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

minus(x, x) → 0
minus(0, x) → 0
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
quot(x, y) → if_quot(minus(x, y), y, le(y, 0), le(y, x))
if_quot(x, y, true, z) → divByZeroError
if_quot(x, y, false, true) → s(quot(x, y))
if_quot(x, y, false, false) → 0

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

minus'(x, x) → 0'
minus'(0', x) → 0'
minus'(x, 0') → x
minus'(s'(x), s'(y)) → minus'(x, y)
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
quot'(x, y) → if_quot'(minus'(x, y), y, le'(y, 0'), le'(y, x))
if_quot'(x, y, true', z) → divByZeroError'
if_quot'(x, y, false', true') → s'(quot'(x, y))
if_quot'(x, y, false', false') → 0'

Rewrite Strategy: INNERMOST

Infered types.

Rules:
minus'(x, x) → 0'
minus'(0', x) → 0'
minus'(x, 0') → x
minus'(s'(x), s'(y)) → minus'(x, y)
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
quot'(x, y) → if_quot'(minus'(x, y), y, le'(y, 0'), le'(y, x))
if_quot'(x, y, true', z) → divByZeroError'
if_quot'(x, y, false', true') → s'(quot'(x, y))
if_quot'(x, y, false', false') → 0'

Types:
minus' :: 0':s':divByZeroError' → 0':s':divByZeroError' → 0':s':divByZeroError'
0' :: 0':s':divByZeroError'
s' :: 0':s':divByZeroError' → 0':s':divByZeroError'
le' :: 0':s':divByZeroError' → 0':s':divByZeroError' → true':false'
true' :: true':false'
false' :: true':false'
quot' :: 0':s':divByZeroError' → 0':s':divByZeroError' → 0':s':divByZeroError'
if_quot' :: 0':s':divByZeroError' → 0':s':divByZeroError' → true':false' → true':false' → 0':s':divByZeroError'
divByZeroError' :: 0':s':divByZeroError'
_hole_0':s':divByZeroError'1 :: 0':s':divByZeroError'
_hole_true':false'2 :: true':false'
_gen_0':s':divByZeroError'3 :: Nat → 0':s':divByZeroError'

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

They will be analysed ascendingly in the following order:
minus' < quot'
le' < quot'

Generator Equations:
_gen_0':s':divByZeroError'3(0) ⇔ 0'
_gen_0':s':divByZeroError'3(+(x, 1)) ⇔ s'(_gen_0':s':divByZeroError'3(x))

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

They will be analysed ascendingly in the following order:
minus' < quot'
le' < quot'

Proved the following rewrite lemma:
minus'(_gen_0':s':divByZeroError'3(_n5), _gen_0':s':divByZeroError'3(_n5)) → _gen_0':s':divByZeroError'3(0), rt ∈ Ω(1 + _n5)

Induction Base:
minus'(_gen_0':s':divByZeroError'3(0), _gen_0':s':divByZeroError'3(0)) →_R^Ω(1)
0'

Induction Step:
minus'(_gen_0':s':divByZeroError'3(+(_$n6, 1)), _gen_0':s':divByZeroError'3(+(_$n6, 1))) →_R^Ω(1)
minus'(_gen_0':s':divByZeroError'3(_$n6), _gen_0':s':divByZeroError'3(_$n6)) →_IH
_gen_0':s':divByZeroError'3(0)

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

Lemmas:
minus'(_gen_0':s':divByZeroError'3(_n5), _gen_0':s':divByZeroError'3(_n5)) → _gen_0':s':divByZeroError'3(0), rt ∈ Ω(1 + _n5)

Generator Equations:
_gen_0':s':divByZeroError'3(0) ⇔ 0'
_gen_0':s':divByZeroError'3(+(x, 1)) ⇔ s'(_gen_0':s':divByZeroError'3(x))

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

They will be analysed ascendingly in the following order:
le' < quot'

Proved the following rewrite lemma:
le'(_gen_0':s':divByZeroError'3(_n1231), _gen_0':s':divByZeroError'3(_n1231)) → true', rt ∈ Ω(1 + _n1231)

Induction Base:
le'(_gen_0':s':divByZeroError'3(0), _gen_0':s':divByZeroError'3(0)) →_R^Ω(1)
true'

Induction Step:
le'(_gen_0':s':divByZeroError'3(+(_$n1232, 1)), _gen_0':s':divByZeroError'3(+(_$n1232, 1))) →_R^Ω(1)
le'(_gen_0':s':divByZeroError'3(_$n1232), _gen_0':s':divByZeroError'3(_$n1232)) →_IH
true'

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

Lemmas:
minus'(_gen_0':s':divByZeroError'3(_n5), _gen_0':s':divByZeroError'3(_n5)) → _gen_0':s':divByZeroError'3(0), rt ∈ Ω(1 + _n5)
le'(_gen_0':s':divByZeroError'3(_n1231), _gen_0':s':divByZeroError'3(_n1231)) → true', rt ∈ Ω(1 + _n1231)

Generator Equations:
_gen_0':s':divByZeroError'3(0) ⇔ 0'
_gen_0':s':divByZeroError'3(+(x, 1)) ⇔ s'(_gen_0':s':divByZeroError'3(x))

The following defined symbols remain to be analysed:
quot'

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

Generator Equations:
_gen_0':s':divByZeroError'3(0) ⇔ 0'
_gen_0':s':divByZeroError'3(+(x, 1)) ⇔ s'(_gen_0':s':divByZeroError'3(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
minus'(_gen_0':s':divByZeroError'3(_n5), _gen_0':s':divByZeroError'3(_n5)) → _gen_0':s':divByZeroError'3(0), rt ∈ Ω(1 + _n5)