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

f(s(s(s(s(s(s(s(s(x)))))))), y, y) → f(id(s(s(s(s(s(s(s(s(x))))))))), y, y)
id(s(x)) → s(id(x))
id(0) → 0

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

f'(s'(s'(s'(s'(s'(s'(s'(s'(x)))))))), y, y) → f'(id'(s'(s'(s'(s'(s'(s'(s'(s'(x))))))))), y, y)
id'(s'(x)) → s'(id'(x))
id'(0') → 0'

Rewrite Strategy: INNERMOST

Infered types.

Rules:
f'(s'(s'(s'(s'(s'(s'(s'(s'(x)))))))), y, y) → f'(id'(s'(s'(s'(s'(s'(s'(s'(s'(x))))))))), y, y)
id'(s'(x)) → s'(id'(x))
id'(0') → 0'

Types:
f' :: s':0' → a → a → f'
s' :: s':0' → s':0'
id' :: s':0' → s':0'
0' :: s':0'
_hole_f'1 :: f'
_hole_s':0'2 :: s':0'
_hole_a3 :: a
_gen_s':0'4 :: Nat → s':0'

Heuristically decided to analyse the following defined symbols:
f', id'

They will be analysed ascendingly in the following order:
id' < f'

Rules:
f'(s'(s'(s'(s'(s'(s'(s'(s'(x)))))))), y, y) → f'(id'(s'(s'(s'(s'(s'(s'(s'(s'(x))))))))), y, y)
id'(s'(x)) → s'(id'(x))
id'(0') → 0'

Types:
f' :: s':0' → a → a → f'
s' :: s':0' → s':0'
id' :: s':0' → s':0'
0' :: s':0'
_hole_f'1 :: f'
_hole_s':0'2 :: s':0'
_hole_a3 :: a
_gen_s':0'4 :: Nat → s':0'

Generator Equations:
_gen_s':0'4(0) ⇔ 0'
_gen_s':0'4(+(x, 1)) ⇔ s'(_gen_s':0'4(x))

The following defined symbols remain to be analysed:
id', f'

They will be analysed ascendingly in the following order:
id' < f'

Proved the following rewrite lemma:
id'(_gen_s':0'4(_n6)) → _gen_s':0'4(_n6), rt ∈ Ω(1 + n6)

Induction Base:
id'(_gen_s':0'4(0)) →RΩ(1)
0'

Induction Step:
id'(_gen_s':0'4(+(_\$n7, 1))) →RΩ(1)
s'(id'(_gen_s':0'4(_\$n7))) →IH
s'(_gen_s':0'4(_\$n7))

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

Rules:
f'(s'(s'(s'(s'(s'(s'(s'(s'(x)))))))), y, y) → f'(id'(s'(s'(s'(s'(s'(s'(s'(s'(x))))))))), y, y)
id'(s'(x)) → s'(id'(x))
id'(0') → 0'

Types:
f' :: s':0' → a → a → f'
s' :: s':0' → s':0'
id' :: s':0' → s':0'
0' :: s':0'
_hole_f'1 :: f'
_hole_s':0'2 :: s':0'
_hole_a3 :: a
_gen_s':0'4 :: Nat → s':0'

Lemmas:
id'(_gen_s':0'4(_n6)) → _gen_s':0'4(_n6), rt ∈ Ω(1 + n6)

Generator Equations:
_gen_s':0'4(0) ⇔ 0'
_gen_s':0'4(+(x, 1)) ⇔ s'(_gen_s':0'4(x))

The following defined symbols remain to be analysed:
f'

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

Rules:
f'(s'(s'(s'(s'(s'(s'(s'(s'(x)))))))), y, y) → f'(id'(s'(s'(s'(s'(s'(s'(s'(s'(x))))))))), y, y)
id'(s'(x)) → s'(id'(x))
id'(0') → 0'

Types:
f' :: s':0' → a → a → f'
s' :: s':0' → s':0'
id' :: s':0' → s':0'
0' :: s':0'
_hole_f'1 :: f'
_hole_s':0'2 :: s':0'
_hole_a3 :: a
_gen_s':0'4 :: Nat → s':0'

Lemmas:
id'(_gen_s':0'4(_n6)) → _gen_s':0'4(_n6), rt ∈ Ω(1 + n6)

Generator Equations:
_gen_s':0'4(0) ⇔ 0'
_gen_s':0'4(+(x, 1)) ⇔ s'(_gen_s':0'4(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
id'(_gen_s':0'4(_n6)) → _gen_s':0'4(_n6), rt ∈ Ω(1 + n6)