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

f(true, x, y) → f(gt(x, y), x, round(s(y)))
round(0) → 0
round(s(0)) → s(s(0))
round(s(s(x))) → s(s(round(x)))
gt(0, v) → false
gt(s(u), 0) → true
gt(s(u), s(v)) → gt(u, v)

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

f'(true', x, y) → f'(gt'(x, y), x, round'(s'(y)))
round'(0') → 0'
round'(s'(0')) → s'(s'(0'))
round'(s'(s'(x))) → s'(s'(round'(x)))
gt'(0', v) → false'
gt'(s'(u), 0') → true'
gt'(s'(u), s'(v)) → gt'(u, v)

Rewrite Strategy: INNERMOST

Infered types.

Rules:
f'(true', x, y) → f'(gt'(x, y), x, round'(s'(y)))
round'(0') → 0'
round'(s'(0')) → s'(s'(0'))
round'(s'(s'(x))) → s'(s'(round'(x)))
gt'(0', v) → false'
gt'(s'(u), 0') → true'
gt'(s'(u), s'(v)) → gt'(u, v)

Types:
f' :: true':false' → s':0' → s':0' → f'
true' :: true':false'
gt' :: s':0' → s':0' → true':false'
round' :: s':0' → s':0'
s' :: s':0' → s':0'
0' :: s':0'
false' :: true':false'
_hole_f'1 :: f'
_hole_true':false'2 :: true':false'
_hole_s':0'3 :: s':0'
_gen_s':0'4 :: Nat → s':0'

Heuristically decided to analyse the following defined symbols:
f', gt', round'

They will be analysed ascendingly in the following order:
gt' < f'
round' < f'

Rules:
f'(true', x, y) → f'(gt'(x, y), x, round'(s'(y)))
round'(0') → 0'
round'(s'(0')) → s'(s'(0'))
round'(s'(s'(x))) → s'(s'(round'(x)))
gt'(0', v) → false'
gt'(s'(u), 0') → true'
gt'(s'(u), s'(v)) → gt'(u, v)

Types:
f' :: true':false' → s':0' → s':0' → f'
true' :: true':false'
gt' :: s':0' → s':0' → true':false'
round' :: s':0' → s':0'
s' :: s':0' → s':0'
0' :: s':0'
false' :: true':false'
_hole_f'1 :: f'
_hole_true':false'2 :: true':false'
_hole_s':0'3 :: s':0'
_gen_s':0'4 :: Nat → s':0'

Generator Equations:
_gen_s':0'4(0) ⇔ 0'
_gen_s':0'4(+(x, 1)) ⇔ s'(_gen_s':0'4(x))

The following defined symbols remain to be analysed:
gt', f', round'

They will be analysed ascendingly in the following order:
gt' < f'
round' < f'

Proved the following rewrite lemma:
gt'(_gen_s':0'4(_n6), _gen_s':0'4(_n6)) → false', rt ∈ Ω(1 + n6)

Induction Base:
gt'(_gen_s':0'4(0), _gen_s':0'4(0)) →RΩ(1)
false'

Induction Step:
gt'(_gen_s':0'4(+(_\$n7, 1)), _gen_s':0'4(+(_\$n7, 1))) →RΩ(1)
gt'(_gen_s':0'4(_\$n7), _gen_s':0'4(_\$n7)) →IH
false'

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

Rules:
f'(true', x, y) → f'(gt'(x, y), x, round'(s'(y)))
round'(0') → 0'
round'(s'(0')) → s'(s'(0'))
round'(s'(s'(x))) → s'(s'(round'(x)))
gt'(0', v) → false'
gt'(s'(u), 0') → true'
gt'(s'(u), s'(v)) → gt'(u, v)

Types:
f' :: true':false' → s':0' → s':0' → f'
true' :: true':false'
gt' :: s':0' → s':0' → true':false'
round' :: s':0' → s':0'
s' :: s':0' → s':0'
0' :: s':0'
false' :: true':false'
_hole_f'1 :: f'
_hole_true':false'2 :: true':false'
_hole_s':0'3 :: s':0'
_gen_s':0'4 :: Nat → s':0'

Lemmas:
gt'(_gen_s':0'4(_n6), _gen_s':0'4(_n6)) → false', rt ∈ Ω(1 + n6)

Generator Equations:
_gen_s':0'4(0) ⇔ 0'
_gen_s':0'4(+(x, 1)) ⇔ s'(_gen_s':0'4(x))

The following defined symbols remain to be analysed:
round', f'

They will be analysed ascendingly in the following order:
round' < f'

Proved the following rewrite lemma:
round'(_gen_s':0'4(*(2, _n324))) → _gen_s':0'4(*(2, _n324)), rt ∈ Ω(1 + n324)

Induction Base:
round'(_gen_s':0'4(*(2, 0))) →RΩ(1)
0'

Induction Step:
round'(_gen_s':0'4(*(2, +(_\$n325, 1)))) →RΩ(1)
s'(s'(round'(_gen_s':0'4(*(2, _\$n325))))) →IH
s'(s'(_gen_s':0'4(*(2, _\$n325))))

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

Rules:
f'(true', x, y) → f'(gt'(x, y), x, round'(s'(y)))
round'(0') → 0'
round'(s'(0')) → s'(s'(0'))
round'(s'(s'(x))) → s'(s'(round'(x)))
gt'(0', v) → false'
gt'(s'(u), 0') → true'
gt'(s'(u), s'(v)) → gt'(u, v)

Types:
f' :: true':false' → s':0' → s':0' → f'
true' :: true':false'
gt' :: s':0' → s':0' → true':false'
round' :: s':0' → s':0'
s' :: s':0' → s':0'
0' :: s':0'
false' :: true':false'
_hole_f'1 :: f'
_hole_true':false'2 :: true':false'
_hole_s':0'3 :: s':0'
_gen_s':0'4 :: Nat → s':0'

Lemmas:
gt'(_gen_s':0'4(_n6), _gen_s':0'4(_n6)) → false', rt ∈ Ω(1 + n6)
round'(_gen_s':0'4(*(2, _n324))) → _gen_s':0'4(*(2, _n324)), rt ∈ Ω(1 + n324)

Generator Equations:
_gen_s':0'4(0) ⇔ 0'
_gen_s':0'4(+(x, 1)) ⇔ s'(_gen_s':0'4(x))

The following defined symbols remain to be analysed:
f'

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

Rules:
f'(true', x, y) → f'(gt'(x, y), x, round'(s'(y)))
round'(0') → 0'
round'(s'(0')) → s'(s'(0'))
round'(s'(s'(x))) → s'(s'(round'(x)))
gt'(0', v) → false'
gt'(s'(u), 0') → true'
gt'(s'(u), s'(v)) → gt'(u, v)

Types:
f' :: true':false' → s':0' → s':0' → f'
true' :: true':false'
gt' :: s':0' → s':0' → true':false'
round' :: s':0' → s':0'
s' :: s':0' → s':0'
0' :: s':0'
false' :: true':false'
_hole_f'1 :: f'
_hole_true':false'2 :: true':false'
_hole_s':0'3 :: s':0'
_gen_s':0'4 :: Nat → s':0'

Lemmas:
gt'(_gen_s':0'4(_n6), _gen_s':0'4(_n6)) → false', rt ∈ Ω(1 + n6)
round'(_gen_s':0'4(*(2, _n324))) → _gen_s':0'4(*(2, _n324)), rt ∈ Ω(1 + n324)

Generator Equations:
_gen_s':0'4(0) ⇔ 0'
_gen_s':0'4(+(x, 1)) ⇔ s'(_gen_s':0'4(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
gt'(_gen_s':0'4(_n6), _gen_s':0'4(_n6)) → false', rt ∈ Ω(1 + n6)