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

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

Rewrite Strategy: INNERMOST

Infered types.

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

Types:
f' :: 0':s' → 0':s' → 0':s' → f'
g' :: 0':s' → 0':s'
s' :: 0':s' → 0':s'
0' :: 0':s'
_hole_f'1 :: f'
_hole_0':s'2 :: 0':s'
_gen_0':s'3 :: Nat → 0':s'

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

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

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

Types:
f' :: 0':s' → 0':s' → 0':s' → f'
g' :: 0':s' → 0':s'
s' :: 0':s' → 0':s'
0' :: 0':s'
_hole_f'1 :: f'
_hole_0':s'2 :: 0':s'
_gen_0':s'3 :: Nat → 0':s'

Generator Equations:
_gen_0':s'3(0) ⇔ 0'
_gen_0':s'3(+(x, 1)) ⇔ s'(_gen_0':s'3(x))

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

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

Proved the following rewrite lemma:
g'(_gen_0':s'3(_n5)) → _gen_0':s'3(_n5), rt ∈ Ω(1 + n5)

Induction Base:
g'(_gen_0':s'3(0)) →RΩ(1)
0'

Induction Step:
g'(_gen_0':s'3(+(_$n6, 1))) →RΩ(1)
s'(g'(_gen_0':s'3(_$n6))) →IH
s'(_gen_0':s'3(_$n6))

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

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

Types:
f' :: 0':s' → 0':s' → 0':s' → f'
g' :: 0':s' → 0':s'
s' :: 0':s' → 0':s'
0' :: 0':s'
_hole_f'1 :: f'
_hole_0':s'2 :: 0':s'
_gen_0':s'3 :: Nat → 0':s'

Lemmas:
g'(_gen_0':s'3(_n5)) → _gen_0':s'3(_n5), rt ∈ Ω(1 + n5)

Generator Equations:
_gen_0':s'3(0) ⇔ 0'
_gen_0':s'3(+(x, 1)) ⇔ s'(_gen_0':s'3(x))

The following defined symbols remain to be analysed:
f'

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

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

Types:
f' :: 0':s' → 0':s' → 0':s' → f'
g' :: 0':s' → 0':s'
s' :: 0':s' → 0':s'
0' :: 0':s'
_hole_f'1 :: f'
_hole_0':s'2 :: 0':s'
_gen_0':s'3 :: Nat → 0':s'

Lemmas:
g'(_gen_0':s'3(_n5)) → _gen_0':s'3(_n5), rt ∈ Ω(1 + n5)

Generator Equations:
_gen_0':s'3(0) ⇔ 0'
_gen_0':s'3(+(x, 1)) ⇔ s'(_gen_0':s'3(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
g'(_gen_0':s'3(_n5)) → _gen_0':s'3(_n5), rt ∈ Ω(1 + n5)