WORST_CASE(Omega(n^1),?) proof of /export/starexec/sandbox2/benchmark/theBenchmark.trs # AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). (0) CpxRelTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (4) typed CpxTrs (5) OrderProof [LOWER BOUND(ID), 4 ms] (6) typed CpxTrs (7) RewriteLemmaProof [LOWER BOUND(ID), 475 ms] (8) BEST (9) proven lower bound (10) LowerBoundPropagationProof [FINISHED, 0 ms] (11) BOUNDS(n^1, INF) (12) typed CpxTrs (13) RewriteLemmaProof [LOWER BOUND(ID), 50 ms] (14) BOUNDS(1, INF) ---------------------------------------- (0) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: ZEROS -> c ZEROS -> c1 U11'(tt, z0) -> c2(U12'(tt, activate(z0)), ACTIVATE(z0)) U12'(tt, z0) -> c3(LENGTH(activate(z0)), ACTIVATE(z0)) U21'(tt, z0, z1, z2) -> c4(U22'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z0)) U21'(tt, z0, z1, z2) -> c5(U22'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z1)) U21'(tt, z0, z1, z2) -> c6(U22'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z2)) U22'(tt, z0, z1, z2) -> c7(U23'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z0)) U22'(tt, z0, z1, z2) -> c8(U23'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z1)) U22'(tt, z0, z1, z2) -> c9(U23'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z2)) U23'(tt, z0, z1, z2) -> c10(ACTIVATE(z2)) U23'(tt, z0, z1, z2) -> c11(ACTIVATE(z1)) U23'(tt, z0, z1, z2) -> c12(ACTIVATE(z0)) LENGTH(nil) -> c13 LENGTH(cons(z0, z1)) -> c14(U11'(tt, activate(z1)), ACTIVATE(z1)) TAKE(0, z0) -> c15 TAKE(s(z0), cons(z1, z2)) -> c16(U21'(tt, activate(z2), z0, z1), ACTIVATE(z2)) TAKE(z0, z1) -> c17 ACTIVATE(n__zeros) -> c18(ZEROS) ACTIVATE(n__take(z0, z1)) -> c19(TAKE(z0, z1)) ACTIVATE(z0) -> c20 The (relative) TRS S consists of the following rules: zeros -> cons(0, n__zeros) zeros -> n__zeros U11(tt, z0) -> U12(tt, activate(z0)) U12(tt, z0) -> s(length(activate(z0))) U21(tt, z0, z1, z2) -> U22(tt, activate(z0), activate(z1), activate(z2)) U22(tt, z0, z1, z2) -> U23(tt, activate(z0), activate(z1), activate(z2)) U23(tt, z0, z1, z2) -> cons(activate(z2), n__take(activate(z1), activate(z0))) length(nil) -> 0 length(cons(z0, z1)) -> U11(tt, activate(z1)) take(0, z0) -> nil take(s(z0), cons(z1, z2)) -> U21(tt, activate(z2), z0, z1) take(z0, z1) -> n__take(z0, z1) activate(n__zeros) -> zeros activate(n__take(z0, z1)) -> take(z0, z1) activate(z0) -> z0 Rewrite Strategy: INNERMOST ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: ZEROS -> c ZEROS -> c1 U11'(tt, z0) -> c2(U12'(tt, activate(z0)), ACTIVATE(z0)) U12'(tt, z0) -> c3(LENGTH(activate(z0)), ACTIVATE(z0)) U21'(tt, z0, z1, z2) -> c4(U22'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z0)) U21'(tt, z0, z1, z2) -> c5(U22'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z1)) U21'(tt, z0, z1, z2) -> c6(U22'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z2)) U22'(tt, z0, z1, z2) -> c7(U23'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z0)) U22'(tt, z0, z1, z2) -> c8(U23'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z1)) U22'(tt, z0, z1, z2) -> c9(U23'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z2)) U23'(tt, z0, z1, z2) -> c10(ACTIVATE(z2)) U23'(tt, z0, z1, z2) -> c11(ACTIVATE(z1)) U23'(tt, z0, z1, z2) -> c12(ACTIVATE(z0)) LENGTH(nil) -> c13 LENGTH(cons(z0, z1)) -> c14(U11'(tt, activate(z1)), ACTIVATE(z1)) TAKE(0', z0) -> c15 TAKE(s(z0), cons(z1, z2)) -> c16(U21'(tt, activate(z2), z0, z1), ACTIVATE(z2)) TAKE(z0, z1) -> c17 ACTIVATE(n__zeros) -> c18(ZEROS) ACTIVATE(n__take(z0, z1)) -> c19(TAKE(z0, z1)) ACTIVATE(z0) -> c20 The (relative) TRS S consists of the following rules: zeros -> cons(0', n__zeros) zeros -> n__zeros U11(tt, z0) -> U12(tt, activate(z0)) U12(tt, z0) -> s(length(activate(z0))) U21(tt, z0, z1, z2) -> U22(tt, activate(z0), activate(z1), activate(z2)) U22(tt, z0, z1, z2) -> U23(tt, activate(z0), activate(z1), activate(z2)) U23(tt, z0, z1, z2) -> cons(activate(z2), n__take(activate(z1), activate(z0))) length(nil) -> 0' length(cons(z0, z1)) -> U11(tt, activate(z1)) take(0', z0) -> nil take(s(z0), cons(z1, z2)) -> U21(tt, activate(z2), z0, z1) take(z0, z1) -> n__take(z0, z1) activate(n__zeros) -> zeros activate(n__take(z0, z1)) -> take(z0, z1) activate(z0) -> z0 Rewrite Strategy: INNERMOST ---------------------------------------- (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (4) Obligation: Innermost TRS: Rules: ZEROS -> c ZEROS -> c1 U11'(tt, z0) -> c2(U12'(tt, activate(z0)), ACTIVATE(z0)) U12'(tt, z0) -> c3(LENGTH(activate(z0)), ACTIVATE(z0)) U21'(tt, z0, z1, z2) -> c4(U22'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z0)) U21'(tt, z0, z1, z2) -> c5(U22'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z1)) U21'(tt, z0, z1, z2) -> c6(U22'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z2)) U22'(tt, z0, z1, z2) -> c7(U23'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z0)) U22'(tt, z0, z1, z2) -> c8(U23'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z1)) U22'(tt, z0, z1, z2) -> c9(U23'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z2)) U23'(tt, z0, z1, z2) -> c10(ACTIVATE(z2)) U23'(tt, z0, z1, z2) -> c11(ACTIVATE(z1)) U23'(tt, z0, z1, z2) -> c12(ACTIVATE(z0)) LENGTH(nil) -> c13 LENGTH(cons(z0, z1)) -> c14(U11'(tt, activate(z1)), ACTIVATE(z1)) TAKE(0', z0) -> c15 TAKE(s(z0), cons(z1, z2)) -> c16(U21'(tt, activate(z2), z0, z1), ACTIVATE(z2)) TAKE(z0, z1) -> c17 ACTIVATE(n__zeros) -> c18(ZEROS) ACTIVATE(n__take(z0, z1)) -> c19(TAKE(z0, z1)) ACTIVATE(z0) -> c20 zeros -> cons(0', n__zeros) zeros -> n__zeros U11(tt, z0) -> U12(tt, activate(z0)) U12(tt, z0) -> s(length(activate(z0))) U21(tt, z0, z1, z2) -> U22(tt, activate(z0), activate(z1), activate(z2)) U22(tt, z0, z1, z2) -> U23(tt, activate(z0), activate(z1), activate(z2)) U23(tt, z0, z1, z2) -> cons(activate(z2), n__take(activate(z1), activate(z0))) length(nil) -> 0' length(cons(z0, z1)) -> U11(tt, activate(z1)) take(0', z0) -> nil take(s(z0), cons(z1, z2)) -> U21(tt, activate(z2), z0, z1) take(z0, z1) -> n__take(z0, z1) activate(n__zeros) -> zeros activate(n__take(z0, z1)) -> take(z0, z1) activate(z0) -> z0 Types: ZEROS :: c:c1 c :: c:c1 c1 :: c:c1 U11' :: tt -> nil:cons:0':s:n__zeros:n__take -> c2 tt :: tt c2 :: c3 -> c18:c19:c20 -> c2 U12' :: tt -> nil:cons:0':s:n__zeros:n__take -> c3 activate :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take ACTIVATE :: nil:cons:0':s:n__zeros:n__take -> c18:c19:c20 c3 :: c13:c14 -> c18:c19:c20 -> c3 LENGTH :: nil:cons:0':s:n__zeros:n__take -> c13:c14 U21' :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> c4:c5:c6 c4 :: c7:c8:c9 -> c18:c19:c20 -> c4:c5:c6 U22' :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> c7:c8:c9 c5 :: c7:c8:c9 -> c18:c19:c20 -> c4:c5:c6 c6 :: c7:c8:c9 -> c18:c19:c20 -> c4:c5:c6 c7 :: c10:c11:c12 -> c18:c19:c20 -> c7:c8:c9 U23' :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> c10:c11:c12 c8 :: c10:c11:c12 -> c18:c19:c20 -> c7:c8:c9 c9 :: c10:c11:c12 -> c18:c19:c20 -> c7:c8:c9 c10 :: c18:c19:c20 -> c10:c11:c12 c11 :: c18:c19:c20 -> c10:c11:c12 c12 :: c18:c19:c20 -> c10:c11:c12 nil :: nil:cons:0':s:n__zeros:n__take c13 :: c13:c14 cons :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take c14 :: c2 -> c18:c19:c20 -> c13:c14 TAKE :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> c15:c16:c17 0' :: nil:cons:0':s:n__zeros:n__take c15 :: c15:c16:c17 s :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take c16 :: c4:c5:c6 -> c18:c19:c20 -> c15:c16:c17 c17 :: c15:c16:c17 n__zeros :: nil:cons:0':s:n__zeros:n__take c18 :: c:c1 -> c18:c19:c20 n__take :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take c19 :: c15:c16:c17 -> c18:c19:c20 c20 :: c18:c19:c20 zeros :: nil:cons:0':s:n__zeros:n__take U11 :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take U12 :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take length :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take U21 :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take U22 :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take U23 :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take take :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take hole_c:c11_24 :: c:c1 hole_c22_24 :: c2 hole_tt3_24 :: tt hole_nil:cons:0':s:n__zeros:n__take4_24 :: nil:cons:0':s:n__zeros:n__take hole_c35_24 :: c3 hole_c18:c19:c206_24 :: c18:c19:c20 hole_c13:c147_24 :: c13:c14 hole_c4:c5:c68_24 :: c4:c5:c6 hole_c7:c8:c99_24 :: c7:c8:c9 hole_c10:c11:c1210_24 :: c10:c11:c12 hole_c15:c16:c1711_24 :: c15:c16:c17 gen_nil:cons:0':s:n__zeros:n__take12_24 :: Nat -> nil:cons:0':s:n__zeros:n__take ---------------------------------------- (5) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: activate, ACTIVATE, LENGTH, length They will be analysed ascendingly in the following order: activate < ACTIVATE activate < LENGTH activate < length ACTIVATE < LENGTH ---------------------------------------- (6) Obligation: Innermost TRS: Rules: ZEROS -> c ZEROS -> c1 U11'(tt, z0) -> c2(U12'(tt, activate(z0)), ACTIVATE(z0)) U12'(tt, z0) -> c3(LENGTH(activate(z0)), ACTIVATE(z0)) U21'(tt, z0, z1, z2) -> c4(U22'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z0)) U21'(tt, z0, z1, z2) -> c5(U22'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z1)) U21'(tt, z0, z1, z2) -> c6(U22'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z2)) U22'(tt, z0, z1, z2) -> c7(U23'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z0)) U22'(tt, z0, z1, z2) -> c8(U23'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z1)) U22'(tt, z0, z1, z2) -> c9(U23'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z2)) U23'(tt, z0, z1, z2) -> c10(ACTIVATE(z2)) U23'(tt, z0, z1, z2) -> c11(ACTIVATE(z1)) U23'(tt, z0, z1, z2) -> c12(ACTIVATE(z0)) LENGTH(nil) -> c13 LENGTH(cons(z0, z1)) -> c14(U11'(tt, activate(z1)), ACTIVATE(z1)) TAKE(0', z0) -> c15 TAKE(s(z0), cons(z1, z2)) -> c16(U21'(tt, activate(z2), z0, z1), ACTIVATE(z2)) TAKE(z0, z1) -> c17 ACTIVATE(n__zeros) -> c18(ZEROS) ACTIVATE(n__take(z0, z1)) -> c19(TAKE(z0, z1)) ACTIVATE(z0) -> c20 zeros -> cons(0', n__zeros) zeros -> n__zeros U11(tt, z0) -> U12(tt, activate(z0)) U12(tt, z0) -> s(length(activate(z0))) U21(tt, z0, z1, z2) -> U22(tt, activate(z0), activate(z1), activate(z2)) U22(tt, z0, z1, z2) -> U23(tt, activate(z0), activate(z1), activate(z2)) U23(tt, z0, z1, z2) -> cons(activate(z2), n__take(activate(z1), activate(z0))) length(nil) -> 0' length(cons(z0, z1)) -> U11(tt, activate(z1)) take(0', z0) -> nil take(s(z0), cons(z1, z2)) -> U21(tt, activate(z2), z0, z1) take(z0, z1) -> n__take(z0, z1) activate(n__zeros) -> zeros activate(n__take(z0, z1)) -> take(z0, z1) activate(z0) -> z0 Types: ZEROS :: c:c1 c :: c:c1 c1 :: c:c1 U11' :: tt -> nil:cons:0':s:n__zeros:n__take -> c2 tt :: tt c2 :: c3 -> c18:c19:c20 -> c2 U12' :: tt -> nil:cons:0':s:n__zeros:n__take -> c3 activate :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take ACTIVATE :: nil:cons:0':s:n__zeros:n__take -> c18:c19:c20 c3 :: c13:c14 -> c18:c19:c20 -> c3 LENGTH :: nil:cons:0':s:n__zeros:n__take -> c13:c14 U21' :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> c4:c5:c6 c4 :: c7:c8:c9 -> c18:c19:c20 -> c4:c5:c6 U22' :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> c7:c8:c9 c5 :: c7:c8:c9 -> c18:c19:c20 -> c4:c5:c6 c6 :: c7:c8:c9 -> c18:c19:c20 -> c4:c5:c6 c7 :: c10:c11:c12 -> c18:c19:c20 -> c7:c8:c9 U23' :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> c10:c11:c12 c8 :: c10:c11:c12 -> c18:c19:c20 -> c7:c8:c9 c9 :: c10:c11:c12 -> c18:c19:c20 -> c7:c8:c9 c10 :: c18:c19:c20 -> c10:c11:c12 c11 :: c18:c19:c20 -> c10:c11:c12 c12 :: c18:c19:c20 -> c10:c11:c12 nil :: nil:cons:0':s:n__zeros:n__take c13 :: c13:c14 cons :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take c14 :: c2 -> c18:c19:c20 -> c13:c14 TAKE :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> c15:c16:c17 0' :: nil:cons:0':s:n__zeros:n__take c15 :: c15:c16:c17 s :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take c16 :: c4:c5:c6 -> c18:c19:c20 -> c15:c16:c17 c17 :: c15:c16:c17 n__zeros :: nil:cons:0':s:n__zeros:n__take c18 :: c:c1 -> c18:c19:c20 n__take :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take c19 :: c15:c16:c17 -> c18:c19:c20 c20 :: c18:c19:c20 zeros :: nil:cons:0':s:n__zeros:n__take U11 :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take U12 :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take length :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take U21 :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take U22 :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take U23 :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take take :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take hole_c:c11_24 :: c:c1 hole_c22_24 :: c2 hole_tt3_24 :: tt hole_nil:cons:0':s:n__zeros:n__take4_24 :: nil:cons:0':s:n__zeros:n__take hole_c35_24 :: c3 hole_c18:c19:c206_24 :: c18:c19:c20 hole_c13:c147_24 :: c13:c14 hole_c4:c5:c68_24 :: c4:c5:c6 hole_c7:c8:c99_24 :: c7:c8:c9 hole_c10:c11:c1210_24 :: c10:c11:c12 hole_c15:c16:c1711_24 :: c15:c16:c17 gen_nil:cons:0':s:n__zeros:n__take12_24 :: Nat -> nil:cons:0':s:n__zeros:n__take Generator Equations: gen_nil:cons:0':s:n__zeros:n__take12_24(0) <=> nil gen_nil:cons:0':s:n__zeros:n__take12_24(+(x, 1)) <=> cons(nil, gen_nil:cons:0':s:n__zeros:n__take12_24(x)) The following defined symbols remain to be analysed: activate, ACTIVATE, LENGTH, length They will be analysed ascendingly in the following order: activate < ACTIVATE activate < LENGTH activate < length ACTIVATE < LENGTH ---------------------------------------- (7) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: LENGTH(gen_nil:cons:0':s:n__zeros:n__take12_24(n38_24)) -> *13_24, rt in Omega(n38_24) Induction Base: LENGTH(gen_nil:cons:0':s:n__zeros:n__take12_24(0)) Induction Step: LENGTH(gen_nil:cons:0':s:n__zeros:n__take12_24(+(n38_24, 1))) ->_R^Omega(1) c14(U11'(tt, activate(gen_nil:cons:0':s:n__zeros:n__take12_24(n38_24))), ACTIVATE(gen_nil:cons:0':s:n__zeros:n__take12_24(n38_24))) ->_R^Omega(0) c14(U11'(tt, gen_nil:cons:0':s:n__zeros:n__take12_24(n38_24)), ACTIVATE(gen_nil:cons:0':s:n__zeros:n__take12_24(n38_24))) ->_R^Omega(1) c14(c2(U12'(tt, activate(gen_nil:cons:0':s:n__zeros:n__take12_24(n38_24))), ACTIVATE(gen_nil:cons:0':s:n__zeros:n__take12_24(n38_24))), ACTIVATE(gen_nil:cons:0':s:n__zeros:n__take12_24(n38_24))) ->_R^Omega(0) c14(c2(U12'(tt, gen_nil:cons:0':s:n__zeros:n__take12_24(n38_24)), ACTIVATE(gen_nil:cons:0':s:n__zeros:n__take12_24(n38_24))), ACTIVATE(gen_nil:cons:0':s:n__zeros:n__take12_24(n38_24))) ->_R^Omega(1) c14(c2(c3(LENGTH(activate(gen_nil:cons:0':s:n__zeros:n__take12_24(n38_24))), ACTIVATE(gen_nil:cons:0':s:n__zeros:n__take12_24(n38_24))), ACTIVATE(gen_nil:cons:0':s:n__zeros:n__take12_24(n38_24))), ACTIVATE(gen_nil:cons:0':s:n__zeros:n__take12_24(n38_24))) ->_R^Omega(0) c14(c2(c3(LENGTH(gen_nil:cons:0':s:n__zeros:n__take12_24(n38_24)), ACTIVATE(gen_nil:cons:0':s:n__zeros:n__take12_24(n38_24))), ACTIVATE(gen_nil:cons:0':s:n__zeros:n__take12_24(n38_24))), ACTIVATE(gen_nil:cons:0':s:n__zeros:n__take12_24(n38_24))) ->_IH c14(c2(c3(*13_24, ACTIVATE(gen_nil:cons:0':s:n__zeros:n__take12_24(n38_24))), ACTIVATE(gen_nil:cons:0':s:n__zeros:n__take12_24(n38_24))), ACTIVATE(gen_nil:cons:0':s:n__zeros:n__take12_24(n38_24))) ->_R^Omega(1) c14(c2(c3(*13_24, c20), ACTIVATE(gen_nil:cons:0':s:n__zeros:n__take12_24(n38_24))), ACTIVATE(gen_nil:cons:0':s:n__zeros:n__take12_24(n38_24))) ->_R^Omega(1) c14(c2(c3(*13_24, c20), c20), ACTIVATE(gen_nil:cons:0':s:n__zeros:n__take12_24(n38_24))) ->_R^Omega(1) c14(c2(c3(*13_24, c20), c20), c20) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (8) Complex Obligation (BEST) ---------------------------------------- (9) Obligation: Proved the lower bound n^1 for the following obligation: Innermost TRS: Rules: ZEROS -> c ZEROS -> c1 U11'(tt, z0) -> c2(U12'(tt, activate(z0)), ACTIVATE(z0)) U12'(tt, z0) -> c3(LENGTH(activate(z0)), ACTIVATE(z0)) U21'(tt, z0, z1, z2) -> c4(U22'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z0)) U21'(tt, z0, z1, z2) -> c5(U22'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z1)) U21'(tt, z0, z1, z2) -> c6(U22'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z2)) U22'(tt, z0, z1, z2) -> c7(U23'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z0)) U22'(tt, z0, z1, z2) -> c8(U23'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z1)) U22'(tt, z0, z1, z2) -> c9(U23'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z2)) U23'(tt, z0, z1, z2) -> c10(ACTIVATE(z2)) U23'(tt, z0, z1, z2) -> c11(ACTIVATE(z1)) U23'(tt, z0, z1, z2) -> c12(ACTIVATE(z0)) LENGTH(nil) -> c13 LENGTH(cons(z0, z1)) -> c14(U11'(tt, activate(z1)), ACTIVATE(z1)) TAKE(0', z0) -> c15 TAKE(s(z0), cons(z1, z2)) -> c16(U21'(tt, activate(z2), z0, z1), ACTIVATE(z2)) TAKE(z0, z1) -> c17 ACTIVATE(n__zeros) -> c18(ZEROS) ACTIVATE(n__take(z0, z1)) -> c19(TAKE(z0, z1)) ACTIVATE(z0) -> c20 zeros -> cons(0', n__zeros) zeros -> n__zeros U11(tt, z0) -> U12(tt, activate(z0)) U12(tt, z0) -> s(length(activate(z0))) U21(tt, z0, z1, z2) -> U22(tt, activate(z0), activate(z1), activate(z2)) U22(tt, z0, z1, z2) -> U23(tt, activate(z0), activate(z1), activate(z2)) U23(tt, z0, z1, z2) -> cons(activate(z2), n__take(activate(z1), activate(z0))) length(nil) -> 0' length(cons(z0, z1)) -> U11(tt, activate(z1)) take(0', z0) -> nil take(s(z0), cons(z1, z2)) -> U21(tt, activate(z2), z0, z1) take(z0, z1) -> n__take(z0, z1) activate(n__zeros) -> zeros activate(n__take(z0, z1)) -> take(z0, z1) activate(z0) -> z0 Types: ZEROS :: c:c1 c :: c:c1 c1 :: c:c1 U11' :: tt -> nil:cons:0':s:n__zeros:n__take -> c2 tt :: tt c2 :: c3 -> c18:c19:c20 -> c2 U12' :: tt -> nil:cons:0':s:n__zeros:n__take -> c3 activate :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take ACTIVATE :: nil:cons:0':s:n__zeros:n__take -> c18:c19:c20 c3 :: c13:c14 -> c18:c19:c20 -> c3 LENGTH :: nil:cons:0':s:n__zeros:n__take -> c13:c14 U21' :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> c4:c5:c6 c4 :: c7:c8:c9 -> c18:c19:c20 -> c4:c5:c6 U22' :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> c7:c8:c9 c5 :: c7:c8:c9 -> c18:c19:c20 -> c4:c5:c6 c6 :: c7:c8:c9 -> c18:c19:c20 -> c4:c5:c6 c7 :: c10:c11:c12 -> c18:c19:c20 -> c7:c8:c9 U23' :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> c10:c11:c12 c8 :: c10:c11:c12 -> c18:c19:c20 -> c7:c8:c9 c9 :: c10:c11:c12 -> c18:c19:c20 -> c7:c8:c9 c10 :: c18:c19:c20 -> c10:c11:c12 c11 :: c18:c19:c20 -> c10:c11:c12 c12 :: c18:c19:c20 -> c10:c11:c12 nil :: nil:cons:0':s:n__zeros:n__take c13 :: c13:c14 cons :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take c14 :: c2 -> c18:c19:c20 -> c13:c14 TAKE :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> c15:c16:c17 0' :: nil:cons:0':s:n__zeros:n__take c15 :: c15:c16:c17 s :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take c16 :: c4:c5:c6 -> c18:c19:c20 -> c15:c16:c17 c17 :: c15:c16:c17 n__zeros :: nil:cons:0':s:n__zeros:n__take c18 :: c:c1 -> c18:c19:c20 n__take :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take c19 :: c15:c16:c17 -> c18:c19:c20 c20 :: c18:c19:c20 zeros :: nil:cons:0':s:n__zeros:n__take U11 :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take U12 :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take length :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take U21 :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take U22 :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take U23 :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take take :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take hole_c:c11_24 :: c:c1 hole_c22_24 :: c2 hole_tt3_24 :: tt hole_nil:cons:0':s:n__zeros:n__take4_24 :: nil:cons:0':s:n__zeros:n__take hole_c35_24 :: c3 hole_c18:c19:c206_24 :: c18:c19:c20 hole_c13:c147_24 :: c13:c14 hole_c4:c5:c68_24 :: c4:c5:c6 hole_c7:c8:c99_24 :: c7:c8:c9 hole_c10:c11:c1210_24 :: c10:c11:c12 hole_c15:c16:c1711_24 :: c15:c16:c17 gen_nil:cons:0':s:n__zeros:n__take12_24 :: Nat -> nil:cons:0':s:n__zeros:n__take Generator Equations: gen_nil:cons:0':s:n__zeros:n__take12_24(0) <=> nil gen_nil:cons:0':s:n__zeros:n__take12_24(+(x, 1)) <=> cons(nil, gen_nil:cons:0':s:n__zeros:n__take12_24(x)) The following defined symbols remain to be analysed: LENGTH, length ---------------------------------------- (10) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (11) BOUNDS(n^1, INF) ---------------------------------------- (12) Obligation: Innermost TRS: Rules: ZEROS -> c ZEROS -> c1 U11'(tt, z0) -> c2(U12'(tt, activate(z0)), ACTIVATE(z0)) U12'(tt, z0) -> c3(LENGTH(activate(z0)), ACTIVATE(z0)) U21'(tt, z0, z1, z2) -> c4(U22'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z0)) U21'(tt, z0, z1, z2) -> c5(U22'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z1)) U21'(tt, z0, z1, z2) -> c6(U22'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z2)) U22'(tt, z0, z1, z2) -> c7(U23'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z0)) U22'(tt, z0, z1, z2) -> c8(U23'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z1)) U22'(tt, z0, z1, z2) -> c9(U23'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z2)) U23'(tt, z0, z1, z2) -> c10(ACTIVATE(z2)) U23'(tt, z0, z1, z2) -> c11(ACTIVATE(z1)) U23'(tt, z0, z1, z2) -> c12(ACTIVATE(z0)) LENGTH(nil) -> c13 LENGTH(cons(z0, z1)) -> c14(U11'(tt, activate(z1)), ACTIVATE(z1)) TAKE(0', z0) -> c15 TAKE(s(z0), cons(z1, z2)) -> c16(U21'(tt, activate(z2), z0, z1), ACTIVATE(z2)) TAKE(z0, z1) -> c17 ACTIVATE(n__zeros) -> c18(ZEROS) ACTIVATE(n__take(z0, z1)) -> c19(TAKE(z0, z1)) ACTIVATE(z0) -> c20 zeros -> cons(0', n__zeros) zeros -> n__zeros U11(tt, z0) -> U12(tt, activate(z0)) U12(tt, z0) -> s(length(activate(z0))) U21(tt, z0, z1, z2) -> U22(tt, activate(z0), activate(z1), activate(z2)) U22(tt, z0, z1, z2) -> U23(tt, activate(z0), activate(z1), activate(z2)) U23(tt, z0, z1, z2) -> cons(activate(z2), n__take(activate(z1), activate(z0))) length(nil) -> 0' length(cons(z0, z1)) -> U11(tt, activate(z1)) take(0', z0) -> nil take(s(z0), cons(z1, z2)) -> U21(tt, activate(z2), z0, z1) take(z0, z1) -> n__take(z0, z1) activate(n__zeros) -> zeros activate(n__take(z0, z1)) -> take(z0, z1) activate(z0) -> z0 Types: ZEROS :: c:c1 c :: c:c1 c1 :: c:c1 U11' :: tt -> nil:cons:0':s:n__zeros:n__take -> c2 tt :: tt c2 :: c3 -> c18:c19:c20 -> c2 U12' :: tt -> nil:cons:0':s:n__zeros:n__take -> c3 activate :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take ACTIVATE :: nil:cons:0':s:n__zeros:n__take -> c18:c19:c20 c3 :: c13:c14 -> c18:c19:c20 -> c3 LENGTH :: nil:cons:0':s:n__zeros:n__take -> c13:c14 U21' :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> c4:c5:c6 c4 :: c7:c8:c9 -> c18:c19:c20 -> c4:c5:c6 U22' :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> c7:c8:c9 c5 :: c7:c8:c9 -> c18:c19:c20 -> c4:c5:c6 c6 :: c7:c8:c9 -> c18:c19:c20 -> c4:c5:c6 c7 :: c10:c11:c12 -> c18:c19:c20 -> c7:c8:c9 U23' :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> c10:c11:c12 c8 :: c10:c11:c12 -> c18:c19:c20 -> c7:c8:c9 c9 :: c10:c11:c12 -> c18:c19:c20 -> c7:c8:c9 c10 :: c18:c19:c20 -> c10:c11:c12 c11 :: c18:c19:c20 -> c10:c11:c12 c12 :: c18:c19:c20 -> c10:c11:c12 nil :: nil:cons:0':s:n__zeros:n__take c13 :: c13:c14 cons :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take c14 :: c2 -> c18:c19:c20 -> c13:c14 TAKE :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> c15:c16:c17 0' :: nil:cons:0':s:n__zeros:n__take c15 :: c15:c16:c17 s :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take c16 :: c4:c5:c6 -> c18:c19:c20 -> c15:c16:c17 c17 :: c15:c16:c17 n__zeros :: nil:cons:0':s:n__zeros:n__take c18 :: c:c1 -> c18:c19:c20 n__take :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take c19 :: c15:c16:c17 -> c18:c19:c20 c20 :: c18:c19:c20 zeros :: nil:cons:0':s:n__zeros:n__take U11 :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take U12 :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take length :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take U21 :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take U22 :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take U23 :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take take :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take hole_c:c11_24 :: c:c1 hole_c22_24 :: c2 hole_tt3_24 :: tt hole_nil:cons:0':s:n__zeros:n__take4_24 :: nil:cons:0':s:n__zeros:n__take hole_c35_24 :: c3 hole_c18:c19:c206_24 :: c18:c19:c20 hole_c13:c147_24 :: c13:c14 hole_c4:c5:c68_24 :: c4:c5:c6 hole_c7:c8:c99_24 :: c7:c8:c9 hole_c10:c11:c1210_24 :: c10:c11:c12 hole_c15:c16:c1711_24 :: c15:c16:c17 gen_nil:cons:0':s:n__zeros:n__take12_24 :: Nat -> nil:cons:0':s:n__zeros:n__take Lemmas: LENGTH(gen_nil:cons:0':s:n__zeros:n__take12_24(n38_24)) -> *13_24, rt in Omega(n38_24) Generator Equations: gen_nil:cons:0':s:n__zeros:n__take12_24(0) <=> nil gen_nil:cons:0':s:n__zeros:n__take12_24(+(x, 1)) <=> cons(nil, gen_nil:cons:0':s:n__zeros:n__take12_24(x)) The following defined symbols remain to be analysed: length ---------------------------------------- (13) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: length(gen_nil:cons:0':s:n__zeros:n__take12_24(n1598_24)) -> *13_24, rt in Omega(0) Induction Base: length(gen_nil:cons:0':s:n__zeros:n__take12_24(0)) Induction Step: length(gen_nil:cons:0':s:n__zeros:n__take12_24(+(n1598_24, 1))) ->_R^Omega(0) U11(tt, activate(gen_nil:cons:0':s:n__zeros:n__take12_24(n1598_24))) ->_R^Omega(0) U11(tt, gen_nil:cons:0':s:n__zeros:n__take12_24(n1598_24)) ->_R^Omega(0) U12(tt, activate(gen_nil:cons:0':s:n__zeros:n__take12_24(n1598_24))) ->_R^Omega(0) U12(tt, gen_nil:cons:0':s:n__zeros:n__take12_24(n1598_24)) ->_R^Omega(0) s(length(activate(gen_nil:cons:0':s:n__zeros:n__take12_24(n1598_24)))) ->_R^Omega(0) s(length(gen_nil:cons:0':s:n__zeros:n__take12_24(n1598_24))) ->_IH s(*13_24) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (14) BOUNDS(1, INF)