YES
O(n^2)
TRS:
 {
               f(X) -> n__f(X),
          f(f(a())) -> c(n__f(n__g(n__f(n__a())))),
                a() -> n__a(),
               g(X) -> n__g(X),
        activate(X) -> X,
  activate(n__f(X)) -> f(activate(X)),
  activate(n__g(X)) -> g(activate(X)),
   activate(n__a()) -> a()
 }
 DUP: We consider a non-duplicating system.
  Trs:
   {
                 f(X) -> n__f(X),
            f(f(a())) -> c(n__f(n__g(n__f(n__a())))),
                  a() -> n__a(),
                 g(X) -> n__g(X),
          activate(X) -> X,
    activate(n__f(X)) -> f(activate(X)),
    activate(n__g(X)) -> g(activate(X)),
     activate(n__a()) -> a()
   }
  Matrix Interpretation:
   Interpretation class: triangular
              [X1]    [1 3][X1]   [1]
   [activate]([X0]) = [0 1][X0] + [0]
   
       [X1]    [1 1][X1]   [2]
   [g]([X0]) = [0 1][X0] + [2]
   
         [3]
   [a] = [1]
   
       [X1]    [1 1][X1]   [2]
   [f]([X0]) = [0 1][X0] + [2]
   
            [0]
   [n__a] = [1]
   
          [X1]    [1 1][X1]   [0]
   [n__g]([X0]) = [0 1][X0] + [2]
   
          [X1]    [1 1][X1]   [0]
   [n__f]([X0]) = [0 1][X0] + [2]
   
       [X1]    [1 0][X1]   [1]
   [c]([X0]) = [0 0][X0] + [0]
   
   
   Qed