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

fst(0, Z) → nil
fst(s(X), cons(Y, Z)) → cons(Y, n__fst(activate(X), activate(Z)))
from(X) → cons(X, n__from(s(X)))
add(0, X) → X
add(s(X), Y) → s(n__add(activate(X), Y))
len(nil) → 0
len(cons(X, Z)) → s(n__len(activate(Z)))
fst(X1, X2) → n__fst(X1, X2)
from(X) → n__from(X)
add(X1, X2) → n__add(X1, X2)
len(X) → n__len(X)
activate(n__fst(X1, X2)) → fst(X1, X2)
activate(n__from(X)) → from(X)
activate(n__add(X1, X2)) → add(X1, X2)
activate(n__len(X)) → len(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:

fst'(0', Z) → nil'
fst'(s'(X), cons'(Y, Z)) → cons'(Y, n__fst'(activate'(X), activate'(Z)))
from'(X) → cons'(X, n__from'(s'(X)))
add'(0', X) → X
add'(s'(X), Y) → s'(n__add'(activate'(X), Y))
len'(nil') → 0'
len'(cons'(X, Z)) → s'(n__len'(activate'(Z)))
fst'(X1, X2) → n__fst'(X1, X2)
from'(X) → n__from'(X)
add'(X1, X2) → n__add'(X1, X2)
len'(X) → n__len'(X)
activate'(n__fst'(X1, X2)) → fst'(X1, X2)
activate'(n__from'(X)) → from'(X)
activate'(n__add'(X1, X2)) → add'(X1, X2)
activate'(n__len'(X)) → len'(X)
activate'(X) → X

Rewrite Strategy: INNERMOST

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

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

fst'(0', Z) → nil'
fst'(s'(X), cons'(Z)) → cons'(n__fst'(activate'(X), activate'(Z)))
from' → cons'(n__from')
add'(0', X) → X
add'(s'(X), Y) → s'(n__add'(activate'(X), Y))
len'(nil') → 0'
len'(cons'(Z)) → s'(n__len'(activate'(Z)))
fst'(X1, X2) → n__fst'(X1, X2)
from' → n__from'
add'(X1, X2) → n__add'(X1, X2)
len'(X) → n__len'(X)
activate'(n__fst'(X1, X2)) → fst'(X1, X2)
activate'(n__from') → from'
activate'(n__add'(X1, X2)) → add'(X1, X2)
activate'(n__len'(X)) → len'(X)
activate'(X) → X

Rewrite Strategy: INNERMOST

Infered types.

Rules:
fst'(0', Z) → nil'
fst'(s'(X), cons'(Z)) → cons'(n__fst'(activate'(X), activate'(Z)))
from' → cons'(n__from')
add'(0', X) → X
add'(s'(X), Y) → s'(n__add'(activate'(X), Y))
len'(nil') → 0'
len'(cons'(Z)) → s'(n__len'(activate'(Z)))
fst'(X1, X2) → n__fst'(X1, X2)
from' → n__from'
add'(X1, X2) → n__add'(X1, X2)
len'(X) → n__len'(X)
activate'(n__fst'(X1, X2)) → fst'(X1, X2)
activate'(n__from') → from'
activate'(n__add'(X1, X2)) → add'(X1, X2)
activate'(n__len'(X)) → len'(X)
activate'(X) → X

Types:
fst' :: 0':nil':s':cons':n__fst':n__from':n__add':n__len' → 0':nil':s':cons':n__fst':n__from':n__add':n__len' → 0':nil':s':cons':n__fst':n__from':n__add':n__len'
0' :: 0':nil':s':cons':n__fst':n__from':n__add':n__len'
nil' :: 0':nil':s':cons':n__fst':n__from':n__add':n__len'
s' :: 0':nil':s':cons':n__fst':n__from':n__add':n__len' → 0':nil':s':cons':n__fst':n__from':n__add':n__len'
cons' :: 0':nil':s':cons':n__fst':n__from':n__add':n__len' → 0':nil':s':cons':n__fst':n__from':n__add':n__len'
n__fst' :: 0':nil':s':cons':n__fst':n__from':n__add':n__len' → 0':nil':s':cons':n__fst':n__from':n__add':n__len' → 0':nil':s':cons':n__fst':n__from':n__add':n__len'
activate' :: 0':nil':s':cons':n__fst':n__from':n__add':n__len' → 0':nil':s':cons':n__fst':n__from':n__add':n__len'
from' :: 0':nil':s':cons':n__fst':n__from':n__add':n__len'
n__from' :: 0':nil':s':cons':n__fst':n__from':n__add':n__len'
add' :: 0':nil':s':cons':n__fst':n__from':n__add':n__len' → 0':nil':s':cons':n__fst':n__from':n__add':n__len' → 0':nil':s':cons':n__fst':n__from':n__add':n__len'
n__add' :: 0':nil':s':cons':n__fst':n__from':n__add':n__len' → 0':nil':s':cons':n__fst':n__from':n__add':n__len' → 0':nil':s':cons':n__fst':n__from':n__add':n__len'
len' :: 0':nil':s':cons':n__fst':n__from':n__add':n__len' → 0':nil':s':cons':n__fst':n__from':n__add':n__len'
n__len' :: 0':nil':s':cons':n__fst':n__from':n__add':n__len' → 0':nil':s':cons':n__fst':n__from':n__add':n__len'
_hole_0':nil':s':cons':n__fst':n__from':n__add':n__len'1 :: 0':nil':s':cons':n__fst':n__from':n__add':n__len'
_gen_0':nil':s':cons':n__fst':n__from':n__add':n__len'2 :: Nat → 0':nil':s':cons':n__fst':n__from':n__add':n__len'

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

They will be analysed ascendingly in the following order:
activate' = len'

Generator Equations:
_gen_0':nil':s':cons':n__fst':n__from':n__add':n__len'2(0) ⇔ 0'
_gen_0':nil':s':cons':n__fst':n__from':n__add':n__len'2(+(x, 1)) ⇔ s'(_gen_0':nil':s':cons':n__fst':n__from':n__add':n__len'2(x))

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

They will be analysed ascendingly in the following order:
activate' = len'

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

The following defined symbols remain to be analysed:
activate'

They will be analysed ascendingly in the following order:
activate' = len'

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

No more defined symbols left to analyse.