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

lt(x, 0) → false
lt(0, s(y)) → true
lt(s(x), s(y)) → lt(x, y)
plus(x, 0) → x
plus(x, s(y)) → s(plus(x, y))
quot(x, s(y)) → help(x, s(y), 0)
help(x, s(y), c) → if(lt(c, x), x, s(y), c)
if(true, x, s(y), c) → s(help(x, s(y), plus(c, s(y))))
if(false, x, s(y), c) → 0

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

lt'(x, 0') → false'
lt'(0', s'(y)) → true'
lt'(s'(x), s'(y)) → lt'(x, y)
plus'(x, 0') → x
plus'(x, s'(y)) → s'(plus'(x, y))
quot'(x, s'(y)) → help'(x, s'(y), 0')
help'(x, s'(y), c) → if'(lt'(c, x), x, s'(y), c)
if'(true', x, s'(y), c) → s'(help'(x, s'(y), plus'(c, s'(y))))
if'(false', x, s'(y), c) → 0'

Rewrite Strategy: INNERMOST

Infered types.

Rules:
lt'(x, 0') → false'
lt'(0', s'(y)) → true'
lt'(s'(x), s'(y)) → lt'(x, y)
plus'(x, 0') → x
plus'(x, s'(y)) → s'(plus'(x, y))
quot'(x, s'(y)) → help'(x, s'(y), 0')
help'(x, s'(y), c) → if'(lt'(c, x), x, s'(y), c)
if'(true', x, s'(y), c) → s'(help'(x, s'(y), plus'(c, s'(y))))
if'(false', x, s'(y), c) → 0'

Types:
lt' :: 0':s' → 0':s' → false':true'
0' :: 0':s'
false' :: false':true'
s' :: 0':s' → 0':s'
true' :: false':true'
plus' :: 0':s' → 0':s' → 0':s'
quot' :: 0':s' → 0':s' → 0':s'
help' :: 0':s' → 0':s' → 0':s' → 0':s'
if' :: false':true' → 0':s' → 0':s' → 0':s' → 0':s'
_hole_false':true'1 :: false':true'
_hole_0':s'2 :: 0':s'
_gen_0':s'3 :: Nat → 0':s'

Heuristically decided to analyse the following defined symbols:
lt', plus', help'

They will be analysed ascendingly in the following order:
lt' < help'
plus' < help'

Rules:
lt'(x, 0') → false'
lt'(0', s'(y)) → true'
lt'(s'(x), s'(y)) → lt'(x, y)
plus'(x, 0') → x
plus'(x, s'(y)) → s'(plus'(x, y))
quot'(x, s'(y)) → help'(x, s'(y), 0')
help'(x, s'(y), c) → if'(lt'(c, x), x, s'(y), c)
if'(true', x, s'(y), c) → s'(help'(x, s'(y), plus'(c, s'(y))))
if'(false', x, s'(y), c) → 0'

Types:
lt' :: 0':s' → 0':s' → false':true'
0' :: 0':s'
false' :: false':true'
s' :: 0':s' → 0':s'
true' :: false':true'
plus' :: 0':s' → 0':s' → 0':s'
quot' :: 0':s' → 0':s' → 0':s'
help' :: 0':s' → 0':s' → 0':s' → 0':s'
if' :: false':true' → 0':s' → 0':s' → 0':s' → 0':s'
_hole_false':true'1 :: false':true'
_hole_0':s'2 :: 0':s'
_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:
lt', plus', help'

They will be analysed ascendingly in the following order:
lt' < help'
plus' < help'

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

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

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

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

Rules:
lt'(x, 0') → false'
lt'(0', s'(y)) → true'
lt'(s'(x), s'(y)) → lt'(x, y)
plus'(x, 0') → x
plus'(x, s'(y)) → s'(plus'(x, y))
quot'(x, s'(y)) → help'(x, s'(y), 0')
help'(x, s'(y), c) → if'(lt'(c, x), x, s'(y), c)
if'(true', x, s'(y), c) → s'(help'(x, s'(y), plus'(c, s'(y))))
if'(false', x, s'(y), c) → 0'

Types:
lt' :: 0':s' → 0':s' → false':true'
0' :: 0':s'
false' :: false':true'
s' :: 0':s' → 0':s'
true' :: false':true'
plus' :: 0':s' → 0':s' → 0':s'
quot' :: 0':s' → 0':s' → 0':s'
help' :: 0':s' → 0':s' → 0':s' → 0':s'
if' :: false':true' → 0':s' → 0':s' → 0':s' → 0':s'
_hole_false':true'1 :: false':true'
_hole_0':s'2 :: 0':s'
_gen_0':s'3 :: Nat → 0':s'

Lemmas:
lt'(_gen_0':s'3(_n5), _gen_0':s'3(_n5)) → false', 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:
plus', help'

They will be analysed ascendingly in the following order:
plus' < help'

Proved the following rewrite lemma:
plus'(_gen_0':s'3(a), _gen_0':s'3(_n584)) → _gen_0':s'3(+(_n584, a)), rt ∈ Ω(1 + n584)

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

Induction Step:
plus'(_gen_0':s'3(_a717), _gen_0':s'3(+(_\$n585, 1))) →RΩ(1)
s'(plus'(_gen_0':s'3(_a717), _gen_0':s'3(_\$n585))) →IH
s'(_gen_0':s'3(+(_\$n585, _a717)))

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

Rules:
lt'(x, 0') → false'
lt'(0', s'(y)) → true'
lt'(s'(x), s'(y)) → lt'(x, y)
plus'(x, 0') → x
plus'(x, s'(y)) → s'(plus'(x, y))
quot'(x, s'(y)) → help'(x, s'(y), 0')
help'(x, s'(y), c) → if'(lt'(c, x), x, s'(y), c)
if'(true', x, s'(y), c) → s'(help'(x, s'(y), plus'(c, s'(y))))
if'(false', x, s'(y), c) → 0'

Types:
lt' :: 0':s' → 0':s' → false':true'
0' :: 0':s'
false' :: false':true'
s' :: 0':s' → 0':s'
true' :: false':true'
plus' :: 0':s' → 0':s' → 0':s'
quot' :: 0':s' → 0':s' → 0':s'
help' :: 0':s' → 0':s' → 0':s' → 0':s'
if' :: false':true' → 0':s' → 0':s' → 0':s' → 0':s'
_hole_false':true'1 :: false':true'
_hole_0':s'2 :: 0':s'
_gen_0':s'3 :: Nat → 0':s'

Lemmas:
lt'(_gen_0':s'3(_n5), _gen_0':s'3(_n5)) → false', rt ∈ Ω(1 + n5)
plus'(_gen_0':s'3(a), _gen_0':s'3(_n584)) → _gen_0':s'3(+(_n584, a)), rt ∈ Ω(1 + n584)

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:
help'

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

Rules:
lt'(x, 0') → false'
lt'(0', s'(y)) → true'
lt'(s'(x), s'(y)) → lt'(x, y)
plus'(x, 0') → x
plus'(x, s'(y)) → s'(plus'(x, y))
quot'(x, s'(y)) → help'(x, s'(y), 0')
help'(x, s'(y), c) → if'(lt'(c, x), x, s'(y), c)
if'(true', x, s'(y), c) → s'(help'(x, s'(y), plus'(c, s'(y))))
if'(false', x, s'(y), c) → 0'

Types:
lt' :: 0':s' → 0':s' → false':true'
0' :: 0':s'
false' :: false':true'
s' :: 0':s' → 0':s'
true' :: false':true'
plus' :: 0':s' → 0':s' → 0':s'
quot' :: 0':s' → 0':s' → 0':s'
help' :: 0':s' → 0':s' → 0':s' → 0':s'
if' :: false':true' → 0':s' → 0':s' → 0':s' → 0':s'
_hole_false':true'1 :: false':true'
_hole_0':s'2 :: 0':s'
_gen_0':s'3 :: Nat → 0':s'

Lemmas:
lt'(_gen_0':s'3(_n5), _gen_0':s'3(_n5)) → false', rt ∈ Ω(1 + n5)
plus'(_gen_0':s'3(a), _gen_0':s'3(_n584)) → _gen_0':s'3(+(_n584, a)), rt ∈ Ω(1 + n584)

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:
lt'(_gen_0':s'3(_n5), _gen_0':s'3(_n5)) → false', rt ∈ Ω(1 + n5)