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

a__fst(0, Z) → nil
a__fst(s(X), cons(Y, Z)) → cons(mark(Y), fst(X, Z))
a__from(X) → cons(mark(X), from(s(X)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(add(X, Y))
a__len(nil) → 0
a__len(cons(X, Z)) → s(len(Z))
mark(fst(X1, X2)) → a__fst(mark(X1), mark(X2))
mark(from(X)) → a__from(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(len(X)) → a__len(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fst(X1, X2) → fst(X1, X2)
a__from(X) → from(X)
a__add(X1, X2) → add(X1, X2)
a__len(X) → len(X)

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

a__fst'(0', Z) → nil'
a__fst'(s'(X), cons'(Y, Z)) → cons'(mark'(Y), fst'(X, Z))
a__from'(X) → cons'(mark'(X), from'(s'(X)))
a__add'(0', X) → mark'(X)
a__add'(s'(X), Y) → s'(add'(X, Y))
a__len'(nil') → 0'
a__len'(cons'(X, Z)) → s'(len'(Z))
mark'(fst'(X1, X2)) → a__fst'(mark'(X1), mark'(X2))
mark'(from'(X)) → a__from'(mark'(X))
mark'(add'(X1, X2)) → a__add'(mark'(X1), mark'(X2))
mark'(len'(X)) → a__len'(mark'(X))
mark'(0') → 0'
mark'(s'(X)) → s'(X)
mark'(nil') → nil'
mark'(cons'(X1, X2)) → cons'(mark'(X1), X2)
a__fst'(X1, X2) → fst'(X1, X2)
a__from'(X) → from'(X)
a__add'(X1, X2) → add'(X1, X2)
a__len'(X) → len'(X)

Rewrite Strategy: INNERMOST

Sliced the following arguments:
s'/0
cons'/1

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

a__fst'(0', Z) → nil'
a__fst'(s', cons'(Y)) → cons'(mark'(Y))
a__from'(X) → cons'(mark'(X))
a__add'(0', X) → mark'(X)
a__add'(s', Y) → s'
a__len'(nil') → 0'
a__len'(cons'(X)) → s'
mark'(fst'(X1, X2)) → a__fst'(mark'(X1), mark'(X2))
mark'(from'(X)) → a__from'(mark'(X))
mark'(add'(X1, X2)) → a__add'(mark'(X1), mark'(X2))
mark'(len'(X)) → a__len'(mark'(X))
mark'(0') → 0'
mark'(s') → s'
mark'(nil') → nil'
mark'(cons'(X1)) → cons'(mark'(X1))
a__fst'(X1, X2) → fst'(X1, X2)
a__from'(X) → from'(X)
a__add'(X1, X2) → add'(X1, X2)
a__len'(X) → len'(X)

Rewrite Strategy: INNERMOST

Infered types.

Rules:
a__fst'(0', Z) → nil'
a__fst'(s', cons'(Y)) → cons'(mark'(Y))
a__from'(X) → cons'(mark'(X))
a__add'(0', X) → mark'(X)
a__add'(s', Y) → s'
a__len'(nil') → 0'
a__len'(cons'(X)) → s'
mark'(fst'(X1, X2)) → a__fst'(mark'(X1), mark'(X2))
mark'(from'(X)) → a__from'(mark'(X))
mark'(add'(X1, X2)) → a__add'(mark'(X1), mark'(X2))
mark'(len'(X)) → a__len'(mark'(X))
mark'(0') → 0'
mark'(s') → s'
mark'(nil') → nil'
mark'(cons'(X1)) → cons'(mark'(X1))
a__fst'(X1, X2) → fst'(X1, X2)
a__from'(X) → from'(X)
a__add'(X1, X2) → add'(X1, X2)
a__len'(X) → len'(X)

Types:
a__fst' :: 0':nil':s':cons':fst':from':add':len' → 0':nil':s':cons':fst':from':add':len' → 0':nil':s':cons':fst':from':add':len'
0' :: 0':nil':s':cons':fst':from':add':len'
nil' :: 0':nil':s':cons':fst':from':add':len'
s' :: 0':nil':s':cons':fst':from':add':len'
cons' :: 0':nil':s':cons':fst':from':add':len' → 0':nil':s':cons':fst':from':add':len'
mark' :: 0':nil':s':cons':fst':from':add':len' → 0':nil':s':cons':fst':from':add':len'
a__from' :: 0':nil':s':cons':fst':from':add':len' → 0':nil':s':cons':fst':from':add':len'
a__add' :: 0':nil':s':cons':fst':from':add':len' → 0':nil':s':cons':fst':from':add':len' → 0':nil':s':cons':fst':from':add':len'
a__len' :: 0':nil':s':cons':fst':from':add':len' → 0':nil':s':cons':fst':from':add':len'
fst' :: 0':nil':s':cons':fst':from':add':len' → 0':nil':s':cons':fst':from':add':len' → 0':nil':s':cons':fst':from':add':len'
from' :: 0':nil':s':cons':fst':from':add':len' → 0':nil':s':cons':fst':from':add':len'
add' :: 0':nil':s':cons':fst':from':add':len' → 0':nil':s':cons':fst':from':add':len' → 0':nil':s':cons':fst':from':add':len'
len' :: 0':nil':s':cons':fst':from':add':len' → 0':nil':s':cons':fst':from':add':len'
_hole_0':nil':s':cons':fst':from':add':len'1 :: 0':nil':s':cons':fst':from':add':len'
_gen_0':nil':s':cons':fst':from':add':len'2 :: Nat → 0':nil':s':cons':fst':from':add':len'

Heuristically decided to analyse the following defined symbols:
mark', a__from'

They will be analysed ascendingly in the following order:
mark' = a__from'

Generator Equations:
_gen_0':nil':s':cons':fst':from':add':len'2(0) ⇔ 0'
_gen_0':nil':s':cons':fst':from':add':len'2(+(x, 1)) ⇔ cons'(_gen_0':nil':s':cons':fst':from':add':len'2(x))

The following defined symbols remain to be analysed:
a__from', mark'

They will be analysed ascendingly in the following order:
mark' = a__from'

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

Generator Equations:
_gen_0':nil':s':cons':fst':from':add':len'2(0) ⇔ 0'
_gen_0':nil':s':cons':fst':from':add':len'2(+(x, 1)) ⇔ cons'(_gen_0':nil':s':cons':fst':from':add':len'2(x))

The following defined symbols remain to be analysed:
mark'

They will be analysed ascendingly in the following order:
mark' = a__from'

Proved the following rewrite lemma:
mark'(_gen_0':nil':s':cons':fst':from':add':len'2(_n10431)) → _gen_0':nil':s':cons':fst':from':add':len'2(_n10431), rt ∈ Ω(1 + _n10431)

Induction Base:
mark'(_gen_0':nil':s':cons':fst':from':add':len'2(0)) →_R^Ω(1)
0'

Induction Step:
mark'(_gen_0':nil':s':cons':fst':from':add':len'2(+(_$n10432, 1))) →_R^Ω(1)
cons'(mark'(_gen_0':nil':s':cons':fst':from':add':len'2(_$n10432))) →_IH
cons'(_gen_0':nil':s':cons':fst':from':add':len'2(_$n10432))

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

Lemmas:
mark'(_gen_0':nil':s':cons':fst':from':add':len'2(_n10431)) → _gen_0':nil':s':cons':fst':from':add':len'2(_n10431), rt ∈ Ω(1 + _n10431)

Generator Equations:
_gen_0':nil':s':cons':fst':from':add':len'2(0) ⇔ 0'
_gen_0':nil':s':cons':fst':from':add':len'2(+(x, 1)) ⇔ cons'(_gen_0':nil':s':cons':fst':from':add':len'2(x))

The following defined symbols remain to be analysed:
a__from'

They will be analysed ascendingly in the following order:
mark' = a__from'

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

Lemmas:
mark'(_gen_0':nil':s':cons':fst':from':add':len'2(_n10431)) → _gen_0':nil':s':cons':fst':from':add':len'2(_n10431), rt ∈ Ω(1 + _n10431)

Generator Equations:
_gen_0':nil':s':cons':fst':from':add':len'2(0) ⇔ 0'
_gen_0':nil':s':cons':fst':from':add':len'2(+(x, 1)) ⇔ cons'(_gen_0':nil':s':cons':fst':from':add':len'2(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
mark'(_gen_0':nil':s':cons':fst':from':add':len'2(_n10431)) → _gen_0':nil':s':cons':fst':from':add':len'2(_n10431), rt ∈ Ω(1 + _n10431)