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

Renamed function symbols to avoid clashes with predefined symbol.

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

Sliced the following arguments:
cons'/0

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'(inf'(s'(X)))
a__take'(0', X) → nil'
a__take'(s'(X), cons'(L)) → cons'(take'(X, L))
a__length'(nil') → 0'
a__length'(cons'(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'(X2)) → cons'(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

Infered types.

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'(inf'(s'(X)))
a__take'(0', X) → nil'
a__take'(s'(X), cons'(L)) → cons'(take'(X, L))
a__length'(nil') → 0'
a__length'(cons'(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'(X2)) → cons'(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'
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':eq'1 :: 0':true':s':false':inf':cons':nil':take':length':eq'
_gen_0':true':s':false':inf':cons':nil':take':length':eq'2 :: Nat → 0':true':s':false':inf':cons':nil':take':length':eq'

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

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

Generator Equations:
_gen_0':true':s':false':inf':cons':nil':take':length':eq'2(0) ⇔ 0'
_gen_0':true':s':false':inf':cons':nil':take':length':eq'2(+(x, 1)) ⇔ s'(_gen_0':true':s':false':inf':cons':nil':take':length':eq'2(x))

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

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

Proved the following rewrite lemma:
a__eq'(_gen_0':true':s':false':inf':cons':nil':take':length':eq'2(_n4), _gen_0':true':s':false':inf':cons':nil':take':length':eq'2(_n4)) → true', rt ∈ Ω(1 + _n4)

Induction Base:
a__eq'(_gen_0':true':s':false':inf':cons':nil':take':length':eq'2(0), _gen_0':true':s':false':inf':cons':nil':take':length':eq'2(0)) →_R^Ω(1)
true'

Induction Step:
a__eq'(_gen_0':true':s':false':inf':cons':nil':take':length':eq'2(+(_$n5, 1)), _gen_0':true':s':false':inf':cons':nil':take':length':eq'2(+(_$n5, 1))) →_R^Ω(1)
a__eq'(_gen_0':true':s':false':inf':cons':nil':take':length':eq'2(_$n5), _gen_0':true':s':false':inf':cons':nil':take':length':eq'2(_$n5)) →_IH
true'

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

Lemmas:
a__eq'(_gen_0':true':s':false':inf':cons':nil':take':length':eq'2(_n4), _gen_0':true':s':false':inf':cons':nil':take':length':eq'2(_n4)) → true', rt ∈ Ω(1 + _n4)

The following defined symbols remain to be analysed:
mark'

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

Lemmas:
a__eq'(_gen_0':true':s':false':inf':cons':nil':take':length':eq'2(_n4), _gen_0':true':s':false':inf':cons':nil':take':length':eq'2(_n4)) → true', rt ∈ Ω(1 + _n4)

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
a__eq'(_gen_0':true':s':false':inf':cons':nil':take':length':eq'2(_n4), _gen_0':true':s':false':inf':cons':nil':take':length':eq'2(_n4)) → true', rt ∈ Ω(1 + _n4)