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)