The Runtime Complexity (innermost) 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__eq(0, 0) → true
a__eq(s(X), s(Y)) → a__eq(X, Y)
a__eq(X, Y) → false
a__inf(X) → cons(X, inf(s(X)))
a__take(0, X) → nil
a__take(s(X), cons(Y, L)) → cons(Y, take(X, L))
a__length(nil) → 0
a__length(cons(X, L)) → s(length(L))
mark(eq(X1, X2)) → a__eq(X1, X2)
mark(inf(X)) → a__inf(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(length(X)) → a__length(mark(X))
mark(0) → 0
mark(true) → true
mark(s(X)) → s(X)
mark(false) → false
mark(cons(X1, X2)) → cons(X1, X2)
mark(nil) → nil
a__eq(X1, X2) → eq(X1, X2)
a__inf(X) → inf(X)
a__take(X1, X2) → take(X1, X2)
a__length(X) → length(X)

Rewrite Strategy: INNERMOST

(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__eq(0', 0') → true
a__eq(s(X), s(Y)) → a__eq(X, Y)
a__eq(X, Y) → false
a__inf(X) → cons(X, inf(s(X)))
a__take(0', X) → nil
a__take(s(X), cons(Y, L)) → cons(Y, take(X, L))
a__length(nil) → 0'
a__length(cons(X, L)) → s(length(L))
mark(eq(X1, X2)) → a__eq(X1, X2)
mark(inf(X)) → a__inf(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(length(X)) → a__length(mark(X))
mark(0') → 0'
mark(true) → true
mark(s(X)) → s(X)
mark(false) → false
mark(cons(X1, X2)) → cons(X1, X2)
mark(nil) → nil
a__eq(X1, X2) → eq(X1, X2)
a__inf(X) → inf(X)
a__take(X1, X2) → take(X1, X2)
a__length(X) → length(X)

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
a__eq(0', 0') → true
a__eq(s(X), s(Y)) → a__eq(X, Y)
a__eq(X, Y) → false
a__inf(X) → cons(X, inf(s(X)))
a__take(0', X) → nil
a__take(s(X), cons(Y, L)) → cons(Y, take(X, L))
a__length(nil) → 0'
a__length(cons(X, L)) → s(length(L))
mark(eq(X1, X2)) → a__eq(X1, X2)
mark(inf(X)) → a__inf(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(length(X)) → a__length(mark(X))
mark(0') → 0'
mark(true) → true
mark(s(X)) → s(X)
mark(false) → false
mark(cons(X1, X2)) → cons(X1, X2)
mark(nil) → nil
a__eq(X1, X2) → eq(X1, X2)
a__inf(X) → inf(X)
a__take(X1, X2) → take(X1, X2)
a__length(X) → length(X)

Types:
a__eq :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
0' :: 0':true:s:false:inf:cons:nil:take:length:eq
true :: 0':true:s:false:inf:cons:nil:take:length:eq
s :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
false :: 0':true:s:false:inf:cons:nil:take:length:eq
a__inf :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
cons :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
inf :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
a__take :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
nil :: 0':true:s:false:inf:cons:nil:take:length:eq
take :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
a__length :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
length :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
mark :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
eq :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
hole_0':true:s:false:inf:cons:nil:take:length:eq1_0 :: 0':true:s:false:inf:cons:nil:take:length:eq
gen_0':true:s:false:inf:cons:nil:take:length:eq2_0 :: Nat → 0':true:s:false:inf:cons:nil:take:length:eq

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

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

They will be analysed ascendingly in the following order:
a__eq < mark

(6) Obligation:

Innermost TRS:
Rules:
a__eq(0', 0') → true
a__eq(s(X), s(Y)) → a__eq(X, Y)
a__eq(X, Y) → false
a__inf(X) → cons(X, inf(s(X)))
a__take(0', X) → nil
a__take(s(X), cons(Y, L)) → cons(Y, take(X, L))
a__length(nil) → 0'
a__length(cons(X, L)) → s(length(L))
mark(eq(X1, X2)) → a__eq(X1, X2)
mark(inf(X)) → a__inf(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(length(X)) → a__length(mark(X))
mark(0') → 0'
mark(true) → true
mark(s(X)) → s(X)
mark(false) → false
mark(cons(X1, X2)) → cons(X1, X2)
mark(nil) → nil
a__eq(X1, X2) → eq(X1, X2)
a__inf(X) → inf(X)
a__take(X1, X2) → take(X1, X2)
a__length(X) → length(X)

Types:
a__eq :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
0' :: 0':true:s:false:inf:cons:nil:take:length:eq
true :: 0':true:s:false:inf:cons:nil:take:length:eq
s :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
false :: 0':true:s:false:inf:cons:nil:take:length:eq
a__inf :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
cons :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
inf :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
a__take :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
nil :: 0':true:s:false:inf:cons:nil:take:length:eq
take :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
a__length :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
length :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
mark :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
eq :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
hole_0':true:s:false:inf:cons:nil:take:length:eq1_0 :: 0':true:s:false:inf:cons:nil:take:length:eq
gen_0':true:s:false:inf:cons:nil:take:length:eq2_0 :: Nat → 0':true:s:false:inf:cons:nil:take:length:eq

Generator Equations:
gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(0) ⇔ 0'
gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(+(x, 1)) ⇔ s(gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(x))

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

They will be analysed ascendingly in the following order:
a__eq < mark

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

Proved the following rewrite lemma:
a__eq(gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(n4_0), gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(n4_0)) → true, rt ∈ Ω(1 + n4₀)

Induction Base:
a__eq(gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(0), gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(0)) →_R^Ω(1)
true

Induction Step:
a__eq(gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(+(n4_0, 1)), gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(+(n4_0, 1))) →_R^Ω(1)
a__eq(gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(n4_0), gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(n4_0)) →_IH
true

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

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost TRS:
Rules:
a__eq(0', 0') → true
a__eq(s(X), s(Y)) → a__eq(X, Y)
a__eq(X, Y) → false
a__inf(X) → cons(X, inf(s(X)))
a__take(0', X) → nil
a__take(s(X), cons(Y, L)) → cons(Y, take(X, L))
a__length(nil) → 0'
a__length(cons(X, L)) → s(length(L))
mark(eq(X1, X2)) → a__eq(X1, X2)
mark(inf(X)) → a__inf(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(length(X)) → a__length(mark(X))
mark(0') → 0'
mark(true) → true
mark(s(X)) → s(X)
mark(false) → false
mark(cons(X1, X2)) → cons(X1, X2)
mark(nil) → nil
a__eq(X1, X2) → eq(X1, X2)
a__inf(X) → inf(X)
a__take(X1, X2) → take(X1, X2)
a__length(X) → length(X)

Types:
a__eq :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
0' :: 0':true:s:false:inf:cons:nil:take:length:eq
true :: 0':true:s:false:inf:cons:nil:take:length:eq
s :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
false :: 0':true:s:false:inf:cons:nil:take:length:eq
a__inf :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
cons :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
inf :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
a__take :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
nil :: 0':true:s:false:inf:cons:nil:take:length:eq
take :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
a__length :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
length :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
mark :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
eq :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
hole_0':true:s:false:inf:cons:nil:take:length:eq1_0 :: 0':true:s:false:inf:cons:nil:take:length:eq
gen_0':true:s:false:inf:cons:nil:take:length:eq2_0 :: Nat → 0':true:s:false:inf:cons:nil:take:length:eq

Lemmas:
a__eq(gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(n4_0), gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(n4_0)) → true, rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(0) ⇔ 0'
gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(+(x, 1)) ⇔ s(gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(x))

The following defined symbols remain to be analysed:
mark

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

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

(11) Obligation:

Innermost TRS:
Rules:
a__eq(0', 0') → true
a__eq(s(X), s(Y)) → a__eq(X, Y)
a__eq(X, Y) → false
a__inf(X) → cons(X, inf(s(X)))
a__take(0', X) → nil
a__take(s(X), cons(Y, L)) → cons(Y, take(X, L))
a__length(nil) → 0'
a__length(cons(X, L)) → s(length(L))
mark(eq(X1, X2)) → a__eq(X1, X2)
mark(inf(X)) → a__inf(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(length(X)) → a__length(mark(X))
mark(0') → 0'
mark(true) → true
mark(s(X)) → s(X)
mark(false) → false
mark(cons(X1, X2)) → cons(X1, X2)
mark(nil) → nil
a__eq(X1, X2) → eq(X1, X2)
a__inf(X) → inf(X)
a__take(X1, X2) → take(X1, X2)
a__length(X) → length(X)

Types:
a__eq :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
0' :: 0':true:s:false:inf:cons:nil:take:length:eq
true :: 0':true:s:false:inf:cons:nil:take:length:eq
s :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
false :: 0':true:s:false:inf:cons:nil:take:length:eq
a__inf :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
cons :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
inf :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
a__take :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
nil :: 0':true:s:false:inf:cons:nil:take:length:eq
take :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
a__length :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
length :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
mark :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
eq :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
hole_0':true:s:false:inf:cons:nil:take:length:eq1_0 :: 0':true:s:false:inf:cons:nil:take:length:eq
gen_0':true:s:false:inf:cons:nil:take:length:eq2_0 :: Nat → 0':true:s:false:inf:cons:nil:take:length:eq

Lemmas:
a__eq(gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(n4_0), gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(n4_0)) → true, rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(0) ⇔ 0'
gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(+(x, 1)) ⇔ s(gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(x))

No more defined symbols left to analyse.

(12) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
a__eq(gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(n4_0), gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(n4_0)) → true, rt ∈ Ω(1 + n4₀)

(13) BOUNDS(n^1, INF)

(14) Obligation:

Innermost TRS:
Rules:
a__eq(0', 0') → true
a__eq(s(X), s(Y)) → a__eq(X, Y)
a__eq(X, Y) → false
a__inf(X) → cons(X, inf(s(X)))
a__take(0', X) → nil
a__take(s(X), cons(Y, L)) → cons(Y, take(X, L))
a__length(nil) → 0'
a__length(cons(X, L)) → s(length(L))
mark(eq(X1, X2)) → a__eq(X1, X2)
mark(inf(X)) → a__inf(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(length(X)) → a__length(mark(X))
mark(0') → 0'
mark(true) → true
mark(s(X)) → s(X)
mark(false) → false
mark(cons(X1, X2)) → cons(X1, X2)
mark(nil) → nil
a__eq(X1, X2) → eq(X1, X2)
a__inf(X) → inf(X)
a__take(X1, X2) → take(X1, X2)
a__length(X) → length(X)

Types:
a__eq :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
0' :: 0':true:s:false:inf:cons:nil:take:length:eq
true :: 0':true:s:false:inf:cons:nil:take:length:eq
s :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
false :: 0':true:s:false:inf:cons:nil:take:length:eq
a__inf :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
cons :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
inf :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
a__take :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
nil :: 0':true:s:false:inf:cons:nil:take:length:eq
take :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
a__length :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
length :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
mark :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
eq :: 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq → 0':true:s:false:inf:cons:nil:take:length:eq
hole_0':true:s:false:inf:cons:nil:take:length:eq1_0 :: 0':true:s:false:inf:cons:nil:take:length:eq
gen_0':true:s:false:inf:cons:nil:take:length:eq2_0 :: Nat → 0':true:s:false:inf:cons:nil:take:length:eq

Lemmas:
a__eq(gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(n4_0), gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(n4_0)) → true, rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(0) ⇔ 0'
gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(+(x, 1)) ⇔ s(gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
a__eq(gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(n4_0), gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(n4_0)) → true, rt ∈ Ω(1 + n4₀)

(16) BOUNDS(n^1, INF)