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:

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

Rewrite Strategy: FULL

(1) CpxTrsToCpxRelTrsProof (BOTH BOUNDS(ID, ID) transformation)

Transformed TRS to relative TRS where S is empty.

(2) Obligation:

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

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

S is empty.
Rewrite Strategy: FULL

(3) DecreasingLoopProof (EQUIVALENT transformation)

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

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

(4) BOUNDS(2^n, INF)