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

Infered types.

(6) Obligation:

Innermost TRS:
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)

Types:
ack :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → 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

(7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
ack

(8) Obligation:

Innermost TRS:
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)

Types:
ack :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → 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(Nil, gen_Cons:Nil2_0(x))

The following defined symbols remain to be analysed:
ack

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol ack.

(10) Obligation:

Innermost TRS:
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)

Types:
ack :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → 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(Nil, gen_Cons:Nil2_0(x))

No more defined symbols left to analyse.