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

from(X) → cons(X, n__from(n__s(X)))
head(cons(X, XS)) → X
2nd(cons(X, XS)) → head(activate(XS))
take(0, XS) → nil
take(s(N), cons(X, XS)) → cons(X, n__take(N, activate(XS)))
sel(0, cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, activate(XS))
from(X) → n__from(X)
s(X) → n__s(X)
take(X1, X2) → n__take(X1, X2)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__take(X1, X2)) → take(activate(X1), activate(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'(n__s'(X)))
head'(cons'(X, XS)) → X
2nd'(cons'(X, XS)) → head'(activate'(XS))
take'(0', XS) → nil'
take'(s'(N), cons'(X, XS)) → cons'(X, n__take'(N, activate'(XS)))
sel'(0', cons'(X, XS)) → X
sel'(s'(N), cons'(X, XS)) → sel'(N, activate'(XS))
from'(X) → n__from'(X)
s'(X) → n__s'(X)
take'(X1, X2) → n__take'(X1, X2)
activate'(n__from'(X)) → from'(activate'(X))
activate'(n__s'(X)) → s'(activate'(X))
activate'(n__take'(X1, X2)) → take'(activate'(X1), activate'(X2))
activate'(X) → X

Rewrite Strategy: INNERMOST

Infered types.

Rules:
from'(X) → cons'(X, n__from'(n__s'(X)))
head'(cons'(X, XS)) → X
2nd'(cons'(X, XS)) → head'(activate'(XS))
take'(0', XS) → nil'
take'(s'(N), cons'(X, XS)) → cons'(X, n__take'(N, activate'(XS)))
sel'(0', cons'(X, XS)) → X
sel'(s'(N), cons'(X, XS)) → sel'(N, activate'(XS))
from'(X) → n__from'(X)
s'(X) → n__s'(X)
take'(X1, X2) → n__take'(X1, X2)
activate'(n__from'(X)) → from'(activate'(X))
activate'(n__s'(X)) → s'(activate'(X))
activate'(n__take'(X1, X2)) → take'(activate'(X1), activate'(X2))
activate'(X) → X

Types:
from' :: n__s':n__from':cons':0':nil':n__take' → n__s':n__from':cons':0':nil':n__take'
cons' :: n__s':n__from':cons':0':nil':n__take' → n__s':n__from':cons':0':nil':n__take' → n__s':n__from':cons':0':nil':n__take'
n__from' :: n__s':n__from':cons':0':nil':n__take' → n__s':n__from':cons':0':nil':n__take'
n__s' :: n__s':n__from':cons':0':nil':n__take' → n__s':n__from':cons':0':nil':n__take'
head' :: n__s':n__from':cons':0':nil':n__take' → n__s':n__from':cons':0':nil':n__take'
2nd' :: n__s':n__from':cons':0':nil':n__take' → n__s':n__from':cons':0':nil':n__take'
activate' :: n__s':n__from':cons':0':nil':n__take' → n__s':n__from':cons':0':nil':n__take'
take' :: n__s':n__from':cons':0':nil':n__take' → n__s':n__from':cons':0':nil':n__take' → n__s':n__from':cons':0':nil':n__take'
0' :: n__s':n__from':cons':0':nil':n__take'
nil' :: n__s':n__from':cons':0':nil':n__take'
s' :: n__s':n__from':cons':0':nil':n__take' → n__s':n__from':cons':0':nil':n__take'
n__take' :: n__s':n__from':cons':0':nil':n__take' → n__s':n__from':cons':0':nil':n__take' → n__s':n__from':cons':0':nil':n__take'
sel' :: n__s':n__from':cons':0':nil':n__take' → n__s':n__from':cons':0':nil':n__take' → n__s':n__from':cons':0':nil':n__take'
_hole_n__s':n__from':cons':0':nil':n__take'1 :: n__s':n__from':cons':0':nil':n__take'
_gen_n__s':n__from':cons':0':nil':n__take'2 :: Nat → n__s':n__from':cons':0':nil':n__take'

Heuristically decided to analyse the following defined symbols:
activate', sel'

They will be analysed ascendingly in the following order:
activate' < sel'

Generator Equations:
_gen_n__s':n__from':cons':0':nil':n__take'2(0) ⇔ 0'
_gen_n__s':n__from':cons':0':nil':n__take'2(+(x, 1)) ⇔ cons'(0', _gen_n__s':n__from':cons':0':nil':n__take'2(x))

The following defined symbols remain to be analysed:
activate', sel'

They will be analysed ascendingly in the following order:
activate' < sel'

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

Generator Equations:
_gen_n__s':n__from':cons':0':nil':n__take'2(0) ⇔ 0'
_gen_n__s':n__from':cons':0':nil':n__take'2(+(x, 1)) ⇔ cons'(0', _gen_n__s':n__from':cons':0':nil':n__take'2(x))

The following defined symbols remain to be analysed:
sel'

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

Generator Equations:
_gen_n__s':n__from':cons':0':nil':n__take'2(0) ⇔ 0'
_gen_n__s':n__from':cons':0':nil':n__take'2(+(x, 1)) ⇔ cons'(0', _gen_n__s':n__from':cons':0':nil':n__take'2(x))

No more defined symbols left to analyse.