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

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(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(minus(s(x), s(y)), s(y)))

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

Rewrite Strategy: INNERMOST

Infered types.

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

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

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

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

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

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

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

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

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

Proved the following rewrite lemma:
minus'(_gen_0':s'3(_n5), _gen_0':s'3(_n5)) → _gen_0':s'3(0), rt ∈ Ω(1 + n5)

Induction Base:
minus'(_gen_0':s'3(0), _gen_0':s'3(0)) →RΩ(1)
_gen_0':s'3(0)

Induction Step:
minus'(_gen_0':s'3(+(_\$n6, 1)), _gen_0':s'3(+(_\$n6, 1))) →RΩ(1)
minus'(_gen_0':s'3(_\$n6), _gen_0':s'3(_\$n6)) →IH
_gen_0':s'3(0)

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

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

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

Lemmas:
minus'(_gen_0':s'3(_n5), _gen_0':s'3(_n5)) → _gen_0':s'3(0), rt ∈ Ω(1 + n5)

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

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

Proved the following rewrite lemma:
le'(_gen_0':s'3(_n434), _gen_0':s'3(_n434)) → true', rt ∈ Ω(1 + n434)

Induction Base:
le'(_gen_0':s'3(0), _gen_0':s'3(0)) →RΩ(1)
true'

Induction Step:
le'(_gen_0':s'3(+(_\$n435, 1)), _gen_0':s'3(+(_\$n435, 1))) →RΩ(1)
le'(_gen_0':s'3(_\$n435), _gen_0':s'3(_\$n435)) →IH
true'

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

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

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

Lemmas:
minus'(_gen_0':s'3(_n5), _gen_0':s'3(_n5)) → _gen_0':s'3(0), rt ∈ Ω(1 + n5)
le'(_gen_0':s'3(_n434), _gen_0':s'3(_n434)) → true', rt ∈ Ω(1 + n434)

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

The following defined symbols remain to be analysed:
quot'

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

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

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

Lemmas:
minus'(_gen_0':s'3(_n5), _gen_0':s'3(_n5)) → _gen_0':s'3(0), rt ∈ Ω(1 + n5)
le'(_gen_0':s'3(_n434), _gen_0':s'3(_n434)) → true', rt ∈ Ω(1 + n434)

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

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
minus'(_gen_0':s'3(_n5), _gen_0':s'3(_n5)) → _gen_0':s'3(0), rt ∈ Ω(1 + n5)