/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Left Termination of the query pattern goal(g) w.r.t. the given Prolog program could not be shown: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2)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) NonTerminationLoopProof [COMPLETE, 0 ms] (12) NO (13) PrologToPiTRSProof [SOUND, 0 ms] (14) PiTRS (15) DependencyPairsProof [EQUIVALENT, 0 ms] (16) PiDP (17) DependencyGraphProof [EQUIVALENT, 0 ms] (18) PiDP (19) UsableRulesProof [EQUIVALENT, 0 ms] (20) PiDP (21) PiDPToQDPProof [SOUND, 0 ms] (22) QDP (23) NonTerminationLoopProof [COMPLETE, 0 ms] (24) NO (25) PrologToDTProblemTransformerProof [SOUND, 0 ms] (26) TRIPLES (27) TriplesToPiDPProof [SOUND, 0 ms] (28) PiDP (29) DependencyGraphProof [EQUIVALENT, 0 ms] (30) PiDP (31) PiDPToQDPProof [SOUND, 0 ms] (32) QDP (33) NonTerminationLoopProof [COMPLETE, 0 ms] (34) NO (35) PrologToTRSTransformerProof [SOUND, 0 ms] (36) QTRS (37) QTRSRRRProof [EQUIVALENT, 70 ms] (38) QTRS (39) QTRSRRRProof [EQUIVALENT, 4 ms] (40) QTRS (41) QTRSRRRProof [EQUIVALENT, 4 ms] (42) QTRS (43) Overlay + Local Confluence [EQUIVALENT, 0 ms] (44) QTRS (45) DependencyPairsProof [EQUIVALENT, 0 ms] (46) QDP (47) UsableRulesProof [EQUIVALENT, 0 ms] (48) QDP (49) QReductionProof [EQUIVALENT, 0 ms] (50) QDP (51) QTRSRRRProof [EQUIVALENT, 3 ms] (52) QTRS (53) PrologToIRSwTTransformerProof [SOUND, 0 ms] (54) IRSwT (55) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (56) IRSwT (57) IntTRSCompressionProof [EQUIVALENT, 32 ms] (58) IRSwT (59) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (60) IRSwT (61) IRSwTTerminationDigraphProof [EQUIVALENT, 6 ms] (62) IRSwT (63) FilterProof [EQUIVALENT, 0 ms] (64) IntTRS (65) IntTRSNonPeriodicNontermProof [COMPLETE, 4 ms] (66) 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 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 ---------------------------------------- (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 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 ---------------------------------------- (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 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 ---------------------------------------- (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 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 ---------------------------------------- (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 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 ---------------------------------------- (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) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by semiunifying a rule from P directly. s = P_IN_A evaluates to t =P_IN_A Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence The DP semiunifies directly so there is only one rewrite step from P_IN_A to P_IN_A. ---------------------------------------- (12) NO ---------------------------------------- (13) 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 ---------------------------------------- (14) 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) ---------------------------------------- (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: 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 ---------------------------------------- (16) 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 ---------------------------------------- (17) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 7 less nodes. ---------------------------------------- (18) 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 ---------------------------------------- (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_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 ---------------------------------------- (21) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (22) Obligation: Q DP problem: The TRS P consists of the following rules: P_IN_A -> P_IN_A R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (23) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by semiunifying a rule from P directly. s = P_IN_A evaluates to t =P_IN_A Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence The DP semiunifies directly so there is only one rewrite step from P_IN_A to P_IN_A. ---------------------------------------- (24) NO ---------------------------------------- (25) 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": { "58": { "goal": [{ "clause": 3, "scope": 1, "term": "(goal T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "type": "Nodes", "150": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (p X7) (q T6))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T6"], "free": ["X7"], "exprvars": [] } }, "151": { "goal": [{ "clause": -1, "scope": -1, "term": "(p X7)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X7"], "exprvars": [] } }, "162": { "goal": [{ "clause": -1, "scope": -1, "term": "(p X14)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X14"], "exprvars": [] } }, "152": { "goal": [{ "clause": -1, "scope": -1, "term": "(q T6)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T6"], "free": [], "exprvars": [] } }, "153": { "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": [] } }, "164": { "goal": [{ "clause": 2, "scope": 5, "term": "(q T6)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T6"], "free": [], "exprvars": [] } }, "154": { "goal": [{ "clause": 0, "scope": 4, "term": "(p X7)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X7"], "exprvars": [] } }, "1": { "goal": [{ "clause": -1, "scope": -1, "term": "(goal T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "122": { "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": [] } }, "166": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "123": { "goal": [{ "clause": 0, "scope": 2, "term": "(',' (p T3) (q T3))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } }, "156": { "goal": [{ "clause": 1, "scope": 4, "term": "(p X7)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X7"], "exprvars": [] } }, "167": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "124": { "goal": [{ "clause": 1, "scope": 2, "term": "(',' (p T3) (q T3))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } }, "168": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "125": { "goal": [{ "clause": -1, "scope": -1, "term": "(q (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "158": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "126": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "159": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "127": { "goal": [{ "clause": 2, "scope": 3, "term": "(q (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "128": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "74": { "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": 58, "label": "CASE" }, { "from": 58, "to": 74, "label": "ONLY EVAL with clause\ngoal(X2) :- ','(p(X2), q(X2)).\nand substitutionT1 -> T3,\nX2 -> T3" }, { "from": 74, "to": 122, "label": "CASE" }, { "from": 122, "to": 123, "label": "PARALLEL" }, { "from": 122, "to": 124, "label": "PARALLEL" }, { "from": 123, "to": 125, "label": "EVAL with clause\np(a).\nand substitutionT3 -> a" }, { "from": 123, "to": 126, "label": "EVAL-BACKTRACK" }, { "from": 124, "to": 150, "label": "ONLY EVAL with clause\np(X6) :- p(X7).\nand substitutionT3 -> T6,\nX6 -> T6" }, { "from": 125, "to": 127, "label": "CASE" }, { "from": 127, "to": 128, "label": "BACKTRACK\nfor clause: q(b)because of non-unification" }, { "from": 150, "to": 151, "label": "SPLIT 1" }, { "from": 150, "to": 152, "label": "SPLIT 2\nreplacements:X7 -> T7" }, { "from": 151, "to": 153, "label": "CASE" }, { "from": 152, "to": 164, "label": "CASE" }, { "from": 153, "to": 154, "label": "PARALLEL" }, { "from": 153, "to": 156, "label": "PARALLEL" }, { "from": 154, "to": 158, "label": "ONLY EVAL with clause\np(a).\nand substitutionX7 -> a" }, { "from": 156, "to": 162, "label": "ONLY EVAL with clause\np(X13) :- p(X14).\nand substitutionX7 -> X15,\nX13 -> X15" }, { "from": 158, "to": 159, "label": "SUCCESS" }, { "from": 162, "to": 151, "label": "INSTANCE with matching:\nX7 -> X14" }, { "from": 164, "to": 166, "label": "EVAL with clause\nq(b).\nand substitutionT6 -> b" }, { "from": 164, "to": 167, "label": "EVAL-BACKTRACK" }, { "from": 166, "to": 168, "label": "SUCCESS" } ], "type": "Graph" } } ---------------------------------------- (26) Obligation: Triples: pA(X1) :- pA(X2). goalB(X1) :- pA(X2). Clauses: pcA(a). pcA(X1) :- pcA(X2). Afs: goalB(x1) = goalB(x1) ---------------------------------------- (27) 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 ---------------------------------------- (28) 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 ---------------------------------------- (29) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes. ---------------------------------------- (30) 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 ---------------------------------------- (31) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (32) Obligation: Q DP problem: The TRS P consists of the following rules: PA_IN_A -> PA_IN_A R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (33) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by semiunifying a rule from P directly. s = PA_IN_A evaluates to t =PA_IN_A Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence The DP semiunifies directly so there is only one rewrite step from PA_IN_A to PA_IN_A. ---------------------------------------- (34) NO ---------------------------------------- (35) PrologToTRSTransformerProof (SOUND) Transformed Prolog program to TRS. { "root": 46, "program": { "directives": [], "clauses": [ [ "(p (a))", null ], [ "(p X)", "(p Y)" ], [ "(q (b))", null ], [ "(goal X)", "(',' (p X) (q X))" ] ] }, "graph": { "nodes": { "46": { "goal": [{ "clause": -1, "scope": -1, "term": "(goal T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "47": { "goal": [{ "clause": 3, "scope": 1, "term": "(goal T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "48": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (p T4) (q T4))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } }, "170": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "160": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "171": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "161": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "163": { "goal": [{ "clause": -1, "scope": -1, "term": "(p X15)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X15"], "exprvars": [] } }, "142": { "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": [] } }, "143": { "goal": [{ "clause": 0, "scope": 2, "term": "(p T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } }, "165": { "goal": [{ "clause": 2, "scope": 4, "term": "(q T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } }, "144": { "goal": [{ "clause": 1, "scope": 2, "term": "(p T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } }, "155": { "goal": [{ "clause": 0, "scope": 3, "term": "(p X8)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X8"], "exprvars": [] } }, "145": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "146": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "157": { "goal": [{ "clause": 1, "scope": 3, "term": "(p X8)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X8"], "exprvars": [] } }, "147": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "169": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "115": { "goal": [{ "clause": -1, "scope": -1, "term": "(p T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } }, "148": { "goal": [{ "clause": -1, "scope": -1, "term": "(p X8)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X8"], "exprvars": [] } }, "116": { "goal": [{ "clause": -1, "scope": -1, "term": "(q T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } }, "149": { "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": [] } } }, "edges": [ { "from": 46, "to": 47, "label": "CASE" }, { "from": 47, "to": 48, "label": "ONLY EVAL with clause\ngoal(X3) :- ','(p(X3), q(X3)).\nand substitutionT1 -> T4,\nX3 -> T4" }, { "from": 48, "to": 115, "label": "SPLIT 1" }, { "from": 48, "to": 116, "label": "SPLIT 2\nnew knowledge:\nT4 is ground" }, { "from": 115, "to": 142, "label": "CASE" }, { "from": 116, "to": 165, "label": "CASE" }, { "from": 142, "to": 143, "label": "PARALLEL" }, { "from": 142, "to": 144, "label": "PARALLEL" }, { "from": 143, "to": 145, "label": "EVAL with clause\np(a).\nand substitutionT4 -> a" }, { "from": 143, "to": 146, "label": "EVAL-BACKTRACK" }, { "from": 144, "to": 148, "label": "ONLY EVAL with clause\np(X7) :- p(X8).\nand substitutionT4 -> T7,\nX7 -> T7" }, { "from": 145, "to": 147, "label": "SUCCESS" }, { "from": 148, "to": 149, "label": "CASE" }, { "from": 149, "to": 155, "label": "PARALLEL" }, { "from": 149, "to": 157, "label": "PARALLEL" }, { "from": 155, "to": 160, "label": "ONLY EVAL with clause\np(a).\nand substitutionX8 -> a" }, { "from": 157, "to": 163, "label": "ONLY EVAL with clause\np(X14) :- p(X15).\nand substitutionX8 -> X16,\nX14 -> X16" }, { "from": 160, "to": 161, "label": "SUCCESS" }, { "from": 163, "to": 148, "label": "INSTANCE with matching:\nX8 -> X15" }, { "from": 165, "to": 169, "label": "EVAL with clause\nq(b).\nand substitutionT4 -> b" }, { "from": 165, "to": 170, "label": "EVAL-BACKTRACK" }, { "from": 169, "to": 171, "label": "SUCCESS" } ], "type": "Graph" } } ---------------------------------------- (36) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f46_in(T4) -> U1(f48_in(T4), T4) U1(f48_out1, T4) -> f46_out1 f148_in -> f148_out1 f148_in -> U2(f148_in) U2(f148_out1) -> f148_out1 f115_in(a) -> f115_out1 f115_in(T7) -> U3(f148_in, T7) U3(f148_out1, T7) -> f115_out1 f116_in(b) -> f116_out1 f48_in(T4) -> U4(f115_in(T4), T4) U4(f115_out1, T4) -> U5(f116_in(T4), T4) U5(f116_out1, T4) -> f48_out1 Q is empty. ---------------------------------------- (37) QTRSRRRProof (EQUIVALENT) Used ordering: f46_in/1(YES) U1/2(YES,YES) f48_in/1(YES) f48_out1/0) f46_out1/0) f148_in/0) f148_out1/0) U2/1)YES( f115_in/1(YES) a/0) f115_out1/0) U3/2(YES,YES) f116_in/1(YES) b/0) f116_out1/0) U4/2(YES,YES) U5/2(YES,YES) Quasi precedence: f46_in_1 > U1_2 > [f46_out1, a, f115_out1, f116_in_1] f46_in_1 > f48_in_1 > f115_in_1 > [f148_in, f148_out1] > [f46_out1, a, f115_out1, f116_in_1] f46_in_1 > f48_in_1 > f115_in_1 > U3_2 > [f46_out1, a, f115_out1, f116_in_1] f46_in_1 > f48_in_1 > U4_2 > U5_2 > [f46_out1, a, f115_out1, f116_in_1] b > f116_out1 > f48_out1 > [f46_out1, a, f115_out1, f116_in_1] Status: f46_in_1: multiset status U1_2: multiset status f48_in_1: [1] f48_out1: multiset status f46_out1: multiset status f148_in: multiset status f148_out1: multiset status f115_in_1: multiset status a: multiset status f115_out1: multiset status U3_2: multiset status f116_in_1: multiset status b: multiset status f116_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: f46_in(T4) -> U1(f48_in(T4), T4) U1(f48_out1, T4) -> f46_out1 f115_in(a) -> f115_out1 f115_in(T7) -> U3(f148_in, T7) U3(f148_out1, T7) -> f115_out1 f116_in(b) -> f116_out1 f48_in(T4) -> U4(f115_in(T4), T4) U4(f115_out1, T4) -> U5(f116_in(T4), T4) U5(f116_out1, T4) -> f48_out1 ---------------------------------------- (38) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f148_in -> f148_out1 f148_in -> U2(f148_in) U2(f148_out1) -> f148_out1 Q is empty. ---------------------------------------- (39) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U2(x_1)) = x_1 POL(f148_in) = 2 POL(f148_out1) = 0 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: f148_in -> f148_out1 ---------------------------------------- (40) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f148_in -> U2(f148_in) U2(f148_out1) -> f148_out1 Q is empty. ---------------------------------------- (41) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U2(x_1)) = 2*x_1 POL(f148_in) = 0 POL(f148_out1) = 1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U2(f148_out1) -> f148_out1 ---------------------------------------- (42) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f148_in -> U2(f148_in) Q is empty. ---------------------------------------- (43) Overlay + Local Confluence (EQUIVALENT) The TRS is overlay and locally confluent. By [NOC] we can switch to innermost. ---------------------------------------- (44) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f148_in -> U2(f148_in) The set Q consists of the following terms: f148_in ---------------------------------------- (45) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (46) Obligation: Q DP problem: The TRS P consists of the following rules: F148_IN -> F148_IN The TRS R consists of the following rules: f148_in -> U2(f148_in) The set Q consists of the following terms: f148_in We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (47) 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. ---------------------------------------- (48) Obligation: Q DP problem: The TRS P consists of the following rules: F148_IN -> F148_IN R is empty. The set Q consists of the following terms: f148_in We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (49) 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]. f148_in ---------------------------------------- (50) Obligation: Q DP problem: The TRS P consists of the following rules: F148_IN -> F148_IN R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (51) QTRSRRRProof (EQUIVALENT) Used ordering: f148_in/0) f148_out1/0) U2/1)YES( Quasi precedence: f148_in > f148_out1 Status: f148_in: multiset status f148_out1: multiset status With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: f148_in -> f148_out1 ---------------------------------------- (52) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f148_in -> U2(f148_in) U2(f148_out1) -> f148_out1 Q is empty. ---------------------------------------- (53) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 2, "program": { "directives": [], "clauses": [ [ "(p (a))", null ], [ "(p X)", "(p Y)" ], [ "(q (b))", null ], [ "(goal X)", "(',' (p X) (q X))" ] ] }, "graph": { "nodes": { "38": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (p T4) (q T4))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } }, "type": "Nodes", "172": { "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": [] } }, "173": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "130": { "goal": [{ "clause": 0, "scope": 2, "term": "(p T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } }, "141": { "goal": [{ "clause": -1, "scope": -1, "term": "(p X15)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X15"], "exprvars": [] } }, "174": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "131": { "goal": [{ "clause": 1, "scope": 2, "term": "(p T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } }, "175": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "132": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "133": { "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": [] } }, "134": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "135": { "goal": [{ "clause": -1, "scope": -1, "term": "(p X8)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X8"], "exprvars": [] } }, "103": { "goal": [{ "clause": -1, "scope": -1, "term": "(p T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } }, "136": { "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": [] } }, "5": { "goal": [{ "clause": 3, "scope": 1, "term": "(goal T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "137": { "goal": [{ "clause": 0, "scope": 3, "term": "(p X8)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X8"], "exprvars": [] } }, "138": { "goal": [{ "clause": 1, "scope": 3, "term": "(p X8)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X8"], "exprvars": [] } }, "106": { "goal": [{ "clause": -1, "scope": -1, "term": "(q T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } }, "139": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "129": { "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": [] } } }, "edges": [ { "from": 2, "to": 5, "label": "CASE" }, { "from": 5, "to": 38, "label": "ONLY EVAL with clause\ngoal(X3) :- ','(p(X3), q(X3)).\nand substitutionT1 -> T4,\nX3 -> T4" }, { "from": 38, "to": 103, "label": "SPLIT 1" }, { "from": 38, "to": 106, "label": "SPLIT 2\nnew knowledge:\nT4 is ground" }, { "from": 103, "to": 129, "label": "CASE" }, { "from": 106, "to": 172, "label": "CASE" }, { "from": 129, "to": 130, "label": "PARALLEL" }, { "from": 129, "to": 131, "label": "PARALLEL" }, { "from": 130, "to": 132, "label": "EVAL with clause\np(a).\nand substitutionT4 -> a" }, { "from": 130, "to": 133, "label": "EVAL-BACKTRACK" }, { "from": 131, "to": 135, "label": "ONLY EVAL with clause\np(X7) :- p(X8).\nand substitutionT4 -> T7,\nX7 -> T7" }, { "from": 132, "to": 134, "label": "SUCCESS" }, { "from": 135, "to": 136, "label": "CASE" }, { "from": 136, "to": 137, "label": "PARALLEL" }, { "from": 136, "to": 138, "label": "PARALLEL" }, { "from": 137, "to": 139, "label": "ONLY EVAL with clause\np(a).\nand substitutionX8 -> a" }, { "from": 138, "to": 141, "label": "ONLY EVAL with clause\np(X14) :- p(X15).\nand substitutionX8 -> X16,\nX14 -> X16" }, { "from": 139, "to": 140, "label": "SUCCESS" }, { "from": 141, "to": 135, "label": "INSTANCE with matching:\nX8 -> X15" }, { "from": 172, "to": 173, "label": "EVAL with clause\nq(b).\nand substitutionT4 -> b" }, { "from": 172, "to": 174, "label": "EVAL-BACKTRACK" }, { "from": 173, "to": 175, "label": "SUCCESS" } ], "type": "Graph" } } ---------------------------------------- (54) Obligation: Rules: f136_out -> f135_out :|: TRUE f135_in -> f136_in :|: TRUE f135_out -> f141_out :|: TRUE f141_in -> f135_in :|: TRUE f138_in -> f141_in :|: TRUE f141_out -> f138_out :|: TRUE f136_in -> f138_in :|: TRUE f138_out -> f136_out :|: TRUE f136_in -> f137_in :|: TRUE f137_out -> f136_out :|: TRUE f5_out(T1) -> f2_out(T1) :|: TRUE f2_in(x) -> f5_in(x) :|: TRUE f38_out(T4) -> f5_out(T4) :|: TRUE f5_in(x1) -> f38_in(x1) :|: TRUE f106_out(x2) -> f38_out(x2) :|: TRUE f38_in(x3) -> f103_in(x3) :|: TRUE f103_out(x4) -> f106_in(x4) :|: TRUE f103_in(x5) -> f129_in(x5) :|: TRUE f129_out(x6) -> f103_out(x6) :|: TRUE f129_in(x7) -> f131_in(x7) :|: TRUE f131_out(x8) -> f129_out(x8) :|: TRUE f129_in(x9) -> f130_in(x9) :|: TRUE f130_out(x10) -> f129_out(x10) :|: TRUE f135_out -> f131_out(T7) :|: TRUE f131_in(x11) -> f135_in :|: TRUE Start term: f2_in(T1) ---------------------------------------- (55) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f135_in -> f136_in :|: TRUE f141_in -> f135_in :|: TRUE f138_in -> f141_in :|: TRUE f136_in -> f138_in :|: TRUE ---------------------------------------- (56) Obligation: Rules: f135_in -> f136_in :|: TRUE f141_in -> f135_in :|: TRUE f138_in -> f141_in :|: TRUE f136_in -> f138_in :|: TRUE ---------------------------------------- (57) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (58) Obligation: Rules: f138_in -> f138_in :|: TRUE ---------------------------------------- (59) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (60) Obligation: Rules: f138_in -> f138_in :|: TRUE ---------------------------------------- (61) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f138_in -> f138_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (62) Obligation: Termination digraph: Nodes: (1) f138_in -> f138_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (63) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f138_in() Replaced non-predefined constructor symbols by 0. ---------------------------------------- (64) Obligation: Rules: f138_in -> f138_in :|: TRUE ---------------------------------------- (65) IntTRSNonPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc) -> f(1) :|: pc = 1 && TRUE Proved unsatisfiability of the following formula, indicating that the system is never left after entering: ((run2_0 = ((1 * 1)) and (((run1_0 * 1)) = ((1 * 1)) and T)) and !(((run2_0 * 1)) = ((1 * 1)) and T)) Proved satisfiability of the following formula, indicating that the system is entered at least once: (run2_0 = ((1 * 1)) and (((run1_0 * 1)) = ((1 * 1)) and T)) ---------------------------------------- (66) NO