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

div(0, y) → 0
div(x, y) → quot(x, y, y)
quot(0, s(y), z) → 0
quot(s(x), s(y), z) → quot(x, y, z)
quot(x, 0, s(z)) → s(div(x, s(z)))

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

div'(0', y) → 0'
div'(x, y) → quot'(x, y, y)
quot'(0', s'(y), z) → 0'
quot'(s'(x), s'(y), z) → quot'(x, y, z)
quot'(x, 0', s'(z)) → s'(div'(x, s'(z)))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
div'(0', y) → 0'
div'(x, y) → quot'(x, y, y)
quot'(0', s'(y), z) → 0'
quot'(s'(x), s'(y), z) → quot'(x, y, z)
quot'(x, 0', s'(z)) → s'(div'(x, s'(z)))

Types:
div' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
quot' :: 0':s' → 0':s' → 0':s' → 0':s'
s' :: 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_gen_0':s'2 :: Nat → 0':s'

Heuristically decided to analyse the following defined symbols:
div', quot'

They will be analysed ascendingly in the following order:
div' = quot'

Rules:
div'(0', y) → 0'
div'(x, y) → quot'(x, y, y)
quot'(0', s'(y), z) → 0'
quot'(s'(x), s'(y), z) → quot'(x, y, z)
quot'(x, 0', s'(z)) → s'(div'(x, s'(z)))

Types:
div' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
quot' :: 0':s' → 0':s' → 0':s' → 0':s'
s' :: 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_gen_0':s'2 :: Nat → 0':s'

Generator Equations:
_gen_0':s'2(0) ⇔ 0'
_gen_0':s'2(+(x, 1)) ⇔ s'(_gen_0':s'2(x))

The following defined symbols remain to be analysed:
quot', div'

They will be analysed ascendingly in the following order:
div' = quot'

Proved the following rewrite lemma:
quot'(_gen_0':s'2(_n4), _gen_0':s'2(+(1, _n4)), _gen_0':s'2(c)) → _gen_0':s'2(0), rt ∈ Ω(1 + n4)

Induction Base:
quot'(_gen_0':s'2(0), _gen_0':s'2(+(1, 0)), _gen_0':s'2(c)) →RΩ(1)
0'

Induction Step:
quot'(_gen_0':s'2(+(_\$n5, 1)), _gen_0':s'2(+(1, +(_\$n5, 1))), _gen_0':s'2(_c237)) →RΩ(1)
quot'(_gen_0':s'2(_\$n5), _gen_0':s'2(+(1, _\$n5)), _gen_0':s'2(_c237)) →IH
_gen_0':s'2(0)

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

Rules:
div'(0', y) → 0'
div'(x, y) → quot'(x, y, y)
quot'(0', s'(y), z) → 0'
quot'(s'(x), s'(y), z) → quot'(x, y, z)
quot'(x, 0', s'(z)) → s'(div'(x, s'(z)))

Types:
div' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
quot' :: 0':s' → 0':s' → 0':s' → 0':s'
s' :: 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_gen_0':s'2 :: Nat → 0':s'

Lemmas:
quot'(_gen_0':s'2(_n4), _gen_0':s'2(+(1, _n4)), _gen_0':s'2(c)) → _gen_0':s'2(0), rt ∈ Ω(1 + n4)

Generator Equations:
_gen_0':s'2(0) ⇔ 0'
_gen_0':s'2(+(x, 1)) ⇔ s'(_gen_0':s'2(x))

The following defined symbols remain to be analysed:
div'

They will be analysed ascendingly in the following order:
div' = quot'

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

Rules:
div'(0', y) → 0'
div'(x, y) → quot'(x, y, y)
quot'(0', s'(y), z) → 0'
quot'(s'(x), s'(y), z) → quot'(x, y, z)
quot'(x, 0', s'(z)) → s'(div'(x, s'(z)))

Types:
div' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
quot' :: 0':s' → 0':s' → 0':s' → 0':s'
s' :: 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_gen_0':s'2 :: Nat → 0':s'

Lemmas:
quot'(_gen_0':s'2(_n4), _gen_0':s'2(+(1, _n4)), _gen_0':s'2(c)) → _gen_0':s'2(0), rt ∈ Ω(1 + n4)

Generator Equations:
_gen_0':s'2(0) ⇔ 0'
_gen_0':s'2(+(x, 1)) ⇔ s'(_gen_0':s'2(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
quot'(_gen_0':s'2(_n4), _gen_0':s'2(+(1, _n4)), _gen_0':s'2(c)) → _gen_0':s'2(0), rt ∈ Ω(1 + n4)