YES

Problem:
 f(f(X)) -> c(n__f(g(n__f(X))))
 c(X) -> d(activate(X))
 h(X) -> c(n__d(X))
 f(X) -> n__f(X)
 d(X) -> n__d(X)
 activate(n__f(X)) -> f(X)
 activate(n__d(X)) -> d(X)
 activate(X) -> X

Proof:
 WPO Processor:
  algebra: Sum
  weight function:
   w0 = 0
   w(h) = 2
   w(activate) = w(c) = w(f) = 1
   w(n__d) = w(d) = w(g) = w(n__f) = 0
  status function:
   st(n__d) = st(h) = st(d) = st(activate) = st(c) = st(g) = st(n__f) = st(f) = [0]
  precedence:
   c > d ~ activate > n__d ~ h ~ g ~ n__f ~ f
  problem:
   
  Qed