/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern f(g,a,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0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, 0 ms] (6) PiDP (7) PiDPToQDPProof [SOUND, 0 ms] (8) QDP (9) UsableRulesReductionPairsProof [EQUIVALENT, 37 ms] (10) QDP (11) DependencyGraphProof [EQUIVALENT, 0 ms] (12) TRUE ---------------------------------------- (0) Obligation: Clauses: f(0, Y, 0). f(s(X), Y, Z) :- ','(f(X, Y, U), f(U, Y, Z)). Query: f(g,a,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: f_in_3: (b,f,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: f_in_gaa(0, Y, 0) -> f_out_gaa(0, Y, 0) f_in_gaa(s(X), Y, Z) -> U1_gaa(X, Y, Z, f_in_gaa(X, Y, U)) U1_gaa(X, Y, Z, f_out_gaa(X, Y, U)) -> U2_gaa(X, Y, Z, U, f_in_gaa(U, Y, Z)) U2_gaa(X, Y, Z, U, f_out_gaa(U, Y, Z)) -> f_out_gaa(s(X), Y, Z) The argument filtering Pi contains the following mapping: f_in_gaa(x1, x2, x3) = f_in_gaa(x1) 0 = 0 f_out_gaa(x1, x2, x3) = f_out_gaa(x3) s(x1) = s(x1) U1_gaa(x1, x2, x3, x4) = U1_gaa(x4) U2_gaa(x1, x2, x3, x4, x5) = U2_gaa(x5) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: f_in_gaa(0, Y, 0) -> f_out_gaa(0, Y, 0) f_in_gaa(s(X), Y, Z) -> U1_gaa(X, Y, Z, f_in_gaa(X, Y, U)) U1_gaa(X, Y, Z, f_out_gaa(X, Y, U)) -> U2_gaa(X, Y, Z, U, f_in_gaa(U, Y, Z)) U2_gaa(X, Y, Z, U, f_out_gaa(U, Y, Z)) -> f_out_gaa(s(X), Y, Z) The argument filtering Pi contains the following mapping: f_in_gaa(x1, x2, x3) = f_in_gaa(x1) 0 = 0 f_out_gaa(x1, x2, x3) = f_out_gaa(x3) s(x1) = s(x1) U1_gaa(x1, x2, x3, x4) = U1_gaa(x4) U2_gaa(x1, x2, x3, x4, x5) = U2_gaa(x5) ---------------------------------------- (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: F_IN_GAA(s(X), Y, Z) -> U1_GAA(X, Y, Z, f_in_gaa(X, Y, U)) F_IN_GAA(s(X), Y, Z) -> F_IN_GAA(X, Y, U) U1_GAA(X, Y, Z, f_out_gaa(X, Y, U)) -> U2_GAA(X, Y, Z, U, f_in_gaa(U, Y, Z)) U1_GAA(X, Y, Z, f_out_gaa(X, Y, U)) -> F_IN_GAA(U, Y, Z) The TRS R consists of the following rules: f_in_gaa(0, Y, 0) -> f_out_gaa(0, Y, 0) f_in_gaa(s(X), Y, Z) -> U1_gaa(X, Y, Z, f_in_gaa(X, Y, U)) U1_gaa(X, Y, Z, f_out_gaa(X, Y, U)) -> U2_gaa(X, Y, Z, U, f_in_gaa(U, Y, Z)) U2_gaa(X, Y, Z, U, f_out_gaa(U, Y, Z)) -> f_out_gaa(s(X), Y, Z) The argument filtering Pi contains the following mapping: f_in_gaa(x1, x2, x3) = f_in_gaa(x1) 0 = 0 f_out_gaa(x1, x2, x3) = f_out_gaa(x3) s(x1) = s(x1) U1_gaa(x1, x2, x3, x4) = U1_gaa(x4) U2_gaa(x1, x2, x3, x4, x5) = U2_gaa(x5) F_IN_GAA(x1, x2, x3) = F_IN_GAA(x1) U1_GAA(x1, x2, x3, x4) = U1_GAA(x4) U2_GAA(x1, x2, x3, x4, x5) = U2_GAA(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: F_IN_GAA(s(X), Y, Z) -> U1_GAA(X, Y, Z, f_in_gaa(X, Y, U)) F_IN_GAA(s(X), Y, Z) -> F_IN_GAA(X, Y, U) U1_GAA(X, Y, Z, f_out_gaa(X, Y, U)) -> U2_GAA(X, Y, Z, U, f_in_gaa(U, Y, Z)) U1_GAA(X, Y, Z, f_out_gaa(X, Y, U)) -> F_IN_GAA(U, Y, Z) The TRS R consists of the following rules: f_in_gaa(0, Y, 0) -> f_out_gaa(0, Y, 0) f_in_gaa(s(X), Y, Z) -> U1_gaa(X, Y, Z, f_in_gaa(X, Y, U)) U1_gaa(X, Y, Z, f_out_gaa(X, Y, U)) -> U2_gaa(X, Y, Z, U, f_in_gaa(U, Y, Z)) U2_gaa(X, Y, Z, U, f_out_gaa(U, Y, Z)) -> f_out_gaa(s(X), Y, Z) The argument filtering Pi contains the following mapping: f_in_gaa(x1, x2, x3) = f_in_gaa(x1) 0 = 0 f_out_gaa(x1, x2, x3) = f_out_gaa(x3) s(x1) = s(x1) U1_gaa(x1, x2, x3, x4) = U1_gaa(x4) U2_gaa(x1, x2, x3, x4, x5) = U2_gaa(x5) F_IN_GAA(x1, x2, x3) = F_IN_GAA(x1) U1_GAA(x1, x2, x3, x4) = U1_GAA(x4) U2_GAA(x1, x2, x3, x4, x5) = U2_GAA(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node. ---------------------------------------- (6) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_GAA(X, Y, Z, f_out_gaa(X, Y, U)) -> F_IN_GAA(U, Y, Z) F_IN_GAA(s(X), Y, Z) -> U1_GAA(X, Y, Z, f_in_gaa(X, Y, U)) F_IN_GAA(s(X), Y, Z) -> F_IN_GAA(X, Y, U) The TRS R consists of the following rules: f_in_gaa(0, Y, 0) -> f_out_gaa(0, Y, 0) f_in_gaa(s(X), Y, Z) -> U1_gaa(X, Y, Z, f_in_gaa(X, Y, U)) U1_gaa(X, Y, Z, f_out_gaa(X, Y, U)) -> U2_gaa(X, Y, Z, U, f_in_gaa(U, Y, Z)) U2_gaa(X, Y, Z, U, f_out_gaa(U, Y, Z)) -> f_out_gaa(s(X), Y, Z) The argument filtering Pi contains the following mapping: f_in_gaa(x1, x2, x3) = f_in_gaa(x1) 0 = 0 f_out_gaa(x1, x2, x3) = f_out_gaa(x3) s(x1) = s(x1) U1_gaa(x1, x2, x3, x4) = U1_gaa(x4) U2_gaa(x1, x2, x3, x4, x5) = U2_gaa(x5) F_IN_GAA(x1, x2, x3) = F_IN_GAA(x1) U1_GAA(x1, x2, x3, x4) = U1_GAA(x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (7) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (8) Obligation: Q DP problem: The TRS P consists of the following rules: U1_GAA(f_out_gaa(U)) -> F_IN_GAA(U) F_IN_GAA(s(X)) -> U1_GAA(f_in_gaa(X)) F_IN_GAA(s(X)) -> F_IN_GAA(X) The TRS R consists of the following rules: f_in_gaa(0) -> f_out_gaa(0) f_in_gaa(s(X)) -> U1_gaa(f_in_gaa(X)) U1_gaa(f_out_gaa(U)) -> U2_gaa(f_in_gaa(U)) U2_gaa(f_out_gaa(Z)) -> f_out_gaa(Z) The set Q consists of the following terms: f_in_gaa(x0) U1_gaa(x0) U2_gaa(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (9) UsableRulesReductionPairsProof (EQUIVALENT) By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well. The following dependency pairs can be deleted: F_IN_GAA(s(X)) -> U1_GAA(f_in_gaa(X)) F_IN_GAA(s(X)) -> F_IN_GAA(X) The following rules are removed from R: f_in_gaa(s(X)) -> U1_gaa(f_in_gaa(X)) Used ordering: POLO with Polynomial interpretation [POLO]: POL(0) = 0 POL(F_IN_GAA(x_1)) = 2*x_1 POL(U1_GAA(x_1)) = x_1 POL(U1_gaa(x_1)) = 2*x_1 POL(U2_gaa(x_1)) = x_1 POL(f_in_gaa(x_1)) = 2*x_1 POL(f_out_gaa(x_1)) = 2*x_1 POL(s(x_1)) = 2*x_1 ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: U1_GAA(f_out_gaa(U)) -> F_IN_GAA(U) The TRS R consists of the following rules: f_in_gaa(0) -> f_out_gaa(0) U1_gaa(f_out_gaa(U)) -> U2_gaa(f_in_gaa(U)) U2_gaa(f_out_gaa(Z)) -> f_out_gaa(Z) The set Q consists of the following terms: f_in_gaa(x0) U1_gaa(x0) U2_gaa(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (11) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (12) TRUE