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

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

Rewrite Strategy: INNERMOST

Infered types.

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

Types:
f' :: s':0' → s':0'
s' :: s':0' → s':0'
p' :: 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:
f'

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

Types:
f' :: s':0' → s':0'
s' :: s':0' → s':0'
p' :: 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'

Proved the following rewrite lemma:
f'(_gen_s':0'2(+(1, _n4))) → _*3, rt ∈ Ω(n4)

Induction Base:
f'(_gen_s':0'2(+(1, 0)))

Induction Step:
f'(_gen_s':0'2(+(1, +(_$n5, 1)))) →RΩ(1)
s'(f'(f'(p'(s'(_gen_s':0'2(+(1, _$n5))))))) →RΩ(1)
s'(f'(f'(_gen_s':0'2(+(1, _$n5))))) →IH
s'(f'(_*3))

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

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

Types:
f' :: s':0' → s':0'
s' :: s':0' → s':0'
p' :: 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(+(1, _n4))) → _*3, rt ∈ Ω(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(+(1, _n4))) → _*3, rt ∈ Ω(n4)