/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern goal(g) w.r.t. the given Prolog program could not be shown: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2) PiTRS (3) DependencyPairsProof [EQUIVALENT, 0 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) PiDP (7) UsableRulesProof [EQUIVALENT, 0 ms] (8) PiDP (9) PiDPToQDPProof [SOUND, 0 ms] (10) QDP (11) PrologToPiTRSProof [SOUND, 0 ms] (12) PiTRS (13) DependencyPairsProof [EQUIVALENT, 0 ms] (14) PiDP (15) DependencyGraphProof [EQUIVALENT, 0 ms] (16) PiDP (17) UsableRulesProof [EQUIVALENT, 0 ms] (18) PiDP (19) PiDPToQDPProof [SOUND, 0 ms] (20) QDP (21) PrologToTRSTransformerProof [SOUND, 0 ms] (22) QTRS (23) QTRSRRRProof [EQUIVALENT, 71 ms] (24) QTRS (25) QTRSRRRProof [EQUIVALENT, 0 ms] (26) QTRS (27) QTRSRRRProof [EQUIVALENT, 2 ms] (28) QTRS (29) Overlay + Local Confluence [EQUIVALENT, 0 ms] (30) QTRS (31) DependencyPairsProof [EQUIVALENT, 0 ms] (32) QDP (33) UsableRulesProof [EQUIVALENT, 0 ms] (34) QDP (35) QReductionProof [EQUIVALENT, 0 ms] (36) QDP (37) PrologToDTProblemTransformerProof [SOUND, 0 ms] (38) TRIPLES (39) TriplesToPiDPProof [SOUND, 0 ms] (40) PiDP (41) DependencyGraphProof [EQUIVALENT, 0 ms] (42) PiDP (43) PiDPToQDPProof [SOUND, 0 ms] (44) QDP (45) PrologToIRSwTTransformerProof [SOUND, 0 ms] (46) IRSwT (47) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (48) IRSwT (49) IntTRSCompressionProof [EQUIVALENT, 0 ms] (50) IRSwT (51) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (52) IRSwT (53) IRSwTTerminationDigraphProof [EQUIVALENT, 7 ms] (54) IRSwT (55) FilterProof [EQUIVALENT, 0 ms] (56) IntTRS (57) IntTRSPeriodicNontermProof [COMPLETE, 10 ms] (58) NO ---------------------------------------- (0) Obligation: Clauses: p(a). p(X) :- p(Y). q(b). goal(X) :- ','(p(X), q(X)). Query: goal(g) ---------------------------------------- (1) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: goal_in_1: (b) p_in_1: (b) (f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: goal_in_g(X) -> U2_g(X, p_in_g(X)) p_in_g(a) -> p_out_g(a) p_in_g(X) -> U1_g(X, p_in_a(Y)) p_in_a(a) -> p_out_a(a) p_in_a(X) -> U1_a(X, p_in_a(Y)) U1_a(X, p_out_a(Y)) -> p_out_a(X) U1_g(X, p_out_a(Y)) -> p_out_g(X) U2_g(X, p_out_g(X)) -> U3_g(X, q_in_g(X)) q_in_g(b) -> q_out_g(b) U3_g(X, q_out_g(X)) -> goal_out_g(X) The argument filtering Pi contains the following mapping: goal_in_g(x1) = goal_in_g(x1) U2_g(x1, x2) = U2_g(x1, x2) p_in_g(x1) = p_in_g(x1) a = a p_out_g(x1) = p_out_g(x1) U1_g(x1, x2) = U1_g(x1, x2) p_in_a(x1) = p_in_a p_out_a(x1) = p_out_a U1_a(x1, x2) = U1_a(x2) U3_g(x1, x2) = U3_g(x1, x2) q_in_g(x1) = q_in_g(x1) b = b q_out_g(x1) = q_out_g(x1) goal_out_g(x1) = goal_out_g(x1) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: goal_in_g(X) -> U2_g(X, p_in_g(X)) p_in_g(a) -> p_out_g(a) p_in_g(X) -> U1_g(X, p_in_a(Y)) p_in_a(a) -> p_out_a(a) p_in_a(X) -> U1_a(X, p_in_a(Y)) U1_a(X, p_out_a(Y)) -> p_out_a(X) U1_g(X, p_out_a(Y)) -> p_out_g(X) U2_g(X, p_out_g(X)) -> U3_g(X, q_in_g(X)) q_in_g(b) -> q_out_g(b) U3_g(X, q_out_g(X)) -> goal_out_g(X) The argument filtering Pi contains the following mapping: goal_in_g(x1) = goal_in_g(x1) U2_g(x1, x2) = U2_g(x1, x2) p_in_g(x1) = p_in_g(x1) a = a p_out_g(x1) = p_out_g(x1) U1_g(x1, x2) = U1_g(x1, x2) p_in_a(x1) = p_in_a p_out_a(x1) = p_out_a U1_a(x1, x2) = U1_a(x2) U3_g(x1, x2) = U3_g(x1, x2) q_in_g(x1) = q_in_g(x1) b = b q_out_g(x1) = q_out_g(x1) goal_out_g(x1) = goal_out_g(x1) ---------------------------------------- (3) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: Pi DP problem: The TRS P consists of the following rules: GOAL_IN_G(X) -> U2_G(X, p_in_g(X)) GOAL_IN_G(X) -> P_IN_G(X) P_IN_G(X) -> U1_G(X, p_in_a(Y)) P_IN_G(X) -> P_IN_A(Y) P_IN_A(X) -> U1_A(X, p_in_a(Y)) P_IN_A(X) -> P_IN_A(Y) U2_G(X, p_out_g(X)) -> U3_G(X, q_in_g(X)) U2_G(X, p_out_g(X)) -> Q_IN_G(X) The TRS R consists of the following rules: goal_in_g(X) -> U2_g(X, p_in_g(X)) p_in_g(a) -> p_out_g(a) p_in_g(X) -> U1_g(X, p_in_a(Y)) p_in_a(a) -> p_out_a(a) p_in_a(X) -> U1_a(X, p_in_a(Y)) U1_a(X, p_out_a(Y)) -> p_out_a(X) U1_g(X, p_out_a(Y)) -> p_out_g(X) U2_g(X, p_out_g(X)) -> U3_g(X, q_in_g(X)) q_in_g(b) -> q_out_g(b) U3_g(X, q_out_g(X)) -> goal_out_g(X) The argument filtering Pi contains the following mapping: goal_in_g(x1) = goal_in_g(x1) U2_g(x1, x2) = U2_g(x1, x2) p_in_g(x1) = p_in_g(x1) a = a p_out_g(x1) = p_out_g(x1) U1_g(x1, x2) = U1_g(x1, x2) p_in_a(x1) = p_in_a p_out_a(x1) = p_out_a U1_a(x1, x2) = U1_a(x2) U3_g(x1, x2) = U3_g(x1, x2) q_in_g(x1) = q_in_g(x1) b = b q_out_g(x1) = q_out_g(x1) goal_out_g(x1) = goal_out_g(x1) GOAL_IN_G(x1) = GOAL_IN_G(x1) U2_G(x1, x2) = U2_G(x1, x2) P_IN_G(x1) = P_IN_G(x1) U1_G(x1, x2) = U1_G(x1, x2) P_IN_A(x1) = P_IN_A U1_A(x1, x2) = U1_A(x2) U3_G(x1, x2) = U3_G(x1, x2) Q_IN_G(x1) = Q_IN_G(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: GOAL_IN_G(X) -> U2_G(X, p_in_g(X)) GOAL_IN_G(X) -> P_IN_G(X) P_IN_G(X) -> U1_G(X, p_in_a(Y)) P_IN_G(X) -> P_IN_A(Y) P_IN_A(X) -> U1_A(X, p_in_a(Y)) P_IN_A(X) -> P_IN_A(Y) U2_G(X, p_out_g(X)) -> U3_G(X, q_in_g(X)) U2_G(X, p_out_g(X)) -> Q_IN_G(X) The TRS R consists of the following rules: goal_in_g(X) -> U2_g(X, p_in_g(X)) p_in_g(a) -> p_out_g(a) p_in_g(X) -> U1_g(X, p_in_a(Y)) p_in_a(a) -> p_out_a(a) p_in_a(X) -> U1_a(X, p_in_a(Y)) U1_a(X, p_out_a(Y)) -> p_out_a(X) U1_g(X, p_out_a(Y)) -> p_out_g(X) U2_g(X, p_out_g(X)) -> U3_g(X, q_in_g(X)) q_in_g(b) -> q_out_g(b) U3_g(X, q_out_g(X)) -> goal_out_g(X) The argument filtering Pi contains the following mapping: goal_in_g(x1) = goal_in_g(x1) U2_g(x1, x2) = U2_g(x1, x2) p_in_g(x1) = p_in_g(x1) a = a p_out_g(x1) = p_out_g(x1) U1_g(x1, x2) = U1_g(x1, x2) p_in_a(x1) = p_in_a p_out_a(x1) = p_out_a U1_a(x1, x2) = U1_a(x2) U3_g(x1, x2) = U3_g(x1, x2) q_in_g(x1) = q_in_g(x1) b = b q_out_g(x1) = q_out_g(x1) goal_out_g(x1) = goal_out_g(x1) GOAL_IN_G(x1) = GOAL_IN_G(x1) U2_G(x1, x2) = U2_G(x1, x2) P_IN_G(x1) = P_IN_G(x1) U1_G(x1, x2) = U1_G(x1, x2) P_IN_A(x1) = P_IN_A U1_A(x1, x2) = U1_A(x2) U3_G(x1, x2) = U3_G(x1, x2) Q_IN_G(x1) = Q_IN_G(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 7 less nodes. ---------------------------------------- (6) Obligation: Pi DP problem: The TRS P consists of the following rules: P_IN_A(X) -> P_IN_A(Y) The TRS R consists of the following rules: goal_in_g(X) -> U2_g(X, p_in_g(X)) p_in_g(a) -> p_out_g(a) p_in_g(X) -> U1_g(X, p_in_a(Y)) p_in_a(a) -> p_out_a(a) p_in_a(X) -> U1_a(X, p_in_a(Y)) U1_a(X, p_out_a(Y)) -> p_out_a(X) U1_g(X, p_out_a(Y)) -> p_out_g(X) U2_g(X, p_out_g(X)) -> U3_g(X, q_in_g(X)) q_in_g(b) -> q_out_g(b) U3_g(X, q_out_g(X)) -> goal_out_g(X) The argument filtering Pi contains the following mapping: goal_in_g(x1) = goal_in_g(x1) U2_g(x1, x2) = U2_g(x1, x2) p_in_g(x1) = p_in_g(x1) a = a p_out_g(x1) = p_out_g(x1) U1_g(x1, x2) = U1_g(x1, x2) p_in_a(x1) = p_in_a p_out_a(x1) = p_out_a U1_a(x1, x2) = U1_a(x2) U3_g(x1, x2) = U3_g(x1, x2) q_in_g(x1) = q_in_g(x1) b = b q_out_g(x1) = q_out_g(x1) goal_out_g(x1) = goal_out_g(x1) P_IN_A(x1) = P_IN_A We have to consider all (P,R,Pi)-chains ---------------------------------------- (7) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (8) Obligation: Pi DP problem: The TRS P consists of the following rules: P_IN_A(X) -> P_IN_A(Y) R is empty. The argument filtering Pi contains the following mapping: P_IN_A(x1) = P_IN_A We have to consider all (P,R,Pi)-chains ---------------------------------------- (9) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: P_IN_A -> P_IN_A R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (11) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: goal_in_1: (b) p_in_1: (b) (f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: goal_in_g(X) -> U2_g(X, p_in_g(X)) p_in_g(a) -> p_out_g(a) p_in_g(X) -> U1_g(X, p_in_a(Y)) p_in_a(a) -> p_out_a(a) p_in_a(X) -> U1_a(X, p_in_a(Y)) U1_a(X, p_out_a(Y)) -> p_out_a(X) U1_g(X, p_out_a(Y)) -> p_out_g(X) U2_g(X, p_out_g(X)) -> U3_g(X, q_in_g(X)) q_in_g(b) -> q_out_g(b) U3_g(X, q_out_g(X)) -> goal_out_g(X) The argument filtering Pi contains the following mapping: goal_in_g(x1) = goal_in_g(x1) U2_g(x1, x2) = U2_g(x1, x2) p_in_g(x1) = p_in_g(x1) a = a p_out_g(x1) = p_out_g U1_g(x1, x2) = U1_g(x2) p_in_a(x1) = p_in_a p_out_a(x1) = p_out_a U1_a(x1, x2) = U1_a(x2) U3_g(x1, x2) = U3_g(x2) q_in_g(x1) = q_in_g(x1) b = b q_out_g(x1) = q_out_g goal_out_g(x1) = goal_out_g Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (12) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: goal_in_g(X) -> U2_g(X, p_in_g(X)) p_in_g(a) -> p_out_g(a) p_in_g(X) -> U1_g(X, p_in_a(Y)) p_in_a(a) -> p_out_a(a) p_in_a(X) -> U1_a(X, p_in_a(Y)) U1_a(X, p_out_a(Y)) -> p_out_a(X) U1_g(X, p_out_a(Y)) -> p_out_g(X) U2_g(X, p_out_g(X)) -> U3_g(X, q_in_g(X)) q_in_g(b) -> q_out_g(b) U3_g(X, q_out_g(X)) -> goal_out_g(X) The argument filtering Pi contains the following mapping: goal_in_g(x1) = goal_in_g(x1) U2_g(x1, x2) = U2_g(x1, x2) p_in_g(x1) = p_in_g(x1) a = a p_out_g(x1) = p_out_g U1_g(x1, x2) = U1_g(x2) p_in_a(x1) = p_in_a p_out_a(x1) = p_out_a U1_a(x1, x2) = U1_a(x2) U3_g(x1, x2) = U3_g(x2) q_in_g(x1) = q_in_g(x1) b = b q_out_g(x1) = q_out_g goal_out_g(x1) = goal_out_g ---------------------------------------- (13) 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: GOAL_IN_G(X) -> U2_G(X, p_in_g(X)) GOAL_IN_G(X) -> P_IN_G(X) P_IN_G(X) -> U1_G(X, p_in_a(Y)) P_IN_G(X) -> P_IN_A(Y) P_IN_A(X) -> U1_A(X, p_in_a(Y)) P_IN_A(X) -> P_IN_A(Y) U2_G(X, p_out_g(X)) -> U3_G(X, q_in_g(X)) U2_G(X, p_out_g(X)) -> Q_IN_G(X) The TRS R consists of the following rules: goal_in_g(X) -> U2_g(X, p_in_g(X)) p_in_g(a) -> p_out_g(a) p_in_g(X) -> U1_g(X, p_in_a(Y)) p_in_a(a) -> p_out_a(a) p_in_a(X) -> U1_a(X, p_in_a(Y)) U1_a(X, p_out_a(Y)) -> p_out_a(X) U1_g(X, p_out_a(Y)) -> p_out_g(X) U2_g(X, p_out_g(X)) -> U3_g(X, q_in_g(X)) q_in_g(b) -> q_out_g(b) U3_g(X, q_out_g(X)) -> goal_out_g(X) The argument filtering Pi contains the following mapping: goal_in_g(x1) = goal_in_g(x1) U2_g(x1, x2) = U2_g(x1, x2) p_in_g(x1) = p_in_g(x1) a = a p_out_g(x1) = p_out_g U1_g(x1, x2) = U1_g(x2) p_in_a(x1) = p_in_a p_out_a(x1) = p_out_a U1_a(x1, x2) = U1_a(x2) U3_g(x1, x2) = U3_g(x2) q_in_g(x1) = q_in_g(x1) b = b q_out_g(x1) = q_out_g goal_out_g(x1) = goal_out_g GOAL_IN_G(x1) = GOAL_IN_G(x1) U2_G(x1, x2) = U2_G(x1, x2) P_IN_G(x1) = P_IN_G(x1) U1_G(x1, x2) = U1_G(x2) P_IN_A(x1) = P_IN_A U1_A(x1, x2) = U1_A(x2) U3_G(x1, x2) = U3_G(x2) Q_IN_G(x1) = Q_IN_G(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: GOAL_IN_G(X) -> U2_G(X, p_in_g(X)) GOAL_IN_G(X) -> P_IN_G(X) P_IN_G(X) -> U1_G(X, p_in_a(Y)) P_IN_G(X) -> P_IN_A(Y) P_IN_A(X) -> U1_A(X, p_in_a(Y)) P_IN_A(X) -> P_IN_A(Y) U2_G(X, p_out_g(X)) -> U3_G(X, q_in_g(X)) U2_G(X, p_out_g(X)) -> Q_IN_G(X) The TRS R consists of the following rules: goal_in_g(X) -> U2_g(X, p_in_g(X)) p_in_g(a) -> p_out_g(a) p_in_g(X) -> U1_g(X, p_in_a(Y)) p_in_a(a) -> p_out_a(a) p_in_a(X) -> U1_a(X, p_in_a(Y)) U1_a(X, p_out_a(Y)) -> p_out_a(X) U1_g(X, p_out_a(Y)) -> p_out_g(X) U2_g(X, p_out_g(X)) -> U3_g(X, q_in_g(X)) q_in_g(b) -> q_out_g(b) U3_g(X, q_out_g(X)) -> goal_out_g(X) The argument filtering Pi contains the following mapping: goal_in_g(x1) = goal_in_g(x1) U2_g(x1, x2) = U2_g(x1, x2) p_in_g(x1) = p_in_g(x1) a = a p_out_g(x1) = p_out_g U1_g(x1, x2) = U1_g(x2) p_in_a(x1) = p_in_a p_out_a(x1) = p_out_a U1_a(x1, x2) = U1_a(x2) U3_g(x1, x2) = U3_g(x2) q_in_g(x1) = q_in_g(x1) b = b q_out_g(x1) = q_out_g goal_out_g(x1) = goal_out_g GOAL_IN_G(x1) = GOAL_IN_G(x1) U2_G(x1, x2) = U2_G(x1, x2) P_IN_G(x1) = P_IN_G(x1) U1_G(x1, x2) = U1_G(x2) P_IN_A(x1) = P_IN_A U1_A(x1, x2) = U1_A(x2) U3_G(x1, x2) = U3_G(x2) Q_IN_G(x1) = Q_IN_G(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (15) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 7 less nodes. ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: P_IN_A(X) -> P_IN_A(Y) The TRS R consists of the following rules: goal_in_g(X) -> U2_g(X, p_in_g(X)) p_in_g(a) -> p_out_g(a) p_in_g(X) -> U1_g(X, p_in_a(Y)) p_in_a(a) -> p_out_a(a) p_in_a(X) -> U1_a(X, p_in_a(Y)) U1_a(X, p_out_a(Y)) -> p_out_a(X) U1_g(X, p_out_a(Y)) -> p_out_g(X) U2_g(X, p_out_g(X)) -> U3_g(X, q_in_g(X)) q_in_g(b) -> q_out_g(b) U3_g(X, q_out_g(X)) -> goal_out_g(X) The argument filtering Pi contains the following mapping: goal_in_g(x1) = goal_in_g(x1) U2_g(x1, x2) = U2_g(x1, x2) p_in_g(x1) = p_in_g(x1) a = a p_out_g(x1) = p_out_g U1_g(x1, x2) = U1_g(x2) p_in_a(x1) = p_in_a p_out_a(x1) = p_out_a U1_a(x1, x2) = U1_a(x2) U3_g(x1, x2) = U3_g(x2) q_in_g(x1) = q_in_g(x1) b = b q_out_g(x1) = q_out_g goal_out_g(x1) = goal_out_g P_IN_A(x1) = P_IN_A We have to consider all (P,R,Pi)-chains ---------------------------------------- (17) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (18) Obligation: Pi DP problem: The TRS P consists of the following rules: P_IN_A(X) -> P_IN_A(Y) R is empty. The argument filtering Pi contains the following mapping: P_IN_A(x1) = P_IN_A We have to consider all (P,R,Pi)-chains ---------------------------------------- (19) PiDPToQDPProof (SOUND) 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: P_IN_A -> P_IN_A R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (21) PrologToTRSTransformerProof (SOUND) Transformed Prolog program to TRS. { "root": 2, "program": { "directives": [], "clauses": [ [ "(p (a))", null ], [ "(p X)", "(p Y)" ], [ "(q (b))", null ], [ "(goal X)", "(',' (p X) (q X))" ] ] }, "graph": { "nodes": { "23": { "goal": [{ "clause": 3, "scope": 1, "term": "(goal T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "24": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (p T4) (q T4))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } }, "type": "Nodes", "150": { "goal": [{ "clause": 2, "scope": 4, "term": "(q T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } }, "140": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "141": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "120": { "goal": [{ "clause": 0, "scope": 2, "term": "(p T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } }, "142": { "goal": [{ "clause": -1, "scope": -1, "term": "(p X8)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X8"], "exprvars": [] } }, "153": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "121": { "goal": [{ "clause": 1, "scope": 2, "term": "(p T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } }, "143": { "goal": [ { "clause": 0, "scope": 3, "term": "(p X8)" }, { "clause": 1, "scope": 3, "term": "(p X8)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X8"], "exprvars": [] } }, "144": { "goal": [{ "clause": 0, "scope": 3, "term": "(p X8)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X8"], "exprvars": [] } }, "155": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "2": { "goal": [{ "clause": -1, "scope": -1, "term": "(goal T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "145": { "goal": [{ "clause": 1, "scope": 3, "term": "(p X8)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X8"], "exprvars": [] } }, "156": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "146": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "114": { "goal": [ { "clause": 0, "scope": 2, "term": "(p T4)" }, { "clause": 1, "scope": 2, "term": "(p T4)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } }, "147": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "148": { "goal": [{ "clause": -1, "scope": -1, "term": "(p X15)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X15"], "exprvars": [] } }, "139": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "52": { "goal": [{ "clause": -1, "scope": -1, "term": "(p T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } }, "54": { "goal": [{ "clause": -1, "scope": -1, "term": "(q T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } } }, "edges": [ { "from": 2, "to": 23, "label": "CASE" }, { "from": 23, "to": 24, "label": "ONLY EVAL with clause\ngoal(X3) :- ','(p(X3), q(X3)).\nand substitutionT1 -> T4,\nX3 -> T4" }, { "from": 24, "to": 52, "label": "SPLIT 1" }, { "from": 24, "to": 54, "label": "SPLIT 2\nnew knowledge:\nT4 is ground" }, { "from": 52, "to": 114, "label": "CASE" }, { "from": 54, "to": 150, "label": "CASE" }, { "from": 114, "to": 120, "label": "PARALLEL" }, { "from": 114, "to": 121, "label": "PARALLEL" }, { "from": 120, "to": 139, "label": "EVAL with clause\np(a).\nand substitutionT4 -> a" }, { "from": 120, "to": 140, "label": "EVAL-BACKTRACK" }, { "from": 121, "to": 142, "label": "ONLY EVAL with clause\np(X7) :- p(X8).\nand substitutionT4 -> T7,\nX7 -> T7" }, { "from": 139, "to": 141, "label": "SUCCESS" }, { "from": 142, "to": 143, "label": "CASE" }, { "from": 143, "to": 144, "label": "PARALLEL" }, { "from": 143, "to": 145, "label": "PARALLEL" }, { "from": 144, "to": 146, "label": "ONLY EVAL with clause\np(a).\nand substitutionX8 -> a" }, { "from": 145, "to": 148, "label": "ONLY EVAL with clause\np(X14) :- p(X15).\nand substitutionX8 -> X16,\nX14 -> X16" }, { "from": 146, "to": 147, "label": "SUCCESS" }, { "from": 148, "to": 142, "label": "INSTANCE with matching:\nX8 -> X15" }, { "from": 150, "to": 153, "label": "EVAL with clause\nq(b).\nand substitutionT4 -> b" }, { "from": 150, "to": 155, "label": "EVAL-BACKTRACK" }, { "from": 153, "to": 156, "label": "SUCCESS" } ], "type": "Graph" } } ---------------------------------------- (22) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f2_in(T4) -> U1(f24_in(T4), T4) U1(f24_out1, T4) -> f2_out1 f142_in -> f142_out1 f142_in -> U2(f142_in) U2(f142_out1) -> f142_out1 f52_in(a) -> f52_out1 f52_in(T7) -> U3(f142_in, T7) U3(f142_out1, T7) -> f52_out1 f54_in(b) -> f54_out1 f24_in(T4) -> U4(f52_in(T4), T4) U4(f52_out1, T4) -> U5(f54_in(T4), T4) U5(f54_out1, T4) -> f24_out1 Q is empty. ---------------------------------------- (23) QTRSRRRProof (EQUIVALENT) Used ordering: f2_in/1(YES) U1/2(YES,YES) f24_in/1(YES) f24_out1/0) f2_out1/0) f142_in/0) f142_out1/0) U2/1)YES( f52_in/1(YES) a/0) f52_out1/0) U3/2(YES,YES) f54_in/1(YES) b/0) f54_out1/0) U4/2(YES,YES) U5/2(YES,YES) Quasi precedence: f2_in_1 > U1_2 > [f2_out1, a, f52_out1, f54_in_1] f2_in_1 > f24_in_1 > f52_in_1 > [f142_in, f142_out1] > [f2_out1, a, f52_out1, f54_in_1] f2_in_1 > f24_in_1 > f52_in_1 > U3_2 > [f2_out1, a, f52_out1, f54_in_1] f2_in_1 > f24_in_1 > U4_2 > U5_2 > [f2_out1, a, f52_out1, f54_in_1] b > f54_out1 > f24_out1 > [f2_out1, a, f52_out1, f54_in_1] Status: f2_in_1: multiset status U1_2: multiset status f24_in_1: [1] f24_out1: multiset status f2_out1: multiset status f142_in: multiset status f142_out1: multiset status f52_in_1: multiset status a: multiset status f52_out1: multiset status U3_2: multiset status f54_in_1: multiset status b: multiset status f54_out1: multiset status U4_2: multiset status U5_2: multiset status With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: f2_in(T4) -> U1(f24_in(T4), T4) U1(f24_out1, T4) -> f2_out1 f52_in(a) -> f52_out1 f52_in(T7) -> U3(f142_in, T7) U3(f142_out1, T7) -> f52_out1 f54_in(b) -> f54_out1 f24_in(T4) -> U4(f52_in(T4), T4) U4(f52_out1, T4) -> U5(f54_in(T4), T4) U5(f54_out1, T4) -> f24_out1 ---------------------------------------- (24) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f142_in -> f142_out1 f142_in -> U2(f142_in) U2(f142_out1) -> f142_out1 Q is empty. ---------------------------------------- (25) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U2(x_1)) = x_1 POL(f142_in) = 2 POL(f142_out1) = 0 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: f142_in -> f142_out1 ---------------------------------------- (26) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f142_in -> U2(f142_in) U2(f142_out1) -> f142_out1 Q is empty. ---------------------------------------- (27) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U2(x_1)) = 2*x_1 POL(f142_in) = 0 POL(f142_out1) = 1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U2(f142_out1) -> f142_out1 ---------------------------------------- (28) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f142_in -> U2(f142_in) Q is empty. ---------------------------------------- (29) Overlay + Local Confluence (EQUIVALENT) The TRS is overlay and locally confluent. By [NOC] we can switch to innermost. ---------------------------------------- (30) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f142_in -> U2(f142_in) The set Q consists of the following terms: f142_in ---------------------------------------- (31) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (32) Obligation: Q DP problem: The TRS P consists of the following rules: F142_IN -> F142_IN The TRS R consists of the following rules: f142_in -> U2(f142_in) The set Q consists of the following terms: f142_in We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (33) 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. ---------------------------------------- (34) Obligation: Q DP problem: The TRS P consists of the following rules: F142_IN -> F142_IN R is empty. The set Q consists of the following terms: f142_in We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (35) 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]. f142_in ---------------------------------------- (36) Obligation: Q DP problem: The TRS P consists of the following rules: F142_IN -> F142_IN R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (37) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 1, "program": { "directives": [], "clauses": [ [ "(p (a))", null ], [ "(p X)", "(p Y)" ], [ "(q (b))", null ], [ "(goal X)", "(',' (p X) (q X))" ] ] }, "graph": { "nodes": { "55": { "goal": [ { "clause": 0, "scope": 2, "term": "(',' (p T3) (q T3))" }, { "clause": 1, "scope": 2, "term": "(',' (p T3) (q T3))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } }, "type": "Nodes", "151": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "130": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (p X7) (q T6))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T6"], "free": ["X7"], "exprvars": [] } }, "152": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "131": { "goal": [{ "clause": -1, "scope": -1, "term": "(p X7)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X7"], "exprvars": [] } }, "132": { "goal": [{ "clause": -1, "scope": -1, "term": "(q T6)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T6"], "free": [], "exprvars": [] } }, "154": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1": { "goal": [{ "clause": -1, "scope": -1, "term": "(goal T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "133": { "goal": [ { "clause": 0, "scope": 4, "term": "(p X7)" }, { "clause": 1, "scope": 4, "term": "(p X7)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X7"], "exprvars": [] } }, "101": { "goal": [{ "clause": 0, "scope": 2, "term": "(',' (p T3) (q T3))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } }, "123": { "goal": [{ "clause": -1, "scope": -1, "term": "(q (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "134": { "goal": [{ "clause": 0, "scope": 4, "term": "(p X7)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X7"], "exprvars": [] } }, "102": { "goal": [{ "clause": 1, "scope": 2, "term": "(',' (p T3) (q T3))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } }, "135": { "goal": [{ "clause": 1, "scope": 4, "term": "(p X7)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X7"], "exprvars": [] } }, "136": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "137": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "6": { "goal": [{ "clause": 3, "scope": 1, "term": "(goal T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "127": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "138": { "goal": [{ "clause": -1, "scope": -1, "term": "(p X14)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X14"], "exprvars": [] } }, "149": { "goal": [{ "clause": 2, "scope": 5, "term": "(q T6)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T6"], "free": [], "exprvars": [] } }, "128": { "goal": [{ "clause": 2, "scope": 3, "term": "(q (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "129": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "21": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (p T3) (q T3))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 6, "label": "CASE" }, { "from": 6, "to": 21, "label": "ONLY EVAL with clause\ngoal(X2) :- ','(p(X2), q(X2)).\nand substitutionT1 -> T3,\nX2 -> T3" }, { "from": 21, "to": 55, "label": "CASE" }, { "from": 55, "to": 101, "label": "PARALLEL" }, { "from": 55, "to": 102, "label": "PARALLEL" }, { "from": 101, "to": 123, "label": "EVAL with clause\np(a).\nand substitutionT3 -> a" }, { "from": 101, "to": 127, "label": "EVAL-BACKTRACK" }, { "from": 102, "to": 130, "label": "ONLY EVAL with clause\np(X6) :- p(X7).\nand substitutionT3 -> T6,\nX6 -> T6" }, { "from": 123, "to": 128, "label": "CASE" }, { "from": 128, "to": 129, "label": "BACKTRACK\nfor clause: q(b)because of non-unification" }, { "from": 130, "to": 131, "label": "SPLIT 1" }, { "from": 130, "to": 132, "label": "SPLIT 2\nreplacements:X7 -> T7" }, { "from": 131, "to": 133, "label": "CASE" }, { "from": 132, "to": 149, "label": "CASE" }, { "from": 133, "to": 134, "label": "PARALLEL" }, { "from": 133, "to": 135, "label": "PARALLEL" }, { "from": 134, "to": 136, "label": "ONLY EVAL with clause\np(a).\nand substitutionX7 -> a" }, { "from": 135, "to": 138, "label": "ONLY EVAL with clause\np(X13) :- p(X14).\nand substitutionX7 -> X15,\nX13 -> X15" }, { "from": 136, "to": 137, "label": "SUCCESS" }, { "from": 138, "to": 131, "label": "INSTANCE with matching:\nX7 -> X14" }, { "from": 149, "to": 151, "label": "EVAL with clause\nq(b).\nand substitutionT6 -> b" }, { "from": 149, "to": 152, "label": "EVAL-BACKTRACK" }, { "from": 151, "to": 154, "label": "SUCCESS" } ], "type": "Graph" } } ---------------------------------------- (38) Obligation: Triples: pA(X1) :- pA(X2). goalB(X1) :- pA(X2). Clauses: pcA(a). pcA(X1) :- pcA(X2). Afs: goalB(x1) = goalB(x1) ---------------------------------------- (39) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: goalB_in_1: (b) pA_in_1: (f) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: GOALB_IN_G(X1) -> U2_G(X1, pA_in_a(X2)) GOALB_IN_G(X1) -> PA_IN_A(X2) PA_IN_A(X1) -> U1_A(X1, pA_in_a(X2)) PA_IN_A(X1) -> PA_IN_A(X2) R is empty. The argument filtering Pi contains the following mapping: pA_in_a(x1) = pA_in_a GOALB_IN_G(x1) = GOALB_IN_G(x1) U2_G(x1, x2) = U2_G(x1, x2) PA_IN_A(x1) = PA_IN_A U1_A(x1, x2) = U1_A(x2) We have to consider all (P,R,Pi)-chains Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES ---------------------------------------- (40) Obligation: Pi DP problem: The TRS P consists of the following rules: GOALB_IN_G(X1) -> U2_G(X1, pA_in_a(X2)) GOALB_IN_G(X1) -> PA_IN_A(X2) PA_IN_A(X1) -> U1_A(X1, pA_in_a(X2)) PA_IN_A(X1) -> PA_IN_A(X2) R is empty. The argument filtering Pi contains the following mapping: pA_in_a(x1) = pA_in_a GOALB_IN_G(x1) = GOALB_IN_G(x1) U2_G(x1, x2) = U2_G(x1, x2) PA_IN_A(x1) = PA_IN_A U1_A(x1, x2) = U1_A(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (41) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes. ---------------------------------------- (42) Obligation: Pi DP problem: The TRS P consists of the following rules: PA_IN_A(X1) -> PA_IN_A(X2) R is empty. The argument filtering Pi contains the following mapping: PA_IN_A(x1) = PA_IN_A We have to consider all (P,R,Pi)-chains ---------------------------------------- (43) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (44) Obligation: Q DP problem: The TRS P consists of the following rules: PA_IN_A -> PA_IN_A R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (45) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 3, "program": { "directives": [], "clauses": [ [ "(p (a))", null ], [ "(p X)", "(p Y)" ], [ "(q (b))", null ], [ "(goal X)", "(',' (p X) (q X))" ] ] }, "graph": { "nodes": { "22": { "goal": [{ "clause": 3, "scope": 1, "term": "(goal T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "180": { "goal": [{ "clause": 2, "scope": 4, "term": "(q T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } }, "181": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "182": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "183": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "174": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "175": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "110": { "goal": [{ "clause": 1, "scope": 2, "term": "(p T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } }, "176": { "goal": [{ "clause": -1, "scope": -1, "term": "(p X15)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X15"], "exprvars": [] } }, "111": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "122": { "goal": [{ "clause": -1, "scope": -1, "term": "(p X8)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X8"], "exprvars": [] } }, "112": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "3": { "goal": [{ "clause": -1, "scope": -1, "term": "(goal T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "113": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "124": { "goal": [ { "clause": 0, "scope": 3, "term": "(p X8)" }, { "clause": 1, "scope": 3, "term": "(p X8)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X8"], "exprvars": [] } }, "125": { "goal": [{ "clause": 0, "scope": 3, "term": "(p X8)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X8"], "exprvars": [] } }, "126": { "goal": [{ "clause": 1, "scope": 3, "term": "(p X8)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X8"], "exprvars": [] } }, "108": { "goal": [ { "clause": 0, "scope": 2, "term": "(p T4)" }, { "clause": 1, "scope": 2, "term": "(p T4)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } }, "109": { "goal": [{ "clause": 0, "scope": 2, "term": "(p T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } }, "50": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (p T4) (q T4))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } }, "51": { "goal": [{ "clause": -1, "scope": -1, "term": "(p T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } }, "53": { "goal": [{ "clause": -1, "scope": -1, "term": "(q T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } } }, "edges": [ { "from": 3, "to": 22, "label": "CASE" }, { "from": 22, "to": 50, "label": "ONLY EVAL with clause\ngoal(X3) :- ','(p(X3), q(X3)).\nand substitutionT1 -> T4,\nX3 -> T4" }, { "from": 50, "to": 51, "label": "SPLIT 1" }, { "from": 50, "to": 53, "label": "SPLIT 2\nnew knowledge:\nT4 is ground" }, { "from": 51, "to": 108, "label": "CASE" }, { "from": 53, "to": 180, "label": "CASE" }, { "from": 108, "to": 109, "label": "PARALLEL" }, { "from": 108, "to": 110, "label": "PARALLEL" }, { "from": 109, "to": 111, "label": "EVAL with clause\np(a).\nand substitutionT4 -> a" }, { "from": 109, "to": 112, "label": "EVAL-BACKTRACK" }, { "from": 110, "to": 122, "label": "ONLY EVAL with clause\np(X7) :- p(X8).\nand substitutionT4 -> T7,\nX7 -> T7" }, { "from": 111, "to": 113, "label": "SUCCESS" }, { "from": 122, "to": 124, "label": "CASE" }, { "from": 124, "to": 125, "label": "PARALLEL" }, { "from": 124, "to": 126, "label": "PARALLEL" }, { "from": 125, "to": 174, "label": "ONLY EVAL with clause\np(a).\nand substitutionX8 -> a" }, { "from": 126, "to": 176, "label": "ONLY EVAL with clause\np(X14) :- p(X15).\nand substitutionX8 -> X16,\nX14 -> X16" }, { "from": 174, "to": 175, "label": "SUCCESS" }, { "from": 176, "to": 122, "label": "INSTANCE with matching:\nX8 -> X15" }, { "from": 180, "to": 181, "label": "EVAL with clause\nq(b).\nand substitutionT4 -> b" }, { "from": 180, "to": 182, "label": "EVAL-BACKTRACK" }, { "from": 181, "to": 183, "label": "SUCCESS" } ], "type": "Graph" } } ---------------------------------------- (46) Obligation: Rules: f124_out -> f122_out :|: TRUE f122_in -> f124_in :|: TRUE f126_in -> f176_in :|: TRUE f176_out -> f126_out :|: TRUE f124_in -> f125_in :|: TRUE f125_out -> f124_out :|: TRUE f124_in -> f126_in :|: TRUE f126_out -> f124_out :|: TRUE f176_in -> f122_in :|: TRUE f122_out -> f176_out :|: TRUE f3_in(T1) -> f22_in(T1) :|: TRUE f22_out(x) -> f3_out(x) :|: TRUE f22_in(T4) -> f50_in(T4) :|: TRUE f50_out(x1) -> f22_out(x1) :|: TRUE f53_out(x2) -> f50_out(x2) :|: TRUE f50_in(x3) -> f51_in(x3) :|: TRUE f51_out(x4) -> f53_in(x4) :|: TRUE f108_out(x5) -> f51_out(x5) :|: TRUE f51_in(x6) -> f108_in(x6) :|: TRUE f108_in(x7) -> f109_in(x7) :|: TRUE f110_out(x8) -> f108_out(x8) :|: TRUE f109_out(x9) -> f108_out(x9) :|: TRUE f108_in(x10) -> f110_in(x10) :|: TRUE f122_out -> f110_out(T7) :|: TRUE f110_in(x11) -> f122_in :|: TRUE Start term: f3_in(T1) ---------------------------------------- (47) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f122_in -> f124_in :|: TRUE f126_in -> f176_in :|: TRUE f124_in -> f126_in :|: TRUE f176_in -> f122_in :|: TRUE ---------------------------------------- (48) Obligation: Rules: f122_in -> f124_in :|: TRUE f126_in -> f176_in :|: TRUE f124_in -> f126_in :|: TRUE f176_in -> f122_in :|: TRUE ---------------------------------------- (49) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (50) Obligation: Rules: f122_in -> f122_in :|: TRUE ---------------------------------------- (51) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (52) Obligation: Rules: f122_in -> f122_in :|: TRUE ---------------------------------------- (53) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f122_in -> f122_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (54) Obligation: Termination digraph: Nodes: (1) f122_in -> f122_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (55) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f122_in() Replaced non-predefined constructor symbols by 0. ---------------------------------------- (56) Obligation: Rules: f122_in -> f122_in :|: TRUE ---------------------------------------- (57) IntTRSPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc) -> f(1) :|: pc = 1 && TRUE Witness term starting non-terminating reduction: f(1) ---------------------------------------- (58) NO