(0) Obligation:

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

f(X) → g(n__h(f(X)))
h(X) → n__h(X)
activate(n__h(X)) → h(X)
activate(X) → X

Rewrite Strategy: FULL

(1) InfiniteLowerBoundProof (EQUIVALENT transformation)

The loop following loop proves infinite runtime complexity:
The rewrite sequence
f(X) →+ g(n__h(f(X)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
The pumping substitution is [ ].
The result substitution is [ ].