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:

plus(s(s(x)), y) → s(plus(x, s(y)))
plus(x, s(s(y))) → s(plus(s(x), y))
plus(s(0), y) → s(y)
plus(0, y) → y
ack(0, y) → s(y)
ack(s(x), 0) → ack(x, s(0))
ack(s(x), s(y)) → ack(x, plus(y, 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:

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

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
plus(s(s(x)), y) → s(plus(x, s(y)))
plus(x, s(s(y))) → s(plus(s(x), y))
plus(s(0'), y) → s(y)
plus(0', y) → y
ack(0', y) → s(y)
ack(s(x), 0') → ack(x, s(0'))
ack(s(x), s(y)) → ack(x, plus(y, ack(s(x), y)))

Types:
plus :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
ack :: s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

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

Heuristically decided to analyse the following defined symbols:
plus, ack

They will be analysed ascendingly in the following order:
plus < ack

(6) Obligation:

Innermost TRS:
Rules:
plus(s(s(x)), y) → s(plus(x, s(y)))
plus(x, s(s(y))) → s(plus(s(x), y))
plus(s(0'), y) → s(y)
plus(0', y) → y
ack(0', y) → s(y)
ack(s(x), 0') → ack(x, s(0'))
ack(s(x), s(y)) → ack(x, plus(y, ack(s(x), y)))

Types:
plus :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
ack :: s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Generator Equations:
gen_s:0'2_0(0) ⇔ 0'
gen_s:0'2_0(+(x, 1)) ⇔ s(gen_s:0'2_0(x))

The following defined symbols remain to be analysed:
plus, ack

They will be analysed ascendingly in the following order:
plus < ack

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

Proved the following rewrite lemma:
plus(gen_s:0'2_0(+(1, *(2, n4_0))), gen_s:0'2_0(b)) → gen_s:0'2_0(+(+(1, *(2, n4_0)), b)), rt ∈ Ω(1 + n4₀)

Induction Base:
plus(gen_s:0'2_0(+(1, *(2, 0))), gen_s:0'2_0(b)) →_R^Ω(1)
s(gen_s:0'2_0(b))

Induction Step:
plus(gen_s:0'2_0(+(1, *(2, +(n4_0, 1)))), gen_s:0'2_0(b)) →_R^Ω(1)
s(plus(gen_s:0'2_0(+(1, *(2, n4_0))), s(gen_s:0'2_0(b)))) →_IH
s(gen_s:0'2_0(+(+(1, +(b, 1)), *(2, c5_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:
plus(s(s(x)), y) → s(plus(x, s(y)))
plus(x, s(s(y))) → s(plus(s(x), y))
plus(s(0'), y) → s(y)
plus(0', y) → y
ack(0', y) → s(y)
ack(s(x), 0') → ack(x, s(0'))
ack(s(x), s(y)) → ack(x, plus(y, ack(s(x), y)))

Types:
plus :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
ack :: s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Lemmas:
plus(gen_s:0'2_0(+(1, *(2, n4_0))), gen_s:0'2_0(b)) → gen_s:0'2_0(+(+(1, *(2, n4_0)), b)), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_s:0'2_0(0) ⇔ 0'
gen_s:0'2_0(+(x, 1)) ⇔ s(gen_s:0'2_0(x))

The following defined symbols remain to be analysed:
ack

(10) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
ack(gen_s:0'2_0(1), gen_s:0'2_0(+(1, n861_0))) → *3_0, rt ∈ Ω(n861₀)

Induction Base:
ack(gen_s:0'2_0(1), gen_s:0'2_0(+(1, 0)))

Induction Step:
ack(gen_s:0'2_0(1), gen_s:0'2_0(+(1, +(n861_0, 1)))) →_R^Ω(1)
ack(gen_s:0'2_0(0), plus(gen_s:0'2_0(+(1, n861_0)), ack(s(gen_s:0'2_0(0)), gen_s:0'2_0(+(1, n861_0))))) →_IH
ack(gen_s:0'2_0(0), plus(gen_s:0'2_0(+(1, n861_0)), *3_0))

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

(11) Complex Obligation (BEST)

(12) Obligation:

Innermost TRS:
Rules:
plus(s(s(x)), y) → s(plus(x, s(y)))
plus(x, s(s(y))) → s(plus(s(x), y))
plus(s(0'), y) → s(y)
plus(0', y) → y
ack(0', y) → s(y)
ack(s(x), 0') → ack(x, s(0'))
ack(s(x), s(y)) → ack(x, plus(y, ack(s(x), y)))

Types:
plus :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
ack :: s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Lemmas:
plus(gen_s:0'2_0(+(1, *(2, n4_0))), gen_s:0'2_0(b)) → gen_s:0'2_0(+(+(1, *(2, n4_0)), b)), rt ∈ Ω(1 + n4₀)
ack(gen_s:0'2_0(1), gen_s:0'2_0(+(1, n861_0))) → *3_0, rt ∈ Ω(n861₀)

Generator Equations:
gen_s:0'2_0(0) ⇔ 0'
gen_s:0'2_0(+(x, 1)) ⇔ s(gen_s:0'2_0(x))

No more defined symbols left to analyse.

(13) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
plus(gen_s:0'2_0(+(1, *(2, n4_0))), gen_s:0'2_0(b)) → gen_s:0'2_0(+(+(1, *(2, n4_0)), b)), rt ∈ Ω(1 + n4₀)

(14) BOUNDS(n^1, INF)

(15) Obligation:

Innermost TRS:
Rules:
plus(s(s(x)), y) → s(plus(x, s(y)))
plus(x, s(s(y))) → s(plus(s(x), y))
plus(s(0'), y) → s(y)
plus(0', y) → y
ack(0', y) → s(y)
ack(s(x), 0') → ack(x, s(0'))
ack(s(x), s(y)) → ack(x, plus(y, ack(s(x), y)))

Types:
plus :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
ack :: s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Lemmas:
plus(gen_s:0'2_0(+(1, *(2, n4_0))), gen_s:0'2_0(b)) → gen_s:0'2_0(+(+(1, *(2, n4_0)), b)), rt ∈ Ω(1 + n4₀)
ack(gen_s:0'2_0(1), gen_s:0'2_0(+(1, n861_0))) → *3_0, rt ∈ Ω(n861₀)

Generator Equations:
gen_s:0'2_0(0) ⇔ 0'
gen_s:0'2_0(+(x, 1)) ⇔ s(gen_s:0'2_0(x))

No more defined symbols left to analyse.

(16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
plus(gen_s:0'2_0(+(1, *(2, n4_0))), gen_s:0'2_0(b)) → gen_s:0'2_0(+(+(1, *(2, n4_0)), b)), rt ∈ Ω(1 + n4₀)

(17) BOUNDS(n^1, INF)

(18) Obligation:

Innermost TRS:
Rules:
plus(s(s(x)), y) → s(plus(x, s(y)))
plus(x, s(s(y))) → s(plus(s(x), y))
plus(s(0'), y) → s(y)
plus(0', y) → y
ack(0', y) → s(y)
ack(s(x), 0') → ack(x, s(0'))
ack(s(x), s(y)) → ack(x, plus(y, ack(s(x), y)))

Types:
plus :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
ack :: s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Lemmas:
plus(gen_s:0'2_0(+(1, *(2, n4_0))), gen_s:0'2_0(b)) → gen_s:0'2_0(+(+(1, *(2, n4_0)), b)), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_s:0'2_0(0) ⇔ 0'
gen_s:0'2_0(+(x, 1)) ⇔ s(gen_s:0'2_0(x))

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
plus(gen_s:0'2_0(+(1, *(2, n4_0))), gen_s:0'2_0(b)) → gen_s:0'2_0(+(+(1, *(2, n4_0)), b)), rt ∈ Ω(1 + n4₀)

(20) BOUNDS(n^1, INF)