YES
O(n^4)
TRS:
 {
               f(X) -> n__f(X),
          f(f(a())) -> c(n__f(g(f(a())))),
        activate(X) -> X,
  activate(n__f(X)) -> f(X)
 }
 DUP: We consider a non-duplicating system.
  Trs:
   {
                 f(X) -> n__f(X),
            f(f(a())) -> c(n__f(g(f(a())))),
          activate(X) -> X,
    activate(n__f(X)) -> f(X)
   }
  Matrix Interpretation:
   Interpretation class: triangular
              [X3]    [1 1 0 0][X3]   [1]
              [X2]    [0 1 0 0][X2]   [0]
   [activate]([X1]) = [0 0 1 0][X1] + [0]
              [X0]    [0 0 0 1][X0]   [0]
   
         [0]
         [0]
   [a] = [0]
         [0]
   
       [X3]    [1 0 0 0][X3]   [1]
       [X2]    [0 0 0 0][X2]   [1]
   [f]([X1]) = [0 0 0 0][X1] + [0]
       [X0]    [0 0 0 0][X0]   [0]
   
       [X3]    [1 0 0 0][X3]   [0]
       [X2]    [0 0 0 0][X2]   [0]
   [g]([X1]) = [0 0 0 0][X1] + [0]
       [X0]    [0 0 0 0][X0]   [0]
   
          [X3]    [1 0 0 0][X3]   [0]
          [X2]    [0 0 0 0][X2]   [1]
   [n__f]([X1]) = [0 0 0 0][X1] + [0]
          [X0]    [0 0 0 0][X0]   [0]
   
       [X3]    [1 0 0 0][X3]   [0]
       [X2]    [0 0 0 0][X2]   [0]
   [c]([X1]) = [0 0 0 0][X1] + [0]
       [X0]    [0 0 0 0][X0]   [0]
   
   
   Qed