The Runtime Complexity (full) 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_in(0, n) → ack_out(s(n))
ack_in(s(m), 0) → u11(ack_in(m, s(0)))
u11(ack_out(n)) → ack_out(n)
ack_in(s(m), s(n)) → u21(ack_in(s(m), n), m)
u21(ack_out(n), m) → u22(ack_in(m, n))
u22(ack_out(n)) → ack_out(n)

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
ack_in(s(s(m36_3)), 0) →⁺ u11(u21(ack_in(s(m36_3), 0), m36_3))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
The pumping substitution is [m36_3 / s(m36_3)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)