YES

We show the termination of the TRS R:

  f(f(a())) -> f(g(n__f(a())))
  f(X) -> n__f(X)
  activate(n__f(X)) -> f(X)
  activate(X) -> X

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: f#(f(a())) -> f#(g(n__f(a())))
p2: activate#(n__f(X)) -> f#(X)

and R consists of:

r1: f(f(a())) -> f(g(n__f(a())))
r2: f(X) -> n__f(X)
r3: activate(n__f(X)) -> f(X)
r4: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  (no SCCs)