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

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

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

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

Rewrite Strategy: INNERMOST

Infered types.

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

Types:
times' :: 0':s' → 0':s' → 0':s'
help' :: 0':s' → 0':s' → 0':s' → 0':s'
0' :: 0':s'
if' :: true':false' → 0':s' → 0':s' → 0':s' → 0':s'
lt' :: 0':s' → 0':s' → true':false'
true' :: true':false'
plus' :: 0':s' → 0':s' → 0':s'
s' :: 0':s' → 0':s'
false' :: true':false'
_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:
help', lt', plus'

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

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

Types:
times' :: 0':s' → 0':s' → 0':s'
help' :: 0':s' → 0':s' → 0':s' → 0':s'
0' :: 0':s'
if' :: true':false' → 0':s' → 0':s' → 0':s' → 0':s'
lt' :: 0':s' → 0':s' → true':false'
true' :: true':false'
plus' :: 0':s' → 0':s' → 0':s'
s' :: 0':s' → 0':s'
false' :: true':false'
_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:
lt', help', plus'

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(+(1, _n5))) → true', rt ∈ Ω(1 + n5)

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

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

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

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

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

Lemmas:
lt'(_gen_0':s'3(_n5), _gen_0':s'3(+(1, _n5))) → true', 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(_n811)) → _gen_0':s'3(+(_n811, a)), rt ∈ Ω(1 + n811)

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(_a988), _gen_0':s'3(+(_\$n812, 1))) →RΩ(1)
s'(plus'(_gen_0':s'3(_a988), _gen_0':s'3(_\$n812))) →IH
s'(_gen_0':s'3(+(_\$n812, _a988)))

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

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

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

Lemmas:
lt'(_gen_0':s'3(_n5), _gen_0':s'3(+(1, _n5))) → true', rt ∈ Ω(1 + n5)
plus'(_gen_0':s'3(a), _gen_0':s'3(_n811)) → _gen_0':s'3(+(_n811, a)), rt ∈ Ω(1 + n811)

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

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

Lemmas:
lt'(_gen_0':s'3(_n5), _gen_0':s'3(+(1, _n5))) → true', rt ∈ Ω(1 + n5)
plus'(_gen_0':s'3(a), _gen_0':s'3(_n811)) → _gen_0':s'3(+(_n811, a)), rt ∈ Ω(1 + n811)

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(+(1, _n5))) → true', rt ∈ Ω(1 + n5)