/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern p() w.r.t. the given Prolog program could not be shown: (0) Prolog (1) CutEliminatorProof [SOUND, 0 ms] (2) Prolog (3) PrologToPiTRSProof [SOUND, 0 ms] (4) PiTRS (5) DependencyPairsProof [EQUIVALENT, 0 ms] (6) PiDP (7) DependencyGraphProof [EQUIVALENT, 0 ms] (8) PiDP (9) UsableRulesProof [EQUIVALENT, 0 ms] (10) PiDP (11) PiDPToQDPProof [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 [EQUIVALENT, 0 ms] (22) QDP (23) PrologToDTProblemTransformerProof [SOUND, 0 ms] (24) TRIPLES (25) TriplesToPiDPProof [SOUND, 0 ms] (26) PiDP (27) DependencyGraphProof [EQUIVALENT, 0 ms] (28) PiDP (29) PiDPToQDPProof [EQUIVALENT, 0 ms] (30) QDP (31) PrologToTRSTransformerProof [SOUND, 0 ms] (32) QTRS (33) QTRSRRRProof [EQUIVALENT, 30 ms] (34) QTRS (35) QTRSRRRProof [EQUIVALENT, 6 ms] (36) QTRS (37) QTRSRRRProof [EQUIVALENT, 0 ms] (38) QTRS (39) QTRSRRRProof [EQUIVALENT, 0 ms] (40) QTRS (41) Overlay + Local Confluence [EQUIVALENT, 0 ms] (42) QTRS (43) DependencyPairsProof [EQUIVALENT, 0 ms] (44) QDP (45) UsableRulesProof [EQUIVALENT, 0 ms] (46) QDP (47) QReductionProof [EQUIVALENT, 0 ms] (48) QDP (49) PrologToIRSwTTransformerProof [SOUND, 0 ms] (50) IRSwT (51) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (52) IRSwT (53) IntTRSCompressionProof [EQUIVALENT, 44 ms] (54) IRSwT (55) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (56) IRSwT (57) IRSwTTerminationDigraphProof [EQUIVALENT, 6 ms] (58) IRSwT (59) FilterProof [EQUIVALENT, 0 ms] (60) IntTRS (61) IntTRSPeriodicNontermProof [COMPLETE, 5 ms] (62) NO ---------------------------------------- (0) Obligation: Clauses: p :- r. r :- ','(!, q). r :- q. q. q :- r. Query: p() ---------------------------------------- (1) CutEliminatorProof (SOUND) Eliminated all cuts by simply ignoring them[PROLOG]. ---------------------------------------- (2) Obligation: Clauses: p :- r. r :- q. r :- q. q. q :- r. Query: p() ---------------------------------------- (3) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: p_in_ -> U1_(r_in_) r_in_ -> U2_(q_in_) q_in_ -> q_out_ q_in_ -> U3_(r_in_) U3_(r_out_) -> q_out_ U2_(q_out_) -> r_out_ U1_(r_out_) -> p_out_ Pi is empty. Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (4) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: p_in_ -> U1_(r_in_) r_in_ -> U2_(q_in_) q_in_ -> q_out_ q_in_ -> U3_(r_in_) U3_(r_out_) -> q_out_ U2_(q_out_) -> r_out_ U1_(r_out_) -> p_out_ Pi is empty. ---------------------------------------- (5) 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: P_IN_ -> U1_^1(r_in_) P_IN_ -> R_IN_ R_IN_ -> U2_^1(q_in_) R_IN_ -> Q_IN_ Q_IN_ -> U3_^1(r_in_) Q_IN_ -> R_IN_ The TRS R consists of the following rules: p_in_ -> U1_(r_in_) r_in_ -> U2_(q_in_) q_in_ -> q_out_ q_in_ -> U3_(r_in_) U3_(r_out_) -> q_out_ U2_(q_out_) -> r_out_ U1_(r_out_) -> p_out_ Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (6) Obligation: Pi DP problem: The TRS P consists of the following rules: P_IN_ -> U1_^1(r_in_) P_IN_ -> R_IN_ R_IN_ -> U2_^1(q_in_) R_IN_ -> Q_IN_ Q_IN_ -> U3_^1(r_in_) Q_IN_ -> R_IN_ The TRS R consists of the following rules: p_in_ -> U1_(r_in_) r_in_ -> U2_(q_in_) q_in_ -> q_out_ q_in_ -> U3_(r_in_) U3_(r_out_) -> q_out_ U2_(q_out_) -> r_out_ U1_(r_out_) -> p_out_ Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (7) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 4 less nodes. ---------------------------------------- (8) Obligation: Pi DP problem: The TRS P consists of the following rules: R_IN_ -> Q_IN_ Q_IN_ -> R_IN_ The TRS R consists of the following rules: p_in_ -> U1_(r_in_) r_in_ -> U2_(q_in_) q_in_ -> q_out_ q_in_ -> U3_(r_in_) U3_(r_out_) -> q_out_ U2_(q_out_) -> r_out_ U1_(r_out_) -> p_out_ Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (9) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (10) Obligation: Pi DP problem: The TRS P consists of the following rules: R_IN_ -> Q_IN_ Q_IN_ -> R_IN_ R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (11) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: R_IN_ -> Q_IN_ Q_IN_ -> R_IN_ R is empty. Q is empty. We have to consider all (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: Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: p_in_ -> U1_(r_in_) r_in_ -> U2_(q_in_) q_in_ -> q_out_ q_in_ -> U3_(r_in_) U3_(r_out_) -> q_out_ U2_(q_out_) -> r_out_ U1_(r_out_) -> p_out_ Pi is empty. Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (14) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: p_in_ -> U1_(r_in_) r_in_ -> U2_(q_in_) q_in_ -> q_out_ q_in_ -> U3_(r_in_) U3_(r_out_) -> q_out_ U2_(q_out_) -> r_out_ U1_(r_out_) -> p_out_ Pi is empty. ---------------------------------------- (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: P_IN_ -> U1_^1(r_in_) P_IN_ -> R_IN_ R_IN_ -> U2_^1(q_in_) R_IN_ -> Q_IN_ Q_IN_ -> U3_^1(r_in_) Q_IN_ -> R_IN_ The TRS R consists of the following rules: p_in_ -> U1_(r_in_) r_in_ -> U2_(q_in_) q_in_ -> q_out_ q_in_ -> U3_(r_in_) U3_(r_out_) -> q_out_ U2_(q_out_) -> r_out_ U1_(r_out_) -> p_out_ Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: P_IN_ -> U1_^1(r_in_) P_IN_ -> R_IN_ R_IN_ -> U2_^1(q_in_) R_IN_ -> Q_IN_ Q_IN_ -> U3_^1(r_in_) Q_IN_ -> R_IN_ The TRS R consists of the following rules: p_in_ -> U1_(r_in_) r_in_ -> U2_(q_in_) q_in_ -> q_out_ q_in_ -> U3_(r_in_) U3_(r_out_) -> q_out_ U2_(q_out_) -> r_out_ U1_(r_out_) -> p_out_ 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 4 less nodes. ---------------------------------------- (18) Obligation: Pi DP problem: The TRS P consists of the following rules: R_IN_ -> Q_IN_ Q_IN_ -> R_IN_ The TRS R consists of the following rules: p_in_ -> U1_(r_in_) r_in_ -> U2_(q_in_) q_in_ -> q_out_ q_in_ -> U3_(r_in_) U3_(r_out_) -> q_out_ U2_(q_out_) -> r_out_ U1_(r_out_) -> p_out_ Pi is empty. 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: R_IN_ -> Q_IN_ Q_IN_ -> R_IN_ R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (21) PiDPToQDPProof (EQUIVALENT) 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: R_IN_ -> Q_IN_ Q_IN_ -> R_IN_ R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (23) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 1, "program": { "directives": [], "clauses": [ [ "(p)", "(r)" ], [ "(r)", "(',' (!) (q))" ], [ "(r)", "(q)" ], [ "(q)", null ], [ "(q)", "(r)" ] ] }, "graph": { "nodes": { "11": { "goal": [{ "clause": -1, "scope": -1, "term": "(r)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "44": { "goal": [{ "clause": 3, "scope": 3, "term": "(q)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "55": { "goal": [{ "clause": -1, "scope": -1, "term": "(r)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "12": { "goal": [ { "clause": 1, "scope": 2, "term": "(r)" }, { "clause": 2, "scope": 2, "term": "(r)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "45": { "goal": [{ "clause": 4, "scope": 3, "term": "(q)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "13": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (!_2) (q))" }, { "clause": 2, "scope": 2, "term": "(r)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "14": { "goal": [{ "clause": -1, "scope": -1, "term": "(q)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "47": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "49": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "1": { "goal": [{ "clause": -1, "scope": -1, "term": "(p)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "2": { "goal": [{ "clause": 0, "scope": 1, "term": "(p)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "43": { "goal": [ { "clause": 3, "scope": 3, "term": "(q)" }, { "clause": 4, "scope": 3, "term": "(q)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 2, "label": "CASE" }, { "from": 2, "to": 11, "label": "ONLY EVAL with clause\np :- r.\nand substitution" }, { "from": 11, "to": 12, "label": "CASE" }, { "from": 12, "to": 13, "label": "ONLY EVAL with clause\nr :- ','(!_2, q).\nand substitution" }, { "from": 13, "to": 14, "label": "CUT" }, { "from": 14, "to": 43, "label": "CASE" }, { "from": 43, "to": 44, "label": "PARALLEL" }, { "from": 43, "to": 45, "label": "PARALLEL" }, { "from": 44, "to": 47, "label": "ONLY EVAL with clause\nq.\nand substitution" }, { "from": 45, "to": 55, "label": "ONLY EVAL with clause\nq :- r.\nand substitution" }, { "from": 47, "to": 49, "label": "SUCCESS" }, { "from": 55, "to": 11, "label": "INSTANCE" } ], "type": "Graph" } } ---------------------------------------- (24) Obligation: Triples: rA :- rA. pB :- rA. Clauses: rcA. rcA :- rcA. Afs: pB = pB ---------------------------------------- (25) 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: PB_IN_ -> U2_^1(rA_in_) PB_IN_ -> RA_IN_ RA_IN_ -> U1_^1(rA_in_) RA_IN_ -> RA_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 ---------------------------------------- (26) Obligation: Pi DP problem: The TRS P consists of the following rules: PB_IN_ -> U2_^1(rA_in_) PB_IN_ -> RA_IN_ RA_IN_ -> U1_^1(rA_in_) RA_IN_ -> RA_IN_ R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (27) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes. ---------------------------------------- (28) Obligation: Pi DP problem: The TRS P consists of the following rules: RA_IN_ -> RA_IN_ R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (29) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (30) Obligation: Q DP problem: The TRS P consists of the following rules: RA_IN_ -> RA_IN_ R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (31) PrologToTRSTransformerProof (SOUND) Transformed Prolog program to TRS. { "root": 18, "program": { "directives": [], "clauses": [ [ "(p)", "(r)" ], [ "(r)", "(',' (!) (q))" ], [ "(r)", "(q)" ], [ "(q)", null ], [ "(q)", "(r)" ] ] }, "graph": { "nodes": { "66": { "goal": [ { "clause": 3, "scope": 3, "term": "(q)" }, { "clause": 4, "scope": 3, "term": "(q)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "68": { "goal": [{ "clause": 3, "scope": 3, "term": "(q)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "69": { "goal": [{ "clause": 4, "scope": 3, "term": "(q)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "26": { "goal": [{ "clause": -1, "scope": -1, "term": "(r)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "18": { "goal": [{ "clause": -1, "scope": -1, "term": "(p)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "19": { "goal": [{ "clause": 0, "scope": 1, "term": "(p)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "71": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "73": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "63": { "goal": [ { "clause": 1, "scope": 2, "term": "(r)" }, { "clause": 2, "scope": 2, "term": "(r)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "64": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (!_2) (q))" }, { "clause": 2, "scope": 2, "term": "(r)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "75": { "goal": [{ "clause": -1, "scope": -1, "term": "(r)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "65": { "goal": [{ "clause": -1, "scope": -1, "term": "(q)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 18, "to": 19, "label": "CASE" }, { "from": 19, "to": 26, "label": "ONLY EVAL with clause\np :- r.\nand substitution" }, { "from": 26, "to": 63, "label": "CASE" }, { "from": 63, "to": 64, "label": "ONLY EVAL with clause\nr :- ','(!_2, q).\nand substitution" }, { "from": 64, "to": 65, "label": "CUT" }, { "from": 65, "to": 66, "label": "CASE" }, { "from": 66, "to": 68, "label": "PARALLEL" }, { "from": 66, "to": 69, "label": "PARALLEL" }, { "from": 68, "to": 71, "label": "ONLY EVAL with clause\nq.\nand substitution" }, { "from": 69, "to": 75, "label": "ONLY EVAL with clause\nq :- r.\nand substitution" }, { "from": 71, "to": 73, "label": "SUCCESS" }, { "from": 75, "to": 26, "label": "INSTANCE" } ], "type": "Graph" } } ---------------------------------------- (32) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f18_in -> U1(f26_in) U1(f26_out1) -> f18_out1 f26_in -> f26_out1 f26_in -> U2(f26_in) U2(f26_out1) -> f26_out1 Q is empty. ---------------------------------------- (33) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U1(x_1)) = 2*x_1 POL(U2(x_1)) = 2*x_1 POL(f18_in) = 2 POL(f18_out1) = 0 POL(f26_in) = 0 POL(f26_out1) = 0 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: f18_in -> U1(f26_in) ---------------------------------------- (34) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: U1(f26_out1) -> f18_out1 f26_in -> f26_out1 f26_in -> U2(f26_in) U2(f26_out1) -> f26_out1 Q is empty. ---------------------------------------- (35) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U1(x_1)) = 2 + 2*x_1 POL(U2(x_1)) = 2*x_1 POL(f18_out1) = 0 POL(f26_in) = 0 POL(f26_out1) = 0 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U1(f26_out1) -> f18_out1 ---------------------------------------- (36) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f26_in -> f26_out1 f26_in -> U2(f26_in) U2(f26_out1) -> f26_out1 Q is empty. ---------------------------------------- (37) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U2(x_1)) = x_1 POL(f26_in) = 2 POL(f26_out1) = 0 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: f26_in -> f26_out1 ---------------------------------------- (38) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f26_in -> U2(f26_in) U2(f26_out1) -> f26_out1 Q is empty. ---------------------------------------- (39) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U2(x_1)) = 2*x_1 POL(f26_in) = 0 POL(f26_out1) = 1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U2(f26_out1) -> f26_out1 ---------------------------------------- (40) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f26_in -> U2(f26_in) Q is empty. ---------------------------------------- (41) Overlay + Local Confluence (EQUIVALENT) The TRS is overlay and locally confluent. By [NOC] we can switch to innermost. ---------------------------------------- (42) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f26_in -> U2(f26_in) The set Q consists of the following terms: f26_in ---------------------------------------- (43) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (44) Obligation: Q DP problem: The TRS P consists of the following rules: F26_IN -> F26_IN The TRS R consists of the following rules: f26_in -> U2(f26_in) The set Q consists of the following terms: f26_in We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (45) 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. ---------------------------------------- (46) Obligation: Q DP problem: The TRS P consists of the following rules: F26_IN -> F26_IN R is empty. The set Q consists of the following terms: f26_in We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (47) 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]. f26_in ---------------------------------------- (48) Obligation: Q DP problem: The TRS P consists of the following rules: F26_IN -> F26_IN R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (49) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 15, "program": { "directives": [], "clauses": [ [ "(p)", "(r)" ], [ "(r)", "(',' (!) (q))" ], [ "(r)", "(q)" ], [ "(q)", null ], [ "(q)", "(r)" ] ] }, "graph": { "nodes": { "77": { "goal": [{ "clause": 4, "scope": 3, "term": "(q)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "67": { "goal": [ { "clause": 1, "scope": 2, "term": "(r)" }, { "clause": 2, "scope": 2, "term": "(r)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "78": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "79": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "15": { "goal": [{ "clause": -1, "scope": -1, "term": "(p)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "16": { "goal": [{ "clause": 0, "scope": 1, "term": "(p)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "17": { "goal": [{ "clause": -1, "scope": -1, "term": "(r)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "80": { "goal": [{ "clause": -1, "scope": -1, "term": "(r)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "70": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (!_2) (q))" }, { "clause": 2, "scope": 2, "term": "(r)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "72": { "goal": [{ "clause": -1, "scope": -1, "term": "(q)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "74": { "goal": [ { "clause": 3, "scope": 3, "term": "(q)" }, { "clause": 4, "scope": 3, "term": "(q)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "76": { "goal": [{ "clause": 3, "scope": 3, "term": "(q)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 15, "to": 16, "label": "CASE" }, { "from": 16, "to": 17, "label": "ONLY EVAL with clause\np :- r.\nand substitution" }, { "from": 17, "to": 67, "label": "CASE" }, { "from": 67, "to": 70, "label": "ONLY EVAL with clause\nr :- ','(!_2, q).\nand substitution" }, { "from": 70, "to": 72, "label": "CUT" }, { "from": 72, "to": 74, "label": "CASE" }, { "from": 74, "to": 76, "label": "PARALLEL" }, { "from": 74, "to": 77, "label": "PARALLEL" }, { "from": 76, "to": 78, "label": "ONLY EVAL with clause\nq.\nand substitution" }, { "from": 77, "to": 80, "label": "ONLY EVAL with clause\nq :- r.\nand substitution" }, { "from": 78, "to": 79, "label": "SUCCESS" }, { "from": 80, "to": 17, "label": "INSTANCE" } ], "type": "Graph" } } ---------------------------------------- (50) Obligation: Rules: f70_out -> f67_out :|: TRUE f67_in -> f70_in :|: TRUE f72_out -> f70_out :|: TRUE f70_in -> f72_in :|: TRUE f80_in -> f17_in :|: TRUE f17_out -> f80_out :|: TRUE f72_in -> f74_in :|: TRUE f74_out -> f72_out :|: TRUE f17_in -> f67_in :|: TRUE f67_out -> f17_out :|: TRUE f80_out -> f77_out :|: TRUE f77_in -> f80_in :|: TRUE f74_in -> f76_in :|: TRUE f77_out -> f74_out :|: TRUE f76_out -> f74_out :|: TRUE f74_in -> f77_in :|: TRUE f16_out -> f15_out :|: TRUE f15_in -> f16_in :|: TRUE f17_out -> f16_out :|: TRUE f16_in -> f17_in :|: TRUE Start term: f15_in ---------------------------------------- (51) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f67_in -> f70_in :|: TRUE f70_in -> f72_in :|: TRUE f80_in -> f17_in :|: TRUE f72_in -> f74_in :|: TRUE f17_in -> f67_in :|: TRUE f77_in -> f80_in :|: TRUE f74_in -> f77_in :|: TRUE ---------------------------------------- (52) Obligation: Rules: f67_in -> f70_in :|: TRUE f70_in -> f72_in :|: TRUE f80_in -> f17_in :|: TRUE f72_in -> f74_in :|: TRUE f17_in -> f67_in :|: TRUE f77_in -> f80_in :|: TRUE f74_in -> f77_in :|: TRUE ---------------------------------------- (53) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (54) Obligation: Rules: f77_in -> f77_in :|: TRUE ---------------------------------------- (55) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (56) Obligation: Rules: f77_in -> f77_in :|: TRUE ---------------------------------------- (57) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f77_in -> f77_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (58) Obligation: Termination digraph: Nodes: (1) f77_in -> f77_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (59) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f77_in() Replaced non-predefined constructor symbols by 0. ---------------------------------------- (60) Obligation: Rules: f77_in -> f77_in :|: TRUE ---------------------------------------- (61) IntTRSPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc) -> f(1) :|: pc = 1 && TRUE Witness term starting non-terminating reduction: f(1) ---------------------------------------- (62) NO