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

from(X) → cons(X, n__from(s(X)))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, activate(Z))
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)))
sel'(0', cons'(X, Y)) → X
sel'(s'(X), cons'(Y, Z)) → sel'(X, activate'(Z))
from'(X) → n__from'(X)
activate'(n__from'(X)) → from'(X)
activate'(X) → X

Rewrite Strategy: INNERMOST

Infered types.

Rules:
from'(X) → cons'(X, n__from'(s'(X)))
sel'(0', cons'(X, Y)) → X
sel'(s'(X), cons'(Y, Z)) → sel'(X, activate'(Z))
from'(X) → n__from'(X)
activate'(n__from'(X)) → from'(X)
activate'(X) → X

Types:
from' :: s':0' → n__from':cons'
cons' :: s':0' → n__from':cons' → n__from':cons'
n__from' :: s':0' → n__from':cons'
s' :: s':0' → s':0'
sel' :: s':0' → n__from':cons' → s':0'
0' :: s':0'
activate' :: n__from':cons' → n__from':cons'
_hole_n__from':cons'1 :: n__from':cons'
_hole_s':0'2 :: s':0'
_gen_n__from':cons'3 :: Nat → n__from':cons'
_gen_s':0'4 :: Nat → s':0'

Heuristically decided to analyse the following defined symbols:
sel'

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

The following defined symbols remain to be analysed:
sel'

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

The following conjecture could not be proven:

sel'(_gen_s':0'4(_n6), _gen_n__from':cons'3(1)) →^? _*5

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

No more defined symbols left to analyse.