The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(2^n, INF).

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

f(g(x)) → f(a(g(g(f(x))), g(f(x))))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2ⁿ):
The rewrite sequence
f(g(x)) →⁺ f(a(g(g(f(x))), g(f(x))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0,0].
The pumping substitution is [x / g(x)].
The result substitution is [ ].

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

(2) BOUNDS(2^n, INF)