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

f(x) → s(x)
f(s(s(x))) → s(f(f(x)))

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

f'(x) → s'(x)
f'(s'(s'(x))) → s'(f'(f'(x)))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
f'(x) → s'(x)
f'(s'(s'(x))) → s'(f'(f'(x)))

Types:
f' :: s' → s'
s' :: s' → s'
_hole_s'1 :: s'
_gen_s'2 :: Nat → s'

Heuristically decided to analyse the following defined symbols:
f'

Rules:
f'(x) → s'(x)
f'(s'(s'(x))) → s'(f'(f'(x)))

Types:
f' :: s' → s'
s' :: s' → s'
_hole_s'1 :: s'
_gen_s'2 :: Nat → s'

Generator Equations:
_gen_s'2(0) ⇔ _hole_s'1
_gen_s'2(+(x, 1)) ⇔ s'(_gen_s'2(x))

The following defined symbols remain to be analysed:
f'

Proved the following rewrite lemma:
f'(_gen_s'2(+(2, *(2, _n4)))) → _*3, rt ∈ Ω(n4)

Induction Base:
f'(_gen_s'2(+(2, *(2, 0))))

Induction Step:
f'(_gen_s'2(+(2, *(2, +(_\$n5, 1))))) →RΩ(1)
s'(f'(f'(_gen_s'2(+(2, *(2, _\$n5)))))) →IH
s'(f'(_*3))

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

Rules:
f'(x) → s'(x)
f'(s'(s'(x))) → s'(f'(f'(x)))

Types:
f' :: s' → s'
s' :: s' → s'
_hole_s'1 :: s'
_gen_s'2 :: Nat → s'

Lemmas:
f'(_gen_s'2(+(2, *(2, _n4)))) → _*3, rt ∈ Ω(n4)

Generator Equations:
_gen_s'2(0) ⇔ _hole_s'1
_gen_s'2(+(x, 1)) ⇔ s'(_gen_s'2(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
f'(_gen_s'2(+(2, *(2, _n4)))) → _*3, rt ∈ Ω(n4)