/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern q() w.r.t. the given Prolog program could not be shown: (0) Prolog (1) CutEliminatorProof [SOUND, 0 ms] (2) Prolog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, 3 ms] (22) QDP (23) PrologToTRSTransformerProof [SOUND, 0 ms] (24) QTRS (25) QTRSRRRProof [EQUIVALENT, 21 ms] (26) QTRS (27) Overlay + Local Confluence [EQUIVALENT, 0 ms] (28) QTRS (29) DependencyPairsProof [EQUIVALENT, 0 ms] (30) QDP (31) UsableRulesProof [EQUIVALENT, 0 ms] (32) QDP (33) QReductionProof [EQUIVALENT, 0 ms] (34) QDP (35) PrologToDTProblemTransformerProof [SOUND, 0 ms] (36) TRIPLES (37) TriplesToPiDPProof [SOUND, 0 ms] (38) PiDP (39) DependencyGraphProof [EQUIVALENT, 0 ms] (40) PiDP (41) PiDPToQDPProof [EQUIVALENT, 0 ms] (42) QDP (43) PrologToIRSwTTransformerProof [SOUND, 0 ms] (44) IRSwT (45) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (46) IRSwT (47) IntTRSCompressionProof [EQUIVALENT, 6 ms] (48) IRSwT (49) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (50) IRSwT (51) IRSwTTerminationDigraphProof [EQUIVALENT, 6 ms] (52) IRSwT (53) FilterProof [EQUIVALENT, 0 ms] (54) IntTRS (55) IntTRSPeriodicNontermProof [COMPLETE, 0 ms] (56) NO ---------------------------------------- (0) Obligation: Clauses: q :- p(X, X). p(a, b) :- !. p(a, a) :- p(a, a). Query: q() ---------------------------------------- (1) CutEliminatorProof (SOUND) Eliminated all cuts by simply ignoring them[PROLOG]. ---------------------------------------- (2) Obligation: Clauses: q :- p(X, X). p(a, b). p(a, a) :- p(a, a). Query: q() ---------------------------------------- (3) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: p_in_2: (f,f) (b,b) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: q_in_ -> U1_(p_in_aa(X, X)) p_in_aa(a, b) -> p_out_aa(a, b) p_in_aa(a, a) -> U2_aa(p_in_gg(a, a)) p_in_gg(a, b) -> p_out_gg(a, b) p_in_gg(a, a) -> U2_gg(p_in_gg(a, a)) U2_gg(p_out_gg(a, a)) -> p_out_gg(a, a) U2_aa(p_out_gg(a, a)) -> p_out_aa(a, a) U1_(p_out_aa(X, X)) -> q_out_ The argument filtering Pi contains the following mapping: q_in_ = q_in_ U1_(x1) = U1_(x1) p_in_aa(x1, x2) = p_in_aa p_out_aa(x1, x2) = p_out_aa(x1, x2) U2_aa(x1) = U2_aa(x1) p_in_gg(x1, x2) = p_in_gg(x1, x2) a = a b = b p_out_gg(x1, x2) = p_out_gg U2_gg(x1) = U2_gg(x1) q_out_ = q_out_ Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (4) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: q_in_ -> U1_(p_in_aa(X, X)) p_in_aa(a, b) -> p_out_aa(a, b) p_in_aa(a, a) -> U2_aa(p_in_gg(a, a)) p_in_gg(a, b) -> p_out_gg(a, b) p_in_gg(a, a) -> U2_gg(p_in_gg(a, a)) U2_gg(p_out_gg(a, a)) -> p_out_gg(a, a) U2_aa(p_out_gg(a, a)) -> p_out_aa(a, a) U1_(p_out_aa(X, X)) -> q_out_ The argument filtering Pi contains the following mapping: q_in_ = q_in_ U1_(x1) = U1_(x1) p_in_aa(x1, x2) = p_in_aa p_out_aa(x1, x2) = p_out_aa(x1, x2) U2_aa(x1) = U2_aa(x1) p_in_gg(x1, x2) = p_in_gg(x1, x2) a = a b = b p_out_gg(x1, x2) = p_out_gg U2_gg(x1) = U2_gg(x1) q_out_ = q_out_ ---------------------------------------- (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: Q_IN_ -> U1_^1(p_in_aa(X, X)) Q_IN_ -> P_IN_AA(X, X) P_IN_AA(a, a) -> U2_AA(p_in_gg(a, a)) P_IN_AA(a, a) -> P_IN_GG(a, a) P_IN_GG(a, a) -> U2_GG(p_in_gg(a, a)) P_IN_GG(a, a) -> P_IN_GG(a, a) The TRS R consists of the following rules: q_in_ -> U1_(p_in_aa(X, X)) p_in_aa(a, b) -> p_out_aa(a, b) p_in_aa(a, a) -> U2_aa(p_in_gg(a, a)) p_in_gg(a, b) -> p_out_gg(a, b) p_in_gg(a, a) -> U2_gg(p_in_gg(a, a)) U2_gg(p_out_gg(a, a)) -> p_out_gg(a, a) U2_aa(p_out_gg(a, a)) -> p_out_aa(a, a) U1_(p_out_aa(X, X)) -> q_out_ The argument filtering Pi contains the following mapping: q_in_ = q_in_ U1_(x1) = U1_(x1) p_in_aa(x1, x2) = p_in_aa p_out_aa(x1, x2) = p_out_aa(x1, x2) U2_aa(x1) = U2_aa(x1) p_in_gg(x1, x2) = p_in_gg(x1, x2) a = a b = b p_out_gg(x1, x2) = p_out_gg U2_gg(x1) = U2_gg(x1) q_out_ = q_out_ Q_IN_ = Q_IN_ U1_^1(x1) = U1_^1(x1) P_IN_AA(x1, x2) = P_IN_AA U2_AA(x1) = U2_AA(x1) P_IN_GG(x1, x2) = P_IN_GG(x1, x2) U2_GG(x1) = U2_GG(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (6) Obligation: Pi DP problem: The TRS P consists of the following rules: Q_IN_ -> U1_^1(p_in_aa(X, X)) Q_IN_ -> P_IN_AA(X, X) P_IN_AA(a, a) -> U2_AA(p_in_gg(a, a)) P_IN_AA(a, a) -> P_IN_GG(a, a) P_IN_GG(a, a) -> U2_GG(p_in_gg(a, a)) P_IN_GG(a, a) -> P_IN_GG(a, a) The TRS R consists of the following rules: q_in_ -> U1_(p_in_aa(X, X)) p_in_aa(a, b) -> p_out_aa(a, b) p_in_aa(a, a) -> U2_aa(p_in_gg(a, a)) p_in_gg(a, b) -> p_out_gg(a, b) p_in_gg(a, a) -> U2_gg(p_in_gg(a, a)) U2_gg(p_out_gg(a, a)) -> p_out_gg(a, a) U2_aa(p_out_gg(a, a)) -> p_out_aa(a, a) U1_(p_out_aa(X, X)) -> q_out_ The argument filtering Pi contains the following mapping: q_in_ = q_in_ U1_(x1) = U1_(x1) p_in_aa(x1, x2) = p_in_aa p_out_aa(x1, x2) = p_out_aa(x1, x2) U2_aa(x1) = U2_aa(x1) p_in_gg(x1, x2) = p_in_gg(x1, x2) a = a b = b p_out_gg(x1, x2) = p_out_gg U2_gg(x1) = U2_gg(x1) q_out_ = q_out_ Q_IN_ = Q_IN_ U1_^1(x1) = U1_^1(x1) P_IN_AA(x1, x2) = P_IN_AA U2_AA(x1) = U2_AA(x1) P_IN_GG(x1, x2) = P_IN_GG(x1, x2) U2_GG(x1) = U2_GG(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (7) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 5 less nodes. ---------------------------------------- (8) Obligation: Pi DP problem: The TRS P consists of the following rules: P_IN_GG(a, a) -> P_IN_GG(a, a) The TRS R consists of the following rules: q_in_ -> U1_(p_in_aa(X, X)) p_in_aa(a, b) -> p_out_aa(a, b) p_in_aa(a, a) -> U2_aa(p_in_gg(a, a)) p_in_gg(a, b) -> p_out_gg(a, b) p_in_gg(a, a) -> U2_gg(p_in_gg(a, a)) U2_gg(p_out_gg(a, a)) -> p_out_gg(a, a) U2_aa(p_out_gg(a, a)) -> p_out_aa(a, a) U1_(p_out_aa(X, X)) -> q_out_ The argument filtering Pi contains the following mapping: q_in_ = q_in_ U1_(x1) = U1_(x1) p_in_aa(x1, x2) = p_in_aa p_out_aa(x1, x2) = p_out_aa(x1, x2) U2_aa(x1) = U2_aa(x1) p_in_gg(x1, x2) = p_in_gg(x1, x2) a = a b = b p_out_gg(x1, x2) = p_out_gg U2_gg(x1) = U2_gg(x1) q_out_ = q_out_ P_IN_GG(x1, x2) = P_IN_GG(x1, x2) 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: P_IN_GG(a, a) -> P_IN_GG(a, a) 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: P_IN_GG(a, a) -> P_IN_GG(a, a) 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: p_in_2: (f,f) (b,b) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: q_in_ -> U1_(p_in_aa(X, X)) p_in_aa(a, b) -> p_out_aa(a, b) p_in_aa(a, a) -> U2_aa(p_in_gg(a, a)) p_in_gg(a, b) -> p_out_gg(a, b) p_in_gg(a, a) -> U2_gg(p_in_gg(a, a)) U2_gg(p_out_gg(a, a)) -> p_out_gg(a, a) U2_aa(p_out_gg(a, a)) -> p_out_aa(a, a) U1_(p_out_aa(X, X)) -> q_out_ The argument filtering Pi contains the following mapping: q_in_ = q_in_ U1_(x1) = U1_(x1) p_in_aa(x1, x2) = p_in_aa p_out_aa(x1, x2) = p_out_aa(x1, x2) U2_aa(x1) = U2_aa(x1) p_in_gg(x1, x2) = p_in_gg(x1, x2) a = a b = b p_out_gg(x1, x2) = p_out_gg(x1, x2) U2_gg(x1) = U2_gg(x1) q_out_ = q_out_ Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (14) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: q_in_ -> U1_(p_in_aa(X, X)) p_in_aa(a, b) -> p_out_aa(a, b) p_in_aa(a, a) -> U2_aa(p_in_gg(a, a)) p_in_gg(a, b) -> p_out_gg(a, b) p_in_gg(a, a) -> U2_gg(p_in_gg(a, a)) U2_gg(p_out_gg(a, a)) -> p_out_gg(a, a) U2_aa(p_out_gg(a, a)) -> p_out_aa(a, a) U1_(p_out_aa(X, X)) -> q_out_ The argument filtering Pi contains the following mapping: q_in_ = q_in_ U1_(x1) = U1_(x1) p_in_aa(x1, x2) = p_in_aa p_out_aa(x1, x2) = p_out_aa(x1, x2) U2_aa(x1) = U2_aa(x1) p_in_gg(x1, x2) = p_in_gg(x1, x2) a = a b = b p_out_gg(x1, x2) = p_out_gg(x1, x2) U2_gg(x1) = U2_gg(x1) q_out_ = q_out_ ---------------------------------------- (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: Q_IN_ -> U1_^1(p_in_aa(X, X)) Q_IN_ -> P_IN_AA(X, X) P_IN_AA(a, a) -> U2_AA(p_in_gg(a, a)) P_IN_AA(a, a) -> P_IN_GG(a, a) P_IN_GG(a, a) -> U2_GG(p_in_gg(a, a)) P_IN_GG(a, a) -> P_IN_GG(a, a) The TRS R consists of the following rules: q_in_ -> U1_(p_in_aa(X, X)) p_in_aa(a, b) -> p_out_aa(a, b) p_in_aa(a, a) -> U2_aa(p_in_gg(a, a)) p_in_gg(a, b) -> p_out_gg(a, b) p_in_gg(a, a) -> U2_gg(p_in_gg(a, a)) U2_gg(p_out_gg(a, a)) -> p_out_gg(a, a) U2_aa(p_out_gg(a, a)) -> p_out_aa(a, a) U1_(p_out_aa(X, X)) -> q_out_ The argument filtering Pi contains the following mapping: q_in_ = q_in_ U1_(x1) = U1_(x1) p_in_aa(x1, x2) = p_in_aa p_out_aa(x1, x2) = p_out_aa(x1, x2) U2_aa(x1) = U2_aa(x1) p_in_gg(x1, x2) = p_in_gg(x1, x2) a = a b = b p_out_gg(x1, x2) = p_out_gg(x1, x2) U2_gg(x1) = U2_gg(x1) q_out_ = q_out_ Q_IN_ = Q_IN_ U1_^1(x1) = U1_^1(x1) P_IN_AA(x1, x2) = P_IN_AA U2_AA(x1) = U2_AA(x1) P_IN_GG(x1, x2) = P_IN_GG(x1, x2) U2_GG(x1) = U2_GG(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: Q_IN_ -> U1_^1(p_in_aa(X, X)) Q_IN_ -> P_IN_AA(X, X) P_IN_AA(a, a) -> U2_AA(p_in_gg(a, a)) P_IN_AA(a, a) -> P_IN_GG(a, a) P_IN_GG(a, a) -> U2_GG(p_in_gg(a, a)) P_IN_GG(a, a) -> P_IN_GG(a, a) The TRS R consists of the following rules: q_in_ -> U1_(p_in_aa(X, X)) p_in_aa(a, b) -> p_out_aa(a, b) p_in_aa(a, a) -> U2_aa(p_in_gg(a, a)) p_in_gg(a, b) -> p_out_gg(a, b) p_in_gg(a, a) -> U2_gg(p_in_gg(a, a)) U2_gg(p_out_gg(a, a)) -> p_out_gg(a, a) U2_aa(p_out_gg(a, a)) -> p_out_aa(a, a) U1_(p_out_aa(X, X)) -> q_out_ The argument filtering Pi contains the following mapping: q_in_ = q_in_ U1_(x1) = U1_(x1) p_in_aa(x1, x2) = p_in_aa p_out_aa(x1, x2) = p_out_aa(x1, x2) U2_aa(x1) = U2_aa(x1) p_in_gg(x1, x2) = p_in_gg(x1, x2) a = a b = b p_out_gg(x1, x2) = p_out_gg(x1, x2) U2_gg(x1) = U2_gg(x1) q_out_ = q_out_ Q_IN_ = Q_IN_ U1_^1(x1) = U1_^1(x1) P_IN_AA(x1, x2) = P_IN_AA U2_AA(x1) = U2_AA(x1) P_IN_GG(x1, x2) = P_IN_GG(x1, x2) U2_GG(x1) = U2_GG(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (17) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 5 less nodes. ---------------------------------------- (18) Obligation: Pi DP problem: The TRS P consists of the following rules: P_IN_GG(a, a) -> P_IN_GG(a, a) The TRS R consists of the following rules: q_in_ -> U1_(p_in_aa(X, X)) p_in_aa(a, b) -> p_out_aa(a, b) p_in_aa(a, a) -> U2_aa(p_in_gg(a, a)) p_in_gg(a, b) -> p_out_gg(a, b) p_in_gg(a, a) -> U2_gg(p_in_gg(a, a)) U2_gg(p_out_gg(a, a)) -> p_out_gg(a, a) U2_aa(p_out_gg(a, a)) -> p_out_aa(a, a) U1_(p_out_aa(X, X)) -> q_out_ The argument filtering Pi contains the following mapping: q_in_ = q_in_ U1_(x1) = U1_(x1) p_in_aa(x1, x2) = p_in_aa p_out_aa(x1, x2) = p_out_aa(x1, x2) U2_aa(x1) = U2_aa(x1) p_in_gg(x1, x2) = p_in_gg(x1, x2) a = a b = b p_out_gg(x1, x2) = p_out_gg(x1, x2) U2_gg(x1) = U2_gg(x1) q_out_ = q_out_ P_IN_GG(x1, x2) = P_IN_GG(x1, x2) 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: P_IN_GG(a, a) -> P_IN_GG(a, a) 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: P_IN_GG(a, a) -> P_IN_GG(a, a) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (23) PrologToTRSTransformerProof (SOUND) Transformed Prolog program to TRS. { "root": 2, "program": { "directives": [], "clauses": [ [ "(q)", "(p X X)" ], [ "(p (a) (b))", "(!)" ], [ "(p (a) (a))", "(p (a) (a))" ] ] }, "graph": { "nodes": { "11": { "goal": [{ "clause": -1, "scope": -1, "term": "(p X2 X2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X2"], "exprvars": [] } }, "2": { "goal": [{ "clause": -1, "scope": -1, "term": "(q)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "24": { "goal": [ { "clause": 1, "scope": 2, "term": "(p X2 X2)" }, { "clause": 2, "scope": 2, "term": "(p X2 X2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X2"], "exprvars": [] } }, "35": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (a) (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "3": { "goal": [{ "clause": 0, "scope": 1, "term": "(q)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "37": { "goal": [ { "clause": 1, "scope": 3, "term": "(p (a) (a))" }, { "clause": 2, "scope": 3, "term": "(p (a) (a))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "39": { "goal": [{ "clause": 2, "scope": 3, "term": "(p (a) (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "29": { "goal": [{ "clause": 2, "scope": 2, "term": "(p X2 X2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X2"], "exprvars": [] } }, "type": "Nodes", "42": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (a) (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 2, "to": 3, "label": "CASE" }, { "from": 3, "to": 11, "label": "ONLY EVAL with clause\nq :- p(X2, X2).\nand substitution" }, { "from": 11, "to": 24, "label": "CASE" }, { "from": 24, "to": 29, "label": "BACKTRACK\nfor clause: p(a, b) :- !because of non-unification" }, { "from": 29, "to": 35, "label": "ONLY EVAL with clause\np(a, a) :- p(a, a).\nand substitutionX2 -> a" }, { "from": 35, "to": 37, "label": "CASE" }, { "from": 37, "to": 39, "label": "BACKTRACK\nfor clause: p(a, b) :- !because of non-unification" }, { "from": 39, "to": 42, "label": "ONLY EVAL with clause\np(a, a) :- p(a, a).\nand substitution" }, { "from": 42, "to": 35, "label": "INSTANCE" } ], "type": "Graph" } } ---------------------------------------- (24) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f2_in -> U1(f35_in) U1(f35_out1) -> f2_out1 f35_in -> U2(f35_in) U2(f35_out1) -> f35_out1 Q is empty. ---------------------------------------- (25) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U1(x_1)) = 2*x_1 POL(U2(x_1)) = 2*x_1 POL(f2_in) = 1 POL(f2_out1) = 0 POL(f35_in) = 0 POL(f35_out1) = 2 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: f2_in -> U1(f35_in) U1(f35_out1) -> f2_out1 U2(f35_out1) -> f35_out1 ---------------------------------------- (26) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f35_in -> U2(f35_in) Q is empty. ---------------------------------------- (27) Overlay + Local Confluence (EQUIVALENT) The TRS is overlay and locally confluent. By [NOC] we can switch to innermost. ---------------------------------------- (28) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f35_in -> U2(f35_in) The set Q consists of the following terms: f35_in ---------------------------------------- (29) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (30) Obligation: Q DP problem: The TRS P consists of the following rules: F35_IN -> F35_IN The TRS R consists of the following rules: f35_in -> U2(f35_in) The set Q consists of the following terms: f35_in We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (31) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (32) Obligation: Q DP problem: The TRS P consists of the following rules: F35_IN -> F35_IN R is empty. The set Q consists of the following terms: f35_in We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (33) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. f35_in ---------------------------------------- (34) Obligation: Q DP problem: The TRS P consists of the following rules: F35_IN -> F35_IN R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (35) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 16, "program": { "directives": [], "clauses": [ [ "(q)", "(p X X)" ], [ "(p (a) (b))", "(!)" ], [ "(p (a) (a))", "(p (a) (a))" ] ] }, "graph": { "nodes": { "36": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (a) (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "16": { "goal": [{ "clause": -1, "scope": -1, "term": "(q)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "38": { "goal": [ { "clause": 1, "scope": 3, "term": "(p (a) (a))" }, { "clause": 2, "scope": 3, "term": "(p (a) (a))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "17": { "goal": [{ "clause": 0, "scope": 1, "term": "(q)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "28": { "goal": [{ "clause": -1, "scope": -1, "term": "(p X1 X1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X1"], "exprvars": [] } }, "type": "Nodes", "40": { "goal": [{ "clause": 2, "scope": 3, "term": "(p (a) (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "31": { "goal": [ { "clause": 1, "scope": 2, "term": "(p X1 X1)" }, { "clause": 2, "scope": 2, "term": "(p X1 X1)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X1"], "exprvars": [] } }, "32": { "goal": [{ "clause": 2, "scope": 2, "term": "(p X1 X1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X1"], "exprvars": [] } }, "43": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (a) (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 16, "to": 17, "label": "CASE" }, { "from": 17, "to": 28, "label": "ONLY EVAL with clause\nq :- p(X1, X1).\nand substitution" }, { "from": 28, "to": 31, "label": "CASE" }, { "from": 31, "to": 32, "label": "BACKTRACK\nfor clause: p(a, b) :- !because of non-unification" }, { "from": 32, "to": 36, "label": "ONLY EVAL with clause\np(a, a) :- p(a, a).\nand substitutionX1 -> a" }, { "from": 36, "to": 38, "label": "CASE" }, { "from": 38, "to": 40, "label": "BACKTRACK\nfor clause: p(a, b) :- !because of non-unification" }, { "from": 40, "to": 43, "label": "ONLY EVAL with clause\np(a, a) :- p(a, a).\nand substitution" }, { "from": 43, "to": 36, "label": "INSTANCE" } ], "type": "Graph" } } ---------------------------------------- (36) Obligation: Triples: pA :- pA. qB :- pA. Clauses: pcA :- pcA. Afs: qB = qB ---------------------------------------- (37) 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 ---------------------------------------- (38) 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 ---------------------------------------- (39) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes. ---------------------------------------- (40) 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 ---------------------------------------- (41) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (42) 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. ---------------------------------------- (43) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 1, "program": { "directives": [], "clauses": [ [ "(q)", "(p X X)" ], [ "(p (a) (b))", "(!)" ], [ "(p (a) (a))", "(p (a) (a))" ] ] }, "graph": { "nodes": { "33": { "goal": [ { "clause": 1, "scope": 2, "term": "(p X2 X2)" }, { "clause": 2, "scope": 2, "term": "(p X2 X2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X2"], "exprvars": [] } }, "1": { "goal": [{ "clause": -1, "scope": -1, "term": "(q)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "12": { "goal": [{ "clause": 0, "scope": 1, "term": "(q)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "34": { "goal": [{ "clause": 2, "scope": 2, "term": "(p X2 X2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X2"], "exprvars": [] } }, "45": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (a) (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "13": { "goal": [{ "clause": -1, "scope": -1, "term": "(p X2 X2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X2"], "exprvars": [] } }, "46": { "goal": [ { "clause": 1, "scope": 3, "term": "(p (a) (a))" }, { "clause": 2, "scope": 3, "term": "(p (a) (a))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "47": { "goal": [{ "clause": 2, "scope": 3, "term": "(p (a) (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "48": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (a) (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes" }, "edges": [ { "from": 1, "to": 12, "label": "CASE" }, { "from": 12, "to": 13, "label": "ONLY EVAL with clause\nq :- p(X2, X2).\nand substitution" }, { "from": 13, "to": 33, "label": "CASE" }, { "from": 33, "to": 34, "label": "BACKTRACK\nfor clause: p(a, b) :- !because of non-unification" }, { "from": 34, "to": 45, "label": "ONLY EVAL with clause\np(a, a) :- p(a, a).\nand substitutionX2 -> a" }, { "from": 45, "to": 46, "label": "CASE" }, { "from": 46, "to": 47, "label": "BACKTRACK\nfor clause: p(a, b) :- !because of non-unification" }, { "from": 47, "to": 48, "label": "ONLY EVAL with clause\np(a, a) :- p(a, a).\nand substitution" }, { "from": 48, "to": 45, "label": "INSTANCE" } ], "type": "Graph" } } ---------------------------------------- (44) Obligation: Rules: f46_in -> f47_in :|: TRUE f47_out -> f46_out :|: TRUE f45_in -> f46_in :|: TRUE f46_out -> f45_out :|: TRUE f48_out -> f47_out :|: TRUE f47_in -> f48_in :|: TRUE f45_out -> f48_out :|: TRUE f48_in -> f45_in :|: TRUE f1_in -> f12_in :|: TRUE f12_out -> f1_out :|: TRUE f12_in -> f13_in :|: TRUE f13_out -> f12_out :|: TRUE f13_in -> f33_in :|: TRUE f33_out -> f13_out :|: TRUE f34_out -> f33_out :|: TRUE f33_in -> f34_in :|: TRUE f34_in -> f45_in :|: TRUE f45_out -> f34_out :|: TRUE Start term: f1_in ---------------------------------------- (45) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f46_in -> f47_in :|: TRUE f45_in -> f46_in :|: TRUE f47_in -> f48_in :|: TRUE f48_in -> f45_in :|: TRUE ---------------------------------------- (46) Obligation: Rules: f46_in -> f47_in :|: TRUE f45_in -> f46_in :|: TRUE f47_in -> f48_in :|: TRUE f48_in -> f45_in :|: TRUE ---------------------------------------- (47) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (48) Obligation: Rules: f45_in -> f45_in :|: TRUE ---------------------------------------- (49) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (50) Obligation: Rules: f45_in -> f45_in :|: TRUE ---------------------------------------- (51) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f45_in -> f45_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (52) Obligation: Termination digraph: Nodes: (1) f45_in -> f45_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (53) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f45_in() Replaced non-predefined constructor symbols by 0. ---------------------------------------- (54) Obligation: Rules: f45_in -> f45_in :|: TRUE ---------------------------------------- (55) IntTRSPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc) -> f(1) :|: pc = 1 && TRUE Witness term starting non-terminating reduction: f(1) ---------------------------------------- (56) NO