [EQUIVALENT, 0 ms] (32) IntTRS (33) IntTRSPeriodicNontermProof [COMPLETE, 1 ms] (34) NO ---------------------------------------- (0) Obligation: Clauses: q :- p(!). p(X) :- ','(X, failure(a)). p(X) :- p(X). failure(b). Query: q() ---------------------------------------- (1) PrologToTRSTransformerProof (SOUND) Transformed Prolog program to TRS. { "root": 3, "program": { "directives": [], "clauses": [ [ "(q)", "(p (!))" ], [ "(p X)", "(',' X (failure (a)))" ], [ "(p X)", "(p X)" ], [ "(failure (b))", null ] ] }, "graph": { "nodes": { "22": { "goal": [{ "clause": 2, "scope": 2, "term": "(p (!))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "34": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "25": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (call (!)) (failure (a)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "36": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (!))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "27": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (!_3) (failure (a)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "18": { "goal": [ { "clause": 1, "scope": 2, "term": "(p (!))" }, { "clause": 2, "scope": 2, "term": "(p (!))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "29": { "goal": [{ "clause": -1, "scope": -1, "term": "(failure (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "3": { "goal": [{ "clause": -1, "scope": -1, "term": "(q)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "4": { "goal": [{ "clause": 0, "scope": 1, "term": "(q)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "8": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (!))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "20": { "goal": [{ "clause": 1, "scope": 2, "term": "(p (!))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "32": { "goal": [{ "clause": 3, "scope": 4, "term": "(failure (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 3, "to": 4, "label": "CASE" }, { "from": 4, "to": 8, "label": "ONLY EVAL with clause\nq :- p(!).\nand substitution" }, { "from": 8, "to": 18, "label": "CASE" }, { "from": 18, "to": 20, "label": "PARALLEL" }, { "from": 18, "to": 22, "label": "PARALLEL" }, { "from": 20, "to": 25, "label": "ONLY EVAL with clause\np(X7) :- ','(call(X7), failure(a)).\nand substitutionX7 -> !" }, { "from": 22, "to": 36, "label": "ONLY EVAL with clause\np(X11) :- p(X11).\nand substitutionX11 -> !" }, { "from": 25, "to": 27, "label": "CALL" }, { "from": 27, "to": 29, "label": "CUT" }, { "from": 29, "to": 32, "label": "CASE" }, { "from": 32, "to": 34, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 36, "to": 8, "label": "INSTANCE" } ], "type": "Graph" } } ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f3_in -> U1(f8_in) U1(f8_out1) -> f3_out1 f8_in -> U2(f8_in) U2(f8_out1) -> f8_out1 Q is empty. ---------------------------------------- (3) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U1(x_1)) = 2*x_1 POL(U2(x_1)) = 2*x_1 POL(f3_in) = 1 POL(f3_out1) = 0 POL(f8_in) = 0 POL(f8_out1) = 2 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: f3_in -> U1(f8_in) U1(f8_out1) -> f3_out1 U2(f8_out1) -> f8_out1 ---------------------------------------- (4) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f8_in -> U2(f8_in) Q is empty. ---------------------------------------- (5) Overlay + Local Confluence (EQUIVALENT) The TRS is overlay and locally confluent. By [NOC] we can switch to innermost. ---------------------------------------- (6) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f8_in -> U2(f8_in) The set Q consists of the following terms: f8_in ---------------------------------------- (7) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (8) Obligation: Q DP problem: The TRS P consists of the following rules: F8_IN -> F8_IN The TRS R consists of the following rules: f8_in -> U2(f8_in) The set Q consists of the following terms: f8_in 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: F8_IN -> F8_IN R is empty. The set Q consists of the following terms: f8_in 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]. f8_in ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: F8_IN -> F8_IN R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (13) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 1, "program": { "directives": [], "clauses": [ [ "(q)", "(p (!))" ], [ "(p X)", "(',' X (failure (a)))" ], [ "(p X)", "(p X)" ], [ "(failure (b))", null ] ] }, "graph": { "nodes": { "11": { "goal": [{ "clause": 1, "scope": 2, "term": "(p (!))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "12": { "goal": [{ "clause": 2, "scope": 2, "term": "(p (!))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "13": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (call (!)) (failure (a)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "24": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (!))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "14": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (!_3) (failure (a)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "15": { "goal": [{ "clause": -1, "scope": -1, "term": "(failure (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "16": { "goal": [{ "clause": 3, "scope": 4, "term": "(failure (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "17": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "1": { "goal": [{ "clause": -1, "scope": -1, "term": "(q)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "6": { "goal": [{ "clause": 0, "scope": 1, "term": "(q)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "7": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (!))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "10": { "goal": [ { "clause": 1, "scope": 2, "term": "(p (!))" }, { "clause": 2, "scope": 2, "term": "(p (!))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 6, "label": "CASE" }, { "from": 6, "to": 7, "label": "ONLY EVAL with clause\nq :- p(!).\nand substitution" }, { "from": 7, "to": 10, "label": "CASE" }, { "from": 10, "to": 11, "label": "PARALLEL" }, { "from": 10, "to": 12, "label": "PARALLEL" }, { "from": 11, "to": 13, "label": "ONLY EVAL with clause\np(X5) :- ','(call(X5), failure(a)).\nand substitutionX5 -> !" }, { "from": 12, "to": 24, "label": "ONLY EVAL with clause\np(X9) :- p(X9).\nand substitutionX9 -> !" }, { "from": 13, "to": 14, "label": "CALL" }, { "from": 14, "to": 15, "label": "CUT" }, { "from": 15, "to": 16, "label": "CASE" }, { "from": 16, "to": 17, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 24, "to": 7, "label": "INSTANCE" } ], "type": "Graph" } } ---------------------------------------- (14) Obligation: Triples: pA :- pA. qB :- pA. Clauses: pcA :- pcA. Afs: qB = qB ---------------------------------------- (15) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: QB_IN_ -> U2_^1(pA_in_) QB_IN_ -> PA_IN_ PA_IN_ -> U1_^1(pA_in_) PA_IN_ -> PA_IN_ R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: QB_IN_ -> U2_^1(pA_in_) QB_IN_ -> PA_IN_ PA_IN_ -> U1_^1(pA_in_) PA_IN_ -> PA_IN_ R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (17) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes. ---------------------------------------- (18) Obligation: Pi DP problem: The TRS P consists of the following rules: PA_IN_ -> PA_IN_ R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (19) PiDPToQDPProof (EQUIVALENT) 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: PA_IN_ -> PA_IN_ R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (21) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 2, "program": { "directives": [], "clauses": [ [ "(q)", "(p (!))" ], [ "(p X)", "(',' X (failure (a)))" ], [ "(p X)", "(p X)" ], [ "(failure (b))", null ] ] }, "graph": { "nodes": { "33": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "23": { "goal": [{ "clause": 2, "scope": 2, "term": "(p (!))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "35": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (!))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "26": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (call (!)) (failure (a)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "28": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (!_3) (failure (a)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "19": { "goal": [ { "clause": 1, "scope": 2, "term": "(p (!))" }, { "clause": 2, "scope": 2, "term": "(p (!))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "2": { "goal": [{ "clause": -1, "scope": -1, "term": "(q)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "5": { "goal": [{ "clause": 0, "scope": 1, "term": "(q)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "9": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (!))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "30": { "goal": [{ "clause": -1, "scope": -1, "term": "(failure (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "31": { "goal": [{ "clause": 3, "scope": 4, "term": "(failure (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "21": { "goal": [{ "clause": 1, "scope": 2, "term": "(p (!))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 2, "to": 5, "label": "CASE" }, { "from": 5, "to": 9, "label": "ONLY EVAL with clause\nq :- p(!).\nand substitution" }, { "from": 9, "to": 19, "label": "CASE" }, { "from": 19, "to": 21, "label": "PARALLEL" }, { "from": 19, "to": 23, "label": "PARALLEL" }, { "from": 21, "to": 26, "label": "ONLY EVAL with clause\np(X7) :- ','(call(X7), failure(a)).\nand substitutionX7 -> !" }, { "from": 23, "to": 35, "label": "ONLY EVAL with clause\np(X11) :- p(X11).\nand substitutionX11 -> !" }, { "from": 26, "to": 28, "label": "CALL" }, { "from": 28, "to": 30, "label": "CUT" }, { "from": 30, "to": 31, "label": "CASE" }, { "from": 31, "to": 33, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 35, "to": 9, "label": "INSTANCE" } ], "type": "Graph" } } ---------------------------------------- (22) Obligation: Rules: f19_in -> f21_in :|: TRUE f21_out -> f19_out :|: TRUE f23_out -> f19_out :|: TRUE f19_in -> f23_in :|: TRUE f9_in -> f19_in :|: TRUE f19_out -> f9_out :|: TRUE f35_in -> f9_in :|: TRUE f9_out -> f35_out :|: TRUE f35_out -> f23_out :|: TRUE f23_in -> f35_in :|: TRUE f2_in -> f5_in :|: TRUE f5_out -> f2_out :|: TRUE f5_in -> f9_in :|: TRUE f9_out -> f5_out :|: TRUE Start term: f2_in ---------------------------------------- (23) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f19_in -> f23_in :|: TRUE f9_in -> f19_in :|: TRUE f35_in -> f9_in :|: TRUE f23_in -> f35_in :|: TRUE ---------------------------------------- (24) Obligation: Rules: f19_in -> f23_in :|: TRUE f9_in -> f19_in :|: TRUE f35_in -> f9_in :|: TRUE f23_in -> f35_in :|: TRUE ---------------------------------------- (25) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (26) Obligation: Rules: f35_in -> f35_in :|: TRUE ---------------------------------------- (27) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (28) Obligation: Rules: f35_in -> f35_in :|: TRUE ---------------------------------------- (29) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f35_in -> f35_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (30) Obligation: Termination digraph: Nodes: (1) f35_in -> f35_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (31) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f35_in() Replaced non-predefined constructor symbols by 0. ---------------------------------------- (32) Obligation: Rules: f35_in -> f35_in :|: TRUE ---------------------------------------- (33) IntTRSPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc) -> f(1) :|: pc = 1 && TRUE Witness term starting non-terminating reduction: f(1) ---------------------------------------- (34) NO