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

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x, x) → 0
minus(x, 0) → x
minus(0, x) → 0
minus(s(x), s(y)) → minus(x, y)
isZero(0) → true
isZero(s(x)) → false
mod(x, y) → if_mod(isZero(y), le(y, x), x, y, minus(x, y))
if_mod(true, b, x, y, z) → divByZeroError
if_mod(false, false, x, y, z) → x
if_mod(false, true, x, y, z) → mod(z, y)

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
minus'(x, x) → 0'
minus'(x, 0') → x
minus'(0', x) → 0'
minus'(s'(x), s'(y)) → minus'(x, y)
isZero'(0') → true'
isZero'(s'(x)) → false'
mod'(x, y) → if_mod'(isZero'(y), le'(y, x), x, y, minus'(x, y))
if_mod'(true', b, x, y, z) → divByZeroError'
if_mod'(false', false', x, y, z) → x
if_mod'(false', true', x, y, z) → mod'(z, y)

Rewrite Strategy: INNERMOST

Infered types.

Rules:
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
minus'(x, x) → 0'
minus'(x, 0') → x
minus'(0', x) → 0'
minus'(s'(x), s'(y)) → minus'(x, y)
isZero'(0') → true'
isZero'(s'(x)) → false'
mod'(x, y) → if_mod'(isZero'(y), le'(y, x), x, y, minus'(x, y))
if_mod'(true', b, x, y, z) → divByZeroError'
if_mod'(false', false', x, y, z) → x
if_mod'(false', true', x, y, z) → mod'(z, y)

Types:
le' :: 0':s':divByZeroError' → 0':s':divByZeroError' → true':false'
0' :: 0':s':divByZeroError'
true' :: true':false'
s' :: 0':s':divByZeroError' → 0':s':divByZeroError'
false' :: true':false'
minus' :: 0':s':divByZeroError' → 0':s':divByZeroError' → 0':s':divByZeroError'
isZero' :: 0':s':divByZeroError' → true':false'
mod' :: 0':s':divByZeroError' → 0':s':divByZeroError' → 0':s':divByZeroError'
if_mod' :: true':false' → true':false' → 0':s':divByZeroError' → 0':s':divByZeroError' → 0':s':divByZeroError' → 0':s':divByZeroError'
divByZeroError' :: 0':s':divByZeroError'
_hole_true':false'1 :: true':false'
_hole_0':s':divByZeroError'2 :: 0':s':divByZeroError'
_gen_0':s':divByZeroError'3 :: Nat → 0':s':divByZeroError'

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

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

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', minus', mod'

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

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

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(+(_$n6, 1)), _gen_0':s':divByZeroError'3(+(_$n6, 1))) →_R^Ω(1)
le'(_gen_0':s':divByZeroError'3(_$n6), _gen_0':s':divByZeroError'3(_$n6)) →_IH
true'

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

Lemmas:
le'(_gen_0':s':divByZeroError'3(_n5), _gen_0':s':divByZeroError'3(_n5)) → true', 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:
minus', mod'

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

Proved the following rewrite lemma:
minus'(_gen_0':s':divByZeroError'3(_n704), _gen_0':s':divByZeroError'3(_n704)) → _gen_0':s':divByZeroError'3(0), rt ∈ Ω(1 + _n704)

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(+(_$n705, 1)), _gen_0':s':divByZeroError'3(+(_$n705, 1))) →_R^Ω(1)
minus'(_gen_0':s':divByZeroError'3(_$n705), _gen_0':s':divByZeroError'3(_$n705)) →_IH
_gen_0':s':divByZeroError'3(0)

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

Lemmas:
le'(_gen_0':s':divByZeroError'3(_n5), _gen_0':s':divByZeroError'3(_n5)) → true', rt ∈ Ω(1 + _n5)
minus'(_gen_0':s':divByZeroError'3(_n704), _gen_0':s':divByZeroError'3(_n704)) → _gen_0':s':divByZeroError'3(0), rt ∈ Ω(1 + _n704)

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:
mod'

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

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