/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern split(g,g,a,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) UndefinedPredicateHandlerProof [SOUND, 0 ms] (2) Prolog (3) PrologToPiTRSProof [SOUND, 19 ms] (4) PiTRS (5) DependencyPairsProof [EQUIVALENT, 0 ms] (6) PiDP (7) DependencyGraphProof [EQUIVALENT, 0 ms] (8) PiDP (9) UsableRulesProof [EQUIVALENT, 0 ms] (10) PiDP (11) PiDPToQDPProof [SOUND, 0 ms] (12) QDP (13) QDPSizeChangeProof [EQUIVALENT, 0 ms] (14) YES ---------------------------------------- (0) Obligation: Clauses: split(L, 0, [], L). split(.(X, Xs), N, .(X, Ys), Zs) :- ','(>(N, 0), ','(is(N1, -(N, 1)), split(Xs, N1, Ys, Zs))). Query: split(g,g,a,a) ---------------------------------------- (1) UndefinedPredicateHandlerProof (SOUND) Added facts for all undefined predicates [PROLOG]. ---------------------------------------- (2) Obligation: Clauses: split(L, 0, [], L). split(.(X, Xs), N, .(X, Ys), Zs) :- ','(>(N, 0), ','(is(N1, -(N, 1)), split(Xs, N1, Ys, Zs))). >(X0, X1). is(X0, X1). Query: split(g,g,a,a) ---------------------------------------- (3) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: split_in_4: (b,b,f,f) (b,f,f,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: split_in_ggaa(L, 0, [], L) -> split_out_ggaa(L, 0, [], L) split_in_ggaa(.(X, Xs), N, .(X, Ys), Zs) -> U1_ggaa(X, Xs, N, Ys, Zs, >_in_gg(N, 0)) >_in_gg(X0, X1) -> >_out_gg(X0, X1) U1_ggaa(X, Xs, N, Ys, Zs, >_out_gg(N, 0)) -> U2_ggaa(X, Xs, N, Ys, Zs, is_in_ag(N1, -(N, 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U2_ggaa(X, Xs, N, Ys, Zs, is_out_ag(N1, -(N, 1))) -> U3_ggaa(X, Xs, N, Ys, Zs, N1, split_in_gaaa(Xs, N1, Ys, Zs)) split_in_gaaa(L, 0, [], L) -> split_out_gaaa(L, 0, [], L) split_in_gaaa(.(X, Xs), N, .(X, Ys), Zs) -> U1_gaaa(X, Xs, N, Ys, Zs, >_in_ag(N, 0)) >_in_ag(X0, X1) -> >_out_ag(X0, X1) U1_gaaa(X, Xs, N, Ys, Zs, >_out_ag(N, 0)) -> U2_gaaa(X, Xs, N, Ys, Zs, is_in_ag(N1, -(N, 1))) U2_gaaa(X, Xs, N, Ys, Zs, is_out_ag(N1, -(N, 1))) -> U3_gaaa(X, Xs, N, Ys, Zs, N1, split_in_gaaa(Xs, N1, Ys, Zs)) U3_gaaa(X, Xs, N, Ys, Zs, N1, split_out_gaaa(Xs, N1, Ys, Zs)) -> split_out_gaaa(.(X, Xs), N, .(X, Ys), Zs) U3_ggaa(X, Xs, N, Ys, Zs, N1, split_out_gaaa(Xs, N1, Ys, Zs)) -> split_out_ggaa(.(X, Xs), N, .(X, Ys), Zs) The argument filtering Pi contains the following mapping: split_in_ggaa(x1, x2, x3, x4) = split_in_ggaa(x1, x2) 0 = 0 split_out_ggaa(x1, x2, x3, x4) = split_out_ggaa(x1, x2, x3, x4) .(x1, x2) = .(x1, x2) U1_ggaa(x1, x2, x3, x4, x5, x6) = U1_ggaa(x1, x2, x3, x6) >_in_gg(x1, x2) = >_in_gg(x1, x2) >_out_gg(x1, x2) = >_out_gg(x1, x2) U2_ggaa(x1, x2, x3, x4, x5, x6) = U2_ggaa(x1, x2, x3, x6) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag(x2) -(x1, x2) = -(x2) 1 = 1 U3_ggaa(x1, x2, x3, x4, x5, x6, x7) = U3_ggaa(x1, x2, x3, x7) split_in_gaaa(x1, x2, x3, x4) = split_in_gaaa(x1) split_out_gaaa(x1, x2, x3, x4) = split_out_gaaa(x1, x3, x4) U1_gaaa(x1, x2, x3, x4, x5, x6) = U1_gaaa(x1, x2, x6) >_in_ag(x1, x2) = >_in_ag(x2) >_out_ag(x1, x2) = >_out_ag(x2) U2_gaaa(x1, x2, x3, x4, x5, x6) = U2_gaaa(x1, x2, x6) U3_gaaa(x1, x2, x3, x4, x5, x6, x7) = U3_gaaa(x1, x2, x7) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (4) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: split_in_ggaa(L, 0, [], L) -> split_out_ggaa(L, 0, [], L) split_in_ggaa(.(X, Xs), N, .(X, Ys), Zs) -> U1_ggaa(X, Xs, N, Ys, Zs, >_in_gg(N, 0)) >_in_gg(X0, X1) -> >_out_gg(X0, X1) U1_ggaa(X, Xs, N, Ys, Zs, >_out_gg(N, 0)) -> U2_ggaa(X, Xs, N, Ys, Zs, is_in_ag(N1, -(N, 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U2_ggaa(X, Xs, N, Ys, Zs, is_out_ag(N1, -(N, 1))) -> U3_ggaa(X, Xs, N, Ys, Zs, N1, split_in_gaaa(Xs, N1, Ys, Zs)) split_in_gaaa(L, 0, [], L) -> split_out_gaaa(L, 0, [], L) split_in_gaaa(.(X, Xs), N, .(X, Ys), Zs) -> U1_gaaa(X, Xs, N, Ys, Zs, >_in_ag(N, 0)) >_in_ag(X0, X1) -> >_out_ag(X0, X1) U1_gaaa(X, Xs, N, Ys, Zs, >_out_ag(N, 0)) -> U2_gaaa(X, Xs, N, Ys, Zs, is_in_ag(N1, -(N, 1))) U2_gaaa(X, Xs, N, Ys, Zs, is_out_ag(N1, -(N, 1))) -> U3_gaaa(X, Xs, N, Ys, Zs, N1, split_in_gaaa(Xs, N1, Ys, Zs)) U3_gaaa(X, Xs, N, Ys, Zs, N1, split_out_gaaa(Xs, N1, Ys, Zs)) -> split_out_gaaa(.(X, Xs), N, .(X, Ys), Zs) U3_ggaa(X, Xs, N, Ys, Zs, N1, split_out_gaaa(Xs, N1, Ys, Zs)) -> split_out_ggaa(.(X, Xs), N, .(X, Ys), Zs) The argument filtering Pi contains the following mapping: split_in_ggaa(x1, x2, x3, x4) = split_in_ggaa(x1, x2) 0 = 0 split_out_ggaa(x1, x2, x3, x4) = split_out_ggaa(x1, x2, x3, x4) .(x1, x2) = .(x1, x2) U1_ggaa(x1, x2, x3, x4, x5, x6) = U1_ggaa(x1, x2, x3, x6) >_in_gg(x1, x2) = >_in_gg(x1, x2) >_out_gg(x1, x2) = >_out_gg(x1, x2) U2_ggaa(x1, x2, x3, x4, x5, x6) = U2_ggaa(x1, x2, x3, x6) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag(x2) -(x1, x2) = -(x2) 1 = 1 U3_ggaa(x1, x2, x3, x4, x5, x6, x7) = U3_ggaa(x1, x2, x3, x7) split_in_gaaa(x1, x2, x3, x4) = split_in_gaaa(x1) split_out_gaaa(x1, x2, x3, x4) = split_out_gaaa(x1, x3, x4) U1_gaaa(x1, x2, x3, x4, x5, x6) = U1_gaaa(x1, x2, x6) >_in_ag(x1, x2) = >_in_ag(x2) >_out_ag(x1, x2) = >_out_ag(x2) U2_gaaa(x1, x2, x3, x4, x5, x6) = U2_gaaa(x1, x2, x6) U3_gaaa(x1, x2, x3, x4, x5, x6, x7) = U3_gaaa(x1, x2, x7) ---------------------------------------- (5) 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: SPLIT_IN_GGAA(.(X, Xs), N, .(X, Ys), Zs) -> U1_GGAA(X, Xs, N, Ys, Zs, >_in_gg(N, 0)) SPLIT_IN_GGAA(.(X, Xs), N, .(X, Ys), Zs) -> >_IN_GG(N, 0) U1_GGAA(X, Xs, N, Ys, Zs, >_out_gg(N, 0)) -> U2_GGAA(X, Xs, N, Ys, Zs, is_in_ag(N1, -(N, 1))) U1_GGAA(X, Xs, N, Ys, Zs, >_out_gg(N, 0)) -> IS_IN_AG(N1, -(N, 1)) U2_GGAA(X, Xs, N, Ys, Zs, is_out_ag(N1, -(N, 1))) -> U3_GGAA(X, Xs, N, Ys, Zs, N1, split_in_gaaa(Xs, N1, Ys, Zs)) U2_GGAA(X, Xs, N, Ys, Zs, is_out_ag(N1, -(N, 1))) -> SPLIT_IN_GAAA(Xs, N1, Ys, Zs) SPLIT_IN_GAAA(.(X, Xs), N, .(X, Ys), Zs) -> U1_GAAA(X, Xs, N, Ys, Zs, >_in_ag(N, 0)) SPLIT_IN_GAAA(.(X, Xs), N, .(X, Ys), Zs) -> >_IN_AG(N, 0) U1_GAAA(X, Xs, N, Ys, Zs, >_out_ag(N, 0)) -> U2_GAAA(X, Xs, N, Ys, Zs, is_in_ag(N1, -(N, 1))) U1_GAAA(X, Xs, N, Ys, Zs, >_out_ag(N, 0)) -> IS_IN_AG(N1, -(N, 1)) U2_GAAA(X, Xs, N, Ys, Zs, is_out_ag(N1, -(N, 1))) -> U3_GAAA(X, Xs, N, Ys, Zs, N1, split_in_gaaa(Xs, N1, Ys, Zs)) U2_GAAA(X, Xs, N, Ys, Zs, is_out_ag(N1, -(N, 1))) -> SPLIT_IN_GAAA(Xs, N1, Ys, Zs) The TRS R consists of the following rules: split_in_ggaa(L, 0, [], L) -> split_out_ggaa(L, 0, [], L) split_in_ggaa(.(X, Xs), N, .(X, Ys), Zs) -> U1_ggaa(X, Xs, N, Ys, Zs, >_in_gg(N, 0)) >_in_gg(X0, X1) -> >_out_gg(X0, X1) U1_ggaa(X, Xs, N, Ys, Zs, >_out_gg(N, 0)) -> U2_ggaa(X, Xs, N, Ys, Zs, is_in_ag(N1, -(N, 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U2_ggaa(X, Xs, N, Ys, Zs, is_out_ag(N1, -(N, 1))) -> U3_ggaa(X, Xs, N, Ys, Zs, N1, split_in_gaaa(Xs, N1, Ys, Zs)) split_in_gaaa(L, 0, [], L) -> split_out_gaaa(L, 0, [], L) split_in_gaaa(.(X, Xs), N, .(X, Ys), Zs) -> U1_gaaa(X, Xs, N, Ys, Zs, >_in_ag(N, 0)) >_in_ag(X0, X1) -> >_out_ag(X0, X1) U1_gaaa(X, Xs, N, Ys, Zs, >_out_ag(N, 0)) -> U2_gaaa(X, Xs, N, Ys, Zs, is_in_ag(N1, -(N, 1))) U2_gaaa(X, Xs, N, Ys, Zs, is_out_ag(N1, -(N, 1))) -> U3_gaaa(X, Xs, N, Ys, Zs, N1, split_in_gaaa(Xs, N1, Ys, Zs)) U3_gaaa(X, Xs, N, Ys, Zs, N1, split_out_gaaa(Xs, N1, Ys, Zs)) -> split_out_gaaa(.(X, Xs), N, .(X, Ys), Zs) U3_ggaa(X, Xs, N, Ys, Zs, N1, split_out_gaaa(Xs, N1, Ys, Zs)) -> split_out_ggaa(.(X, Xs), N, .(X, Ys), Zs) The argument filtering Pi contains the following mapping: split_in_ggaa(x1, x2, x3, x4) = split_in_ggaa(x1, x2) 0 = 0 split_out_ggaa(x1, x2, x3, x4) = split_out_ggaa(x1, x2, x3, x4) .(x1, x2) = .(x1, x2) U1_ggaa(x1, x2, x3, x4, x5, x6) = U1_ggaa(x1, x2, x3, x6) >_in_gg(x1, x2) = >_in_gg(x1, x2) >_out_gg(x1, x2) = >_out_gg(x1, x2) U2_ggaa(x1, x2, x3, x4, x5, x6) = U2_ggaa(x1, x2, x3, x6) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag(x2) -(x1, x2) = -(x2) 1 = 1 U3_ggaa(x1, x2, x3, x4, x5, x6, x7) = U3_ggaa(x1, x2, x3, x7) split_in_gaaa(x1, x2, x3, x4) = split_in_gaaa(x1) split_out_gaaa(x1, x2, x3, x4) = split_out_gaaa(x1, x3, x4) U1_gaaa(x1, x2, x3, x4, x5, x6) = U1_gaaa(x1, x2, x6) >_in_ag(x1, x2) = >_in_ag(x2) >_out_ag(x1, x2) = >_out_ag(x2) U2_gaaa(x1, x2, x3, x4, x5, x6) = U2_gaaa(x1, x2, x6) U3_gaaa(x1, x2, x3, x4, x5, x6, x7) = U3_gaaa(x1, x2, x7) SPLIT_IN_GGAA(x1, x2, x3, x4) = SPLIT_IN_GGAA(x1, x2) U1_GGAA(x1, x2, x3, x4, x5, x6) = U1_GGAA(x1, x2, x3, x6) >_IN_GG(x1, x2) = >_IN_GG(x1, x2) U2_GGAA(x1, x2, x3, x4, x5, x6) = U2_GGAA(x1, x2, x3, x6) IS_IN_AG(x1, x2) = IS_IN_AG(x2) U3_GGAA(x1, x2, x3, x4, x5, x6, x7) = U3_GGAA(x1, x2, x3, x7) SPLIT_IN_GAAA(x1, x2, x3, x4) = SPLIT_IN_GAAA(x1) U1_GAAA(x1, x2, x3, x4, x5, x6) = U1_GAAA(x1, x2, x6) >_IN_AG(x1, x2) = >_IN_AG(x2) U2_GAAA(x1, x2, x3, x4, x5, x6) = U2_GAAA(x1, x2, x6) U3_GAAA(x1, x2, x3, x4, x5, x6, x7) = U3_GAAA(x1, x2, x7) We have to consider all (P,R,Pi)-chains ---------------------------------------- (6) Obligation: Pi DP problem: The TRS P consists of the following rules: SPLIT_IN_GGAA(.(X, Xs), N, .(X, Ys), Zs) -> U1_GGAA(X, Xs, N, Ys, Zs, >_in_gg(N, 0)) SPLIT_IN_GGAA(.(X, Xs), N, .(X, Ys), Zs) -> >_IN_GG(N, 0) U1_GGAA(X, Xs, N, Ys, Zs, >_out_gg(N, 0)) -> U2_GGAA(X, Xs, N, Ys, Zs, is_in_ag(N1, -(N, 1))) U1_GGAA(X, Xs, N, Ys, Zs, >_out_gg(N, 0)) -> IS_IN_AG(N1, -(N, 1)) U2_GGAA(X, Xs, N, Ys, Zs, is_out_ag(N1, -(N, 1))) -> U3_GGAA(X, Xs, N, Ys, Zs, N1, split_in_gaaa(Xs, N1, Ys, Zs)) U2_GGAA(X, Xs, N, Ys, Zs, is_out_ag(N1, -(N, 1))) -> SPLIT_IN_GAAA(Xs, N1, Ys, Zs) SPLIT_IN_GAAA(.(X, Xs), N, .(X, Ys), Zs) -> U1_GAAA(X, Xs, N, Ys, Zs, >_in_ag(N, 0)) SPLIT_IN_GAAA(.(X, Xs), N, .(X, Ys), Zs) -> >_IN_AG(N, 0) U1_GAAA(X, Xs, N, Ys, Zs, >_out_ag(N, 0)) -> U2_GAAA(X, Xs, N, Ys, Zs, is_in_ag(N1, -(N, 1))) U1_GAAA(X, Xs, N, Ys, Zs, >_out_ag(N, 0)) -> IS_IN_AG(N1, -(N, 1)) U2_GAAA(X, Xs, N, Ys, Zs, is_out_ag(N1, -(N, 1))) -> U3_GAAA(X, Xs, N, Ys, Zs, N1, split_in_gaaa(Xs, N1, Ys, Zs)) U2_GAAA(X, Xs, N, Ys, Zs, is_out_ag(N1, -(N, 1))) -> SPLIT_IN_GAAA(Xs, N1, Ys, Zs) The TRS R consists of the following rules: split_in_ggaa(L, 0, [], L) -> split_out_ggaa(L, 0, [], L) split_in_ggaa(.(X, Xs), N, .(X, Ys), Zs) -> U1_ggaa(X, Xs, N, Ys, Zs, >_in_gg(N, 0)) >_in_gg(X0, X1) -> >_out_gg(X0, X1) U1_ggaa(X, Xs, N, Ys, Zs, >_out_gg(N, 0)) -> U2_ggaa(X, Xs, N, Ys, Zs, is_in_ag(N1, -(N, 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U2_ggaa(X, Xs, N, Ys, Zs, is_out_ag(N1, -(N, 1))) -> U3_ggaa(X, Xs, N, Ys, Zs, N1, split_in_gaaa(Xs, N1, Ys, Zs)) split_in_gaaa(L, 0, [], L) -> split_out_gaaa(L, 0, [], L) split_in_gaaa(.(X, Xs), N, .(X, Ys), Zs) -> U1_gaaa(X, Xs, N, Ys, Zs, >_in_ag(N, 0)) >_in_ag(X0, X1) -> >_out_ag(X0, X1) U1_gaaa(X, Xs, N, Ys, Zs, >_out_ag(N, 0)) -> U2_gaaa(X, Xs, N, Ys, Zs, is_in_ag(N1, -(N, 1))) U2_gaaa(X, Xs, N, Ys, Zs, is_out_ag(N1, -(N, 1))) -> U3_gaaa(X, Xs, N, Ys, Zs, N1, split_in_gaaa(Xs, N1, Ys, Zs)) U3_gaaa(X, Xs, N, Ys, Zs, N1, split_out_gaaa(Xs, N1, Ys, Zs)) -> split_out_gaaa(.(X, Xs), N, .(X, Ys), Zs) U3_ggaa(X, Xs, N, Ys, Zs, N1, split_out_gaaa(Xs, N1, Ys, Zs)) -> split_out_ggaa(.(X, Xs), N, .(X, Ys), Zs) The argument filtering Pi contains the following mapping: split_in_ggaa(x1, x2, x3, x4) = split_in_ggaa(x1, x2) 0 = 0 split_out_ggaa(x1, x2, x3, x4) = split_out_ggaa(x1, x2, x3, x4) .(x1, x2) = .(x1, x2) U1_ggaa(x1, x2, x3, x4, x5, x6) = U1_ggaa(x1, x2, x3, x6) >_in_gg(x1, x2) = >_in_gg(x1, x2) >_out_gg(x1, x2) = >_out_gg(x1, x2) U2_ggaa(x1, x2, x3, x4, x5, x6) = U2_ggaa(x1, x2, x3, x6) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag(x2) -(x1, x2) = -(x2) 1 = 1 U3_ggaa(x1, x2, x3, x4, x5, x6, x7) = U3_ggaa(x1, x2, x3, x7) split_in_gaaa(x1, x2, x3, x4) = split_in_gaaa(x1) split_out_gaaa(x1, x2, x3, x4) = split_out_gaaa(x1, x3, x4) U1_gaaa(x1, x2, x3, x4, x5, x6) = U1_gaaa(x1, x2, x6) >_in_ag(x1, x2) = >_in_ag(x2) >_out_ag(x1, x2) = >_out_ag(x2) U2_gaaa(x1, x2, x3, x4, x5, x6) = U2_gaaa(x1, x2, x6) U3_gaaa(x1, x2, x3, x4, x5, x6, x7) = U3_gaaa(x1, x2, x7) SPLIT_IN_GGAA(x1, x2, x3, x4) = SPLIT_IN_GGAA(x1, x2) U1_GGAA(x1, x2, x3, x4, x5, x6) = U1_GGAA(x1, x2, x3, x6) >_IN_GG(x1, x2) = >_IN_GG(x1, x2) U2_GGAA(x1, x2, x3, x4, x5, x6) = U2_GGAA(x1, x2, x3, x6) IS_IN_AG(x1, x2) = IS_IN_AG(x2) U3_GGAA(x1, x2, x3, x4, x5, x6, x7) = U3_GGAA(x1, x2, x3, x7) SPLIT_IN_GAAA(x1, x2, x3, x4) = SPLIT_IN_GAAA(x1) U1_GAAA(x1, x2, x3, x4, x5, x6) = U1_GAAA(x1, x2, x6) >_IN_AG(x1, x2) = >_IN_AG(x2) U2_GAAA(x1, x2, x3, x4, x5, x6) = U2_GAAA(x1, x2, x6) U3_GAAA(x1, x2, x3, x4, x5, x6, x7) = U3_GAAA(x1, x2, x7) We have to consider all (P,R,Pi)-chains ---------------------------------------- (7) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 9 less nodes. ---------------------------------------- (8) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_GAAA(X, Xs, N, Ys, Zs, >_out_ag(N, 0)) -> U2_GAAA(X, Xs, N, Ys, Zs, is_in_ag(N1, -(N, 1))) U2_GAAA(X, Xs, N, Ys, Zs, is_out_ag(N1, -(N, 1))) -> SPLIT_IN_GAAA(Xs, N1, Ys, Zs) SPLIT_IN_GAAA(.(X, Xs), N, .(X, Ys), Zs) -> U1_GAAA(X, Xs, N, Ys, Zs, >_in_ag(N, 0)) The TRS R consists of the following rules: split_in_ggaa(L, 0, [], L) -> split_out_ggaa(L, 0, [], L) split_in_ggaa(.(X, Xs), N, .(X, Ys), Zs) -> U1_ggaa(X, Xs, N, Ys, Zs, >_in_gg(N, 0)) >_in_gg(X0, X1) -> >_out_gg(X0, X1) U1_ggaa(X, Xs, N, Ys, Zs, >_out_gg(N, 0)) -> U2_ggaa(X, Xs, N, Ys, Zs, is_in_ag(N1, -(N, 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U2_ggaa(X, Xs, N, Ys, Zs, is_out_ag(N1, -(N, 1))) -> U3_ggaa(X, Xs, N, Ys, Zs, N1, split_in_gaaa(Xs, N1, Ys, Zs)) split_in_gaaa(L, 0, [], L) -> split_out_gaaa(L, 0, [], L) split_in_gaaa(.(X, Xs), N, .(X, Ys), Zs) -> U1_gaaa(X, Xs, N, Ys, Zs, >_in_ag(N, 0)) >_in_ag(X0, X1) -> >_out_ag(X0, X1) U1_gaaa(X, Xs, N, Ys, Zs, >_out_ag(N, 0)) -> U2_gaaa(X, Xs, N, Ys, Zs, is_in_ag(N1, -(N, 1))) U2_gaaa(X, Xs, N, Ys, Zs, is_out_ag(N1, -(N, 1))) -> U3_gaaa(X, Xs, N, Ys, Zs, N1, split_in_gaaa(Xs, N1, Ys, Zs)) U3_gaaa(X, Xs, N, Ys, Zs, N1, split_out_gaaa(Xs, N1, Ys, Zs)) -> split_out_gaaa(.(X, Xs), N, .(X, Ys), Zs) U3_ggaa(X, Xs, N, Ys, Zs, N1, split_out_gaaa(Xs, N1, Ys, Zs)) -> split_out_ggaa(.(X, Xs), N, .(X, Ys), Zs) The argument filtering Pi contains the following mapping: split_in_ggaa(x1, x2, x3, x4) = split_in_ggaa(x1, x2) 0 = 0 split_out_ggaa(x1, x2, x3, x4) = split_out_ggaa(x1, x2, x3, x4) .(x1, x2) = .(x1, x2) U1_ggaa(x1, x2, x3, x4, x5, x6) = U1_ggaa(x1, x2, x3, x6) >_in_gg(x1, x2) = >_in_gg(x1, x2) >_out_gg(x1, x2) = >_out_gg(x1, x2) U2_ggaa(x1, x2, x3, x4, x5, x6) = U2_ggaa(x1, x2, x3, x6) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag(x2) -(x1, x2) = -(x2) 1 = 1 U3_ggaa(x1, x2, x3, x4, x5, x6, x7) = U3_ggaa(x1, x2, x3, x7) split_in_gaaa(x1, x2, x3, x4) = split_in_gaaa(x1) split_out_gaaa(x1, x2, x3, x4) = split_out_gaaa(x1, x3, x4) U1_gaaa(x1, x2, x3, x4, x5, x6) = U1_gaaa(x1, x2, x6) >_in_ag(x1, x2) = >_in_ag(x2) >_out_ag(x1, x2) = >_out_ag(x2) U2_gaaa(x1, x2, x3, x4, x5, x6) = U2_gaaa(x1, x2, x6) U3_gaaa(x1, x2, x3, x4, x5, x6, x7) = U3_gaaa(x1, x2, x7) SPLIT_IN_GAAA(x1, x2, x3, x4) = SPLIT_IN_GAAA(x1) U1_GAAA(x1, x2, x3, x4, x5, x6) = U1_GAAA(x1, x2, x6) U2_GAAA(x1, x2, x3, x4, x5, x6) = U2_GAAA(x1, x2, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (9) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (10) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_GAAA(X, Xs, N, Ys, Zs, >_out_ag(N, 0)) -> U2_GAAA(X, Xs, N, Ys, Zs, is_in_ag(N1, -(N, 1))) U2_GAAA(X, Xs, N, Ys, Zs, is_out_ag(N1, -(N, 1))) -> SPLIT_IN_GAAA(Xs, N1, Ys, Zs) SPLIT_IN_GAAA(.(X, Xs), N, .(X, Ys), Zs) -> U1_GAAA(X, Xs, N, Ys, Zs, >_in_ag(N, 0)) The TRS R consists of the following rules: is_in_ag(X0, X1) -> is_out_ag(X0, X1) >_in_ag(X0, X1) -> >_out_ag(X0, X1) The argument filtering Pi contains the following mapping: 0 = 0 .(x1, x2) = .(x1, x2) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag(x2) -(x1, x2) = -(x2) 1 = 1 >_in_ag(x1, x2) = >_in_ag(x2) >_out_ag(x1, x2) = >_out_ag(x2) SPLIT_IN_GAAA(x1, x2, x3, x4) = SPLIT_IN_GAAA(x1) U1_GAAA(x1, x2, x3, x4, x5, x6) = U1_GAAA(x1, x2, x6) U2_GAAA(x1, x2, x3, x4, x5, x6) = U2_GAAA(x1, x2, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (11) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: U1_GAAA(X, Xs, >_out_ag(0)) -> U2_GAAA(X, Xs, is_in_ag(-(1))) U2_GAAA(X, Xs, is_out_ag(-(1))) -> SPLIT_IN_GAAA(Xs) SPLIT_IN_GAAA(.(X, Xs)) -> U1_GAAA(X, Xs, >_in_ag(0)) The TRS R consists of the following rules: is_in_ag(X1) -> is_out_ag(X1) >_in_ag(X1) -> >_out_ag(X1) The set Q consists of the following terms: is_in_ag(x0) >_in_ag(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (13) 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: *U2_GAAA(X, Xs, is_out_ag(-(1))) -> SPLIT_IN_GAAA(Xs) The graph contains the following edges 2 >= 1 *SPLIT_IN_GAAA(.(X, Xs)) -> U1_GAAA(X, Xs, >_in_ag(0)) The graph contains the following edges 1 > 1, 1 > 2 *U1_GAAA(X, Xs, >_out_ag(0)) -> U2_GAAA(X, Xs, is_in_ag(-(1))) The graph contains the following edges 1 >= 1, 2 >= 2 ---------------------------------------- (14) YES