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:

f(x, y) → g(x, y)
g(h(x), y) → h(f(x, y))
g(h(x), y) → h(g(x, y))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
f(h(x3_0), y) →⁺ h(f(x3_0, y))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [x3_0 / h(x3_0)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)