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

ack(0, y) → s(y)
ack(s(x), 0) → ack(x, s(0))
ack(s(x), s(y)) → ack(x, ack(s(x), y))
f(s(x), y) → f(x, s(x))
f(x, s(y)) → f(y, x)
f(x, y) → ack(x, y)
ack(s(x), y) → f(x, x)

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

ack'(0', y) → s'(y)
ack'(s'(x), 0') → ack'(x, s'(0'))
ack'(s'(x), s'(y)) → ack'(x, ack'(s'(x), y))
f'(s'(x), y) → f'(x, s'(x))
f'(x, s'(y)) → f'(y, x)
f'(x, y) → ack'(x, y)
ack'(s'(x), y) → f'(x, x)

Rewrite Strategy: INNERMOST

Infered types.

Rules:
ack'(0', y) → s'(y)
ack'(s'(x), 0') → ack'(x, s'(0'))
ack'(s'(x), s'(y)) → ack'(x, ack'(s'(x), y))
f'(s'(x), y) → f'(x, s'(x))
f'(x, s'(y)) → f'(y, x)
f'(x, y) → ack'(x, y)
ack'(s'(x), y) → f'(x, x)

Types:
ack' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
f' :: 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_gen_0':s'2 :: Nat → 0':s'

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

They will be analysed ascendingly in the following order:
ack' = f'

Rules:
ack'(0', y) → s'(y)
ack'(s'(x), 0') → ack'(x, s'(0'))
ack'(s'(x), s'(y)) → ack'(x, ack'(s'(x), y))
f'(s'(x), y) → f'(x, s'(x))
f'(x, s'(y)) → f'(y, x)
f'(x, y) → ack'(x, y)
ack'(s'(x), y) → f'(x, x)

Types:
ack' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
f' :: 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_gen_0':s'2 :: Nat → 0':s'

Generator Equations:
_gen_0':s'2(0) ⇔ 0'
_gen_0':s'2(+(x, 1)) ⇔ s'(_gen_0':s'2(x))

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

They will be analysed ascendingly in the following order:
ack' = f'

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

The following conjecture could not be proven:

f'(_gen_0':s'2(+(1, _n4)), _gen_0':s'2(b)) →? _*3

Rules:
ack'(0', y) → s'(y)
ack'(s'(x), 0') → ack'(x, s'(0'))
ack'(s'(x), s'(y)) → ack'(x, ack'(s'(x), y))
f'(s'(x), y) → f'(x, s'(x))
f'(x, s'(y)) → f'(y, x)
f'(x, y) → ack'(x, y)
ack'(s'(x), y) → f'(x, x)

Types:
ack' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
f' :: 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_gen_0':s'2 :: Nat → 0':s'

Generator Equations:
_gen_0':s'2(0) ⇔ 0'
_gen_0':s'2(+(x, 1)) ⇔ s'(_gen_0':s'2(x))

The following defined symbols remain to be analysed:
ack'

They will be analysed ascendingly in the following order:
ack' = f'

Proved the following rewrite lemma:
ack'(_gen_0':s'2(1), _gen_0':s'2(+(1, _n6107))) → _*3, rt ∈ Ω(n6107)

Induction Base:
ack'(_gen_0':s'2(1), _gen_0':s'2(+(1, 0)))

Induction Step:
ack'(_gen_0':s'2(1), _gen_0':s'2(+(1, +(_\$n6108, 1)))) →RΩ(1)
ack'(_gen_0':s'2(0), ack'(s'(_gen_0':s'2(0)), _gen_0':s'2(+(1, _\$n6108)))) →IH
ack'(_gen_0':s'2(0), _*3)

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

Rules:
ack'(0', y) → s'(y)
ack'(s'(x), 0') → ack'(x, s'(0'))
ack'(s'(x), s'(y)) → ack'(x, ack'(s'(x), y))
f'(s'(x), y) → f'(x, s'(x))
f'(x, s'(y)) → f'(y, x)
f'(x, y) → ack'(x, y)
ack'(s'(x), y) → f'(x, x)

Types:
ack' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
f' :: 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_gen_0':s'2 :: Nat → 0':s'

Lemmas:
ack'(_gen_0':s'2(1), _gen_0':s'2(+(1, _n6107))) → _*3, rt ∈ Ω(n6107)

Generator Equations:
_gen_0':s'2(0) ⇔ 0'
_gen_0':s'2(+(x, 1)) ⇔ s'(_gen_0':s'2(x))

The following defined symbols remain to be analysed:
f'

They will be analysed ascendingly in the following order:
ack' = f'

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

The following conjecture could not be proven:

f'(_gen_0':s'2(+(1, _n20147)), _gen_0':s'2(b)) →? _*3

Rules:
ack'(0', y) → s'(y)
ack'(s'(x), 0') → ack'(x, s'(0'))
ack'(s'(x), s'(y)) → ack'(x, ack'(s'(x), y))
f'(s'(x), y) → f'(x, s'(x))
f'(x, s'(y)) → f'(y, x)
f'(x, y) → ack'(x, y)
ack'(s'(x), y) → f'(x, x)

Types:
ack' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
f' :: 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_gen_0':s'2 :: Nat → 0':s'

Lemmas:
ack'(_gen_0':s'2(1), _gen_0':s'2(+(1, _n6107))) → _*3, rt ∈ Ω(n6107)

Generator Equations:
_gen_0':s'2(0) ⇔ 0'
_gen_0':s'2(+(x, 1)) ⇔ s'(_gen_0':s'2(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
ack'(_gen_0':s'2(1), _gen_0':s'2(+(1, _n6107))) → _*3, rt ∈ Ω(n6107)