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

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


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

Rewrite Strategy: INNERMOST

Infered types.

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

Types:
quot' :: 0':s' → 0':s' → 0':s' → 0':s'
0' :: 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:
quot'

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

Types:
quot' :: 0':s' → 0':s' → 0':s' → 0':s'
0' :: 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'

Proved the following rewrite lemma:
quot'(_gen_0':s'2(_n4), _gen_0':s'2(+(1, _n4)), _gen_0':s'2(1)) → _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(1)) →RΩ(1)
0'

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

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

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

Types:
quot' :: 0':s' → 0':s' → 0':s' → 0':s'
0' :: 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(1)) → _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(1)) → _gen_0':s'2(0), rt ∈ Ω(1 + n4)