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) CpxTrsToCpxRelTrsProof (BOTH BOUNDS(ID, ID) transformation)

Transformed TRS to relative TRS where S is empty.

(2) 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

(3) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
Cons/0

(4) 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

(5) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
ack(Cons(Cons(xs'14_0)), Nil) →⁺ ack(xs'14_0, ack(Cons(xs'14_0), Nil))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [xs'14_0 / Cons(xs'14_0)].
The result substitution is [ ].

(6) BOUNDS(n^1, INF)