The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__and(tt, X) → mark(X)
a__length(nil) → 0
a__length(cons(N, L)) → s(a__length(mark(L)))
a__take(0, IL) → nil
a__take(s(M), cons(N, IL)) → cons(mark(N), take(M, IL))
mark(zeros) → a__zeros
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(nil) → nil
mark(s(X)) → s(mark(X))
a__zeros → zeros
a__and(X1, X2) → and(X1, X2)
a__length(X) → length(X)
a__take(X1, X2) → take(X1, X2)

Rewrite Strategy: FULL

(1) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(2) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

a__zeros → cons(0', zeros)
a__and(tt, X) → mark(X)
a__length(nil) → 0'
a__length(cons(N, L)) → s(a__length(mark(L)))
a__take(0', IL) → nil
a__take(s(M), cons(N, IL)) → cons(mark(N), take(M, IL))
mark(zeros) → a__zeros
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(tt) → tt
mark(nil) → nil
mark(s(X)) → s(mark(X))
a__zeros → zeros
a__and(X1, X2) → and(X1, X2)
a__length(X) → length(X)
a__take(X1, X2) → take(X1, X2)

S is empty.
Rewrite Strategy: FULL

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(4) Obligation:

TRS:
Rules:
a__zeros → cons(0', zeros)
a__and(tt, X) → mark(X)
a__length(nil) → 0'
a__length(cons(N, L)) → s(a__length(mark(L)))
a__take(0', IL) → nil
a__take(s(M), cons(N, IL)) → cons(mark(N), take(M, IL))
mark(zeros) → a__zeros
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(tt) → tt
mark(nil) → nil
mark(s(X)) → s(mark(X))
a__zeros → zeros
a__and(X1, X2) → and(X1, X2)
a__length(X) → length(X)
a__take(X1, X2) → take(X1, X2)

Types:
a__zeros :: 0':zeros:cons:tt:nil:s:take:and:length
cons :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
0' :: 0':zeros:cons:tt:nil:s:take:and:length
zeros :: 0':zeros:cons:tt:nil:s:take:and:length
a__and :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
tt :: 0':zeros:cons:tt:nil:s:take:and:length
mark :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
a__length :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
nil :: 0':zeros:cons:tt:nil:s:take:and:length
s :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
a__take :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
take :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
and :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
length :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
hole_0':zeros:cons:tt:nil:s:take:and:length1_0 :: 0':zeros:cons:tt:nil:s:take:and:length
gen_0':zeros:cons:tt:nil:s:take:and:length2_0 :: Nat → 0':zeros:cons:tt:nil:s:take:and:length

(5) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
mark, a__length

They will be analysed ascendingly in the following order:
mark = a__length

(6) Obligation:

TRS:
Rules:
a__zeros → cons(0', zeros)
a__and(tt, X) → mark(X)
a__length(nil) → 0'
a__length(cons(N, L)) → s(a__length(mark(L)))
a__take(0', IL) → nil
a__take(s(M), cons(N, IL)) → cons(mark(N), take(M, IL))
mark(zeros) → a__zeros
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(tt) → tt
mark(nil) → nil
mark(s(X)) → s(mark(X))
a__zeros → zeros
a__and(X1, X2) → and(X1, X2)
a__length(X) → length(X)
a__take(X1, X2) → take(X1, X2)

Types:
a__zeros :: 0':zeros:cons:tt:nil:s:take:and:length
cons :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
0' :: 0':zeros:cons:tt:nil:s:take:and:length
zeros :: 0':zeros:cons:tt:nil:s:take:and:length
a__and :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
tt :: 0':zeros:cons:tt:nil:s:take:and:length
mark :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
a__length :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
nil :: 0':zeros:cons:tt:nil:s:take:and:length
s :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
a__take :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
take :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
and :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
length :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
hole_0':zeros:cons:tt:nil:s:take:and:length1_0 :: 0':zeros:cons:tt:nil:s:take:and:length
gen_0':zeros:cons:tt:nil:s:take:and:length2_0 :: Nat → 0':zeros:cons:tt:nil:s:take:and:length

Generator Equations:
gen_0':zeros:cons:tt:nil:s:take:and:length2_0(0) ⇔ 0'
gen_0':zeros:cons:tt:nil:s:take:and:length2_0(+(x, 1)) ⇔ cons(0', gen_0':zeros:cons:tt:nil:s:take:and:length2_0(x))

The following defined symbols remain to be analysed:
a__length, mark

They will be analysed ascendingly in the following order:
mark = a__length

(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
a__length(gen_0':zeros:cons:tt:nil:s:take:and:length2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Induction Base:
a__length(gen_0':zeros:cons:tt:nil:s:take:and:length2_0(+(1, 0)))

Induction Step:
a__length(gen_0':zeros:cons:tt:nil:s:take:and:length2_0(+(1, +(n4_0, 1)))) →_R^Ω(1)
s(a__length(mark(gen_0':zeros:cons:tt:nil:s:take:and:length2_0(+(1, n4_0))))) →_R^Ω(1)
s(a__length(cons(mark(0'), gen_0':zeros:cons:tt:nil:s:take:and:length2_0(n4_0)))) →_R^Ω(1)
s(a__length(cons(0', gen_0':zeros:cons:tt:nil:s:take:and:length2_0(n4_0)))) →_IH
s(*3_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(8) Complex Obligation (BEST)

(9) Obligation:

TRS:
Rules:
a__zeros → cons(0', zeros)
a__and(tt, X) → mark(X)
a__length(nil) → 0'
a__length(cons(N, L)) → s(a__length(mark(L)))
a__take(0', IL) → nil
a__take(s(M), cons(N, IL)) → cons(mark(N), take(M, IL))
mark(zeros) → a__zeros
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(tt) → tt
mark(nil) → nil
mark(s(X)) → s(mark(X))
a__zeros → zeros
a__and(X1, X2) → and(X1, X2)
a__length(X) → length(X)
a__take(X1, X2) → take(X1, X2)

Types:
a__zeros :: 0':zeros:cons:tt:nil:s:take:and:length
cons :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
0' :: 0':zeros:cons:tt:nil:s:take:and:length
zeros :: 0':zeros:cons:tt:nil:s:take:and:length
a__and :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
tt :: 0':zeros:cons:tt:nil:s:take:and:length
mark :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
a__length :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
nil :: 0':zeros:cons:tt:nil:s:take:and:length
s :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
a__take :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
take :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
and :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
length :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
hole_0':zeros:cons:tt:nil:s:take:and:length1_0 :: 0':zeros:cons:tt:nil:s:take:and:length
gen_0':zeros:cons:tt:nil:s:take:and:length2_0 :: Nat → 0':zeros:cons:tt:nil:s:take:and:length

Lemmas:
a__length(gen_0':zeros:cons:tt:nil:s:take:and:length2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_0':zeros:cons:tt:nil:s:take:and:length2_0(0) ⇔ 0'
gen_0':zeros:cons:tt:nil:s:take:and:length2_0(+(x, 1)) ⇔ cons(0', gen_0':zeros:cons:tt:nil:s:take:and:length2_0(x))

The following defined symbols remain to be analysed:
mark

They will be analysed ascendingly in the following order:
mark = a__length

(10) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol mark.

(11) Obligation:

TRS:
Rules:
a__zeros → cons(0', zeros)
a__and(tt, X) → mark(X)
a__length(nil) → 0'
a__length(cons(N, L)) → s(a__length(mark(L)))
a__take(0', IL) → nil
a__take(s(M), cons(N, IL)) → cons(mark(N), take(M, IL))
mark(zeros) → a__zeros
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(tt) → tt
mark(nil) → nil
mark(s(X)) → s(mark(X))
a__zeros → zeros
a__and(X1, X2) → and(X1, X2)
a__length(X) → length(X)
a__take(X1, X2) → take(X1, X2)

Types:
a__zeros :: 0':zeros:cons:tt:nil:s:take:and:length
cons :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
0' :: 0':zeros:cons:tt:nil:s:take:and:length
zeros :: 0':zeros:cons:tt:nil:s:take:and:length
a__and :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
tt :: 0':zeros:cons:tt:nil:s:take:and:length
mark :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
a__length :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
nil :: 0':zeros:cons:tt:nil:s:take:and:length
s :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
a__take :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
take :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
and :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
length :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
hole_0':zeros:cons:tt:nil:s:take:and:length1_0 :: 0':zeros:cons:tt:nil:s:take:and:length
gen_0':zeros:cons:tt:nil:s:take:and:length2_0 :: Nat → 0':zeros:cons:tt:nil:s:take:and:length

Lemmas:
a__length(gen_0':zeros:cons:tt:nil:s:take:and:length2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_0':zeros:cons:tt:nil:s:take:and:length2_0(0) ⇔ 0'
gen_0':zeros:cons:tt:nil:s:take:and:length2_0(+(x, 1)) ⇔ cons(0', gen_0':zeros:cons:tt:nil:s:take:and:length2_0(x))

No more defined symbols left to analyse.

(12) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
a__length(gen_0':zeros:cons:tt:nil:s:take:and:length2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

(13) BOUNDS(n^1, INF)

(14) Obligation:

TRS:
Rules:
a__zeros → cons(0', zeros)
a__and(tt, X) → mark(X)
a__length(nil) → 0'
a__length(cons(N, L)) → s(a__length(mark(L)))
a__take(0', IL) → nil
a__take(s(M), cons(N, IL)) → cons(mark(N), take(M, IL))
mark(zeros) → a__zeros
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(tt) → tt
mark(nil) → nil
mark(s(X)) → s(mark(X))
a__zeros → zeros
a__and(X1, X2) → and(X1, X2)
a__length(X) → length(X)
a__take(X1, X2) → take(X1, X2)

Types:
a__zeros :: 0':zeros:cons:tt:nil:s:take:and:length
cons :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
0' :: 0':zeros:cons:tt:nil:s:take:and:length
zeros :: 0':zeros:cons:tt:nil:s:take:and:length
a__and :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
tt :: 0':zeros:cons:tt:nil:s:take:and:length
mark :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
a__length :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
nil :: 0':zeros:cons:tt:nil:s:take:and:length
s :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
a__take :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
take :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
and :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
length :: 0':zeros:cons:tt:nil:s:take:and:length → 0':zeros:cons:tt:nil:s:take:and:length
hole_0':zeros:cons:tt:nil:s:take:and:length1_0 :: 0':zeros:cons:tt:nil:s:take:and:length
gen_0':zeros:cons:tt:nil:s:take:and:length2_0 :: Nat → 0':zeros:cons:tt:nil:s:take:and:length

Lemmas:
a__length(gen_0':zeros:cons:tt:nil:s:take:and:length2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_0':zeros:cons:tt:nil:s:take:and:length2_0(0) ⇔ 0'
gen_0':zeros:cons:tt:nil:s:take:and:length2_0(+(x, 1)) ⇔ cons(0', gen_0':zeros:cons:tt:nil:s:take:and:length2_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
a__length(gen_0':zeros:cons:tt:nil:s:take:and:length2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

(16) BOUNDS(n^1, INF)