YES

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

Proof:
 DP Processor:
  DPs:
   f#(f(X)) -> c#(n__f(n__g(n__f(X))))
   c#(X) -> activate#(X)
   c#(X) -> d#(activate(X))
   h#(X) -> c#(n__d(X))
   activate#(n__f(X)) -> activate#(X)
   activate#(n__f(X)) -> f#(activate(X))
   activate#(n__g(X)) -> g#(X)
   activate#(n__d(X)) -> d#(X)
  TRS:
   f(f(X)) -> c(n__f(n__g(n__f(X))))
   c(X) -> d(activate(X))
   h(X) -> c(n__d(X))
   f(X) -> n__f(X)
   g(X) -> n__g(X)
   d(X) -> n__d(X)
   activate(n__f(X)) -> f(activate(X))
   activate(n__g(X)) -> g(X)
   activate(n__d(X)) -> d(X)
   activate(X) -> X
  Usable Rule Processor:
   DPs:
    f#(f(X)) -> c#(n__f(n__g(n__f(X))))
    c#(X) -> activate#(X)
    c#(X) -> d#(activate(X))
    h#(X) -> c#(n__d(X))
    activate#(n__f(X)) -> activate#(X)
    activate#(n__f(X)) -> f#(activate(X))
    activate#(n__g(X)) -> g#(X)
    activate#(n__d(X)) -> d#(X)
   TRS:
    activate(n__f(X)) -> f(activate(X))
    activate(n__g(X)) -> g(X)
    activate(n__d(X)) -> d(X)
    activate(X) -> X
    f(f(X)) -> c(n__f(n__g(n__f(X))))
    f(X) -> n__f(X)
    c(X) -> d(activate(X))
    d(X) -> n__d(X)
    g(X) -> n__g(X)
   Matrix Interpretation Processor: dim=1
    
    interpretation:
     [g#](x0) = 0,
     
     [h#](x0) = 16x0 + 17,
     
     [d#](x0) = 0,
     
     [activate#](x0) = 12x0 + 1,
     
     [c#](x0) = 16x0 + 16,
     
     [f#](x0) = 6x0 + 3,
     
     [g](x0) = 0,
     
     [n__d](x0) = 0,
     
     [d](x0) = 0,
     
     [activate](x0) = 8x0,
     
     [c](x0) = 16x0,
     
     [n__g](x0) = 0,
     
     [n__f](x0) = 4x0 + 2,
     
     [f](x0) = 4x0 + 9
    orientation:
     f#(f(X)) = 24X + 57 >= 48 = c#(n__f(n__g(n__f(X))))
     
     c#(X) = 16X + 16 >= 12X + 1 = activate#(X)
     
     c#(X) = 16X + 16 >= 0 = d#(activate(X))
     
     h#(X) = 16X + 17 >= 16 = c#(n__d(X))
     
     activate#(n__f(X)) = 48X + 25 >= 12X + 1 = activate#(X)
     
     activate#(n__f(X)) = 48X + 25 >= 48X + 3 = f#(activate(X))
     
     activate#(n__g(X)) = 1 >= 0 = g#(X)
     
     activate#(n__d(X)) = 1 >= 0 = d#(X)
     
     activate(n__f(X)) = 32X + 16 >= 32X + 9 = f(activate(X))
     
     activate(n__g(X)) = 0 >= 0 = g(X)
     
     activate(n__d(X)) = 0 >= 0 = d(X)
     
     activate(X) = 8X >= X = X
     
     f(f(X)) = 16X + 45 >= 32 = c(n__f(n__g(n__f(X))))
     
     f(X) = 4X + 9 >= 4X + 2 = n__f(X)
     
     c(X) = 16X >= 0 = d(activate(X))
     
     d(X) = 0 >= 0 = n__d(X)
     
     g(X) = 0 >= 0 = n__g(X)
    problem:
     DPs:
      
     TRS:
      activate(n__f(X)) -> f(activate(X))
      activate(n__g(X)) -> g(X)
      activate(n__d(X)) -> d(X)
      activate(X) -> X
      f(f(X)) -> c(n__f(n__g(n__f(X))))
      f(X) -> n__f(X)
      c(X) -> d(activate(X))
      d(X) -> n__d(X)
      g(X) -> n__g(X)
    Qed