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(Cons(x, xs), Nil) → ack(xs, Cons(Nil, Nil))
ack(Cons(x', xs'), Cons(x, xs)) → ack(xs', ack(Cons(x', xs'), xs))
ack(Nil, n) → Cons(Cons(Nil, Nil), n)
goal(m, n) → ack(m, n)

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
ack(Cons(x, Cons(x'30_0, xs'31_0)), Nil) →⁺ ack(xs'31_0, ack(Cons(x'30_0, xs'31_0), Nil))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [xs'31_0 / Cons(x'30_0, xs'31_0)].
The result substitution is [x / x'30_0].

(2) BOUNDS(n^1, INF)

(3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(4) Obligation:

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

ack(Cons(x, xs), Nil) → ack(xs, Cons(Nil, Nil))
ack(Cons(x', xs'), Cons(x, xs)) → ack(xs', ack(Cons(x', xs'), xs))
ack(Nil, n) → Cons(Cons(Nil, Nil), n)
goal(m, n) → ack(m, n)

S is empty.
Rewrite Strategy: INNERMOST

(5) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
Cons/0

(6) Obligation:

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

ack(Cons(xs), Nil) → ack(xs, Cons(Nil))
ack(Cons(xs'), Cons(xs)) → ack(xs', ack(Cons(xs'), xs))
ack(Nil, n) → Cons(n)
goal(m, n) → ack(m, n)

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(8) Obligation:

Innermost TRS:
Rules:
ack(Cons(xs), Nil) → ack(xs, Cons(Nil))
ack(Cons(xs'), Cons(xs)) → ack(xs', ack(Cons(xs'), xs))
ack(Nil, n) → Cons(n)
goal(m, n) → ack(m, n)

Types:
ack :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil → Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat → Cons:Nil

(9) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
ack

(10) Obligation:

Innermost TRS:
Rules:
ack(Cons(xs), Nil) → ack(xs, Cons(Nil))
ack(Cons(xs'), Cons(xs)) → ack(xs', ack(Cons(xs'), xs))
ack(Nil, n) → Cons(n)
goal(m, n) → ack(m, n)

Types:
ack :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil → Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat → Cons:Nil

Generator Equations:
gen_Cons:Nil2_0(0) ⇔ Nil
gen_Cons:Nil2_0(+(x, 1)) ⇔ Cons(gen_Cons:Nil2_0(x))

The following defined symbols remain to be analysed:
ack

(11) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
ack(gen_Cons:Nil2_0(1), gen_Cons:Nil2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Induction Base:
ack(gen_Cons:Nil2_0(1), gen_Cons:Nil2_0(+(1, 0)))

Induction Step:
ack(gen_Cons:Nil2_0(1), gen_Cons:Nil2_0(+(1, +(n4_0, 1)))) →_R^Ω(1)
ack(gen_Cons:Nil2_0(0), ack(Cons(gen_Cons:Nil2_0(0)), gen_Cons:Nil2_0(+(1, n4_0)))) →_IH
ack(gen_Cons:Nil2_0(0), *3_0)

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

(12) Complex Obligation (BEST)

(13) Obligation:

Innermost TRS:
Rules:
ack(Cons(xs), Nil) → ack(xs, Cons(Nil))
ack(Cons(xs'), Cons(xs)) → ack(xs', ack(Cons(xs'), xs))
ack(Nil, n) → Cons(n)
goal(m, n) → ack(m, n)

Types:
ack :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil → Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat → Cons:Nil

Lemmas:
ack(gen_Cons:Nil2_0(1), gen_Cons:Nil2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_Cons:Nil2_0(0) ⇔ Nil
gen_Cons:Nil2_0(+(x, 1)) ⇔ Cons(gen_Cons:Nil2_0(x))

No more defined symbols left to analyse.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
ack(gen_Cons:Nil2_0(1), gen_Cons:Nil2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

Innermost TRS:
Rules:
ack(Cons(xs), Nil) → ack(xs, Cons(Nil))
ack(Cons(xs'), Cons(xs)) → ack(xs', ack(Cons(xs'), xs))
ack(Nil, n) → Cons(n)
goal(m, n) → ack(m, n)

Types:
ack :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil → Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat → Cons:Nil

Lemmas:
ack(gen_Cons:Nil2_0(1), gen_Cons:Nil2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_Cons:Nil2_0(0) ⇔ Nil
gen_Cons:Nil2_0(+(x, 1)) ⇔ Cons(gen_Cons:Nil2_0(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
ack(gen_Cons:Nil2_0(1), gen_Cons:Nil2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

(18) BOUNDS(n^1, INF)