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)
mod(x, 0) → modZeroErro
mod(x, s(y)) → modIter(x, s(y), 0, 0)
modIter(x, s(y), z, u) → if(le(x, z), x, s(y), z, u)
if(true, x, y, z, u) → u
if(false, x, y, z, u) → if2(le(y, s(u)), x, y, s(z), s(u))
if2(false, x, y, z, u) → modIter(x, y, z, u)
if2(true, x, y, z, u) → modIter(x, y, z, 0)

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)
mod'(x, 0') → modZeroErro'
mod'(x, s'(y)) → modIter'(x, s'(y), 0', 0')
modIter'(x, s'(y), z, u) → if'(le'(x, z), x, s'(y), z, u)
if'(true', x, y, z, u) → u
if'(false', x, y, z, u) → if2'(le'(y, s'(u)), x, y, s'(z), s'(u))
if2'(false', x, y, z, u) → modIter'(x, y, z, u)
if2'(true', x, y, z, u) → modIter'(x, y, z, 0')

Rewrite Strategy: INNERMOST

Infered types.

Rules:
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
mod'(x, 0') → modZeroErro'
mod'(x, s'(y)) → modIter'(x, s'(y), 0', 0')
modIter'(x, s'(y), z, u) → if'(le'(x, z), x, s'(y), z, u)
if'(true', x, y, z, u) → u
if'(false', x, y, z, u) → if2'(le'(y, s'(u)), x, y, s'(z), s'(u))
if2'(false', x, y, z, u) → modIter'(x, y, z, u)
if2'(true', x, y, z, u) → modIter'(x, y, z, 0')

Types:
le' :: 0':s':modZeroErro' → 0':s':modZeroErro' → true':false'
0' :: 0':s':modZeroErro'
true' :: true':false'
s' :: 0':s':modZeroErro' → 0':s':modZeroErro'
false' :: true':false'
mod' :: 0':s':modZeroErro' → 0':s':modZeroErro' → 0':s':modZeroErro'
modZeroErro' :: 0':s':modZeroErro'
modIter' :: 0':s':modZeroErro' → 0':s':modZeroErro' → 0':s':modZeroErro' → 0':s':modZeroErro' → 0':s':modZeroErro'
if' :: true':false' → 0':s':modZeroErro' → 0':s':modZeroErro' → 0':s':modZeroErro' → 0':s':modZeroErro' → 0':s':modZeroErro'
if2' :: true':false' → 0':s':modZeroErro' → 0':s':modZeroErro' → 0':s':modZeroErro' → 0':s':modZeroErro' → 0':s':modZeroErro'
_hole_true':false'1 :: true':false'
_hole_0':s':modZeroErro'2 :: 0':s':modZeroErro'
_gen_0':s':modZeroErro'3 :: Nat → 0':s':modZeroErro'

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

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

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

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

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

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

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

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

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

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

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

The following defined symbols remain to be analysed:
modIter'

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

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

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

No more defined symbols left to analyse.

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