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
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [g#](x0) = 0,
    
    [h#](x0) = 18,
    
    [d#](x0) = 11,
    
    [activate#](x0) = 6x0 + 16,
    
    [c#](x0) = 6x0 + 17,
    
    [f#](x0) = x0 + 20,
    
    [g](x0) = 0,
    
    [n__d](x0) = 0,
    
    [h](x0) = 0,
    
    [d](x0) = 0,
    
    [activate](x0) = 4x0,
    
    [c](x0) = 0,
    
    [n__g](x0) = 0,
    
    [n__f](x0) = 4x0 + 1,
    
    [f](x0) = 4x0 + 4
   orientation:
    f#(f(X)) = 4X + 24 >= 23 = c#(n__f(n__g(n__f(X))))
    
    c#(X) = 6X + 17 >= 6X + 16 = activate#(X)
    
    c#(X) = 6X + 17 >= 11 = d#(activate(X))
    
    h#(X) = 18 >= 17 = c#(n__d(X))
    
    activate#(n__f(X)) = 24X + 22 >= 6X + 16 = activate#(X)
    
    activate#(n__f(X)) = 24X + 22 >= 4X + 20 = f#(activate(X))
    
    activate#(n__g(X)) = 16 >= 0 = g#(X)
    
    activate#(n__d(X)) = 16 >= 11 = d#(X)
    
    f(f(X)) = 16X + 20 >= 0 = c(n__f(n__g(n__f(X))))
    
    c(X) = 0 >= 0 = d(activate(X))
    
    h(X) = 0 >= 0 = c(n__d(X))
    
    f(X) = 4X + 4 >= 4X + 1 = n__f(X)
    
    g(X) = 0 >= 0 = n__g(X)
    
    d(X) = 0 >= 0 = n__d(X)
    
    activate(n__f(X)) = 16X + 4 >= 16X + 4 = f(activate(X))
    
    activate(n__g(X)) = 0 >= 0 = g(X)
    
    activate(n__d(X)) = 0 >= 0 = d(X)
    
    activate(X) = 4X >= X = X
   problem:
    DPs:
     
    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
   Qed