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

first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
from(X) → cons(X, n__from(n__s(X)))
first(X1, X2) → n__first(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(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:

first'(0', X) → nil'
first'(s'(X), cons'(Y, Z)) → cons'(Y, n__first'(X, activate'(Z)))
from'(X) → cons'(X, n__from'(n__s'(X)))
first'(X1, X2) → n__first'(X1, X2)
from'(X) → n__from'(X)
s'(X) → n__s'(X)
activate'(n__first'(X1, X2)) → first'(activate'(X1), activate'(X2))
activate'(n__from'(X)) → from'(activate'(X))
activate'(n__s'(X)) → s'(activate'(X))
activate'(X) → X

Rewrite Strategy: INNERMOST

Sliced the following arguments:
cons'/0

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

first'(0', X) → nil'
first'(s'(X), cons'(Z)) → cons'(n__first'(X, activate'(Z)))
from'(X) → cons'(n__from'(n__s'(X)))
first'(X1, X2) → n__first'(X1, X2)
from'(X) → n__from'(X)
s'(X) → n__s'(X)
activate'(n__first'(X1, X2)) → first'(activate'(X1), activate'(X2))
activate'(n__from'(X)) → from'(activate'(X))
activate'(n__s'(X)) → s'(activate'(X))
activate'(X) → X

Rewrite Strategy: INNERMOST

Infered types.

Rules:
first'(0', X) → nil'
first'(s'(X), cons'(Z)) → cons'(n__first'(X, activate'(Z)))
from'(X) → cons'(n__from'(n__s'(X)))
first'(X1, X2) → n__first'(X1, X2)
from'(X) → n__from'(X)
s'(X) → n__s'(X)
activate'(n__first'(X1, X2)) → first'(activate'(X1), activate'(X2))
activate'(n__from'(X)) → from'(activate'(X))
activate'(n__s'(X)) → s'(activate'(X))
activate'(X) → X

Types:
first' :: 0':nil':cons':n__first':n__s':n__from' → 0':nil':cons':n__first':n__s':n__from' → 0':nil':cons':n__first':n__s':n__from'
0' :: 0':nil':cons':n__first':n__s':n__from'
nil' :: 0':nil':cons':n__first':n__s':n__from'
s' :: 0':nil':cons':n__first':n__s':n__from' → 0':nil':cons':n__first':n__s':n__from'
cons' :: 0':nil':cons':n__first':n__s':n__from' → 0':nil':cons':n__first':n__s':n__from'
n__first' :: 0':nil':cons':n__first':n__s':n__from' → 0':nil':cons':n__first':n__s':n__from' → 0':nil':cons':n__first':n__s':n__from'
activate' :: 0':nil':cons':n__first':n__s':n__from' → 0':nil':cons':n__first':n__s':n__from'
from' :: 0':nil':cons':n__first':n__s':n__from' → 0':nil':cons':n__first':n__s':n__from'
n__from' :: 0':nil':cons':n__first':n__s':n__from' → 0':nil':cons':n__first':n__s':n__from'
n__s' :: 0':nil':cons':n__first':n__s':n__from' → 0':nil':cons':n__first':n__s':n__from'
_hole_0':nil':cons':n__first':n__s':n__from'1 :: 0':nil':cons':n__first':n__s':n__from'
_gen_0':nil':cons':n__first':n__s':n__from'2 :: Nat → 0':nil':cons':n__first':n__s':n__from'

Heuristically decided to analyse the following defined symbols:
activate'

Rules:
first'(0', X) → nil'
first'(s'(X), cons'(Z)) → cons'(n__first'(X, activate'(Z)))
from'(X) → cons'(n__from'(n__s'(X)))
first'(X1, X2) → n__first'(X1, X2)
from'(X) → n__from'(X)
s'(X) → n__s'(X)
activate'(n__first'(X1, X2)) → first'(activate'(X1), activate'(X2))
activate'(n__from'(X)) → from'(activate'(X))
activate'(n__s'(X)) → s'(activate'(X))
activate'(X) → X

Types:
first' :: 0':nil':cons':n__first':n__s':n__from' → 0':nil':cons':n__first':n__s':n__from' → 0':nil':cons':n__first':n__s':n__from'
0' :: 0':nil':cons':n__first':n__s':n__from'
nil' :: 0':nil':cons':n__first':n__s':n__from'
s' :: 0':nil':cons':n__first':n__s':n__from' → 0':nil':cons':n__first':n__s':n__from'
cons' :: 0':nil':cons':n__first':n__s':n__from' → 0':nil':cons':n__first':n__s':n__from'
n__first' :: 0':nil':cons':n__first':n__s':n__from' → 0':nil':cons':n__first':n__s':n__from' → 0':nil':cons':n__first':n__s':n__from'
activate' :: 0':nil':cons':n__first':n__s':n__from' → 0':nil':cons':n__first':n__s':n__from'
from' :: 0':nil':cons':n__first':n__s':n__from' → 0':nil':cons':n__first':n__s':n__from'
n__from' :: 0':nil':cons':n__first':n__s':n__from' → 0':nil':cons':n__first':n__s':n__from'
n__s' :: 0':nil':cons':n__first':n__s':n__from' → 0':nil':cons':n__first':n__s':n__from'
_hole_0':nil':cons':n__first':n__s':n__from'1 :: 0':nil':cons':n__first':n__s':n__from'
_gen_0':nil':cons':n__first':n__s':n__from'2 :: Nat → 0':nil':cons':n__first':n__s':n__from'

Generator Equations:
_gen_0':nil':cons':n__first':n__s':n__from'2(0) ⇔ 0'
_gen_0':nil':cons':n__first':n__s':n__from'2(+(x, 1)) ⇔ cons'(_gen_0':nil':cons':n__first':n__s':n__from'2(x))

The following defined symbols remain to be analysed:
activate'

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

Rules:
first'(0', X) → nil'
first'(s'(X), cons'(Z)) → cons'(n__first'(X, activate'(Z)))
from'(X) → cons'(n__from'(n__s'(X)))
first'(X1, X2) → n__first'(X1, X2)
from'(X) → n__from'(X)
s'(X) → n__s'(X)
activate'(n__first'(X1, X2)) → first'(activate'(X1), activate'(X2))
activate'(n__from'(X)) → from'(activate'(X))
activate'(n__s'(X)) → s'(activate'(X))
activate'(X) → X

Types:
first' :: 0':nil':cons':n__first':n__s':n__from' → 0':nil':cons':n__first':n__s':n__from' → 0':nil':cons':n__first':n__s':n__from'
0' :: 0':nil':cons':n__first':n__s':n__from'
nil' :: 0':nil':cons':n__first':n__s':n__from'
s' :: 0':nil':cons':n__first':n__s':n__from' → 0':nil':cons':n__first':n__s':n__from'
cons' :: 0':nil':cons':n__first':n__s':n__from' → 0':nil':cons':n__first':n__s':n__from'
n__first' :: 0':nil':cons':n__first':n__s':n__from' → 0':nil':cons':n__first':n__s':n__from' → 0':nil':cons':n__first':n__s':n__from'
activate' :: 0':nil':cons':n__first':n__s':n__from' → 0':nil':cons':n__first':n__s':n__from'
from' :: 0':nil':cons':n__first':n__s':n__from' → 0':nil':cons':n__first':n__s':n__from'
n__from' :: 0':nil':cons':n__first':n__s':n__from' → 0':nil':cons':n__first':n__s':n__from'
n__s' :: 0':nil':cons':n__first':n__s':n__from' → 0':nil':cons':n__first':n__s':n__from'
_hole_0':nil':cons':n__first':n__s':n__from'1 :: 0':nil':cons':n__first':n__s':n__from'
_gen_0':nil':cons':n__first':n__s':n__from'2 :: Nat → 0':nil':cons':n__first':n__s':n__from'

Generator Equations:
_gen_0':nil':cons':n__first':n__s':n__from'2(0) ⇔ 0'
_gen_0':nil':cons':n__first':n__s':n__from'2(+(x, 1)) ⇔ cons'(_gen_0':nil':cons':n__first':n__s':n__from'2(x))

No more defined symbols left to analyse.