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

from(X) → cons(X, n__from(s(X)))
length(n__nil) → 0
length(n__cons(X, Y)) → s(length1(activate(Y)))
length1(X) → length(activate(X))
from(X) → n__from(X)
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
activate(n__from(X)) → from(X)
activate(n__nil) → nil
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(X) → X

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

from'(X) → cons'(X, n__from'(s'(X)))
length'(n__nil') → 0'
length'(n__cons'(X, Y)) → s'(length1'(activate'(Y)))
length1'(X) → length'(activate'(X))
from'(X) → n__from'(X)
nil' → n__nil'
cons'(X1, X2) → n__cons'(X1, X2)
activate'(n__from'(X)) → from'(X)
activate'(n__nil') → nil'
activate'(n__cons'(X1, X2)) → cons'(X1, X2)
activate'(X) → X

Rewrite Strategy: INNERMOST

Sliced the following arguments:
from'/0
cons'/0
n__from'/0
n__cons'/0

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

from' → cons'(n__from')
length'(n__nil') → 0'
length'(n__cons'(Y)) → s'(length1'(activate'(Y)))
length1'(X) → length'(activate'(X))
from' → n__from'
nil' → n__nil'
cons'(X2) → n__cons'(X2)
activate'(n__from') → from'
activate'(n__nil') → nil'
activate'(n__cons'(X2)) → cons'(X2)
activate'(X) → X

Rewrite Strategy: INNERMOST

Infered types.

Rules:
from' → cons'(n__from')
length'(n__nil') → 0'
length'(n__cons'(Y)) → s'(length1'(activate'(Y)))
length1'(X) → length'(activate'(X))
from' → n__from'
nil' → n__nil'
cons'(X2) → n__cons'(X2)
activate'(n__from') → from'
activate'(n__nil') → nil'
activate'(n__cons'(X2)) → cons'(X2)
activate'(X) → X

Types:
from' :: n__from':n__nil':n__cons'
cons' :: n__from':n__nil':n__cons' → n__from':n__nil':n__cons'
n__from' :: n__from':n__nil':n__cons'
length' :: n__from':n__nil':n__cons' → 0':s'
n__nil' :: n__from':n__nil':n__cons'
0' :: 0':s'
n__cons' :: n__from':n__nil':n__cons' → n__from':n__nil':n__cons'
s' :: 0':s' → 0':s'
length1' :: n__from':n__nil':n__cons' → 0':s'
activate' :: n__from':n__nil':n__cons' → n__from':n__nil':n__cons'
nil' :: n__from':n__nil':n__cons'
_hole_n__from':n__nil':n__cons'1 :: n__from':n__nil':n__cons'
_hole_0':s'2 :: 0':s'
_gen_n__from':n__nil':n__cons'3 :: Nat → n__from':n__nil':n__cons'
_gen_0':s'4 :: Nat → 0':s'

Heuristically decided to analyse the following defined symbols:
length', length1'

They will be analysed ascendingly in the following order:
length' = length1'

Generator Equations:
_gen_n__from':n__nil':n__cons'3(0) ⇔ n__from'
_gen_n__from':n__nil':n__cons'3(+(x, 1)) ⇔ n__cons'(_gen_n__from':n__nil':n__cons'3(x))
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))

The following defined symbols remain to be analysed:
length1', length'

They will be analysed ascendingly in the following order:
length' = length1'

Proved the following rewrite lemma:
length1'(_gen_n__from':n__nil':n__cons'3(+(1, _n6))) → _*5, rt ∈ Ω(_n6)

Induction Base:
length1'(_gen_n__from':n__nil':n__cons'3(+(1, 0)))

Induction Step:
length1'(_gen_n__from':n__nil':n__cons'3(+(1, +(_$n7, 1)))) →_R^Ω(1)
length'(activate'(_gen_n__from':n__nil':n__cons'3(+(1, +(_$n7, 1))))) →_R^Ω(1)
length'(_gen_n__from':n__nil':n__cons'3(+(2, _$n7))) →_R^Ω(1)
s'(length1'(activate'(_gen_n__from':n__nil':n__cons'3(+(1, _$n7))))) →_R^Ω(1)
s'(length1'(_gen_n__from':n__nil':n__cons'3(+(1, _$n7)))) →_IH
s'(_*5)

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

Lemmas:
length1'(_gen_n__from':n__nil':n__cons'3(+(1, _n6))) → _*5, rt ∈ Ω(_n6)

The following defined symbols remain to be analysed:
length'

They will be analysed ascendingly in the following order:
length' = length1'

Proved the following rewrite lemma:
length'(_gen_n__from':n__nil':n__cons'3(+(1, _n4495))) → _*5, rt ∈ Ω(_n4495)

Induction Base:
length'(_gen_n__from':n__nil':n__cons'3(+(1, 0)))

Induction Step:
length'(_gen_n__from':n__nil':n__cons'3(+(1, +(_$n4496, 1)))) →_R^Ω(1)
s'(length1'(activate'(_gen_n__from':n__nil':n__cons'3(+(1, _$n4496))))) →_R^Ω(1)
s'(length1'(_gen_n__from':n__nil':n__cons'3(+(1, _$n4496)))) →_R^Ω(1)
s'(length'(activate'(_gen_n__from':n__nil':n__cons'3(+(1, _$n4496))))) →_R^Ω(1)
s'(length'(_gen_n__from':n__nil':n__cons'3(+(1, _$n4496)))) →_IH
s'(_*5)

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

Lemmas:
length1'(_gen_n__from':n__nil':n__cons'3(+(1, _n6))) → _*5, rt ∈ Ω(_n6)
length'(_gen_n__from':n__nil':n__cons'3(+(1, _n4495))) → _*5, rt ∈ Ω(_n4495)

The following defined symbols remain to be analysed:
length1'

They will be analysed ascendingly in the following order:
length' = length1'

Proved the following rewrite lemma:
length1'(_gen_n__from':n__nil':n__cons'3(+(1, _n8031))) → _*5, rt ∈ Ω(_n8031)

Induction Base:
length1'(_gen_n__from':n__nil':n__cons'3(+(1, 0)))

Induction Step:
length1'(_gen_n__from':n__nil':n__cons'3(+(1, +(_$n8032, 1)))) →_R^Ω(1)
length'(activate'(_gen_n__from':n__nil':n__cons'3(+(1, +(_$n8032, 1))))) →_R^Ω(1)
length'(_gen_n__from':n__nil':n__cons'3(+(2, _$n8032))) →_R^Ω(1)
s'(length1'(activate'(_gen_n__from':n__nil':n__cons'3(+(1, _$n8032))))) →_R^Ω(1)
s'(length1'(_gen_n__from':n__nil':n__cons'3(+(1, _$n8032)))) →_IH
s'(_*5)

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

Lemmas:
length1'(_gen_n__from':n__nil':n__cons'3(+(1, _n8031))) → _*5, rt ∈ Ω(_n8031)
length'(_gen_n__from':n__nil':n__cons'3(+(1, _n4495))) → _*5, rt ∈ Ω(_n4495)

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
length1'(_gen_n__from':n__nil':n__cons'3(+(1, _n8031))) → _*5, rt ∈ Ω(_n8031)