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(a) → b
f(c) → d
f(g(x, y)) → g(f(x), f(y))
f(h(x, y)) → g(h(y, f(x)), h(x, f(y)))
g(x, x) → h(e, x)

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(x, y)) →⁺ g(h(y, f(x)), h(x, f(y)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1].
The pumping substitution is [x / h(x, y)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)