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:

f(h(x), y) → h(f(y, f(x, h(a))))

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

(2) BOUNDS(n^1, INF)