/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern t(g) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2) PiTRS (3) DependencyPairsProof [EQUIVALENT, 2 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) AND (7) PiDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) PiDP (10) PiDPToQDPProof [SOUND, 9 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 (21) PiDP (22) UsableRulesProof [EQUIVALENT, 0 ms] (23) PiDP (24) PiDPToQDPProof [SOUND, 0 ms] (25) QDP (26) QDPSizeChangeProof [EQUIVALENT, 0 ms] (27) YES (28) PiDP (29) UsableRulesProof [EQUIVALENT, 0 ms] (30) PiDP (31) PiDPToQDPProof [SOUND, 0 ms] (32) QDP (33) MRRProof [EQUIVALENT, 2 ms] (34) QDP (35) PisEmptyProof [EQUIVALENT, 0 ms] (36) YES ---------------------------------------- (0) Obligation: Clauses: t(N) :- ','(ll(N, Xs), ','(select(X1, Xs, Xs1), ','(ll(M, Xs1), t(M)))). t(0). ll(s(N), .(X, Xs)) :- ll(N, Xs). ll(0, []). select(X, .(Y, Xs), .(Y, Ys)) :- select(X, Xs, Ys). select(X, .(X, Xs), Xs). Query: t(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: t_in_1: (b) ll_in_2: (b,f) (f,b) select_in_3: (f,b,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: t_in_g(N) -> U1_g(N, ll_in_ga(N, Xs)) ll_in_ga(s(N), .(X, Xs)) -> U5_ga(N, X, Xs, ll_in_ga(N, Xs)) ll_in_ga(0, []) -> ll_out_ga(0, []) U5_ga(N, X, Xs, ll_out_ga(N, Xs)) -> ll_out_ga(s(N), .(X, Xs)) U1_g(N, ll_out_ga(N, Xs)) -> U2_g(N, select_in_aga(X1, Xs, Xs1)) select_in_aga(X, .(Y, Xs), .(Y, Ys)) -> U6_aga(X, Y, Xs, Ys, select_in_aga(X, Xs, Ys)) select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) U6_aga(X, Y, Xs, Ys, select_out_aga(X, Xs, Ys)) -> select_out_aga(X, .(Y, Xs), .(Y, Ys)) U2_g(N, select_out_aga(X1, Xs, Xs1)) -> U3_g(N, ll_in_ag(M, Xs1)) ll_in_ag(s(N), .(X, Xs)) -> U5_ag(N, X, Xs, ll_in_ag(N, Xs)) ll_in_ag(0, []) -> ll_out_ag(0, []) U5_ag(N, X, Xs, ll_out_ag(N, Xs)) -> ll_out_ag(s(N), .(X, Xs)) U3_g(N, ll_out_ag(M, Xs1)) -> U4_g(N, t_in_g(M)) t_in_g(0) -> t_out_g(0) U4_g(N, t_out_g(M)) -> t_out_g(N) The argument filtering Pi contains the following mapping: t_in_g(x1) = t_in_g(x1) U1_g(x1, x2) = U1_g(x2) ll_in_ga(x1, x2) = ll_in_ga(x1) s(x1) = s(x1) U5_ga(x1, x2, x3, x4) = U5_ga(x4) 0 = 0 ll_out_ga(x1, x2) = ll_out_ga(x2) .(x1, x2) = .(x2) U2_g(x1, x2) = U2_g(x2) select_in_aga(x1, x2, x3) = select_in_aga(x2) U6_aga(x1, x2, x3, x4, x5) = U6_aga(x5) select_out_aga(x1, x2, x3) = select_out_aga(x3) U3_g(x1, x2) = U3_g(x2) ll_in_ag(x1, x2) = ll_in_ag(x2) U5_ag(x1, x2, x3, x4) = U5_ag(x4) [] = [] ll_out_ag(x1, x2) = ll_out_ag(x1) U4_g(x1, x2) = U4_g(x2) t_out_g(x1) = t_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: t_in_g(N) -> U1_g(N, ll_in_ga(N, Xs)) ll_in_ga(s(N), .(X, Xs)) -> U5_ga(N, X, Xs, ll_in_ga(N, Xs)) ll_in_ga(0, []) -> ll_out_ga(0, []) U5_ga(N, X, Xs, ll_out_ga(N, Xs)) -> ll_out_ga(s(N), .(X, Xs)) U1_g(N, ll_out_ga(N, Xs)) -> U2_g(N, select_in_aga(X1, Xs, Xs1)) select_in_aga(X, .(Y, Xs), .(Y, Ys)) -> U6_aga(X, Y, Xs, Ys, select_in_aga(X, Xs, Ys)) select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) U6_aga(X, Y, Xs, Ys, select_out_aga(X, Xs, Ys)) -> select_out_aga(X, .(Y, Xs), .(Y, Ys)) U2_g(N, select_out_aga(X1, Xs, Xs1)) -> U3_g(N, ll_in_ag(M, Xs1)) ll_in_ag(s(N), .(X, Xs)) -> U5_ag(N, X, Xs, ll_in_ag(N, Xs)) ll_in_ag(0, []) -> ll_out_ag(0, []) U5_ag(N, X, Xs, ll_out_ag(N, Xs)) -> ll_out_ag(s(N), .(X, Xs)) U3_g(N, ll_out_ag(M, Xs1)) -> U4_g(N, t_in_g(M)) t_in_g(0) -> t_out_g(0) U4_g(N, t_out_g(M)) -> t_out_g(N) The argument filtering Pi contains the following mapping: t_in_g(x1) = t_in_g(x1) U1_g(x1, x2) = U1_g(x2) ll_in_ga(x1, x2) = ll_in_ga(x1) s(x1) = s(x1) U5_ga(x1, x2, x3, x4) = U5_ga(x4) 0 = 0 ll_out_ga(x1, x2) = ll_out_ga(x2) .(x1, x2) = .(x2) U2_g(x1, x2) = U2_g(x2) select_in_aga(x1, x2, x3) = select_in_aga(x2) U6_aga(x1, x2, x3, x4, x5) = U6_aga(x5) select_out_aga(x1, x2, x3) = select_out_aga(x3) U3_g(x1, x2) = U3_g(x2) ll_in_ag(x1, x2) = ll_in_ag(x2) U5_ag(x1, x2, x3, x4) = U5_ag(x4) [] = [] ll_out_ag(x1, x2) = ll_out_ag(x1) U4_g(x1, x2) = U4_g(x2) t_out_g(x1) = t_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: T_IN_G(N) -> U1_G(N, ll_in_ga(N, Xs)) T_IN_G(N) -> LL_IN_GA(N, Xs) LL_IN_GA(s(N), .(X, Xs)) -> U5_GA(N, X, Xs, ll_in_ga(N, Xs)) LL_IN_GA(s(N), .(X, Xs)) -> LL_IN_GA(N, Xs) U1_G(N, ll_out_ga(N, Xs)) -> U2_G(N, select_in_aga(X1, Xs, Xs1)) U1_G(N, ll_out_ga(N, Xs)) -> SELECT_IN_AGA(X1, Xs, Xs1) SELECT_IN_AGA(X, .(Y, Xs), .(Y, Ys)) -> U6_AGA(X, Y, Xs, Ys, select_in_aga(X, Xs, Ys)) SELECT_IN_AGA(X, .(Y, Xs), .(Y, Ys)) -> SELECT_IN_AGA(X, Xs, Ys) U2_G(N, select_out_aga(X1, Xs, Xs1)) -> U3_G(N, ll_in_ag(M, Xs1)) U2_G(N, select_out_aga(X1, Xs, Xs1)) -> LL_IN_AG(M, Xs1) LL_IN_AG(s(N), .(X, Xs)) -> U5_AG(N, X, Xs, ll_in_ag(N, Xs)) LL_IN_AG(s(N), .(X, Xs)) -> LL_IN_AG(N, Xs) U3_G(N, ll_out_ag(M, Xs1)) -> U4_G(N, t_in_g(M)) U3_G(N, ll_out_ag(M, Xs1)) -> T_IN_G(M) The TRS R consists of the following rules: t_in_g(N) -> U1_g(N, ll_in_ga(N, Xs)) ll_in_ga(s(N), .(X, Xs)) -> U5_ga(N, X, Xs, ll_in_ga(N, Xs)) ll_in_ga(0, []) -> ll_out_ga(0, []) U5_ga(N, X, Xs, ll_out_ga(N, Xs)) -> ll_out_ga(s(N), .(X, Xs)) U1_g(N, ll_out_ga(N, Xs)) -> U2_g(N, select_in_aga(X1, Xs, Xs1)) select_in_aga(X, .(Y, Xs), .(Y, Ys)) -> U6_aga(X, Y, Xs, Ys, select_in_aga(X, Xs, Ys)) select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) U6_aga(X, Y, Xs, Ys, select_out_aga(X, Xs, Ys)) -> select_out_aga(X, .(Y, Xs), .(Y, Ys)) U2_g(N, select_out_aga(X1, Xs, Xs1)) -> U3_g(N, ll_in_ag(M, Xs1)) ll_in_ag(s(N), .(X, Xs)) -> U5_ag(N, X, Xs, ll_in_ag(N, Xs)) ll_in_ag(0, []) -> ll_out_ag(0, []) U5_ag(N, X, Xs, ll_out_ag(N, Xs)) -> ll_out_ag(s(N), .(X, Xs)) U3_g(N, ll_out_ag(M, Xs1)) -> U4_g(N, t_in_g(M)) t_in_g(0) -> t_out_g(0) U4_g(N, t_out_g(M)) -> t_out_g(N) The argument filtering Pi contains the following mapping: t_in_g(x1) = t_in_g(x1) U1_g(x1, x2) = U1_g(x2) ll_in_ga(x1, x2) = ll_in_ga(x1) s(x1) = s(x1) U5_ga(x1, x2, x3, x4) = U5_ga(x4) 0 = 0 ll_out_ga(x1, x2) = ll_out_ga(x2) .(x1, x2) = .(x2) U2_g(x1, x2) = U2_g(x2) select_in_aga(x1, x2, x3) = select_in_aga(x2) U6_aga(x1, x2, x3, x4, x5) = U6_aga(x5) select_out_aga(x1, x2, x3) = select_out_aga(x3) U3_g(x1, x2) = U3_g(x2) ll_in_ag(x1, x2) = ll_in_ag(x2) U5_ag(x1, x2, x3, x4) = U5_ag(x4) [] = [] ll_out_ag(x1, x2) = ll_out_ag(x1) U4_g(x1, x2) = U4_g(x2) t_out_g(x1) = t_out_g T_IN_G(x1) = T_IN_G(x1) U1_G(x1, x2) = U1_G(x2) LL_IN_GA(x1, x2) = LL_IN_GA(x1) U5_GA(x1, x2, x3, x4) = U5_GA(x4) U2_G(x1, x2) = U2_G(x2) SELECT_IN_AGA(x1, x2, x3) = SELECT_IN_AGA(x2) U6_AGA(x1, x2, x3, x4, x5) = U6_AGA(x5) U3_G(x1, x2) = U3_G(x2) LL_IN_AG(x1, x2) = LL_IN_AG(x2) U5_AG(x1, x2, x3, x4) = U5_AG(x4) U4_G(x1, x2) = U4_G(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: T_IN_G(N) -> U1_G(N, ll_in_ga(N, Xs)) T_IN_G(N) -> LL_IN_GA(N, Xs) LL_IN_GA(s(N), .(X, Xs)) -> U5_GA(N, X, Xs, ll_in_ga(N, Xs)) LL_IN_GA(s(N), .(X, Xs)) -> LL_IN_GA(N, Xs) U1_G(N, ll_out_ga(N, Xs)) -> U2_G(N, select_in_aga(X1, Xs, Xs1)) U1_G(N, ll_out_ga(N, Xs)) -> SELECT_IN_AGA(X1, Xs, Xs1) SELECT_IN_AGA(X, .(Y, Xs), .(Y, Ys)) -> U6_AGA(X, Y, Xs, Ys, select_in_aga(X, Xs, Ys)) SELECT_IN_AGA(X, .(Y, Xs), .(Y, Ys)) -> SELECT_IN_AGA(X, Xs, Ys) U2_G(N, select_out_aga(X1, Xs, Xs1)) -> U3_G(N, ll_in_ag(M, Xs1)) U2_G(N, select_out_aga(X1, Xs, Xs1)) -> LL_IN_AG(M, Xs1) LL_IN_AG(s(N), .(X, Xs)) -> U5_AG(N, X, Xs, ll_in_ag(N, Xs)) LL_IN_AG(s(N), .(X, Xs)) -> LL_IN_AG(N, Xs) U3_G(N, ll_out_ag(M, Xs1)) -> U4_G(N, t_in_g(M)) U3_G(N, ll_out_ag(M, Xs1)) -> T_IN_G(M) The TRS R consists of the following rules: t_in_g(N) -> U1_g(N, ll_in_ga(N, Xs)) ll_in_ga(s(N), .(X, Xs)) -> U5_ga(N, X, Xs, ll_in_ga(N, Xs)) ll_in_ga(0, []) -> ll_out_ga(0, []) U5_ga(N, X, Xs, ll_out_ga(N, Xs)) -> ll_out_ga(s(N), .(X, Xs)) U1_g(N, ll_out_ga(N, Xs)) -> U2_g(N, select_in_aga(X1, Xs, Xs1)) select_in_aga(X, .(Y, Xs), .(Y, Ys)) -> U6_aga(X, Y, Xs, Ys, select_in_aga(X, Xs, Ys)) select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) U6_aga(X, Y, Xs, Ys, select_out_aga(X, Xs, Ys)) -> select_out_aga(X, .(Y, Xs), .(Y, Ys)) U2_g(N, select_out_aga(X1, Xs, Xs1)) -> U3_g(N, ll_in_ag(M, Xs1)) ll_in_ag(s(N), .(X, Xs)) -> U5_ag(N, X, Xs, ll_in_ag(N, Xs)) ll_in_ag(0, []) -> ll_out_ag(0, []) U5_ag(N, X, Xs, ll_out_ag(N, Xs)) -> ll_out_ag(s(N), .(X, Xs)) U3_g(N, ll_out_ag(M, Xs1)) -> U4_g(N, t_in_g(M)) t_in_g(0) -> t_out_g(0) U4_g(N, t_out_g(M)) -> t_out_g(N) The argument filtering Pi contains the following mapping: t_in_g(x1) = t_in_g(x1) U1_g(x1, x2) = U1_g(x2) ll_in_ga(x1, x2) = ll_in_ga(x1) s(x1) = s(x1) U5_ga(x1, x2, x3, x4) = U5_ga(x4) 0 = 0 ll_out_ga(x1, x2) = ll_out_ga(x2) .(x1, x2) = .(x2) U2_g(x1, x2) = U2_g(x2) select_in_aga(x1, x2, x3) = select_in_aga(x2) U6_aga(x1, x2, x3, x4, x5) = U6_aga(x5) select_out_aga(x1, x2, x3) = select_out_aga(x3) U3_g(x1, x2) = U3_g(x2) ll_in_ag(x1, x2) = ll_in_ag(x2) U5_ag(x1, x2, x3, x4) = U5_ag(x4) [] = [] ll_out_ag(x1, x2) = ll_out_ag(x1) U4_g(x1, x2) = U4_g(x2) t_out_g(x1) = t_out_g T_IN_G(x1) = T_IN_G(x1) U1_G(x1, x2) = U1_G(x2) LL_IN_GA(x1, x2) = LL_IN_GA(x1) U5_GA(x1, x2, x3, x4) = U5_GA(x4) U2_G(x1, x2) = U2_G(x2) SELECT_IN_AGA(x1, x2, x3) = SELECT_IN_AGA(x2) U6_AGA(x1, x2, x3, x4, x5) = U6_AGA(x5) U3_G(x1, x2) = U3_G(x2) LL_IN_AG(x1, x2) = LL_IN_AG(x2) U5_AG(x1, x2, x3, x4) = U5_AG(x4) U4_G(x1, x2) = U4_G(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 7 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: LL_IN_AG(s(N), .(X, Xs)) -> LL_IN_AG(N, Xs) The TRS R consists of the following rules: t_in_g(N) -> U1_g(N, ll_in_ga(N, Xs)) ll_in_ga(s(N), .(X, Xs)) -> U5_ga(N, X, Xs, ll_in_ga(N, Xs)) ll_in_ga(0, []) -> ll_out_ga(0, []) U5_ga(N, X, Xs, ll_out_ga(N, Xs)) -> ll_out_ga(s(N), .(X, Xs)) U1_g(N, ll_out_ga(N, Xs)) -> U2_g(N, select_in_aga(X1, Xs, Xs1)) select_in_aga(X, .(Y, Xs), .(Y, Ys)) -> U6_aga(X, Y, Xs, Ys, select_in_aga(X, Xs, Ys)) select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) U6_aga(X, Y, Xs, Ys, select_out_aga(X, Xs, Ys)) -> select_out_aga(X, .(Y, Xs), .(Y, Ys)) U2_g(N, select_out_aga(X1, Xs, Xs1)) -> U3_g(N, ll_in_ag(M, Xs1)) ll_in_ag(s(N), .(X, Xs)) -> U5_ag(N, X, Xs, ll_in_ag(N, Xs)) ll_in_ag(0, []) -> ll_out_ag(0, []) U5_ag(N, X, Xs, ll_out_ag(N, Xs)) -> ll_out_ag(s(N), .(X, Xs)) U3_g(N, ll_out_ag(M, Xs1)) -> U4_g(N, t_in_g(M)) t_in_g(0) -> t_out_g(0) U4_g(N, t_out_g(M)) -> t_out_g(N) The argument filtering Pi contains the following mapping: t_in_g(x1) = t_in_g(x1) U1_g(x1, x2) = U1_g(x2) ll_in_ga(x1, x2) = ll_in_ga(x1) s(x1) = s(x1) U5_ga(x1, x2, x3, x4) = U5_ga(x4) 0 = 0 ll_out_ga(x1, x2) = ll_out_ga(x2) .(x1, x2) = .(x2) U2_g(x1, x2) = U2_g(x2) select_in_aga(x1, x2, x3) = select_in_aga(x2) U6_aga(x1, x2, x3, x4, x5) = U6_aga(x5) select_out_aga(x1, x2, x3) = select_out_aga(x3) U3_g(x1, x2) = U3_g(x2) ll_in_ag(x1, x2) = ll_in_ag(x2) U5_ag(x1, x2, x3, x4) = U5_ag(x4) [] = [] ll_out_ag(x1, x2) = ll_out_ag(x1) U4_g(x1, x2) = U4_g(x2) t_out_g(x1) = t_out_g LL_IN_AG(x1, x2) = LL_IN_AG(x2) 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: LL_IN_AG(s(N), .(X, Xs)) -> LL_IN_AG(N, Xs) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) .(x1, x2) = .(x2) LL_IN_AG(x1, x2) = LL_IN_AG(x2) 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: LL_IN_AG(.(Xs)) -> LL_IN_AG(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: *LL_IN_AG(.(Xs)) -> LL_IN_AG(Xs) The graph contains the following edges 1 > 1 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: SELECT_IN_AGA(X, .(Y, Xs), .(Y, Ys)) -> SELECT_IN_AGA(X, Xs, Ys) The TRS R consists of the following rules: t_in_g(N) -> U1_g(N, ll_in_ga(N, Xs)) ll_in_ga(s(N), .(X, Xs)) -> U5_ga(N, X, Xs, ll_in_ga(N, Xs)) ll_in_ga(0, []) -> ll_out_ga(0, []) U5_ga(N, X, Xs, ll_out_ga(N, Xs)) -> ll_out_ga(s(N), .(X, Xs)) U1_g(N, ll_out_ga(N, Xs)) -> U2_g(N, select_in_aga(X1, Xs, Xs1)) select_in_aga(X, .(Y, Xs), .(Y, Ys)) -> U6_aga(X, Y, Xs, Ys, select_in_aga(X, Xs, Ys)) select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) U6_aga(X, Y, Xs, Ys, select_out_aga(X, Xs, Ys)) -> select_out_aga(X, .(Y, Xs), .(Y, Ys)) U2_g(N, select_out_aga(X1, Xs, Xs1)) -> U3_g(N, ll_in_ag(M, Xs1)) ll_in_ag(s(N), .(X, Xs)) -> U5_ag(N, X, Xs, ll_in_ag(N, Xs)) ll_in_ag(0, []) -> ll_out_ag(0, []) U5_ag(N, X, Xs, ll_out_ag(N, Xs)) -> ll_out_ag(s(N), .(X, Xs)) U3_g(N, ll_out_ag(M, Xs1)) -> U4_g(N, t_in_g(M)) t_in_g(0) -> t_out_g(0) U4_g(N, t_out_g(M)) -> t_out_g(N) The argument filtering Pi contains the following mapping: t_in_g(x1) = t_in_g(x1) U1_g(x1, x2) = U1_g(x2) ll_in_ga(x1, x2) = ll_in_ga(x1) s(x1) = s(x1) U5_ga(x1, x2, x3, x4) = U5_ga(x4) 0 = 0 ll_out_ga(x1, x2) = ll_out_ga(x2) .(x1, x2) = .(x2) U2_g(x1, x2) = U2_g(x2) select_in_aga(x1, x2, x3) = select_in_aga(x2) U6_aga(x1, x2, x3, x4, x5) = U6_aga(x5) select_out_aga(x1, x2, x3) = select_out_aga(x3) U3_g(x1, x2) = U3_g(x2) ll_in_ag(x1, x2) = ll_in_ag(x2) U5_ag(x1, x2, x3, x4) = U5_ag(x4) [] = [] ll_out_ag(x1, x2) = ll_out_ag(x1) U4_g(x1, x2) = U4_g(x2) t_out_g(x1) = t_out_g SELECT_IN_AGA(x1, x2, x3) = SELECT_IN_AGA(x2) 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: SELECT_IN_AGA(X, .(Y, Xs), .(Y, Ys)) -> SELECT_IN_AGA(X, Xs, Ys) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) SELECT_IN_AGA(x1, x2, x3) = SELECT_IN_AGA(x2) 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: SELECT_IN_AGA(.(Xs)) -> SELECT_IN_AGA(Xs) 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: *SELECT_IN_AGA(.(Xs)) -> SELECT_IN_AGA(Xs) The graph contains the following edges 1 > 1 ---------------------------------------- (20) YES ---------------------------------------- (21) Obligation: Pi DP problem: The TRS P consists of the following rules: LL_IN_GA(s(N), .(X, Xs)) -> LL_IN_GA(N, Xs) The TRS R consists of the following rules: t_in_g(N) -> U1_g(N, ll_in_ga(N, Xs)) ll_in_ga(s(N), .(X, Xs)) -> U5_ga(N, X, Xs, ll_in_ga(N, Xs)) ll_in_ga(0, []) -> ll_out_ga(0, []) U5_ga(N, X, Xs, ll_out_ga(N, Xs)) -> ll_out_ga(s(N), .(X, Xs)) U1_g(N, ll_out_ga(N, Xs)) -> U2_g(N, select_in_aga(X1, Xs, Xs1)) select_in_aga(X, .(Y, Xs), .(Y, Ys)) -> U6_aga(X, Y, Xs, Ys, select_in_aga(X, Xs, Ys)) select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) U6_aga(X, Y, Xs, Ys, select_out_aga(X, Xs, Ys)) -> select_out_aga(X, .(Y, Xs), .(Y, Ys)) U2_g(N, select_out_aga(X1, Xs, Xs1)) -> U3_g(N, ll_in_ag(M, Xs1)) ll_in_ag(s(N), .(X, Xs)) -> U5_ag(N, X, Xs, ll_in_ag(N, Xs)) ll_in_ag(0, []) -> ll_out_ag(0, []) U5_ag(N, X, Xs, ll_out_ag(N, Xs)) -> ll_out_ag(s(N), .(X, Xs)) U3_g(N, ll_out_ag(M, Xs1)) -> U4_g(N, t_in_g(M)) t_in_g(0) -> t_out_g(0) U4_g(N, t_out_g(M)) -> t_out_g(N) The argument filtering Pi contains the following mapping: t_in_g(x1) = t_in_g(x1) U1_g(x1, x2) = U1_g(x2) ll_in_ga(x1, x2) = ll_in_ga(x1) s(x1) = s(x1) U5_ga(x1, x2, x3, x4) = U5_ga(x4) 0 = 0 ll_out_ga(x1, x2) = ll_out_ga(x2) .(x1, x2) = .(x2) U2_g(x1, x2) = U2_g(x2) select_in_aga(x1, x2, x3) = select_in_aga(x2) U6_aga(x1, x2, x3, x4, x5) = U6_aga(x5) select_out_aga(x1, x2, x3) = select_out_aga(x3) U3_g(x1, x2) = U3_g(x2) ll_in_ag(x1, x2) = ll_in_ag(x2) U5_ag(x1, x2, x3, x4) = U5_ag(x4) [] = [] ll_out_ag(x1, x2) = ll_out_ag(x1) U4_g(x1, x2) = U4_g(x2) t_out_g(x1) = t_out_g LL_IN_GA(x1, x2) = LL_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (22) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (23) Obligation: Pi DP problem: The TRS P consists of the following rules: LL_IN_GA(s(N), .(X, Xs)) -> LL_IN_GA(N, Xs) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) .(x1, x2) = .(x2) LL_IN_GA(x1, x2) = LL_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (24) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (25) Obligation: Q DP problem: The TRS P consists of the following rules: LL_IN_GA(s(N)) -> LL_IN_GA(N) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (26) 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: *LL_IN_GA(s(N)) -> LL_IN_GA(N) The graph contains the following edges 1 > 1 ---------------------------------------- (27) YES ---------------------------------------- (28) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_G(N, ll_out_ga(N, Xs)) -> U2_G(N, select_in_aga(X1, Xs, Xs1)) U2_G(N, select_out_aga(X1, Xs, Xs1)) -> U3_G(N, ll_in_ag(M, Xs1)) U3_G(N, ll_out_ag(M, Xs1)) -> T_IN_G(M) T_IN_G(N) -> U1_G(N, ll_in_ga(N, Xs)) The TRS R consists of the following rules: t_in_g(N) -> U1_g(N, ll_in_ga(N, Xs)) ll_in_ga(s(N), .(X, Xs)) -> U5_ga(N, X, Xs, ll_in_ga(N, Xs)) ll_in_ga(0, []) -> ll_out_ga(0, []) U5_ga(N, X, Xs, ll_out_ga(N, Xs)) -> ll_out_ga(s(N), .(X, Xs)) U1_g(N, ll_out_ga(N, Xs)) -> U2_g(N, select_in_aga(X1, Xs, Xs1)) select_in_aga(X, .(Y, Xs), .(Y, Ys)) -> U6_aga(X, Y, Xs, Ys, select_in_aga(X, Xs, Ys)) select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) U6_aga(X, Y, Xs, Ys, select_out_aga(X, Xs, Ys)) -> select_out_aga(X, .(Y, Xs), .(Y, Ys)) U2_g(N, select_out_aga(X1, Xs, Xs1)) -> U3_g(N, ll_in_ag(M, Xs1)) ll_in_ag(s(N), .(X, Xs)) -> U5_ag(N, X, Xs, ll_in_ag(N, Xs)) ll_in_ag(0, []) -> ll_out_ag(0, []) U5_ag(N, X, Xs, ll_out_ag(N, Xs)) -> ll_out_ag(s(N), .(X, Xs)) U3_g(N, ll_out_ag(M, Xs1)) -> U4_g(N, t_in_g(M)) t_in_g(0) -> t_out_g(0) U4_g(N, t_out_g(M)) -> t_out_g(N) The argument filtering Pi contains the following mapping: t_in_g(x1) = t_in_g(x1) U1_g(x1, x2) = U1_g(x2) ll_in_ga(x1, x2) = ll_in_ga(x1) s(x1) = s(x1) U5_ga(x1, x2, x3, x4) = U5_ga(x4) 0 = 0 ll_out_ga(x1, x2) = ll_out_ga(x2) .(x1, x2) = .(x2) U2_g(x1, x2) = U2_g(x2) select_in_aga(x1, x2, x3) = select_in_aga(x2) U6_aga(x1, x2, x3, x4, x5) = U6_aga(x5) select_out_aga(x1, x2, x3) = select_out_aga(x3) U3_g(x1, x2) = U3_g(x2) ll_in_ag(x1, x2) = ll_in_ag(x2) U5_ag(x1, x2, x3, x4) = U5_ag(x4) [] = [] ll_out_ag(x1, x2) = ll_out_ag(x1) U4_g(x1, x2) = U4_g(x2) t_out_g(x1) = t_out_g T_IN_G(x1) = T_IN_G(x1) U1_G(x1, x2) = U1_G(x2) U2_G(x1, x2) = U2_G(x2) U3_G(x1, x2) = U3_G(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (29) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (30) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_G(N, ll_out_ga(N, Xs)) -> U2_G(N, select_in_aga(X1, Xs, Xs1)) U2_G(N, select_out_aga(X1, Xs, Xs1)) -> U3_G(N, ll_in_ag(M, Xs1)) U3_G(N, ll_out_ag(M, Xs1)) -> T_IN_G(M) T_IN_G(N) -> U1_G(N, ll_in_ga(N, Xs)) The TRS R consists of the following rules: select_in_aga(X, .(Y, Xs), .(Y, Ys)) -> U6_aga(X, Y, Xs, Ys, select_in_aga(X, Xs, Ys)) select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) ll_in_ag(s(N), .(X, Xs)) -> U5_ag(N, X, Xs, ll_in_ag(N, Xs)) ll_in_ag(0, []) -> ll_out_ag(0, []) ll_in_ga(s(N), .(X, Xs)) -> U5_ga(N, X, Xs, ll_in_ga(N, Xs)) ll_in_ga(0, []) -> ll_out_ga(0, []) U6_aga(X, Y, Xs, Ys, select_out_aga(X, Xs, Ys)) -> select_out_aga(X, .(Y, Xs), .(Y, Ys)) U5_ag(N, X, Xs, ll_out_ag(N, Xs)) -> ll_out_ag(s(N), .(X, Xs)) U5_ga(N, X, Xs, ll_out_ga(N, Xs)) -> ll_out_ga(s(N), .(X, Xs)) The argument filtering Pi contains the following mapping: ll_in_ga(x1, x2) = ll_in_ga(x1) s(x1) = s(x1) U5_ga(x1, x2, x3, x4) = U5_ga(x4) 0 = 0 ll_out_ga(x1, x2) = ll_out_ga(x2) .(x1, x2) = .(x2) select_in_aga(x1, x2, x3) = select_in_aga(x2) U6_aga(x1, x2, x3, x4, x5) = U6_aga(x5) select_out_aga(x1, x2, x3) = select_out_aga(x3) ll_in_ag(x1, x2) = ll_in_ag(x2) U5_ag(x1, x2, x3, x4) = U5_ag(x4) [] = [] ll_out_ag(x1, x2) = ll_out_ag(x1) T_IN_G(x1) = T_IN_G(x1) U1_G(x1, x2) = U1_G(x2) U2_G(x1, x2) = U2_G(x2) U3_G(x1, x2) = U3_G(x2) 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: U1_G(ll_out_ga(Xs)) -> U2_G(select_in_aga(Xs)) U2_G(select_out_aga(Xs1)) -> U3_G(ll_in_ag(Xs1)) U3_G(ll_out_ag(M)) -> T_IN_G(M) T_IN_G(N) -> U1_G(ll_in_ga(N)) The TRS R consists of the following rules: select_in_aga(.(Xs)) -> U6_aga(select_in_aga(Xs)) select_in_aga(.(Xs)) -> select_out_aga(Xs) ll_in_ag(.(Xs)) -> U5_ag(ll_in_ag(Xs)) ll_in_ag([]) -> ll_out_ag(0) ll_in_ga(s(N)) -> U5_ga(ll_in_ga(N)) ll_in_ga(0) -> ll_out_ga([]) U6_aga(select_out_aga(Ys)) -> select_out_aga(.(Ys)) U5_ag(ll_out_ag(N)) -> ll_out_ag(s(N)) U5_ga(ll_out_ga(Xs)) -> ll_out_ga(.(Xs)) The set Q consists of the following terms: select_in_aga(x0) ll_in_ag(x0) ll_in_ga(x0) U6_aga(x0) U5_ag(x0) U5_ga(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (33) 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_G(ll_out_ga(Xs)) -> U2_G(select_in_aga(Xs)) U2_G(select_out_aga(Xs1)) -> U3_G(ll_in_ag(Xs1)) U3_G(ll_out_ag(M)) -> T_IN_G(M) T_IN_G(N) -> U1_G(ll_in_ga(N)) Strictly oriented rules of the TRS R: select_in_aga(.(Xs)) -> U6_aga(select_in_aga(Xs)) select_in_aga(.(Xs)) -> select_out_aga(Xs) ll_in_ag(.(Xs)) -> U5_ag(ll_in_ag(Xs)) ll_in_ag([]) -> ll_out_ag(0) ll_in_ga(s(N)) -> U5_ga(ll_in_ga(N)) ll_in_ga(0) -> ll_out_ga([]) U6_aga(select_out_aga(Ys)) -> select_out_aga(.(Ys)) U5_ag(ll_out_ag(N)) -> ll_out_ag(s(N)) U5_ga(ll_out_ga(Xs)) -> ll_out_ga(.(Xs)) Used ordering: Knuth-Bendix order [KBO] with precedence:s_1 > select_in_aga_1 > ll_in_ag_1 > 0 > U6_aga_1 > U5_ag_1 > select_out_aga_1 > ll_in_ga_1 > U5_ga_1 > ll_out_ga_1 > ll_out_ag_1 > U1_G_1 > [] > U2_G_1 > U3_G_1 > T_IN_G_1 > ._1 and weight map: []=2 0=1 select_in_aga_1=2 ._1=8 U6_aga_1=8 select_out_aga_1=10 ll_in_ag_1=11 U5_ag_1=8 ll_out_ag_1=7 ll_in_ga_1=3 s_1=8 U5_ga_1=8 ll_out_ga_1=1 U1_G_1=4 U2_G_1=2 U3_G_1=1 T_IN_G_1=8 The variable weight is 1 ---------------------------------------- (34) Obligation: Q DP problem: P is empty. R is empty. The set Q consists of the following terms: select_in_aga(x0) ll_in_ag(x0) ll_in_ga(x0) U6_aga(x0) U5_ag(x0) U5_ga(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (35) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (36) YES