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

from(X) → cons(X, n__from(s(X)))
after(0, XS) → XS
after(s(N), cons(X, XS)) → after(N, activate(XS))
from(X) → n__from(X)
activate(n__from(X)) → from(X)
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)))
after'(0', XS) → XS
after'(s'(N), cons'(X, XS)) → after'(N, activate'(XS))
from'(X) → n__from'(X)
activate'(n__from'(X)) → from'(X)
activate'(X) → X

Rewrite Strategy: INNERMOST

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

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


from' → cons'(n__from')
after'(0', XS) → XS
after'(s'(N), cons'(XS)) → after'(N, activate'(XS))
from' → n__from'
activate'(n__from') → from'
activate'(X) → X

Rewrite Strategy: INNERMOST

Infered types.

Rules:
from' → cons'(n__from')
after'(0', XS) → XS
after'(s'(N), cons'(XS)) → after'(N, activate'(XS))
from' → n__from'
activate'(n__from') → from'
activate'(X) → X

Types:
from' :: n__from':cons'
cons' :: n__from':cons' → n__from':cons'
n__from' :: n__from':cons'
after' :: 0':s' → n__from':cons' → n__from':cons'
0' :: 0':s'
s' :: 0':s' → 0':s'
activate' :: n__from':cons' → n__from':cons'
_hole_n__from':cons'1 :: n__from':cons'
_hole_0':s'2 :: 0':s'
_gen_n__from':cons'3 :: Nat → n__from':cons'
_gen_0':s'4 :: Nat → 0':s'

Heuristically decided to analyse the following defined symbols:
after'

Rules:
from' → cons'(n__from')
after'(0', XS) → XS
after'(s'(N), cons'(XS)) → after'(N, activate'(XS))
from' → n__from'
activate'(n__from') → from'
activate'(X) → X

Types:
from' :: n__from':cons'
cons' :: n__from':cons' → n__from':cons'
n__from' :: n__from':cons'
after' :: 0':s' → n__from':cons' → n__from':cons'
0' :: 0':s'
s' :: 0':s' → 0':s'
activate' :: n__from':cons' → n__from':cons'
_hole_n__from':cons'1 :: n__from':cons'
_hole_0':s'2 :: 0':s'
_gen_n__from':cons'3 :: Nat → n__from':cons'
_gen_0':s'4 :: Nat → 0':s'

Generator Equations:
_gen_n__from':cons'3(0) ⇔ n__from'
_gen_n__from':cons'3(+(x, 1)) ⇔ cons'(_gen_n__from':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:
after'

Proved the following rewrite lemma:
after'(_gen_0':s'4(_n6), _gen_n__from':cons'3(1)) → _gen_n__from':cons'3(1), rt ∈ Ω(1 + n6)

Induction Base:
after'(_gen_0':s'4(0), _gen_n__from':cons'3(1)) →RΩ(1)
_gen_n__from':cons'3(1)

Induction Step:
after'(_gen_0':s'4(+(_$n7, 1)), _gen_n__from':cons'3(1)) →RΩ(1)
after'(_gen_0':s'4(_$n7), activate'(_gen_n__from':cons'3(0))) →RΩ(1)
after'(_gen_0':s'4(_$n7), from') →RΩ(1)
after'(_gen_0':s'4(_$n7), cons'(n__from')) →IH
_gen_n__from':cons'3(1)

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

Rules:
from' → cons'(n__from')
after'(0', XS) → XS
after'(s'(N), cons'(XS)) → after'(N, activate'(XS))
from' → n__from'
activate'(n__from') → from'
activate'(X) → X

Types:
from' :: n__from':cons'
cons' :: n__from':cons' → n__from':cons'
n__from' :: n__from':cons'
after' :: 0':s' → n__from':cons' → n__from':cons'
0' :: 0':s'
s' :: 0':s' → 0':s'
activate' :: n__from':cons' → n__from':cons'
_hole_n__from':cons'1 :: n__from':cons'
_hole_0':s'2 :: 0':s'
_gen_n__from':cons'3 :: Nat → n__from':cons'
_gen_0':s'4 :: Nat → 0':s'

Lemmas:
after'(_gen_0':s'4(_n6), _gen_n__from':cons'3(1)) → _gen_n__from':cons'3(1), rt ∈ Ω(1 + n6)

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

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
after'(_gen_0':s'4(_n6), _gen_n__from':cons'3(1)) → _gen_n__from':cons'3(1), rt ∈ Ω(1 + n6)