/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern goal(g) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0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) AND (7) PiDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) PiDP (10) PiDPToQDPProof [SOUND, 3 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) QDPSizeChangeProof [EQUIVALENT, 0 ms] (20) YES ---------------------------------------- (0) Obligation: Clauses: list([]). list(.(X, XS)) :- list(XS). s2l(s(X), .(Y, Xs)) :- s2l(X, Xs). s2l(0, []). goal(X) :- ','(s2l(X, XS), list(XS)). 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) s2l_in_2: (b,f) list_in_1: (b) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: goal_in_g(X) -> U3_g(X, s2l_in_ga(X, XS)) s2l_in_ga(s(X), .(Y, Xs)) -> U2_ga(X, Y, Xs, s2l_in_ga(X, Xs)) s2l_in_ga(0, []) -> s2l_out_ga(0, []) U2_ga(X, Y, Xs, s2l_out_ga(X, Xs)) -> s2l_out_ga(s(X), .(Y, Xs)) U3_g(X, s2l_out_ga(X, XS)) -> U4_g(X, list_in_g(XS)) list_in_g([]) -> list_out_g([]) list_in_g(.(X, XS)) -> U1_g(X, XS, list_in_g(XS)) U1_g(X, XS, list_out_g(XS)) -> list_out_g(.(X, XS)) U4_g(X, list_out_g(XS)) -> goal_out_g(X) The argument filtering Pi contains the following mapping: goal_in_g(x1) = goal_in_g(x1) U3_g(x1, x2) = U3_g(x2) s2l_in_ga(x1, x2) = s2l_in_ga(x1) s(x1) = s(x1) U2_ga(x1, x2, x3, x4) = U2_ga(x4) 0 = 0 s2l_out_ga(x1, x2) = s2l_out_ga(x2) .(x1, x2) = .(x2) U4_g(x1, x2) = U4_g(x2) list_in_g(x1) = list_in_g(x1) [] = [] list_out_g(x1) = list_out_g U1_g(x1, x2, x3) = U1_g(x3) 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) -> U3_g(X, s2l_in_ga(X, XS)) s2l_in_ga(s(X), .(Y, Xs)) -> U2_ga(X, Y, Xs, s2l_in_ga(X, Xs)) s2l_in_ga(0, []) -> s2l_out_ga(0, []) U2_ga(X, Y, Xs, s2l_out_ga(X, Xs)) -> s2l_out_ga(s(X), .(Y, Xs)) U3_g(X, s2l_out_ga(X, XS)) -> U4_g(X, list_in_g(XS)) list_in_g([]) -> list_out_g([]) list_in_g(.(X, XS)) -> U1_g(X, XS, list_in_g(XS)) U1_g(X, XS, list_out_g(XS)) -> list_out_g(.(X, XS)) U4_g(X, list_out_g(XS)) -> goal_out_g(X) The argument filtering Pi contains the following mapping: goal_in_g(x1) = goal_in_g(x1) U3_g(x1, x2) = U3_g(x2) s2l_in_ga(x1, x2) = s2l_in_ga(x1) s(x1) = s(x1) U2_ga(x1, x2, x3, x4) = U2_ga(x4) 0 = 0 s2l_out_ga(x1, x2) = s2l_out_ga(x2) .(x1, x2) = .(x2) U4_g(x1, x2) = U4_g(x2) list_in_g(x1) = list_in_g(x1) [] = [] list_out_g(x1) = list_out_g U1_g(x1, x2, x3) = U1_g(x3) 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) -> U3_G(X, s2l_in_ga(X, XS)) GOAL_IN_G(X) -> S2L_IN_GA(X, XS) S2L_IN_GA(s(X), .(Y, Xs)) -> U2_GA(X, Y, Xs, s2l_in_ga(X, Xs)) S2L_IN_GA(s(X), .(Y, Xs)) -> S2L_IN_GA(X, Xs) U3_G(X, s2l_out_ga(X, XS)) -> U4_G(X, list_in_g(XS)) U3_G(X, s2l_out_ga(X, XS)) -> LIST_IN_G(XS) LIST_IN_G(.(X, XS)) -> U1_G(X, XS, list_in_g(XS)) LIST_IN_G(.(X, XS)) -> LIST_IN_G(XS) The TRS R consists of the following rules: goal_in_g(X) -> U3_g(X, s2l_in_ga(X, XS)) s2l_in_ga(s(X), .(Y, Xs)) -> U2_ga(X, Y, Xs, s2l_in_ga(X, Xs)) s2l_in_ga(0, []) -> s2l_out_ga(0, []) U2_ga(X, Y, Xs, s2l_out_ga(X, Xs)) -> s2l_out_ga(s(X), .(Y, Xs)) U3_g(X, s2l_out_ga(X, XS)) -> U4_g(X, list_in_g(XS)) list_in_g([]) -> list_out_g([]) list_in_g(.(X, XS)) -> U1_g(X, XS, list_in_g(XS)) U1_g(X, XS, list_out_g(XS)) -> list_out_g(.(X, XS)) U4_g(X, list_out_g(XS)) -> goal_out_g(X) The argument filtering Pi contains the following mapping: goal_in_g(x1) = goal_in_g(x1) U3_g(x1, x2) = U3_g(x2) s2l_in_ga(x1, x2) = s2l_in_ga(x1) s(x1) = s(x1) U2_ga(x1, x2, x3, x4) = U2_ga(x4) 0 = 0 s2l_out_ga(x1, x2) = s2l_out_ga(x2) .(x1, x2) = .(x2) U4_g(x1, x2) = U4_g(x2) list_in_g(x1) = list_in_g(x1) [] = [] list_out_g(x1) = list_out_g U1_g(x1, x2, x3) = U1_g(x3) goal_out_g(x1) = goal_out_g GOAL_IN_G(x1) = GOAL_IN_G(x1) U3_G(x1, x2) = U3_G(x2) S2L_IN_GA(x1, x2) = S2L_IN_GA(x1) U2_GA(x1, x2, x3, x4) = U2_GA(x4) U4_G(x1, x2) = U4_G(x2) LIST_IN_G(x1) = LIST_IN_G(x1) U1_G(x1, x2, x3) = U1_G(x3) 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) -> U3_G(X, s2l_in_ga(X, XS)) GOAL_IN_G(X) -> S2L_IN_GA(X, XS) S2L_IN_GA(s(X), .(Y, Xs)) -> U2_GA(X, Y, Xs, s2l_in_ga(X, Xs)) S2L_IN_GA(s(X), .(Y, Xs)) -> S2L_IN_GA(X, Xs) U3_G(X, s2l_out_ga(X, XS)) -> U4_G(X, list_in_g(XS)) U3_G(X, s2l_out_ga(X, XS)) -> LIST_IN_G(XS) LIST_IN_G(.(X, XS)) -> U1_G(X, XS, list_in_g(XS)) LIST_IN_G(.(X, XS)) -> LIST_IN_G(XS) The TRS R consists of the following rules: goal_in_g(X) -> U3_g(X, s2l_in_ga(X, XS)) s2l_in_ga(s(X), .(Y, Xs)) -> U2_ga(X, Y, Xs, s2l_in_ga(X, Xs)) s2l_in_ga(0, []) -> s2l_out_ga(0, []) U2_ga(X, Y, Xs, s2l_out_ga(X, Xs)) -> s2l_out_ga(s(X), .(Y, Xs)) U3_g(X, s2l_out_ga(X, XS)) -> U4_g(X, list_in_g(XS)) list_in_g([]) -> list_out_g([]) list_in_g(.(X, XS)) -> U1_g(X, XS, list_in_g(XS)) U1_g(X, XS, list_out_g(XS)) -> list_out_g(.(X, XS)) U4_g(X, list_out_g(XS)) -> goal_out_g(X) The argument filtering Pi contains the following mapping: goal_in_g(x1) = goal_in_g(x1) U3_g(x1, x2) = U3_g(x2) s2l_in_ga(x1, x2) = s2l_in_ga(x1) s(x1) = s(x1) U2_ga(x1, x2, x3, x4) = U2_ga(x4) 0 = 0 s2l_out_ga(x1, x2) = s2l_out_ga(x2) .(x1, x2) = .(x2) U4_g(x1, x2) = U4_g(x2) list_in_g(x1) = list_in_g(x1) [] = [] list_out_g(x1) = list_out_g U1_g(x1, x2, x3) = U1_g(x3) goal_out_g(x1) = goal_out_g GOAL_IN_G(x1) = GOAL_IN_G(x1) U3_G(x1, x2) = U3_G(x2) S2L_IN_GA(x1, x2) = S2L_IN_GA(x1) U2_GA(x1, x2, x3, x4) = U2_GA(x4) U4_G(x1, x2) = U4_G(x2) LIST_IN_G(x1) = LIST_IN_G(x1) U1_G(x1, x2, x3) = U1_G(x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: LIST_IN_G(.(X, XS)) -> LIST_IN_G(XS) The TRS R consists of the following rules: goal_in_g(X) -> U3_g(X, s2l_in_ga(X, XS)) s2l_in_ga(s(X), .(Y, Xs)) -> U2_ga(X, Y, Xs, s2l_in_ga(X, Xs)) s2l_in_ga(0, []) -> s2l_out_ga(0, []) U2_ga(X, Y, Xs, s2l_out_ga(X, Xs)) -> s2l_out_ga(s(X), .(Y, Xs)) U3_g(X, s2l_out_ga(X, XS)) -> U4_g(X, list_in_g(XS)) list_in_g([]) -> list_out_g([]) list_in_g(.(X, XS)) -> U1_g(X, XS, list_in_g(XS)) U1_g(X, XS, list_out_g(XS)) -> list_out_g(.(X, XS)) U4_g(X, list_out_g(XS)) -> goal_out_g(X) The argument filtering Pi contains the following mapping: goal_in_g(x1) = goal_in_g(x1) U3_g(x1, x2) = U3_g(x2) s2l_in_ga(x1, x2) = s2l_in_ga(x1) s(x1) = s(x1) U2_ga(x1, x2, x3, x4) = U2_ga(x4) 0 = 0 s2l_out_ga(x1, x2) = s2l_out_ga(x2) .(x1, x2) = .(x2) U4_g(x1, x2) = U4_g(x2) list_in_g(x1) = list_in_g(x1) [] = [] list_out_g(x1) = list_out_g U1_g(x1, x2, x3) = U1_g(x3) goal_out_g(x1) = goal_out_g LIST_IN_G(x1) = LIST_IN_G(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: LIST_IN_G(.(X, XS)) -> LIST_IN_G(XS) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) LIST_IN_G(x1) = LIST_IN_G(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: LIST_IN_G(.(XS)) -> LIST_IN_G(XS) 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: *LIST_IN_G(.(XS)) -> LIST_IN_G(XS) The graph contains the following edges 1 > 1 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: S2L_IN_GA(s(X), .(Y, Xs)) -> S2L_IN_GA(X, Xs) The TRS R consists of the following rules: goal_in_g(X) -> U3_g(X, s2l_in_ga(X, XS)) s2l_in_ga(s(X), .(Y, Xs)) -> U2_ga(X, Y, Xs, s2l_in_ga(X, Xs)) s2l_in_ga(0, []) -> s2l_out_ga(0, []) U2_ga(X, Y, Xs, s2l_out_ga(X, Xs)) -> s2l_out_ga(s(X), .(Y, Xs)) U3_g(X, s2l_out_ga(X, XS)) -> U4_g(X, list_in_g(XS)) list_in_g([]) -> list_out_g([]) list_in_g(.(X, XS)) -> U1_g(X, XS, list_in_g(XS)) U1_g(X, XS, list_out_g(XS)) -> list_out_g(.(X, XS)) U4_g(X, list_out_g(XS)) -> goal_out_g(X) The argument filtering Pi contains the following mapping: goal_in_g(x1) = goal_in_g(x1) U3_g(x1, x2) = U3_g(x2) s2l_in_ga(x1, x2) = s2l_in_ga(x1) s(x1) = s(x1) U2_ga(x1, x2, x3, x4) = U2_ga(x4) 0 = 0 s2l_out_ga(x1, x2) = s2l_out_ga(x2) .(x1, x2) = .(x2) U4_g(x1, x2) = U4_g(x2) list_in_g(x1) = list_in_g(x1) [] = [] list_out_g(x1) = list_out_g U1_g(x1, x2, x3) = U1_g(x3) goal_out_g(x1) = goal_out_g S2L_IN_GA(x1, x2) = S2L_IN_GA(x1) 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: S2L_IN_GA(s(X), .(Y, Xs)) -> S2L_IN_GA(X, Xs) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) .(x1, x2) = .(x2) S2L_IN_GA(x1, x2) = S2L_IN_GA(x1) 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: S2L_IN_GA(s(X)) -> S2L_IN_GA(X) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (19) 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: *S2L_IN_GA(s(X)) -> S2L_IN_GA(X) The graph contains the following edges 1 > 1 ---------------------------------------- (20) YES