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

g(s(x)) → f(x)
f(0) → s(0)
f(s(x)) → s(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:


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

Rewrite Strategy: INNERMOST

Infered types.

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

Types:
g' :: s':0' → s':0'
s' :: s':0' → s':0'
f' :: s':0' → s':0'
0' :: s':0'
_hole_s':0'1 :: s':0'
_gen_s':0'2 :: Nat → s':0'

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

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

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

Types:
g' :: s':0' → s':0'
s' :: s':0' → s':0'
f' :: s':0' → s':0'
0' :: s':0'
_hole_s':0'1 :: s':0'
_gen_s':0'2 :: Nat → s':0'

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

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

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

Proved the following rewrite lemma:
f'(_gen_s':0'2(*(2, _n4))) → _gen_s':0'2(+(1, *(2, _n4))), rt ∈ Ω(1 + n4)

Induction Base:
f'(_gen_s':0'2(*(2, 0))) →RΩ(1)
s'(0')

Induction Step:
f'(_gen_s':0'2(*(2, +(_$n5, 1)))) →RΩ(1)
s'(s'(g'(_gen_s':0'2(+(1, *(2, _$n5)))))) →RΩ(1)
s'(s'(f'(_gen_s':0'2(*(2, _$n5))))) →IH
s'(s'(_gen_s':0'2(+(1, *(2, _$n5)))))

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

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

Types:
g' :: s':0' → s':0'
s' :: s':0' → s':0'
f' :: s':0' → s':0'
0' :: s':0'
_hole_s':0'1 :: s':0'
_gen_s':0'2 :: Nat → s':0'

Lemmas:
f'(_gen_s':0'2(*(2, _n4))) → _gen_s':0'2(+(1, *(2, _n4))), rt ∈ Ω(1 + n4)

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

The following defined symbols remain to be analysed:
g'

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

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

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

Types:
g' :: s':0' → s':0'
s' :: s':0' → s':0'
f' :: s':0' → s':0'
0' :: s':0'
_hole_s':0'1 :: s':0'
_gen_s':0'2 :: Nat → s':0'

Lemmas:
f'(_gen_s':0'2(*(2, _n4))) → _gen_s':0'2(+(1, *(2, _n4))), rt ∈ Ω(1 + n4)

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

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
f'(_gen_s':0'2(*(2, _n4))) → _gen_s':0'2(+(1, *(2, _n4))), rt ∈ Ω(1 + n4)