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

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

Rewrite Strategy: INNERMOST

Infered types.

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

Types:
f' :: s':0' → s':0'
s' :: s':0' → s':0'
g' :: s':0' → g'
0' :: s':0'
_hole_s':0'1 :: s':0'
_hole_g'2 :: g'
_gen_s':0'3 :: Nat → s':0'

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

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

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

Types:
f' :: s':0' → s':0'
s' :: s':0' → s':0'
g' :: s':0' → g'
0' :: s':0'
_hole_s':0'1 :: s':0'
_hole_g'2 :: g'
_gen_s':0'3 :: Nat → s':0'

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

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

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

Proved the following rewrite lemma:
f'(_gen_s':0'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)

Induction Base:
f'(_gen_s':0'3(+(1, 0)))

Induction Step:
f'(_gen_s':0'3(+(1, +(_$n6, 1)))) →RΩ(1)
f'(_gen_s':0'3(+(1, _$n6))) →IH
_*4

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

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

Types:
f' :: s':0' → s':0'
s' :: s':0' → s':0'
g' :: s':0' → g'
0' :: s':0'
_hole_s':0'1 :: s':0'
_hole_g'2 :: g'
_gen_s':0'3 :: Nat → s':0'

Lemmas:
f'(_gen_s':0'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)

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

The following defined symbols remain to be analysed:
g'

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

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

Types:
f' :: s':0' → s':0'
s' :: s':0' → s':0'
g' :: s':0' → g'
0' :: s':0'
_hole_s':0'1 :: s':0'
_hole_g'2 :: g'
_gen_s':0'3 :: Nat → s':0'

Lemmas:
f'(_gen_s':0'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)

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

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
f'(_gen_s':0'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)