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

ackin(s(X), s(Y)) → u21(ackin(s(X), Y), X)
u21(ackout(X), Y) → u22(ackin(Y, X))

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

ackin'(s'(X), s'(Y)) → u21'(ackin'(s'(X), Y), X)
u21'(ackout'(X), Y) → u22'(ackin'(Y, X))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
ackin'(s'(X), s'(Y)) → u21'(ackin'(s'(X), Y), X)
u21'(ackout'(X), Y) → u22'(ackin'(Y, X))

Types:
ackin' :: s' → s' → ackout':u22'
s' :: s' → s'
u21' :: ackout':u22' → s' → ackout':u22'
ackout' :: s' → ackout':u22'
u22' :: ackout':u22' → ackout':u22'
_hole_ackout':u22'1 :: ackout':u22'
_hole_s'2 :: s'
_gen_ackout':u22'3 :: Nat → ackout':u22'
_gen_s'4 :: Nat → s'

Heuristically decided to analyse the following defined symbols:
ackin', u21'

They will be analysed ascendingly in the following order:
ackin' = u21'

Rules:
ackin'(s'(X), s'(Y)) → u21'(ackin'(s'(X), Y), X)
u21'(ackout'(X), Y) → u22'(ackin'(Y, X))

Types:
ackin' :: s' → s' → ackout':u22'
s' :: s' → s'
u21' :: ackout':u22' → s' → ackout':u22'
ackout' :: s' → ackout':u22'
u22' :: ackout':u22' → ackout':u22'
_hole_ackout':u22'1 :: ackout':u22'
_hole_s'2 :: s'
_gen_ackout':u22'3 :: Nat → ackout':u22'
_gen_s'4 :: Nat → s'

Generator Equations:
_gen_ackout':u22'3(0) ⇔ ackout'(_hole_s'2)
_gen_ackout':u22'3(+(x, 1)) ⇔ u22'(_gen_ackout':u22'3(x))
_gen_s'4(0) ⇔ _hole_s'2
_gen_s'4(+(x, 1)) ⇔ s'(_gen_s'4(x))

The following defined symbols remain to be analysed:
u21', ackin'

They will be analysed ascendingly in the following order:
ackin' = u21'

Could not prove a rewrite lemma for the defined symbol u21'.

Rules:
ackin'(s'(X), s'(Y)) → u21'(ackin'(s'(X), Y), X)
u21'(ackout'(X), Y) → u22'(ackin'(Y, X))

Types:
ackin' :: s' → s' → ackout':u22'
s' :: s' → s'
u21' :: ackout':u22' → s' → ackout':u22'
ackout' :: s' → ackout':u22'
u22' :: ackout':u22' → ackout':u22'
_hole_ackout':u22'1 :: ackout':u22'
_hole_s'2 :: s'
_gen_ackout':u22'3 :: Nat → ackout':u22'
_gen_s'4 :: Nat → s'

Generator Equations:
_gen_ackout':u22'3(0) ⇔ ackout'(_hole_s'2)
_gen_ackout':u22'3(+(x, 1)) ⇔ u22'(_gen_ackout':u22'3(x))
_gen_s'4(0) ⇔ _hole_s'2
_gen_s'4(+(x, 1)) ⇔ s'(_gen_s'4(x))

The following defined symbols remain to be analysed:
ackin'

They will be analysed ascendingly in the following order:
ackin' = u21'

Proved the following rewrite lemma:
ackin'(_gen_s'4(1), _gen_s'4(+(1, _n90))) → _*5, rt ∈ Ω(n90)

Induction Base:
ackin'(_gen_s'4(1), _gen_s'4(+(1, 0)))

Induction Step:
ackin'(_gen_s'4(1), _gen_s'4(+(1, +(_\$n91, 1)))) →RΩ(1)
u21'(ackin'(s'(_gen_s'4(0)), _gen_s'4(+(1, _\$n91))), _gen_s'4(0)) →IH
u21'(_*5, _gen_s'4(0))

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

Rules:
ackin'(s'(X), s'(Y)) → u21'(ackin'(s'(X), Y), X)
u21'(ackout'(X), Y) → u22'(ackin'(Y, X))

Types:
ackin' :: s' → s' → ackout':u22'
s' :: s' → s'
u21' :: ackout':u22' → s' → ackout':u22'
ackout' :: s' → ackout':u22'
u22' :: ackout':u22' → ackout':u22'
_hole_ackout':u22'1 :: ackout':u22'
_hole_s'2 :: s'
_gen_ackout':u22'3 :: Nat → ackout':u22'
_gen_s'4 :: Nat → s'

Lemmas:
ackin'(_gen_s'4(1), _gen_s'4(+(1, _n90))) → _*5, rt ∈ Ω(n90)

Generator Equations:
_gen_ackout':u22'3(0) ⇔ ackout'(_hole_s'2)
_gen_ackout':u22'3(+(x, 1)) ⇔ u22'(_gen_ackout':u22'3(x))
_gen_s'4(0) ⇔ _hole_s'2
_gen_s'4(+(x, 1)) ⇔ s'(_gen_s'4(x))

The following defined symbols remain to be analysed:
u21'

They will be analysed ascendingly in the following order:
ackin' = u21'

Could not prove a rewrite lemma for the defined symbol u21'.

Rules:
ackin'(s'(X), s'(Y)) → u21'(ackin'(s'(X), Y), X)
u21'(ackout'(X), Y) → u22'(ackin'(Y, X))

Types:
ackin' :: s' → s' → ackout':u22'
s' :: s' → s'
u21' :: ackout':u22' → s' → ackout':u22'
ackout' :: s' → ackout':u22'
u22' :: ackout':u22' → ackout':u22'
_hole_ackout':u22'1 :: ackout':u22'
_hole_s'2 :: s'
_gen_ackout':u22'3 :: Nat → ackout':u22'
_gen_s'4 :: Nat → s'

Lemmas:
ackin'(_gen_s'4(1), _gen_s'4(+(1, _n90))) → _*5, rt ∈ Ω(n90)

Generator Equations:
_gen_ackout':u22'3(0) ⇔ ackout'(_hole_s'2)
_gen_ackout':u22'3(+(x, 1)) ⇔ u22'(_gen_ackout':u22'3(x))
_gen_s'4(0) ⇔ _hole_s'2
_gen_s'4(+(x, 1)) ⇔ s'(_gen_s'4(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
ackin'(_gen_s'4(1), _gen_s'4(+(1, _n90))) → _*5, rt ∈ Ω(n90)