YES
O(n^4)
TRS:
 {
               c(X) -> d(activate(X)),
               f(X) -> n__f(X),
            f(f(X)) -> c(n__f(n__g(n__f(X)))),
               d(X) -> n__d(X),
        activate(X) -> X,
  activate(n__f(X)) -> f(activate(X)),
  activate(n__g(X)) -> g(X),
  activate(n__d(X)) -> d(X),
               h(X) -> c(n__d(X)),
               g(X) -> n__g(X)
 }
 DUP: We consider a non-duplicating system.
  Trs:
   {
                 c(X) -> d(activate(X)),
                 f(X) -> n__f(X),
              f(f(X)) -> c(n__f(n__g(n__f(X)))),
                 d(X) -> n__d(X),
          activate(X) -> X,
    activate(n__f(X)) -> f(activate(X)),
    activate(n__g(X)) -> g(X),
    activate(n__d(X)) -> d(X),
                 h(X) -> c(n__d(X)),
                 g(X) -> n__g(X)
   }
  Matrix Interpretation:
   Interpretation class: triangular
       [X3]    [1 0 0 0][X3]   [1]
       [X2]    [0 0 0 0][X2]   [0]
   [g]([X1]) = [0 0 0 0][X1] + [0]
       [X0]    [0 0 0 0][X0]   [1]
   
       [X3]    [1 4 2 4][X3]   [7]
       [X2]    [0 0 4 4][X2]   [4]
   [h]([X1]) = [0 0 1 4][X1] + [2]
       [X0]    [0 0 0 0][X0]   [5]
   
          [X3]    [1 0 0 0][X3]   [0]
          [X2]    [0 0 0 0][X2]   [0]
   [n__d]([X1]) = [0 0 0 0][X1] + [0]
          [X0]    [0 0 0 0][X0]   [1]
   
              [X3]    [1 0 0 1][X3]   [1]
              [X2]    [0 1 7 4][X2]   [0]
   [activate]([X1]) = [0 0 1 0][X1] + [0]
              [X0]    [0 0 0 1][X0]   [0]
   
       [X3]    [1 0 0 0][X3]   [1]
       [X2]    [0 0 0 0][X2]   [0]
   [d]([X1]) = [0 0 0 0][X1] + [0]
       [X0]    [0 0 0 0][X0]   [1]
   
       [X3]    [1 0 5 0][X3]   [3]
       [X2]    [0 0 0 4][X2]   [1]
   [f]([X1]) = [0 0 0 0][X1] + [1]
       [X0]    [0 0 0 1][X0]   [6]
   
          [X3]    [1 0 0 0][X3]   [0]
          [X2]    [0 0 0 0][X2]   [0]
   [n__g]([X1]) = [0 0 0 0][X1] + [0]
          [X0]    [0 0 0 0][X0]   [1]
   
          [X3]    [1 0 5 0][X3]   [0]
          [X2]    [0 0 0 4][X2]   [1]
   [n__f]([X1]) = [0 0 0 0][X1] + [1]
          [X0]    [0 0 0 1][X0]   [6]
   
       [X3]    [1 0 0 1][X3]   [3]
       [X2]    [0 0 0 0][X2]   [0]
   [c]([X1]) = [0 0 0 0][X1] + [0]
       [X0]    [0 0 0 0][X0]   [4]
   
   
   Qed