/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.pl /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern even(g) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0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 [EQUIVALENT, 0 ms] (10) QDP (11) QDPSizeChangeProof [EQUIVALENT, 0 ms] (12) YES ---------------------------------------- (0) Obligation: Clauses: even(0). even(s(X)) :- odd(X). odd(s(X)) :- even(X). Query: even(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: even_in_1: (b) odd_in_1: (b) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: even_in_g(0) -> even_out_g(0) even_in_g(s(X)) -> U1_g(X, odd_in_g(X)) odd_in_g(s(X)) -> U2_g(X, even_in_g(X)) U2_g(X, even_out_g(X)) -> odd_out_g(s(X)) U1_g(X, odd_out_g(X)) -> even_out_g(s(X)) The argument filtering Pi contains the following mapping: even_in_g(x1) = even_in_g(x1) 0 = 0 even_out_g(x1) = even_out_g s(x1) = s(x1) U1_g(x1, x2) = U1_g(x2) odd_in_g(x1) = odd_in_g(x1) U2_g(x1, x2) = U2_g(x2) odd_out_g(x1) = odd_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: even_in_g(0) -> even_out_g(0) even_in_g(s(X)) -> U1_g(X, odd_in_g(X)) odd_in_g(s(X)) -> U2_g(X, even_in_g(X)) U2_g(X, even_out_g(X)) -> odd_out_g(s(X)) U1_g(X, odd_out_g(X)) -> even_out_g(s(X)) The argument filtering Pi contains the following mapping: even_in_g(x1) = even_in_g(x1) 0 = 0 even_out_g(x1) = even_out_g s(x1) = s(x1) U1_g(x1, x2) = U1_g(x2) odd_in_g(x1) = odd_in_g(x1) U2_g(x1, x2) = U2_g(x2) odd_out_g(x1) = odd_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: EVEN_IN_G(s(X)) -> U1_G(X, odd_in_g(X)) EVEN_IN_G(s(X)) -> ODD_IN_G(X) ODD_IN_G(s(X)) -> U2_G(X, even_in_g(X)) ODD_IN_G(s(X)) -> EVEN_IN_G(X) The TRS R consists of the following rules: even_in_g(0) -> even_out_g(0) even_in_g(s(X)) -> U1_g(X, odd_in_g(X)) odd_in_g(s(X)) -> U2_g(X, even_in_g(X)) U2_g(X, even_out_g(X)) -> odd_out_g(s(X)) U1_g(X, odd_out_g(X)) -> even_out_g(s(X)) The argument filtering Pi contains the following mapping: even_in_g(x1) = even_in_g(x1) 0 = 0 even_out_g(x1) = even_out_g s(x1) = s(x1) U1_g(x1, x2) = U1_g(x2) odd_in_g(x1) = odd_in_g(x1) U2_g(x1, x2) = U2_g(x2) odd_out_g(x1) = odd_out_g EVEN_IN_G(x1) = EVEN_IN_G(x1) U1_G(x1, x2) = U1_G(x2) ODD_IN_G(x1) = ODD_IN_G(x1) U2_G(x1, x2) = U2_G(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: EVEN_IN_G(s(X)) -> U1_G(X, odd_in_g(X)) EVEN_IN_G(s(X)) -> ODD_IN_G(X) ODD_IN_G(s(X)) -> U2_G(X, even_in_g(X)) ODD_IN_G(s(X)) -> EVEN_IN_G(X) The TRS R consists of the following rules: even_in_g(0) -> even_out_g(0) even_in_g(s(X)) -> U1_g(X, odd_in_g(X)) odd_in_g(s(X)) -> U2_g(X, even_in_g(X)) U2_g(X, even_out_g(X)) -> odd_out_g(s(X)) U1_g(X, odd_out_g(X)) -> even_out_g(s(X)) The argument filtering Pi contains the following mapping: even_in_g(x1) = even_in_g(x1) 0 = 0 even_out_g(x1) = even_out_g s(x1) = s(x1) U1_g(x1, x2) = U1_g(x2) odd_in_g(x1) = odd_in_g(x1) U2_g(x1, x2) = U2_g(x2) odd_out_g(x1) = odd_out_g EVEN_IN_G(x1) = EVEN_IN_G(x1) U1_G(x1, x2) = U1_G(x2) ODD_IN_G(x1) = ODD_IN_G(x1) U2_G(x1, x2) = U2_G(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 2 less nodes. ---------------------------------------- (6) Obligation: Pi DP problem: The TRS P consists of the following rules: EVEN_IN_G(s(X)) -> ODD_IN_G(X) ODD_IN_G(s(X)) -> EVEN_IN_G(X) The TRS R consists of the following rules: even_in_g(0) -> even_out_g(0) even_in_g(s(X)) -> U1_g(X, odd_in_g(X)) odd_in_g(s(X)) -> U2_g(X, even_in_g(X)) U2_g(X, even_out_g(X)) -> odd_out_g(s(X)) U1_g(X, odd_out_g(X)) -> even_out_g(s(X)) The argument filtering Pi contains the following mapping: even_in_g(x1) = even_in_g(x1) 0 = 0 even_out_g(x1) = even_out_g s(x1) = s(x1) U1_g(x1, x2) = U1_g(x2) odd_in_g(x1) = odd_in_g(x1) U2_g(x1, x2) = U2_g(x2) odd_out_g(x1) = odd_out_g EVEN_IN_G(x1) = EVEN_IN_G(x1) ODD_IN_G(x1) = ODD_IN_G(x1) 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: EVEN_IN_G(s(X)) -> ODD_IN_G(X) ODD_IN_G(s(X)) -> EVEN_IN_G(X) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (9) PiDPToQDPProof (EQUIVALENT) 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: EVEN_IN_G(s(X)) -> ODD_IN_G(X) ODD_IN_G(s(X)) -> EVEN_IN_G(X) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (11) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *ODD_IN_G(s(X)) -> EVEN_IN_G(X) The graph contains the following edges 1 > 1 *EVEN_IN_G(s(X)) -> ODD_IN_G(X) The graph contains the following edges 1 > 1 ---------------------------------------- (12) YES