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

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
ack(s(s(x15_1)), 0) →⁺ ack(x15_1, ack(s(x15_1), 0))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [x15_1 / s(x15_1)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)