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:

h(f(x, y)) → f(f(a, h(h(y))), x)

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

(2) BOUNDS(n^1, INF)