/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern slice(g,g,g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) UndefinedPredicateHandlerProof [SOUND, 0 ms] (2) Prolog (3) PrologToPiTRSProof [SOUND, 21 ms] (4) PiTRS (5) DependencyPairsProof [EQUIVALENT, 41 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: slice(.(X, X1), 1, 1, .(X, [])). slice(.(X, Xs), 1, K, .(X, Ys)) :- ','(>(K, 1), ','(is(K1, -(K, 1)), slice(Xs, 1, K1, Ys))). slice(.(X2, Xs), I, K, Ys) :- ','(>(I, 1), ','(is(I1, -(I, 1)), ','(is(K1, -(K, 1)), slice(Xs, I1, K1, Ys)))). Query: slice(g,g,g,a) ---------------------------------------- (1) UndefinedPredicateHandlerProof (SOUND) Added facts for all undefined predicates [PROLOG]. ---------------------------------------- (2) Obligation: Clauses: slice(.(X, X1), 1, 1, .(X, [])). slice(.(X, Xs), 1, K, .(X, Ys)) :- ','(>(K, 1), ','(is(K1, -(K, 1)), slice(Xs, 1, K1, Ys))). slice(.(X2, Xs), I, K, Ys) :- ','(>(I, 1), ','(is(I1, -(I, 1)), ','(is(K1, -(K, 1)), slice(Xs, I1, K1, Ys)))). >(X0, X1). is(X0, X1). Query: slice(g,g,g,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: slice_in_4: (b,b,b,f) (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: slice_in_ggga(.(X, X1), 1, 1, .(X, [])) -> slice_out_ggga(.(X, X1), 1, 1, .(X, [])) slice_in_ggga(.(X, Xs), 1, K, .(X, Ys)) -> U1_ggga(X, Xs, K, Ys, >_in_gg(K, 1)) >_in_gg(X0, X1) -> >_out_gg(X0, X1) U1_ggga(X, Xs, K, Ys, >_out_gg(K, 1)) -> U2_ggga(X, Xs, K, Ys, is_in_ag(K1, -(K, 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U2_ggga(X, Xs, K, Ys, is_out_ag(K1, -(K, 1))) -> U3_ggga(X, Xs, K, Ys, K1, slice_in_ggaa(Xs, 1, K1, Ys)) slice_in_ggaa(.(X, X1), 1, 1, .(X, [])) -> slice_out_ggaa(.(X, X1), 1, 1, .(X, [])) slice_in_ggaa(.(X, Xs), 1, K, .(X, Ys)) -> U1_ggaa(X, Xs, K, Ys, >_in_ag(K, 1)) >_in_ag(X0, X1) -> >_out_ag(X0, X1) U1_ggaa(X, Xs, K, Ys, >_out_ag(K, 1)) -> U2_ggaa(X, Xs, K, Ys, is_in_ag(K1, -(K, 1))) U2_ggaa(X, Xs, K, Ys, is_out_ag(K1, -(K, 1))) -> U3_ggaa(X, Xs, K, Ys, K1, slice_in_ggaa(Xs, 1, K1, Ys)) slice_in_ggaa(.(X2, Xs), I, K, Ys) -> U4_ggaa(X2, Xs, I, K, Ys, >_in_gg(I, 1)) U4_ggaa(X2, Xs, I, K, Ys, >_out_gg(I, 1)) -> U5_ggaa(X2, Xs, I, K, Ys, is_in_ag(I1, -(I, 1))) U5_ggaa(X2, Xs, I, K, Ys, is_out_ag(I1, -(I, 1))) -> U6_ggaa(X2, Xs, I, K, Ys, I1, is_in_ag(K1, -(K, 1))) U6_ggaa(X2, Xs, I, K, Ys, I1, is_out_ag(K1, -(K, 1))) -> U7_ggaa(X2, Xs, I, K, Ys, I1, K1, slice_in_gaaa(Xs, I1, K1, Ys)) slice_in_gaaa(.(X, X1), 1, 1, .(X, [])) -> slice_out_gaaa(.(X, X1), 1, 1, .(X, [])) slice_in_gaaa(.(X, Xs), 1, K, .(X, Ys)) -> U1_gaaa(X, Xs, K, Ys, >_in_ag(K, 1)) U1_gaaa(X, Xs, K, Ys, >_out_ag(K, 1)) -> U2_gaaa(X, Xs, K, Ys, is_in_ag(K1, -(K, 1))) U2_gaaa(X, Xs, K, Ys, is_out_ag(K1, -(K, 1))) -> U3_gaaa(X, Xs, K, Ys, K1, slice_in_ggaa(Xs, 1, K1, Ys)) U3_gaaa(X, Xs, K, Ys, K1, slice_out_ggaa(Xs, 1, K1, Ys)) -> slice_out_gaaa(.(X, Xs), 1, K, .(X, Ys)) slice_in_gaaa(.(X2, Xs), I, K, Ys) -> U4_gaaa(X2, Xs, I, K, Ys, >_in_ag(I, 1)) U4_gaaa(X2, Xs, I, K, Ys, >_out_ag(I, 1)) -> U5_gaaa(X2, Xs, I, K, Ys, is_in_ag(I1, -(I, 1))) U5_gaaa(X2, Xs, I, K, Ys, is_out_ag(I1, -(I, 1))) -> U6_gaaa(X2, Xs, I, K, Ys, I1, is_in_ag(K1, -(K, 1))) U6_gaaa(X2, Xs, I, K, Ys, I1, is_out_ag(K1, -(K, 1))) -> U7_gaaa(X2, Xs, I, K, Ys, I1, K1, slice_in_gaaa(Xs, I1, K1, Ys)) U7_gaaa(X2, Xs, I, K, Ys, I1, K1, slice_out_gaaa(Xs, I1, K1, Ys)) -> slice_out_gaaa(.(X2, Xs), I, K, Ys) U7_ggaa(X2, Xs, I, K, Ys, I1, K1, slice_out_gaaa(Xs, I1, K1, Ys)) -> slice_out_ggaa(.(X2, Xs), I, K, Ys) U3_ggaa(X, Xs, K, Ys, K1, slice_out_ggaa(Xs, 1, K1, Ys)) -> slice_out_ggaa(.(X, Xs), 1, K, .(X, Ys)) U3_ggga(X, Xs, K, Ys, K1, slice_out_ggaa(Xs, 1, K1, Ys)) -> slice_out_ggga(.(X, Xs), 1, K, .(X, Ys)) slice_in_ggga(.(X2, Xs), I, K, Ys) -> U4_ggga(X2, Xs, I, K, Ys, >_in_gg(I, 1)) U4_ggga(X2, Xs, I, K, Ys, >_out_gg(I, 1)) -> U5_ggga(X2, Xs, I, K, Ys, is_in_ag(I1, -(I, 1))) U5_ggga(X2, Xs, I, K, Ys, is_out_ag(I1, -(I, 1))) -> U6_ggga(X2, Xs, I, K, Ys, I1, is_in_ag(K1, -(K, 1))) U6_ggga(X2, Xs, I, K, Ys, I1, is_out_ag(K1, -(K, 1))) -> U7_ggga(X2, Xs, I, K, Ys, I1, K1, slice_in_gaaa(Xs, I1, K1, Ys)) U7_ggga(X2, Xs, I, K, Ys, I1, K1, slice_out_gaaa(Xs, I1, K1, Ys)) -> slice_out_ggga(.(X2, Xs), I, K, Ys) The argument filtering Pi contains the following mapping: slice_in_ggga(x1, x2, x3, x4) = slice_in_ggga(x1, x2, x3) .(x1, x2) = .(x1, x2) 1 = 1 slice_out_ggga(x1, x2, x3, x4) = slice_out_ggga(x4) U1_ggga(x1, x2, x3, x4, x5) = U1_ggga(x1, x2, x5) >_in_gg(x1, x2) = >_in_gg(x1, x2) >_out_gg(x1, x2) = >_out_gg U2_ggga(x1, x2, x3, x4, x5) = U2_ggga(x1, x2, x5) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag -(x1, x2) = -(x2) U3_ggga(x1, x2, x3, x4, x5, x6) = U3_ggga(x1, x6) slice_in_ggaa(x1, x2, x3, x4) = slice_in_ggaa(x1, x2) slice_out_ggaa(x1, x2, x3, x4) = slice_out_ggaa(x4) U1_ggaa(x1, x2, x3, x4, x5) = U1_ggaa(x1, x2, x5) >_in_ag(x1, x2) = >_in_ag(x2) >_out_ag(x1, x2) = >_out_ag U2_ggaa(x1, x2, x3, x4, x5) = U2_ggaa(x1, x2, x5) U3_ggaa(x1, x2, x3, x4, x5, x6) = U3_ggaa(x1, x6) U4_ggaa(x1, x2, x3, x4, x5, x6) = U4_ggaa(x2, x6) U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x2, x6) U6_ggaa(x1, x2, x3, x4, x5, x6, x7) = U6_ggaa(x2, x7) U7_ggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U7_ggaa(x8) slice_in_gaaa(x1, x2, x3, x4) = slice_in_gaaa(x1) slice_out_gaaa(x1, x2, x3, x4) = slice_out_gaaa(x4) U1_gaaa(x1, x2, x3, x4, x5) = U1_gaaa(x1, x2, x5) U2_gaaa(x1, x2, x3, x4, x5) = U2_gaaa(x1, x2, x5) U3_gaaa(x1, x2, x3, x4, x5, x6) = U3_gaaa(x1, x6) U4_gaaa(x1, x2, x3, x4, x5, x6) = U4_gaaa(x2, x6) U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x2, x6) U6_gaaa(x1, x2, x3, x4, x5, x6, x7) = U6_gaaa(x2, x7) U7_gaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U7_gaaa(x8) U4_ggga(x1, x2, x3, x4, x5, x6) = U4_ggga(x2, x6) U5_ggga(x1, x2, x3, x4, x5, x6) = U5_ggga(x2, x6) U6_ggga(x1, x2, x3, x4, x5, x6, x7) = U6_ggga(x2, x7) U7_ggga(x1, x2, x3, x4, x5, x6, x7, x8) = U7_ggga(x8) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (4) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: slice_in_ggga(.(X, X1), 1, 1, .(X, [])) -> slice_out_ggga(.(X, X1), 1, 1, .(X, [])) slice_in_ggga(.(X, Xs), 1, K, .(X, Ys)) -> U1_ggga(X, Xs, K, Ys, >_in_gg(K, 1)) >_in_gg(X0, X1) -> >_out_gg(X0, X1) U1_ggga(X, Xs, K, Ys, >_out_gg(K, 1)) -> U2_ggga(X, Xs, K, Ys, is_in_ag(K1, -(K, 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U2_ggga(X, Xs, K, Ys, is_out_ag(K1, -(K, 1))) -> U3_ggga(X, Xs, K, Ys, K1, slice_in_ggaa(Xs, 1, K1, Ys)) slice_in_ggaa(.(X, X1), 1, 1, .(X, [])) -> slice_out_ggaa(.(X, X1), 1, 1, .(X, [])) slice_in_ggaa(.(X, Xs), 1, K, .(X, Ys)) -> U1_ggaa(X, Xs, K, Ys, >_in_ag(K, 1)) >_in_ag(X0, X1) -> >_out_ag(X0, X1) U1_ggaa(X, Xs, K, Ys, >_out_ag(K, 1)) -> U2_ggaa(X, Xs, K, Ys, is_in_ag(K1, -(K, 1))) U2_ggaa(X, Xs, K, Ys, is_out_ag(K1, -(K, 1))) -> U3_ggaa(X, Xs, K, Ys, K1, slice_in_ggaa(Xs, 1, K1, Ys)) slice_in_ggaa(.(X2, Xs), I, K, Ys) -> U4_ggaa(X2, Xs, I, K, Ys, >_in_gg(I, 1)) U4_ggaa(X2, Xs, I, K, Ys, >_out_gg(I, 1)) -> U5_ggaa(X2, Xs, I, K, Ys, is_in_ag(I1, -(I, 1))) U5_ggaa(X2, Xs, I, K, Ys, is_out_ag(I1, -(I, 1))) -> U6_ggaa(X2, Xs, I, K, Ys, I1, is_in_ag(K1, -(K, 1))) U6_ggaa(X2, Xs, I, K, Ys, I1, is_out_ag(K1, -(K, 1))) -> U7_ggaa(X2, Xs, I, K, Ys, I1, K1, slice_in_gaaa(Xs, I1, K1, Ys)) slice_in_gaaa(.(X, X1), 1, 1, .(X, [])) -> slice_out_gaaa(.(X, X1), 1, 1, .(X, [])) slice_in_gaaa(.(X, Xs), 1, K, .(X, Ys)) -> U1_gaaa(X, Xs, K, Ys, >_in_ag(K, 1)) U1_gaaa(X, Xs, K, Ys, >_out_ag(K, 1)) -> U2_gaaa(X, Xs, K, Ys, is_in_ag(K1, -(K, 1))) U2_gaaa(X, Xs, K, Ys, is_out_ag(K1, -(K, 1))) -> U3_gaaa(X, Xs, K, Ys, K1, slice_in_ggaa(Xs, 1, K1, Ys)) U3_gaaa(X, Xs, K, Ys, K1, slice_out_ggaa(Xs, 1, K1, Ys)) -> slice_out_gaaa(.(X, Xs), 1, K, .(X, Ys)) slice_in_gaaa(.(X2, Xs), I, K, Ys) -> U4_gaaa(X2, Xs, I, K, Ys, >_in_ag(I, 1)) U4_gaaa(X2, Xs, I, K, Ys, >_out_ag(I, 1)) -> U5_gaaa(X2, Xs, I, K, Ys, is_in_ag(I1, -(I, 1))) U5_gaaa(X2, Xs, I, K, Ys, is_out_ag(I1, -(I, 1))) -> U6_gaaa(X2, Xs, I, K, Ys, I1, is_in_ag(K1, -(K, 1))) U6_gaaa(X2, Xs, I, K, Ys, I1, is_out_ag(K1, -(K, 1))) -> U7_gaaa(X2, Xs, I, K, Ys, I1, K1, slice_in_gaaa(Xs, I1, K1, Ys)) U7_gaaa(X2, Xs, I, K, Ys, I1, K1, slice_out_gaaa(Xs, I1, K1, Ys)) -> slice_out_gaaa(.(X2, Xs), I, K, Ys) U7_ggaa(X2, Xs, I, K, Ys, I1, K1, slice_out_gaaa(Xs, I1, K1, Ys)) -> slice_out_ggaa(.(X2, Xs), I, K, Ys) U3_ggaa(X, Xs, K, Ys, K1, slice_out_ggaa(Xs, 1, K1, Ys)) -> slice_out_ggaa(.(X, Xs), 1, K, .(X, Ys)) U3_ggga(X, Xs, K, Ys, K1, slice_out_ggaa(Xs, 1, K1, Ys)) -> slice_out_ggga(.(X, Xs), 1, K, .(X, Ys)) slice_in_ggga(.(X2, Xs), I, K, Ys) -> U4_ggga(X2, Xs, I, K, Ys, >_in_gg(I, 1)) U4_ggga(X2, Xs, I, K, Ys, >_out_gg(I, 1)) -> U5_ggga(X2, Xs, I, K, Ys, is_in_ag(I1, -(I, 1))) U5_ggga(X2, Xs, I, K, Ys, is_out_ag(I1, -(I, 1))) -> U6_ggga(X2, Xs, I, K, Ys, I1, is_in_ag(K1, -(K, 1))) U6_ggga(X2, Xs, I, K, Ys, I1, is_out_ag(K1, -(K, 1))) -> U7_ggga(X2, Xs, I, K, Ys, I1, K1, slice_in_gaaa(Xs, I1, K1, Ys)) U7_ggga(X2, Xs, I, K, Ys, I1, K1, slice_out_gaaa(Xs, I1, K1, Ys)) -> slice_out_ggga(.(X2, Xs), I, K, Ys) The argument filtering Pi contains the following mapping: slice_in_ggga(x1, x2, x3, x4) = slice_in_ggga(x1, x2, x3) .(x1, x2) = .(x1, x2) 1 = 1 slice_out_ggga(x1, x2, x3, x4) = slice_out_ggga(x4) U1_ggga(x1, x2, x3, x4, x5) = U1_ggga(x1, x2, x5) >_in_gg(x1, x2) = >_in_gg(x1, x2) >_out_gg(x1, x2) = >_out_gg U2_ggga(x1, x2, x3, x4, x5) = U2_ggga(x1, x2, x5) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag -(x1, x2) = -(x2) U3_ggga(x1, x2, x3, x4, x5, x6) = U3_ggga(x1, x6) slice_in_ggaa(x1, x2, x3, x4) = slice_in_ggaa(x1, x2) slice_out_ggaa(x1, x2, x3, x4) = slice_out_ggaa(x4) U1_ggaa(x1, x2, x3, x4, x5) = U1_ggaa(x1, x2, x5) >_in_ag(x1, x2) = >_in_ag(x2) >_out_ag(x1, x2) = >_out_ag U2_ggaa(x1, x2, x3, x4, x5) = U2_ggaa(x1, x2, x5) U3_ggaa(x1, x2, x3, x4, x5, x6) = U3_ggaa(x1, x6) U4_ggaa(x1, x2, x3, x4, x5, x6) = U4_ggaa(x2, x6) U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x2, x6) U6_ggaa(x1, x2, x3, x4, x5, x6, x7) = U6_ggaa(x2, x7) U7_ggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U7_ggaa(x8) slice_in_gaaa(x1, x2, x3, x4) = slice_in_gaaa(x1) slice_out_gaaa(x1, x2, x3, x4) = slice_out_gaaa(x4) U1_gaaa(x1, x2, x3, x4, x5) = U1_gaaa(x1, x2, x5) U2_gaaa(x1, x2, x3, x4, x5) = U2_gaaa(x1, x2, x5) U3_gaaa(x1, x2, x3, x4, x5, x6) = U3_gaaa(x1, x6) U4_gaaa(x1, x2, x3, x4, x5, x6) = U4_gaaa(x2, x6) U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x2, x6) U6_gaaa(x1, x2, x3, x4, x5, x6, x7) = U6_gaaa(x2, x7) U7_gaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U7_gaaa(x8) U4_ggga(x1, x2, x3, x4, x5, x6) = U4_ggga(x2, x6) U5_ggga(x1, x2, x3, x4, x5, x6) = U5_ggga(x2, x6) U6_ggga(x1, x2, x3, x4, x5, x6, x7) = U6_ggga(x2, x7) U7_ggga(x1, x2, x3, x4, x5, x6, x7, x8) = U7_ggga(x8) ---------------------------------------- (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: SLICE_IN_GGGA(.(X, Xs), 1, K, .(X, Ys)) -> U1_GGGA(X, Xs, K, Ys, >_in_gg(K, 1)) SLICE_IN_GGGA(.(X, Xs), 1, K, .(X, Ys)) -> >_IN_GG(K, 1) U1_GGGA(X, Xs, K, Ys, >_out_gg(K, 1)) -> U2_GGGA(X, Xs, K, Ys, is_in_ag(K1, -(K, 1))) U1_GGGA(X, Xs, K, Ys, >_out_gg(K, 1)) -> IS_IN_AG(K1, -(K, 1)) U2_GGGA(X, Xs, K, Ys, is_out_ag(K1, -(K, 1))) -> U3_GGGA(X, Xs, K, Ys, K1, slice_in_ggaa(Xs, 1, K1, Ys)) U2_GGGA(X, Xs, K, Ys, is_out_ag(K1, -(K, 1))) -> SLICE_IN_GGAA(Xs, 1, K1, Ys) SLICE_IN_GGAA(.(X, Xs), 1, K, .(X, Ys)) -> U1_GGAA(X, Xs, K, Ys, >_in_ag(K, 1)) SLICE_IN_GGAA(.(X, Xs), 1, K, .(X, Ys)) -> >_IN_AG(K, 1) U1_GGAA(X, Xs, K, Ys, >_out_ag(K, 1)) -> U2_GGAA(X, Xs, K, Ys, is_in_ag(K1, -(K, 1))) U1_GGAA(X, Xs, K, Ys, >_out_ag(K, 1)) -> IS_IN_AG(K1, -(K, 1)) U2_GGAA(X, Xs, K, Ys, is_out_ag(K1, -(K, 1))) -> U3_GGAA(X, Xs, K, Ys, K1, slice_in_ggaa(Xs, 1, K1, Ys)) U2_GGAA(X, Xs, K, Ys, is_out_ag(K1, -(K, 1))) -> SLICE_IN_GGAA(Xs, 1, K1, Ys) SLICE_IN_GGAA(.(X2, Xs), I, K, Ys) -> U4_GGAA(X2, Xs, I, K, Ys, >_in_gg(I, 1)) SLICE_IN_GGAA(.(X2, Xs), I, K, Ys) -> >_IN_GG(I, 1) U4_GGAA(X2, Xs, I, K, Ys, >_out_gg(I, 1)) -> U5_GGAA(X2, Xs, I, K, Ys, is_in_ag(I1, -(I, 1))) U4_GGAA(X2, Xs, I, K, Ys, >_out_gg(I, 1)) -> IS_IN_AG(I1, -(I, 1)) U5_GGAA(X2, Xs, I, K, Ys, is_out_ag(I1, -(I, 1))) -> U6_GGAA(X2, Xs, I, K, Ys, I1, is_in_ag(K1, -(K, 1))) U5_GGAA(X2, Xs, I, K, Ys, is_out_ag(I1, -(I, 1))) -> IS_IN_AG(K1, -(K, 1)) U6_GGAA(X2, Xs, I, K, Ys, I1, is_out_ag(K1, -(K, 1))) -> U7_GGAA(X2, Xs, I, K, Ys, I1, K1, slice_in_gaaa(Xs, I1, K1, Ys)) U6_GGAA(X2, Xs, I, K, Ys, I1, is_out_ag(K1, -(K, 1))) -> SLICE_IN_GAAA(Xs, I1, K1, Ys) SLICE_IN_GAAA(.(X, Xs), 1, K, .(X, Ys)) -> U1_GAAA(X, Xs, K, Ys, >_in_ag(K, 1)) SLICE_IN_GAAA(.(X, Xs), 1, K, .(X, Ys)) -> >_IN_AG(K, 1) U1_GAAA(X, Xs, K, Ys, >_out_ag(K, 1)) -> U2_GAAA(X, Xs, K, Ys, is_in_ag(K1, -(K, 1))) U1_GAAA(X, Xs, K, Ys, >_out_ag(K, 1)) -> IS_IN_AG(K1, -(K, 1)) U2_GAAA(X, Xs, K, Ys, is_out_ag(K1, -(K, 1))) -> U3_GAAA(X, Xs, K, Ys, K1, slice_in_ggaa(Xs, 1, K1, Ys)) U2_GAAA(X, Xs, K, Ys, is_out_ag(K1, -(K, 1))) -> SLICE_IN_GGAA(Xs, 1, K1, Ys) SLICE_IN_GAAA(.(X2, Xs), I, K, Ys) -> U4_GAAA(X2, Xs, I, K, Ys, >_in_ag(I, 1)) SLICE_IN_GAAA(.(X2, Xs), I, K, Ys) -> >_IN_AG(I, 1) U4_GAAA(X2, Xs, I, K, Ys, >_out_ag(I, 1)) -> U5_GAAA(X2, Xs, I, K, Ys, is_in_ag(I1, -(I, 1))) U4_GAAA(X2, Xs, I, K, Ys, >_out_ag(I, 1)) -> IS_IN_AG(I1, -(I, 1)) U5_GAAA(X2, Xs, I, K, Ys, is_out_ag(I1, -(I, 1))) -> U6_GAAA(X2, Xs, I, K, Ys, I1, is_in_ag(K1, -(K, 1))) U5_GAAA(X2, Xs, I, K, Ys, is_out_ag(I1, -(I, 1))) -> IS_IN_AG(K1, -(K, 1)) U6_GAAA(X2, Xs, I, K, Ys, I1, is_out_ag(K1, -(K, 1))) -> U7_GAAA(X2, Xs, I, K, Ys, I1, K1, slice_in_gaaa(Xs, I1, K1, Ys)) U6_GAAA(X2, Xs, I, K, Ys, I1, is_out_ag(K1, -(K, 1))) -> SLICE_IN_GAAA(Xs, I1, K1, Ys) SLICE_IN_GGGA(.(X2, Xs), I, K, Ys) -> U4_GGGA(X2, Xs, I, K, Ys, >_in_gg(I, 1)) SLICE_IN_GGGA(.(X2, Xs), I, K, Ys) -> >_IN_GG(I, 1) U4_GGGA(X2, Xs, I, K, Ys, >_out_gg(I, 1)) -> U5_GGGA(X2, Xs, I, K, Ys, is_in_ag(I1, -(I, 1))) U4_GGGA(X2, Xs, I, K, Ys, >_out_gg(I, 1)) -> IS_IN_AG(I1, -(I, 1)) U5_GGGA(X2, Xs, I, K, Ys, is_out_ag(I1, -(I, 1))) -> U6_GGGA(X2, Xs, I, K, Ys, I1, is_in_ag(K1, -(K, 1))) U5_GGGA(X2, Xs, I, K, Ys, is_out_ag(I1, -(I, 1))) -> IS_IN_AG(K1, -(K, 1)) U6_GGGA(X2, Xs, I, K, Ys, I1, is_out_ag(K1, -(K, 1))) -> U7_GGGA(X2, Xs, I, K, Ys, I1, K1, slice_in_gaaa(Xs, I1, K1, Ys)) U6_GGGA(X2, Xs, I, K, Ys, I1, is_out_ag(K1, -(K, 1))) -> SLICE_IN_GAAA(Xs, I1, K1, Ys) The TRS R consists of the following rules: slice_in_ggga(.(X, X1), 1, 1, .(X, [])) -> slice_out_ggga(.(X, X1), 1, 1, .(X, [])) slice_in_ggga(.(X, Xs), 1, K, .(X, Ys)) -> U1_ggga(X, Xs, K, Ys, >_in_gg(K, 1)) >_in_gg(X0, X1) -> >_out_gg(X0, X1) U1_ggga(X, Xs, K, Ys, >_out_gg(K, 1)) -> U2_ggga(X, Xs, K, Ys, is_in_ag(K1, -(K, 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U2_ggga(X, Xs, K, Ys, is_out_ag(K1, -(K, 1))) -> U3_ggga(X, Xs, K, Ys, K1, slice_in_ggaa(Xs, 1, K1, Ys)) slice_in_ggaa(.(X, X1), 1, 1, .(X, [])) -> slice_out_ggaa(.(X, X1), 1, 1, .(X, [])) slice_in_ggaa(.(X, Xs), 1, K, .(X, Ys)) -> U1_ggaa(X, Xs, K, Ys, >_in_ag(K, 1)) >_in_ag(X0, X1) -> >_out_ag(X0, X1) U1_ggaa(X, Xs, K, Ys, >_out_ag(K, 1)) -> U2_ggaa(X, Xs, K, Ys, is_in_ag(K1, -(K, 1))) U2_ggaa(X, Xs, K, Ys, is_out_ag(K1, -(K, 1))) -> U3_ggaa(X, Xs, K, Ys, K1, slice_in_ggaa(Xs, 1, K1, Ys)) slice_in_ggaa(.(X2, Xs), I, K, Ys) -> U4_ggaa(X2, Xs, I, K, Ys, >_in_gg(I, 1)) U4_ggaa(X2, Xs, I, K, Ys, >_out_gg(I, 1)) -> U5_ggaa(X2, Xs, I, K, Ys, is_in_ag(I1, -(I, 1))) U5_ggaa(X2, Xs, I, K, Ys, is_out_ag(I1, -(I, 1))) -> U6_ggaa(X2, Xs, I, K, Ys, I1, is_in_ag(K1, -(K, 1))) U6_ggaa(X2, Xs, I, K, Ys, I1, is_out_ag(K1, -(K, 1))) -> U7_ggaa(X2, Xs, I, K, Ys, I1, K1, slice_in_gaaa(Xs, I1, K1, Ys)) slice_in_gaaa(.(X, X1), 1, 1, .(X, [])) -> slice_out_gaaa(.(X, X1), 1, 1, .(X, [])) slice_in_gaaa(.(X, Xs), 1, K, .(X, Ys)) -> U1_gaaa(X, Xs, K, Ys, >_in_ag(K, 1)) U1_gaaa(X, Xs, K, Ys, >_out_ag(K, 1)) -> U2_gaaa(X, Xs, K, Ys, is_in_ag(K1, -(K, 1))) U2_gaaa(X, Xs, K, Ys, is_out_ag(K1, -(K, 1))) -> U3_gaaa(X, Xs, K, Ys, K1, slice_in_ggaa(Xs, 1, K1, Ys)) U3_gaaa(X, Xs, K, Ys, K1, slice_out_ggaa(Xs, 1, K1, Ys)) -> slice_out_gaaa(.(X, Xs), 1, K, .(X, Ys)) slice_in_gaaa(.(X2, Xs), I, K, Ys) -> U4_gaaa(X2, Xs, I, K, Ys, >_in_ag(I, 1)) U4_gaaa(X2, Xs, I, K, Ys, >_out_ag(I, 1)) -> U5_gaaa(X2, Xs, I, K, Ys, is_in_ag(I1, -(I, 1))) U5_gaaa(X2, Xs, I, K, Ys, is_out_ag(I1, -(I, 1))) -> U6_gaaa(X2, Xs, I, K, Ys, I1, is_in_ag(K1, -(K, 1))) U6_gaaa(X2, Xs, I, K, Ys, I1, is_out_ag(K1, -(K, 1))) -> U7_gaaa(X2, Xs, I, K, Ys, I1, K1, slice_in_gaaa(Xs, I1, K1, Ys)) U7_gaaa(X2, Xs, I, K, Ys, I1, K1, slice_out_gaaa(Xs, I1, K1, Ys)) -> slice_out_gaaa(.(X2, Xs), I, K, Ys) U7_ggaa(X2, Xs, I, K, Ys, I1, K1, slice_out_gaaa(Xs, I1, K1, Ys)) -> slice_out_ggaa(.(X2, Xs), I, K, Ys) U3_ggaa(X, Xs, K, Ys, K1, slice_out_ggaa(Xs, 1, K1, Ys)) -> slice_out_ggaa(.(X, Xs), 1, K, .(X, Ys)) U3_ggga(X, Xs, K, Ys, K1, slice_out_ggaa(Xs, 1, K1, Ys)) -> slice_out_ggga(.(X, Xs), 1, K, .(X, Ys)) slice_in_ggga(.(X2, Xs), I, K, Ys) -> U4_ggga(X2, Xs, I, K, Ys, >_in_gg(I, 1)) U4_ggga(X2, Xs, I, K, Ys, >_out_gg(I, 1)) -> U5_ggga(X2, Xs, I, K, Ys, is_in_ag(I1, -(I, 1))) U5_ggga(X2, Xs, I, K, Ys, is_out_ag(I1, -(I, 1))) -> U6_ggga(X2, Xs, I, K, Ys, I1, is_in_ag(K1, -(K, 1))) U6_ggga(X2, Xs, I, K, Ys, I1, is_out_ag(K1, -(K, 1))) -> U7_ggga(X2, Xs, I, K, Ys, I1, K1, slice_in_gaaa(Xs, I1, K1, Ys)) U7_ggga(X2, Xs, I, K, Ys, I1, K1, slice_out_gaaa(Xs, I1, K1, Ys)) -> slice_out_ggga(.(X2, Xs), I, K, Ys) The argument filtering Pi contains the following mapping: slice_in_ggga(x1, x2, x3, x4) = slice_in_ggga(x1, x2, x3) .(x1, x2) = .(x1, x2) 1 = 1 slice_out_ggga(x1, x2, x3, x4) = slice_out_ggga(x4) U1_ggga(x1, x2, x3, x4, x5) = U1_ggga(x1, x2, x5) >_in_gg(x1, x2) = >_in_gg(x1, x2) >_out_gg(x1, x2) = >_out_gg U2_ggga(x1, x2, x3, x4, x5) = U2_ggga(x1, x2, x5) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag -(x1, x2) = -(x2) U3_ggga(x1, x2, x3, x4, x5, x6) = U3_ggga(x1, x6) slice_in_ggaa(x1, x2, x3, x4) = slice_in_ggaa(x1, x2) slice_out_ggaa(x1, x2, x3, x4) = slice_out_ggaa(x4) U1_ggaa(x1, x2, x3, x4, x5) = U1_ggaa(x1, x2, x5) >_in_ag(x1, x2) = >_in_ag(x2) >_out_ag(x1, x2) = >_out_ag U2_ggaa(x1, x2, x3, x4, x5) = U2_ggaa(x1, x2, x5) U3_ggaa(x1, x2, x3, x4, x5, x6) = U3_ggaa(x1, x6) U4_ggaa(x1, x2, x3, x4, x5, x6) = U4_ggaa(x2, x6) U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x2, x6) U6_ggaa(x1, x2, x3, x4, x5, x6, x7) = U6_ggaa(x2, x7) U7_ggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U7_ggaa(x8) slice_in_gaaa(x1, x2, x3, x4) = slice_in_gaaa(x1) slice_out_gaaa(x1, x2, x3, x4) = slice_out_gaaa(x4) U1_gaaa(x1, x2, x3, x4, x5) = U1_gaaa(x1, x2, x5) U2_gaaa(x1, x2, x3, x4, x5) = U2_gaaa(x1, x2, x5) U3_gaaa(x1, x2, x3, x4, x5, x6) = U3_gaaa(x1, x6) U4_gaaa(x1, x2, x3, x4, x5, x6) = U4_gaaa(x2, x6) U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x2, x6) U6_gaaa(x1, x2, x3, x4, x5, x6, x7) = U6_gaaa(x2, x7) U7_gaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U7_gaaa(x8) U4_ggga(x1, x2, x3, x4, x5, x6) = U4_ggga(x2, x6) U5_ggga(x1, x2, x3, x4, x5, x6) = U5_ggga(x2, x6) U6_ggga(x1, x2, x3, x4, x5, x6, x7) = U6_ggga(x2, x7) U7_ggga(x1, x2, x3, x4, x5, x6, x7, x8) = U7_ggga(x8) SLICE_IN_GGGA(x1, x2, x3, x4) = SLICE_IN_GGGA(x1, x2, x3) U1_GGGA(x1, x2, x3, x4, x5) = U1_GGGA(x1, x2, x5) >_IN_GG(x1, x2) = >_IN_GG(x1, x2) U2_GGGA(x1, x2, x3, x4, x5) = U2_GGGA(x1, x2, x5) IS_IN_AG(x1, x2) = IS_IN_AG(x2) U3_GGGA(x1, x2, x3, x4, x5, x6) = U3_GGGA(x1, x6) SLICE_IN_GGAA(x1, x2, x3, x4) = SLICE_IN_GGAA(x1, x2) U1_GGAA(x1, x2, x3, x4, x5) = U1_GGAA(x1, x2, x5) >_IN_AG(x1, x2) = >_IN_AG(x2) U2_GGAA(x1, x2, x3, x4, x5) = U2_GGAA(x1, x2, x5) U3_GGAA(x1, x2, x3, x4, x5, x6) = U3_GGAA(x1, x6) U4_GGAA(x1, x2, x3, x4, x5, x6) = U4_GGAA(x2, x6) U5_GGAA(x1, x2, x3, x4, x5, x6) = U5_GGAA(x2, x6) U6_GGAA(x1, x2, x3, x4, x5, x6, x7) = U6_GGAA(x2, x7) U7_GGAA(x1, x2, x3, x4, x5, x6, x7, x8) = U7_GGAA(x8) SLICE_IN_GAAA(x1, x2, x3, x4) = SLICE_IN_GAAA(x1) U1_GAAA(x1, x2, x3, x4, x5) = U1_GAAA(x1, x2, x5) U2_GAAA(x1, x2, x3, x4, x5) = U2_GAAA(x1, x2, x5) U3_GAAA(x1, x2, x3, x4, x5, x6) = U3_GAAA(x1, x6) U4_GAAA(x1, x2, x3, x4, x5, x6) = U4_GAAA(x2, x6) U5_GAAA(x1, x2, x3, x4, x5, x6) = U5_GAAA(x2, x6) U6_GAAA(x1, x2, x3, x4, x5, x6, x7) = U6_GAAA(x2, x7) U7_GAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U7_GAAA(x8) U4_GGGA(x1, x2, x3, x4, x5, x6) = U4_GGGA(x2, x6) U5_GGGA(x1, x2, x3, x4, x5, x6) = U5_GGGA(x2, x6) U6_GGGA(x1, x2, x3, x4, x5, x6, x7) = U6_GGGA(x2, x7) U7_GGGA(x1, x2, x3, x4, x5, x6, x7, x8) = U7_GGGA(x8) We have to consider all (P,R,Pi)-chains ---------------------------------------- (6) Obligation: Pi DP problem: The TRS P consists of the following rules: SLICE_IN_GGGA(.(X, Xs), 1, K, .(X, Ys)) -> U1_GGGA(X, Xs, K, Ys, >_in_gg(K, 1)) SLICE_IN_GGGA(.(X, Xs), 1, K, .(X, Ys)) -> >_IN_GG(K, 1) U1_GGGA(X, Xs, K, Ys, >_out_gg(K, 1)) -> U2_GGGA(X, Xs, K, Ys, is_in_ag(K1, -(K, 1))) U1_GGGA(X, Xs, K, Ys, >_out_gg(K, 1)) -> IS_IN_AG(K1, -(K, 1)) U2_GGGA(X, Xs, K, Ys, is_out_ag(K1, -(K, 1))) -> U3_GGGA(X, Xs, K, Ys, K1, slice_in_ggaa(Xs, 1, K1, Ys)) U2_GGGA(X, Xs, K, Ys, is_out_ag(K1, -(K, 1))) -> SLICE_IN_GGAA(Xs, 1, K1, Ys) SLICE_IN_GGAA(.(X, Xs), 1, K, .(X, Ys)) -> U1_GGAA(X, Xs, K, Ys, >_in_ag(K, 1)) SLICE_IN_GGAA(.(X, Xs), 1, K, .(X, Ys)) -> >_IN_AG(K, 1) U1_GGAA(X, Xs, K, Ys, >_out_ag(K, 1)) -> U2_GGAA(X, Xs, K, Ys, is_in_ag(K1, -(K, 1))) U1_GGAA(X, Xs, K, Ys, >_out_ag(K, 1)) -> IS_IN_AG(K1, -(K, 1)) U2_GGAA(X, Xs, K, Ys, is_out_ag(K1, -(K, 1))) -> U3_GGAA(X, Xs, K, Ys, K1, slice_in_ggaa(Xs, 1, K1, Ys)) U2_GGAA(X, Xs, K, Ys, is_out_ag(K1, -(K, 1))) -> SLICE_IN_GGAA(Xs, 1, K1, Ys) SLICE_IN_GGAA(.(X2, Xs), I, K, Ys) -> U4_GGAA(X2, Xs, I, K, Ys, >_in_gg(I, 1)) SLICE_IN_GGAA(.(X2, Xs), I, K, Ys) -> >_IN_GG(I, 1) U4_GGAA(X2, Xs, I, K, Ys, >_out_gg(I, 1)) -> U5_GGAA(X2, Xs, I, K, Ys, is_in_ag(I1, -(I, 1))) U4_GGAA(X2, Xs, I, K, Ys, >_out_gg(I, 1)) -> IS_IN_AG(I1, -(I, 1)) U5_GGAA(X2, Xs, I, K, Ys, is_out_ag(I1, -(I, 1))) -> U6_GGAA(X2, Xs, I, K, Ys, I1, is_in_ag(K1, -(K, 1))) U5_GGAA(X2, Xs, I, K, Ys, is_out_ag(I1, -(I, 1))) -> IS_IN_AG(K1, -(K, 1)) U6_GGAA(X2, Xs, I, K, Ys, I1, is_out_ag(K1, -(K, 1))) -> U7_GGAA(X2, Xs, I, K, Ys, I1, K1, slice_in_gaaa(Xs, I1, K1, Ys)) U6_GGAA(X2, Xs, I, K, Ys, I1, is_out_ag(K1, -(K, 1))) -> SLICE_IN_GAAA(Xs, I1, K1, Ys) SLICE_IN_GAAA(.(X, Xs), 1, K, .(X, Ys)) -> U1_GAAA(X, Xs, K, Ys, >_in_ag(K, 1)) SLICE_IN_GAAA(.(X, Xs), 1, K, .(X, Ys)) -> >_IN_AG(K, 1) U1_GAAA(X, Xs, K, Ys, >_out_ag(K, 1)) -> U2_GAAA(X, Xs, K, Ys, is_in_ag(K1, -(K, 1))) U1_GAAA(X, Xs, K, Ys, >_out_ag(K, 1)) -> IS_IN_AG(K1, -(K, 1)) U2_GAAA(X, Xs, K, Ys, is_out_ag(K1, -(K, 1))) -> U3_GAAA(X, Xs, K, Ys, K1, slice_in_ggaa(Xs, 1, K1, Ys)) U2_GAAA(X, Xs, K, Ys, is_out_ag(K1, -(K, 1))) -> SLICE_IN_GGAA(Xs, 1, K1, Ys) SLICE_IN_GAAA(.(X2, Xs), I, K, Ys) -> U4_GAAA(X2, Xs, I, K, Ys, >_in_ag(I, 1)) SLICE_IN_GAAA(.(X2, Xs), I, K, Ys) -> >_IN_AG(I, 1) U4_GAAA(X2, Xs, I, K, Ys, >_out_ag(I, 1)) -> U5_GAAA(X2, Xs, I, K, Ys, is_in_ag(I1, -(I, 1))) U4_GAAA(X2, Xs, I, K, Ys, >_out_ag(I, 1)) -> IS_IN_AG(I1, -(I, 1)) U5_GAAA(X2, Xs, I, K, Ys, is_out_ag(I1, -(I, 1))) -> U6_GAAA(X2, Xs, I, K, Ys, I1, is_in_ag(K1, -(K, 1))) U5_GAAA(X2, Xs, I, K, Ys, is_out_ag(I1, -(I, 1))) -> IS_IN_AG(K1, -(K, 1)) U6_GAAA(X2, Xs, I, K, Ys, I1, is_out_ag(K1, -(K, 1))) -> U7_GAAA(X2, Xs, I, K, Ys, I1, K1, slice_in_gaaa(Xs, I1, K1, Ys)) U6_GAAA(X2, Xs, I, K, Ys, I1, is_out_ag(K1, -(K, 1))) -> SLICE_IN_GAAA(Xs, I1, K1, Ys) SLICE_IN_GGGA(.(X2, Xs), I, K, Ys) -> U4_GGGA(X2, Xs, I, K, Ys, >_in_gg(I, 1)) SLICE_IN_GGGA(.(X2, Xs), I, K, Ys) -> >_IN_GG(I, 1) U4_GGGA(X2, Xs, I, K, Ys, >_out_gg(I, 1)) -> U5_GGGA(X2, Xs, I, K, Ys, is_in_ag(I1, -(I, 1))) U4_GGGA(X2, Xs, I, K, Ys, >_out_gg(I, 1)) -> IS_IN_AG(I1, -(I, 1)) U5_GGGA(X2, Xs, I, K, Ys, is_out_ag(I1, -(I, 1))) -> U6_GGGA(X2, Xs, I, K, Ys, I1, is_in_ag(K1, -(K, 1))) U5_GGGA(X2, Xs, I, K, Ys, is_out_ag(I1, -(I, 1))) -> IS_IN_AG(K1, -(K, 1)) U6_GGGA(X2, Xs, I, K, Ys, I1, is_out_ag(K1, -(K, 1))) -> U7_GGGA(X2, Xs, I, K, Ys, I1, K1, slice_in_gaaa(Xs, I1, K1, Ys)) U6_GGGA(X2, Xs, I, K, Ys, I1, is_out_ag(K1, -(K, 1))) -> SLICE_IN_GAAA(Xs, I1, K1, Ys) The TRS R consists of the following rules: slice_in_ggga(.(X, X1), 1, 1, .(X, [])) -> slice_out_ggga(.(X, X1), 1, 1, .(X, [])) slice_in_ggga(.(X, Xs), 1, K, .(X, Ys)) -> U1_ggga(X, Xs, K, Ys, >_in_gg(K, 1)) >_in_gg(X0, X1) -> >_out_gg(X0, X1) U1_ggga(X, Xs, K, Ys, >_out_gg(K, 1)) -> U2_ggga(X, Xs, K, Ys, is_in_ag(K1, -(K, 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U2_ggga(X, Xs, K, Ys, is_out_ag(K1, -(K, 1))) -> U3_ggga(X, Xs, K, Ys, K1, slice_in_ggaa(Xs, 1, K1, Ys)) slice_in_ggaa(.(X, X1), 1, 1, .(X, [])) -> slice_out_ggaa(.(X, X1), 1, 1, .(X, [])) slice_in_ggaa(.(X, Xs), 1, K, .(X, Ys)) -> U1_ggaa(X, Xs, K, Ys, >_in_ag(K, 1)) >_in_ag(X0, X1) -> >_out_ag(X0, X1) U1_ggaa(X, Xs, K, Ys, >_out_ag(K, 1)) -> U2_ggaa(X, Xs, K, Ys, is_in_ag(K1, -(K, 1))) U2_ggaa(X, Xs, K, Ys, is_out_ag(K1, -(K, 1))) -> U3_ggaa(X, Xs, K, Ys, K1, slice_in_ggaa(Xs, 1, K1, Ys)) slice_in_ggaa(.(X2, Xs), I, K, Ys) -> U4_ggaa(X2, Xs, I, K, Ys, >_in_gg(I, 1)) U4_ggaa(X2, Xs, I, K, Ys, >_out_gg(I, 1)) -> U5_ggaa(X2, Xs, I, K, Ys, is_in_ag(I1, -(I, 1))) U5_ggaa(X2, Xs, I, K, Ys, is_out_ag(I1, -(I, 1))) -> U6_ggaa(X2, Xs, I, K, Ys, I1, is_in_ag(K1, -(K, 1))) U6_ggaa(X2, Xs, I, K, Ys, I1, is_out_ag(K1, -(K, 1))) -> U7_ggaa(X2, Xs, I, K, Ys, I1, K1, slice_in_gaaa(Xs, I1, K1, Ys)) slice_in_gaaa(.(X, X1), 1, 1, .(X, [])) -> slice_out_gaaa(.(X, X1), 1, 1, .(X, [])) slice_in_gaaa(.(X, Xs), 1, K, .(X, Ys)) -> U1_gaaa(X, Xs, K, Ys, >_in_ag(K, 1)) U1_gaaa(X, Xs, K, Ys, >_out_ag(K, 1)) -> U2_gaaa(X, Xs, K, Ys, is_in_ag(K1, -(K, 1))) U2_gaaa(X, Xs, K, Ys, is_out_ag(K1, -(K, 1))) -> U3_gaaa(X, Xs, K, Ys, K1, slice_in_ggaa(Xs, 1, K1, Ys)) U3_gaaa(X, Xs, K, Ys, K1, slice_out_ggaa(Xs, 1, K1, Ys)) -> slice_out_gaaa(.(X, Xs), 1, K, .(X, Ys)) slice_in_gaaa(.(X2, Xs), I, K, Ys) -> U4_gaaa(X2, Xs, I, K, Ys, >_in_ag(I, 1)) U4_gaaa(X2, Xs, I, K, Ys, >_out_ag(I, 1)) -> U5_gaaa(X2, Xs, I, K, Ys, is_in_ag(I1, -(I, 1))) U5_gaaa(X2, Xs, I, K, Ys, is_out_ag(I1, -(I, 1))) -> U6_gaaa(X2, Xs, I, K, Ys, I1, is_in_ag(K1, -(K, 1))) U6_gaaa(X2, Xs, I, K, Ys, I1, is_out_ag(K1, -(K, 1))) -> U7_gaaa(X2, Xs, I, K, Ys, I1, K1, slice_in_gaaa(Xs, I1, K1, Ys)) U7_gaaa(X2, Xs, I, K, Ys, I1, K1, slice_out_gaaa(Xs, I1, K1, Ys)) -> slice_out_gaaa(.(X2, Xs), I, K, Ys) U7_ggaa(X2, Xs, I, K, Ys, I1, K1, slice_out_gaaa(Xs, I1, K1, Ys)) -> slice_out_ggaa(.(X2, Xs), I, K, Ys) U3_ggaa(X, Xs, K, Ys, K1, slice_out_ggaa(Xs, 1, K1, Ys)) -> slice_out_ggaa(.(X, Xs), 1, K, .(X, Ys)) U3_ggga(X, Xs, K, Ys, K1, slice_out_ggaa(Xs, 1, K1, Ys)) -> slice_out_ggga(.(X, Xs), 1, K, .(X, Ys)) slice_in_ggga(.(X2, Xs), I, K, Ys) -> U4_ggga(X2, Xs, I, K, Ys, >_in_gg(I, 1)) U4_ggga(X2, Xs, I, K, Ys, >_out_gg(I, 1)) -> U5_ggga(X2, Xs, I, K, Ys, is_in_ag(I1, -(I, 1))) U5_ggga(X2, Xs, I, K, Ys, is_out_ag(I1, -(I, 1))) -> U6_ggga(X2, Xs, I, K, Ys, I1, is_in_ag(K1, -(K, 1))) U6_ggga(X2, Xs, I, K, Ys, I1, is_out_ag(K1, -(K, 1))) -> U7_ggga(X2, Xs, I, K, Ys, I1, K1, slice_in_gaaa(Xs, I1, K1, Ys)) U7_ggga(X2, Xs, I, K, Ys, I1, K1, slice_out_gaaa(Xs, I1, K1, Ys)) -> slice_out_ggga(.(X2, Xs), I, K, Ys) The argument filtering Pi contains the following mapping: slice_in_ggga(x1, x2, x3, x4) = slice_in_ggga(x1, x2, x3) .(x1, x2) = .(x1, x2) 1 = 1 slice_out_ggga(x1, x2, x3, x4) = slice_out_ggga(x4) U1_ggga(x1, x2, x3, x4, x5) = U1_ggga(x1, x2, x5) >_in_gg(x1, x2) = >_in_gg(x1, x2) >_out_gg(x1, x2) = >_out_gg U2_ggga(x1, x2, x3, x4, x5) = U2_ggga(x1, x2, x5) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag -(x1, x2) = -(x2) U3_ggga(x1, x2, x3, x4, x5, x6) = U3_ggga(x1, x6) slice_in_ggaa(x1, x2, x3, x4) = slice_in_ggaa(x1, x2) slice_out_ggaa(x1, x2, x3, x4) = slice_out_ggaa(x4) U1_ggaa(x1, x2, x3, x4, x5) = U1_ggaa(x1, x2, x5) >_in_ag(x1, x2) = >_in_ag(x2) >_out_ag(x1, x2) = >_out_ag U2_ggaa(x1, x2, x3, x4, x5) = U2_ggaa(x1, x2, x5) U3_ggaa(x1, x2, x3, x4, x5, x6) = U3_ggaa(x1, x6) U4_ggaa(x1, x2, x3, x4, x5, x6) = U4_ggaa(x2, x6) U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x2, x6) U6_ggaa(x1, x2, x3, x4, x5, x6, x7) = U6_ggaa(x2, x7) U7_ggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U7_ggaa(x8) slice_in_gaaa(x1, x2, x3, x4) = slice_in_gaaa(x1) slice_out_gaaa(x1, x2, x3, x4) = slice_out_gaaa(x4) U1_gaaa(x1, x2, x3, x4, x5) = U1_gaaa(x1, x2, x5) U2_gaaa(x1, x2, x3, x4, x5) = U2_gaaa(x1, x2, x5) U3_gaaa(x1, x2, x3, x4, x5, x6) = U3_gaaa(x1, x6) U4_gaaa(x1, x2, x3, x4, x5, x6) = U4_gaaa(x2, x6) U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x2, x6) U6_gaaa(x1, x2, x3, x4, x5, x6, x7) = U6_gaaa(x2, x7) U7_gaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U7_gaaa(x8) U4_ggga(x1, x2, x3, x4, x5, x6) = U4_ggga(x2, x6) U5_ggga(x1, x2, x3, x4, x5, x6) = U5_ggga(x2, x6) U6_ggga(x1, x2, x3, x4, x5, x6, x7) = U6_ggga(x2, x7) U7_ggga(x1, x2, x3, x4, x5, x6, x7, x8) = U7_ggga(x8) SLICE_IN_GGGA(x1, x2, x3, x4) = SLICE_IN_GGGA(x1, x2, x3) U1_GGGA(x1, x2, x3, x4, x5) = U1_GGGA(x1, x2, x5) >_IN_GG(x1, x2) = >_IN_GG(x1, x2) U2_GGGA(x1, x2, x3, x4, x5) = U2_GGGA(x1, x2, x5) IS_IN_AG(x1, x2) = IS_IN_AG(x2) U3_GGGA(x1, x2, x3, x4, x5, x6) = U3_GGGA(x1, x6) SLICE_IN_GGAA(x1, x2, x3, x4) = SLICE_IN_GGAA(x1, x2) U1_GGAA(x1, x2, x3, x4, x5) = U1_GGAA(x1, x2, x5) >_IN_AG(x1, x2) = >_IN_AG(x2) U2_GGAA(x1, x2, x3, x4, x5) = U2_GGAA(x1, x2, x5) U3_GGAA(x1, x2, x3, x4, x5, x6) = U3_GGAA(x1, x6) U4_GGAA(x1, x2, x3, x4, x5, x6) = U4_GGAA(x2, x6) U5_GGAA(x1, x2, x3, x4, x5, x6) = U5_GGAA(x2, x6) U6_GGAA(x1, x2, x3, x4, x5, x6, x7) = U6_GGAA(x2, x7) U7_GGAA(x1, x2, x3, x4, x5, x6, x7, x8) = U7_GGAA(x8) SLICE_IN_GAAA(x1, x2, x3, x4) = SLICE_IN_GAAA(x1) U1_GAAA(x1, x2, x3, x4, x5) = U1_GAAA(x1, x2, x5) U2_GAAA(x1, x2, x3, x4, x5) = U2_GAAA(x1, x2, x5) U3_GAAA(x1, x2, x3, x4, x5, x6) = U3_GAAA(x1, x6) U4_GAAA(x1, x2, x3, x4, x5, x6) = U4_GAAA(x2, x6) U5_GAAA(x1, x2, x3, x4, x5, x6) = U5_GAAA(x2, x6) U6_GAAA(x1, x2, x3, x4, x5, x6, x7) = U6_GAAA(x2, x7) U7_GAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U7_GAAA(x8) U4_GGGA(x1, x2, x3, x4, x5, x6) = U4_GGGA(x2, x6) U5_GGGA(x1, x2, x3, x4, x5, x6) = U5_GGGA(x2, x6) U6_GGGA(x1, x2, x3, x4, x5, x6, x7) = U6_GGGA(x2, x7) U7_GGGA(x1, x2, x3, x4, x5, x6, x7, x8) = U7_GGGA(x8) We have to consider all (P,R,Pi)-chains ---------------------------------------- (7) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 28 less nodes. ---------------------------------------- (8) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_GGAA(X, Xs, K, Ys, >_out_ag(K, 1)) -> U2_GGAA(X, Xs, K, Ys, is_in_ag(K1, -(K, 1))) U2_GGAA(X, Xs, K, Ys, is_out_ag(K1, -(K, 1))) -> SLICE_IN_GGAA(Xs, 1, K1, Ys) SLICE_IN_GGAA(.(X, Xs), 1, K, .(X, Ys)) -> U1_GGAA(X, Xs, K, Ys, >_in_ag(K, 1)) SLICE_IN_GGAA(.(X2, Xs), I, K, Ys) -> U4_GGAA(X2, Xs, I, K, Ys, >_in_gg(I, 1)) U4_GGAA(X2, Xs, I, K, Ys, >_out_gg(I, 1)) -> U5_GGAA(X2, Xs, I, K, Ys, is_in_ag(I1, -(I, 1))) U5_GGAA(X2, Xs, I, K, Ys, is_out_ag(I1, -(I, 1))) -> U6_GGAA(X2, Xs, I, K, Ys, I1, is_in_ag(K1, -(K, 1))) U6_GGAA(X2, Xs, I, K, Ys, I1, is_out_ag(K1, -(K, 1))) -> SLICE_IN_GAAA(Xs, I1, K1, Ys) SLICE_IN_GAAA(.(X, Xs), 1, K, .(X, Ys)) -> U1_GAAA(X, Xs, K, Ys, >_in_ag(K, 1)) U1_GAAA(X, Xs, K, Ys, >_out_ag(K, 1)) -> U2_GAAA(X, Xs, K, Ys, is_in_ag(K1, -(K, 1))) U2_GAAA(X, Xs, K, Ys, is_out_ag(K1, -(K, 1))) -> SLICE_IN_GGAA(Xs, 1, K1, Ys) SLICE_IN_GAAA(.(X2, Xs), I, K, Ys) -> U4_GAAA(X2, Xs, I, K, Ys, >_in_ag(I, 1)) U4_GAAA(X2, Xs, I, K, Ys, >_out_ag(I, 1)) -> U5_GAAA(X2, Xs, I, K, Ys, is_in_ag(I1, -(I, 1))) U5_GAAA(X2, Xs, I, K, Ys, is_out_ag(I1, -(I, 1))) -> U6_GAAA(X2, Xs, I, K, Ys, I1, is_in_ag(K1, -(K, 1))) U6_GAAA(X2, Xs, I, K, Ys, I1, is_out_ag(K1, -(K, 1))) -> SLICE_IN_GAAA(Xs, I1, K1, Ys) The TRS R consists of the following rules: slice_in_ggga(.(X, X1), 1, 1, .(X, [])) -> slice_out_ggga(.(X, X1), 1, 1, .(X, [])) slice_in_ggga(.(X, Xs), 1, K, .(X, Ys)) -> U1_ggga(X, Xs, K, Ys, >_in_gg(K, 1)) >_in_gg(X0, X1) -> >_out_gg(X0, X1) U1_ggga(X, Xs, K, Ys, >_out_gg(K, 1)) -> U2_ggga(X, Xs, K, Ys, is_in_ag(K1, -(K, 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U2_ggga(X, Xs, K, Ys, is_out_ag(K1, -(K, 1))) -> U3_ggga(X, Xs, K, Ys, K1, slice_in_ggaa(Xs, 1, K1, Ys)) slice_in_ggaa(.(X, X1), 1, 1, .(X, [])) -> slice_out_ggaa(.(X, X1), 1, 1, .(X, [])) slice_in_ggaa(.(X, Xs), 1, K, .(X, Ys)) -> U1_ggaa(X, Xs, K, Ys, >_in_ag(K, 1)) >_in_ag(X0, X1) -> >_out_ag(X0, X1) U1_ggaa(X, Xs, K, Ys, >_out_ag(K, 1)) -> U2_ggaa(X, Xs, K, Ys, is_in_ag(K1, -(K, 1))) U2_ggaa(X, Xs, K, Ys, is_out_ag(K1, -(K, 1))) -> U3_ggaa(X, Xs, K, Ys, K1, slice_in_ggaa(Xs, 1, K1, Ys)) slice_in_ggaa(.(X2, Xs), I, K, Ys) -> U4_ggaa(X2, Xs, I, K, Ys, >_in_gg(I, 1)) U4_ggaa(X2, Xs, I, K, Ys, >_out_gg(I, 1)) -> U5_ggaa(X2, Xs, I, K, Ys, is_in_ag(I1, -(I, 1))) U5_ggaa(X2, Xs, I, K, Ys, is_out_ag(I1, -(I, 1))) -> U6_ggaa(X2, Xs, I, K, Ys, I1, is_in_ag(K1, -(K, 1))) U6_ggaa(X2, Xs, I, K, Ys, I1, is_out_ag(K1, -(K, 1))) -> U7_ggaa(X2, Xs, I, K, Ys, I1, K1, slice_in_gaaa(Xs, I1, K1, Ys)) slice_in_gaaa(.(X, X1), 1, 1, .(X, [])) -> slice_out_gaaa(.(X, X1), 1, 1, .(X, [])) slice_in_gaaa(.(X, Xs), 1, K, .(X, Ys)) -> U1_gaaa(X, Xs, K, Ys, >_in_ag(K, 1)) U1_gaaa(X, Xs, K, Ys, >_out_ag(K, 1)) -> U2_gaaa(X, Xs, K, Ys, is_in_ag(K1, -(K, 1))) U2_gaaa(X, Xs, K, Ys, is_out_ag(K1, -(K, 1))) -> U3_gaaa(X, Xs, K, Ys, K1, slice_in_ggaa(Xs, 1, K1, Ys)) U3_gaaa(X, Xs, K, Ys, K1, slice_out_ggaa(Xs, 1, K1, Ys)) -> slice_out_gaaa(.(X, Xs), 1, K, .(X, Ys)) slice_in_gaaa(.(X2, Xs), I, K, Ys) -> U4_gaaa(X2, Xs, I, K, Ys, >_in_ag(I, 1)) U4_gaaa(X2, Xs, I, K, Ys, >_out_ag(I, 1)) -> U5_gaaa(X2, Xs, I, K, Ys, is_in_ag(I1, -(I, 1))) U5_gaaa(X2, Xs, I, K, Ys, is_out_ag(I1, -(I, 1))) -> U6_gaaa(X2, Xs, I, K, Ys, I1, is_in_ag(K1, -(K, 1))) U6_gaaa(X2, Xs, I, K, Ys, I1, is_out_ag(K1, -(K, 1))) -> U7_gaaa(X2, Xs, I, K, Ys, I1, K1, slice_in_gaaa(Xs, I1, K1, Ys)) U7_gaaa(X2, Xs, I, K, Ys, I1, K1, slice_out_gaaa(Xs, I1, K1, Ys)) -> slice_out_gaaa(.(X2, Xs), I, K, Ys) U7_ggaa(X2, Xs, I, K, Ys, I1, K1, slice_out_gaaa(Xs, I1, K1, Ys)) -> slice_out_ggaa(.(X2, Xs), I, K, Ys) U3_ggaa(X, Xs, K, Ys, K1, slice_out_ggaa(Xs, 1, K1, Ys)) -> slice_out_ggaa(.(X, Xs), 1, K, .(X, Ys)) U3_ggga(X, Xs, K, Ys, K1, slice_out_ggaa(Xs, 1, K1, Ys)) -> slice_out_ggga(.(X, Xs), 1, K, .(X, Ys)) slice_in_ggga(.(X2, Xs), I, K, Ys) -> U4_ggga(X2, Xs, I, K, Ys, >_in_gg(I, 1)) U4_ggga(X2, Xs, I, K, Ys, >_out_gg(I, 1)) -> U5_ggga(X2, Xs, I, K, Ys, is_in_ag(I1, -(I, 1))) U5_ggga(X2, Xs, I, K, Ys, is_out_ag(I1, -(I, 1))) -> U6_ggga(X2, Xs, I, K, Ys, I1, is_in_ag(K1, -(K, 1))) U6_ggga(X2, Xs, I, K, Ys, I1, is_out_ag(K1, -(K, 1))) -> U7_ggga(X2, Xs, I, K, Ys, I1, K1, slice_in_gaaa(Xs, I1, K1, Ys)) U7_ggga(X2, Xs, I, K, Ys, I1, K1, slice_out_gaaa(Xs, I1, K1, Ys)) -> slice_out_ggga(.(X2, Xs), I, K, Ys) The argument filtering Pi contains the following mapping: slice_in_ggga(x1, x2, x3, x4) = slice_in_ggga(x1, x2, x3) .(x1, x2) = .(x1, x2) 1 = 1 slice_out_ggga(x1, x2, x3, x4) = slice_out_ggga(x4) U1_ggga(x1, x2, x3, x4, x5) = U1_ggga(x1, x2, x5) >_in_gg(x1, x2) = >_in_gg(x1, x2) >_out_gg(x1, x2) = >_out_gg U2_ggga(x1, x2, x3, x4, x5) = U2_ggga(x1, x2, x5) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag -(x1, x2) = -(x2) U3_ggga(x1, x2, x3, x4, x5, x6) = U3_ggga(x1, x6) slice_in_ggaa(x1, x2, x3, x4) = slice_in_ggaa(x1, x2) slice_out_ggaa(x1, x2, x3, x4) = slice_out_ggaa(x4) U1_ggaa(x1, x2, x3, x4, x5) = U1_ggaa(x1, x2, x5) >_in_ag(x1, x2) = >_in_ag(x2) >_out_ag(x1, x2) = >_out_ag U2_ggaa(x1, x2, x3, x4, x5) = U2_ggaa(x1, x2, x5) U3_ggaa(x1, x2, x3, x4, x5, x6) = U3_ggaa(x1, x6) U4_ggaa(x1, x2, x3, x4, x5, x6) = U4_ggaa(x2, x6) U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x2, x6) U6_ggaa(x1, x2, x3, x4, x5, x6, x7) = U6_ggaa(x2, x7) U7_ggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U7_ggaa(x8) slice_in_gaaa(x1, x2, x3, x4) = slice_in_gaaa(x1) slice_out_gaaa(x1, x2, x3, x4) = slice_out_gaaa(x4) U1_gaaa(x1, x2, x3, x4, x5) = U1_gaaa(x1, x2, x5) U2_gaaa(x1, x2, x3, x4, x5) = U2_gaaa(x1, x2, x5) U3_gaaa(x1, x2, x3, x4, x5, x6) = U3_gaaa(x1, x6) U4_gaaa(x1, x2, x3, x4, x5, x6) = U4_gaaa(x2, x6) U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x2, x6) U6_gaaa(x1, x2, x3, x4, x5, x6, x7) = U6_gaaa(x2, x7) U7_gaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U7_gaaa(x8) U4_ggga(x1, x2, x3, x4, x5, x6) = U4_ggga(x2, x6) U5_ggga(x1, x2, x3, x4, x5, x6) = U5_ggga(x2, x6) U6_ggga(x1, x2, x3, x4, x5, x6, x7) = U6_ggga(x2, x7) U7_ggga(x1, x2, x3, x4, x5, x6, x7, x8) = U7_ggga(x8) SLICE_IN_GGAA(x1, x2, x3, x4) = SLICE_IN_GGAA(x1, x2) U1_GGAA(x1, x2, x3, x4, x5) = U1_GGAA(x1, x2, x5) U2_GGAA(x1, x2, x3, x4, x5) = U2_GGAA(x1, x2, x5) U4_GGAA(x1, x2, x3, x4, x5, x6) = U4_GGAA(x2, x6) U5_GGAA(x1, x2, x3, x4, x5, x6) = U5_GGAA(x2, x6) U6_GGAA(x1, x2, x3, x4, x5, x6, x7) = U6_GGAA(x2, x7) SLICE_IN_GAAA(x1, x2, x3, x4) = SLICE_IN_GAAA(x1) U1_GAAA(x1, x2, x3, x4, x5) = U1_GAAA(x1, x2, x5) U2_GAAA(x1, x2, x3, x4, x5) = U2_GAAA(x1, x2, x5) U4_GAAA(x1, x2, x3, x4, x5, x6) = U4_GAAA(x2, x6) U5_GAAA(x1, x2, x3, x4, x5, x6) = U5_GAAA(x2, x6) U6_GAAA(x1, x2, x3, x4, x5, x6, x7) = U6_GAAA(x2, x7) 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_GGAA(X, Xs, K, Ys, >_out_ag(K, 1)) -> U2_GGAA(X, Xs, K, Ys, is_in_ag(K1, -(K, 1))) U2_GGAA(X, Xs, K, Ys, is_out_ag(K1, -(K, 1))) -> SLICE_IN_GGAA(Xs, 1, K1, Ys) SLICE_IN_GGAA(.(X, Xs), 1, K, .(X, Ys)) -> U1_GGAA(X, Xs, K, Ys, >_in_ag(K, 1)) SLICE_IN_GGAA(.(X2, Xs), I, K, Ys) -> U4_GGAA(X2, Xs, I, K, Ys, >_in_gg(I, 1)) U4_GGAA(X2, Xs, I, K, Ys, >_out_gg(I, 1)) -> U5_GGAA(X2, Xs, I, K, Ys, is_in_ag(I1, -(I, 1))) U5_GGAA(X2, Xs, I, K, Ys, is_out_ag(I1, -(I, 1))) -> U6_GGAA(X2, Xs, I, K, Ys, I1, is_in_ag(K1, -(K, 1))) U6_GGAA(X2, Xs, I, K, Ys, I1, is_out_ag(K1, -(K, 1))) -> SLICE_IN_GAAA(Xs, I1, K1, Ys) SLICE_IN_GAAA(.(X, Xs), 1, K, .(X, Ys)) -> U1_GAAA(X, Xs, K, Ys, >_in_ag(K, 1)) U1_GAAA(X, Xs, K, Ys, >_out_ag(K, 1)) -> U2_GAAA(X, Xs, K, Ys, is_in_ag(K1, -(K, 1))) U2_GAAA(X, Xs, K, Ys, is_out_ag(K1, -(K, 1))) -> SLICE_IN_GGAA(Xs, 1, K1, Ys) SLICE_IN_GAAA(.(X2, Xs), I, K, Ys) -> U4_GAAA(X2, Xs, I, K, Ys, >_in_ag(I, 1)) U4_GAAA(X2, Xs, I, K, Ys, >_out_ag(I, 1)) -> U5_GAAA(X2, Xs, I, K, Ys, is_in_ag(I1, -(I, 1))) U5_GAAA(X2, Xs, I, K, Ys, is_out_ag(I1, -(I, 1))) -> U6_GAAA(X2, Xs, I, K, Ys, I1, is_in_ag(K1, -(K, 1))) U6_GAAA(X2, Xs, I, K, Ys, I1, is_out_ag(K1, -(K, 1))) -> SLICE_IN_GAAA(Xs, I1, K1, Ys) 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) >_in_gg(X0, X1) -> >_out_gg(X0, X1) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) 1 = 1 >_in_gg(x1, x2) = >_in_gg(x1, x2) >_out_gg(x1, x2) = >_out_gg is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag -(x1, x2) = -(x2) >_in_ag(x1, x2) = >_in_ag(x2) >_out_ag(x1, x2) = >_out_ag SLICE_IN_GGAA(x1, x2, x3, x4) = SLICE_IN_GGAA(x1, x2) U1_GGAA(x1, x2, x3, x4, x5) = U1_GGAA(x1, x2, x5) U2_GGAA(x1, x2, x3, x4, x5) = U2_GGAA(x1, x2, x5) U4_GGAA(x1, x2, x3, x4, x5, x6) = U4_GGAA(x2, x6) U5_GGAA(x1, x2, x3, x4, x5, x6) = U5_GGAA(x2, x6) U6_GGAA(x1, x2, x3, x4, x5, x6, x7) = U6_GGAA(x2, x7) SLICE_IN_GAAA(x1, x2, x3, x4) = SLICE_IN_GAAA(x1) U1_GAAA(x1, x2, x3, x4, x5) = U1_GAAA(x1, x2, x5) U2_GAAA(x1, x2, x3, x4, x5) = U2_GAAA(x1, x2, x5) U4_GAAA(x1, x2, x3, x4, x5, x6) = U4_GAAA(x2, x6) U5_GAAA(x1, x2, x3, x4, x5, x6) = U5_GAAA(x2, x6) U6_GAAA(x1, x2, x3, x4, x5, x6, x7) = U6_GAAA(x2, x7) 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_GGAA(X, Xs, >_out_ag) -> U2_GGAA(X, Xs, is_in_ag(-(1))) U2_GGAA(X, Xs, is_out_ag) -> SLICE_IN_GGAA(Xs, 1) SLICE_IN_GGAA(.(X, Xs), 1) -> U1_GGAA(X, Xs, >_in_ag(1)) SLICE_IN_GGAA(.(X2, Xs), I) -> U4_GGAA(Xs, >_in_gg(I, 1)) U4_GGAA(Xs, >_out_gg) -> U5_GGAA(Xs, is_in_ag(-(1))) U5_GGAA(Xs, is_out_ag) -> U6_GGAA(Xs, is_in_ag(-(1))) U6_GGAA(Xs, is_out_ag) -> SLICE_IN_GAAA(Xs) SLICE_IN_GAAA(.(X, Xs)) -> U1_GAAA(X, Xs, >_in_ag(1)) U1_GAAA(X, Xs, >_out_ag) -> U2_GAAA(X, Xs, is_in_ag(-(1))) U2_GAAA(X, Xs, is_out_ag) -> SLICE_IN_GGAA(Xs, 1) SLICE_IN_GAAA(.(X2, Xs)) -> U4_GAAA(Xs, >_in_ag(1)) U4_GAAA(Xs, >_out_ag) -> U5_GAAA(Xs, is_in_ag(-(1))) U5_GAAA(Xs, is_out_ag) -> U6_GAAA(Xs, is_in_ag(-(1))) U6_GAAA(Xs, is_out_ag) -> SLICE_IN_GAAA(Xs) The TRS R consists of the following rules: is_in_ag(X1) -> is_out_ag >_in_ag(X1) -> >_out_ag >_in_gg(X0, X1) -> >_out_gg The set Q consists of the following terms: is_in_ag(x0) >_in_ag(x0) >_in_gg(x0, x1) 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_GGAA(X, Xs, is_out_ag) -> SLICE_IN_GGAA(Xs, 1) The graph contains the following edges 2 >= 1 *SLICE_IN_GGAA(.(X, Xs), 1) -> U1_GGAA(X, Xs, >_in_ag(1)) The graph contains the following edges 1 > 1, 1 > 2 *SLICE_IN_GGAA(.(X2, Xs), I) -> U4_GGAA(Xs, >_in_gg(I, 1)) The graph contains the following edges 1 > 1 *U1_GGAA(X, Xs, >_out_ag) -> U2_GGAA(X, Xs, is_in_ag(-(1))) The graph contains the following edges 1 >= 1, 2 >= 2 *U2_GAAA(X, Xs, is_out_ag) -> SLICE_IN_GGAA(Xs, 1) The graph contains the following edges 2 >= 1 *U4_GGAA(Xs, >_out_gg) -> U5_GGAA(Xs, is_in_ag(-(1))) The graph contains the following edges 1 >= 1 *U5_GGAA(Xs, is_out_ag) -> U6_GGAA(Xs, is_in_ag(-(1))) The graph contains the following edges 1 >= 1 *U6_GGAA(Xs, is_out_ag) -> SLICE_IN_GAAA(Xs) The graph contains the following edges 1 >= 1 *U1_GAAA(X, Xs, >_out_ag) -> U2_GAAA(X, Xs, is_in_ag(-(1))) The graph contains the following edges 1 >= 1, 2 >= 2 *U6_GAAA(Xs, is_out_ag) -> SLICE_IN_GAAA(Xs) The graph contains the following edges 1 >= 1 *SLICE_IN_GAAA(.(X, Xs)) -> U1_GAAA(X, Xs, >_in_ag(1)) The graph contains the following edges 1 > 1, 1 > 2 *SLICE_IN_GAAA(.(X2, Xs)) -> U4_GAAA(Xs, >_in_ag(1)) The graph contains the following edges 1 > 1 *U4_GAAA(Xs, >_out_ag) -> U5_GAAA(Xs, is_in_ag(-(1))) The graph contains the following edges 1 >= 1 *U5_GAAA(Xs, is_out_ag) -> U6_GAAA(Xs, is_in_ag(-(1))) The graph contains the following edges 1 >= 1 ---------------------------------------- (14) YES