/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.pl /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern normal(g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof [SOUND,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, 2 ms] (6) AND (7) PiDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) PiDP (10) PiDPToQDPProof [SOUND, 0 ms] (11) QDP (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] (13) YES (14) PiDP (15) UsableRulesProof [EQUIVALENT, 0 ms] (16) PiDP (17) PiDPToQDPProof [SOUND, 0 ms] (18) QDP (19) MRRProof [EQUIVALENT, 12 ms] (20) QDP (21) PisEmptyProof [EQUIVALENT, 0 ms] (22) YES ---------------------------------------- (0) Obligation: Clauses: normal(F, N) :- ','(rewrite(F, F1), normal(F1, N)). normal(F, F). rewrite(op(op(A, B), C), op(A, op(B, C))). rewrite(op(A, op(B, C)), op(A, L)) :- rewrite(op(B, C), L). Query: normal(g,a) ---------------------------------------- (1) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: normal_in_2: (b,f) rewrite_in_2: (b,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: normal_in_ga(F, N) -> U1_ga(F, N, rewrite_in_ga(F, F1)) rewrite_in_ga(op(op(A, B), C), op(A, op(B, C))) -> rewrite_out_ga(op(op(A, B), C), op(A, op(B, C))) rewrite_in_ga(op(A, op(B, C)), op(A, L)) -> U3_ga(A, B, C, L, rewrite_in_ga(op(B, C), L)) U3_ga(A, B, C, L, rewrite_out_ga(op(B, C), L)) -> rewrite_out_ga(op(A, op(B, C)), op(A, L)) U1_ga(F, N, rewrite_out_ga(F, F1)) -> U2_ga(F, N, normal_in_ga(F1, N)) normal_in_ga(F, F) -> normal_out_ga(F, F) U2_ga(F, N, normal_out_ga(F1, N)) -> normal_out_ga(F, N) The argument filtering Pi contains the following mapping: normal_in_ga(x1, x2) = normal_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) rewrite_in_ga(x1, x2) = rewrite_in_ga(x1) op(x1, x2) = op(x1, x2) rewrite_out_ga(x1, x2) = rewrite_out_ga(x2) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x5) U2_ga(x1, x2, x3) = U2_ga(x3) normal_out_ga(x1, x2) = normal_out_ga(x2) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: normal_in_ga(F, N) -> U1_ga(F, N, rewrite_in_ga(F, F1)) rewrite_in_ga(op(op(A, B), C), op(A, op(B, C))) -> rewrite_out_ga(op(op(A, B), C), op(A, op(B, C))) rewrite_in_ga(op(A, op(B, C)), op(A, L)) -> U3_ga(A, B, C, L, rewrite_in_ga(op(B, C), L)) U3_ga(A, B, C, L, rewrite_out_ga(op(B, C), L)) -> rewrite_out_ga(op(A, op(B, C)), op(A, L)) U1_ga(F, N, rewrite_out_ga(F, F1)) -> U2_ga(F, N, normal_in_ga(F1, N)) normal_in_ga(F, F) -> normal_out_ga(F, F) U2_ga(F, N, normal_out_ga(F1, N)) -> normal_out_ga(F, N) The argument filtering Pi contains the following mapping: normal_in_ga(x1, x2) = normal_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) rewrite_in_ga(x1, x2) = rewrite_in_ga(x1) op(x1, x2) = op(x1, x2) rewrite_out_ga(x1, x2) = rewrite_out_ga(x2) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x5) U2_ga(x1, x2, x3) = U2_ga(x3) normal_out_ga(x1, x2) = normal_out_ga(x2) ---------------------------------------- (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: NORMAL_IN_GA(F, N) -> U1_GA(F, N, rewrite_in_ga(F, F1)) NORMAL_IN_GA(F, N) -> REWRITE_IN_GA(F, F1) REWRITE_IN_GA(op(A, op(B, C)), op(A, L)) -> U3_GA(A, B, C, L, rewrite_in_ga(op(B, C), L)) REWRITE_IN_GA(op(A, op(B, C)), op(A, L)) -> REWRITE_IN_GA(op(B, C), L) U1_GA(F, N, rewrite_out_ga(F, F1)) -> U2_GA(F, N, normal_in_ga(F1, N)) U1_GA(F, N, rewrite_out_ga(F, F1)) -> NORMAL_IN_GA(F1, N) The TRS R consists of the following rules: normal_in_ga(F, N) -> U1_ga(F, N, rewrite_in_ga(F, F1)) rewrite_in_ga(op(op(A, B), C), op(A, op(B, C))) -> rewrite_out_ga(op(op(A, B), C), op(A, op(B, C))) rewrite_in_ga(op(A, op(B, C)), op(A, L)) -> U3_ga(A, B, C, L, rewrite_in_ga(op(B, C), L)) U3_ga(A, B, C, L, rewrite_out_ga(op(B, C), L)) -> rewrite_out_ga(op(A, op(B, C)), op(A, L)) U1_ga(F, N, rewrite_out_ga(F, F1)) -> U2_ga(F, N, normal_in_ga(F1, N)) normal_in_ga(F, F) -> normal_out_ga(F, F) U2_ga(F, N, normal_out_ga(F1, N)) -> normal_out_ga(F, N) The argument filtering Pi contains the following mapping: normal_in_ga(x1, x2) = normal_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) rewrite_in_ga(x1, x2) = rewrite_in_ga(x1) op(x1, x2) = op(x1, x2) rewrite_out_ga(x1, x2) = rewrite_out_ga(x2) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x5) U2_ga(x1, x2, x3) = U2_ga(x3) normal_out_ga(x1, x2) = normal_out_ga(x2) NORMAL_IN_GA(x1, x2) = NORMAL_IN_GA(x1) U1_GA(x1, x2, x3) = U1_GA(x3) REWRITE_IN_GA(x1, x2) = REWRITE_IN_GA(x1) U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x5) U2_GA(x1, x2, x3) = U2_GA(x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: NORMAL_IN_GA(F, N) -> U1_GA(F, N, rewrite_in_ga(F, F1)) NORMAL_IN_GA(F, N) -> REWRITE_IN_GA(F, F1) REWRITE_IN_GA(op(A, op(B, C)), op(A, L)) -> U3_GA(A, B, C, L, rewrite_in_ga(op(B, C), L)) REWRITE_IN_GA(op(A, op(B, C)), op(A, L)) -> REWRITE_IN_GA(op(B, C), L) U1_GA(F, N, rewrite_out_ga(F, F1)) -> U2_GA(F, N, normal_in_ga(F1, N)) U1_GA(F, N, rewrite_out_ga(F, F1)) -> NORMAL_IN_GA(F1, N) The TRS R consists of the following rules: normal_in_ga(F, N) -> U1_ga(F, N, rewrite_in_ga(F, F1)) rewrite_in_ga(op(op(A, B), C), op(A, op(B, C))) -> rewrite_out_ga(op(op(A, B), C), op(A, op(B, C))) rewrite_in_ga(op(A, op(B, C)), op(A, L)) -> U3_ga(A, B, C, L, rewrite_in_ga(op(B, C), L)) U3_ga(A, B, C, L, rewrite_out_ga(op(B, C), L)) -> rewrite_out_ga(op(A, op(B, C)), op(A, L)) U1_ga(F, N, rewrite_out_ga(F, F1)) -> U2_ga(F, N, normal_in_ga(F1, N)) normal_in_ga(F, F) -> normal_out_ga(F, F) U2_ga(F, N, normal_out_ga(F1, N)) -> normal_out_ga(F, N) The argument filtering Pi contains the following mapping: normal_in_ga(x1, x2) = normal_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) rewrite_in_ga(x1, x2) = rewrite_in_ga(x1) op(x1, x2) = op(x1, x2) rewrite_out_ga(x1, x2) = rewrite_out_ga(x2) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x5) U2_ga(x1, x2, x3) = U2_ga(x3) normal_out_ga(x1, x2) = normal_out_ga(x2) NORMAL_IN_GA(x1, x2) = NORMAL_IN_GA(x1) U1_GA(x1, x2, x3) = U1_GA(x3) REWRITE_IN_GA(x1, x2) = REWRITE_IN_GA(x1) U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x5) U2_GA(x1, x2, x3) = U2_GA(x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 3 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: REWRITE_IN_GA(op(A, op(B, C)), op(A, L)) -> REWRITE_IN_GA(op(B, C), L) The TRS R consists of the following rules: normal_in_ga(F, N) -> U1_ga(F, N, rewrite_in_ga(F, F1)) rewrite_in_ga(op(op(A, B), C), op(A, op(B, C))) -> rewrite_out_ga(op(op(A, B), C), op(A, op(B, C))) rewrite_in_ga(op(A, op(B, C)), op(A, L)) -> U3_ga(A, B, C, L, rewrite_in_ga(op(B, C), L)) U3_ga(A, B, C, L, rewrite_out_ga(op(B, C), L)) -> rewrite_out_ga(op(A, op(B, C)), op(A, L)) U1_ga(F, N, rewrite_out_ga(F, F1)) -> U2_ga(F, N, normal_in_ga(F1, N)) normal_in_ga(F, F) -> normal_out_ga(F, F) U2_ga(F, N, normal_out_ga(F1, N)) -> normal_out_ga(F, N) The argument filtering Pi contains the following mapping: normal_in_ga(x1, x2) = normal_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) rewrite_in_ga(x1, x2) = rewrite_in_ga(x1) op(x1, x2) = op(x1, x2) rewrite_out_ga(x1, x2) = rewrite_out_ga(x2) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x5) U2_ga(x1, x2, x3) = U2_ga(x3) normal_out_ga(x1, x2) = normal_out_ga(x2) REWRITE_IN_GA(x1, x2) = REWRITE_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (8) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (9) Obligation: Pi DP problem: The TRS P consists of the following rules: REWRITE_IN_GA(op(A, op(B, C)), op(A, L)) -> REWRITE_IN_GA(op(B, C), L) R is empty. The argument filtering Pi contains the following mapping: op(x1, x2) = op(x1, x2) REWRITE_IN_GA(x1, x2) = REWRITE_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (10) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: REWRITE_IN_GA(op(A, op(B, C))) -> REWRITE_IN_GA(op(B, C)) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (12) 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: *REWRITE_IN_GA(op(A, op(B, C))) -> REWRITE_IN_GA(op(B, C)) The graph contains the following edges 1 > 1 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_GA(F, N, rewrite_out_ga(F, F1)) -> NORMAL_IN_GA(F1, N) NORMAL_IN_GA(F, N) -> U1_GA(F, N, rewrite_in_ga(F, F1)) The TRS R consists of the following rules: normal_in_ga(F, N) -> U1_ga(F, N, rewrite_in_ga(F, F1)) rewrite_in_ga(op(op(A, B), C), op(A, op(B, C))) -> rewrite_out_ga(op(op(A, B), C), op(A, op(B, C))) rewrite_in_ga(op(A, op(B, C)), op(A, L)) -> U3_ga(A, B, C, L, rewrite_in_ga(op(B, C), L)) U3_ga(A, B, C, L, rewrite_out_ga(op(B, C), L)) -> rewrite_out_ga(op(A, op(B, C)), op(A, L)) U1_ga(F, N, rewrite_out_ga(F, F1)) -> U2_ga(F, N, normal_in_ga(F1, N)) normal_in_ga(F, F) -> normal_out_ga(F, F) U2_ga(F, N, normal_out_ga(F1, N)) -> normal_out_ga(F, N) The argument filtering Pi contains the following mapping: normal_in_ga(x1, x2) = normal_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) rewrite_in_ga(x1, x2) = rewrite_in_ga(x1) op(x1, x2) = op(x1, x2) rewrite_out_ga(x1, x2) = rewrite_out_ga(x2) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x5) U2_ga(x1, x2, x3) = U2_ga(x3) normal_out_ga(x1, x2) = normal_out_ga(x2) NORMAL_IN_GA(x1, x2) = NORMAL_IN_GA(x1) U1_GA(x1, x2, x3) = U1_GA(x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (15) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_GA(F, N, rewrite_out_ga(F, F1)) -> NORMAL_IN_GA(F1, N) NORMAL_IN_GA(F, N) -> U1_GA(F, N, rewrite_in_ga(F, F1)) The TRS R consists of the following rules: rewrite_in_ga(op(op(A, B), C), op(A, op(B, C))) -> rewrite_out_ga(op(op(A, B), C), op(A, op(B, C))) rewrite_in_ga(op(A, op(B, C)), op(A, L)) -> U3_ga(A, B, C, L, rewrite_in_ga(op(B, C), L)) U3_ga(A, B, C, L, rewrite_out_ga(op(B, C), L)) -> rewrite_out_ga(op(A, op(B, C)), op(A, L)) The argument filtering Pi contains the following mapping: rewrite_in_ga(x1, x2) = rewrite_in_ga(x1) op(x1, x2) = op(x1, x2) rewrite_out_ga(x1, x2) = rewrite_out_ga(x2) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x5) NORMAL_IN_GA(x1, x2) = NORMAL_IN_GA(x1) U1_GA(x1, x2, x3) = U1_GA(x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (17) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: U1_GA(rewrite_out_ga(F1)) -> NORMAL_IN_GA(F1) NORMAL_IN_GA(F) -> U1_GA(rewrite_in_ga(F)) The TRS R consists of the following rules: rewrite_in_ga(op(op(A, B), C)) -> rewrite_out_ga(op(A, op(B, C))) rewrite_in_ga(op(A, op(B, C))) -> U3_ga(A, rewrite_in_ga(op(B, C))) U3_ga(A, rewrite_out_ga(L)) -> rewrite_out_ga(op(A, L)) The set Q consists of the following terms: rewrite_in_ga(x0) U3_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (19) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented dependency pairs: U1_GA(rewrite_out_ga(F1)) -> NORMAL_IN_GA(F1) NORMAL_IN_GA(F) -> U1_GA(rewrite_in_ga(F)) Used ordering: Polynomial interpretation [POLO]: POL(NORMAL_IN_GA(x_1)) = 1 + x_1 POL(U1_GA(x_1)) = x_1 POL(U3_ga(x_1, x_2)) = 2 + 2*x_1 + x_2 POL(op(x_1, x_2)) = 2 + 2*x_1 + x_2 POL(rewrite_in_ga(x_1)) = x_1 POL(rewrite_out_ga(x_1)) = 2 + x_1 ---------------------------------------- (20) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: rewrite_in_ga(op(op(A, B), C)) -> rewrite_out_ga(op(A, op(B, C))) rewrite_in_ga(op(A, op(B, C))) -> U3_ga(A, rewrite_in_ga(op(B, C))) U3_ga(A, rewrite_out_ga(L)) -> rewrite_out_ga(op(A, L)) The set Q consists of the following terms: rewrite_in_ga(x0) U3_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (21) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (22) YES