The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) Obligation:

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))

Rewrite Strategy: INNERMOST

(1) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(2) Obligation:

Runtime Complexity Relative 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))

S is empty.
Rewrite Strategy: INNERMOST

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(4) Obligation:

Innermost TRS:
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))

Types:
ack :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

(5) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
ack

(6) Obligation:

Innermost TRS:
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))

Types:
ack :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

The following defined symbols remain to be analysed:
ack

(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
ack(gen_0':s2_0(1), gen_0':s2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Induction Base:
ack(gen_0':s2_0(1), gen_0':s2_0(+(1, 0)))

Induction Step:
ack(gen_0':s2_0(1), gen_0':s2_0(+(1, +(n4_0, 1)))) →_R^Ω(1)
ack(gen_0':s2_0(0), ack(s(gen_0':s2_0(0)), gen_0':s2_0(+(1, n4_0)))) →_IH
ack(gen_0':s2_0(0), *3_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost TRS:
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))

Types:
ack :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
ack(gen_0':s2_0(1), gen_0':s2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

No more defined symbols left to analyse.

(10) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
ack(gen_0':s2_0(1), gen_0':s2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

(11) BOUNDS(n^1, INF)

(12) Obligation:

Innermost TRS:
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))

Types:
ack :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
ack(gen_0':s2_0(1), gen_0':s2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

No more defined symbols left to analyse.

(13) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
ack(gen_0':s2_0(1), gen_0':s2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

(14) BOUNDS(n^1, INF)