(0) Obligation:

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

f(g(X), Y) → f(X, n__f(g(X), activate(Y)))
f(X1, X2) → n__f(X1, X2)
activate(n__f(X1, X2)) → f(X1, X2)
activate(X) → X

Rewrite Strategy: FULL

(1) InfiniteLowerBoundProof (EQUIVALENT transformation)

The loop following loop proves infinite runtime complexity:
The rewrite sequence
f(g(g(X22_0)), Y) →+ f(X22_0, n__f(g(X22_0), f(g(g(X22_0)), activate(Y))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1,1].
The pumping substitution is [ ].
The result substitution is [Y / activate(Y)].