YES

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

Proof:
 DP Processor:
  DPs:
   f#(f(X)) -> c#(n__f(g(n__f(X))))
   c#(X) -> activate#(X)
   c#(X) -> d#(activate(X))
   h#(X) -> c#(n__d(X))
   activate#(n__f(X)) -> f#(X)
   activate#(n__d(X)) -> d#(X)
  TRS:
   f(f(X)) -> c(n__f(g(n__f(X))))
   c(X) -> d(activate(X))
   h(X) -> c(n__d(X))
   f(X) -> n__f(X)
   d(X) -> n__d(X)
   activate(n__f(X)) -> f(X)
   activate(n__d(X)) -> d(X)
   activate(X) -> X
  Arctic Interpretation Processor:
   dimension: 1
   interpretation:
    [h#](x0) = 4x0 + 7,
    
    [d#](x0) = 0,
    
    [activate#](x0) = 1x0 + 1,
    
    [c#](x0) = 2x0 + 4,
    
    [f#](x0) = 2x0 + 4,
    
    [n__d](x0) = x0,
    
    [h](x0) = 12x0 + 12,
    
    [d](x0) = 1x0,
    
    [activate](x0) = 3x0 + 0,
    
    [c](x0) = 8x0 + 8,
    
    [g](x0) = 1,
    
    [n__f](x0) = 5x0 + 12,
    
    [f](x0) = 7x0 + 14
   orientation:
    f#(f(X)) = 9X + 16 >= 14 = c#(n__f(g(n__f(X))))
    
    c#(X) = 2X + 4 >= 1X + 1 = activate#(X)
    
    c#(X) = 2X + 4 >= 0 = d#(activate(X))
    
    h#(X) = 4X + 7 >= 2X + 4 = c#(n__d(X))
    
    activate#(n__f(X)) = 6X + 13 >= 2X + 4 = f#(X)
    
    activate#(n__d(X)) = 1X + 1 >= 0 = d#(X)
    
    f(f(X)) = 14X + 21 >= 20 = c(n__f(g(n__f(X))))
    
    c(X) = 8X + 8 >= 4X + 1 = d(activate(X))
    
    h(X) = 12X + 12 >= 8X + 8 = c(n__d(X))
    
    f(X) = 7X + 14 >= 5X + 12 = n__f(X)
    
    d(X) = 1X >= X = n__d(X)
    
    activate(n__f(X)) = 8X + 15 >= 7X + 14 = f(X)
    
    activate(n__d(X)) = 3X + 0 >= 1X = d(X)
    
    activate(X) = 3X + 0 >= X = X
   problem:
    DPs:
     
    TRS:
     f(f(X)) -> c(n__f(g(n__f(X))))
     c(X) -> d(activate(X))
     h(X) -> c(n__d(X))
     f(X) -> n__f(X)
     d(X) -> n__d(X)
     activate(n__f(X)) -> f(X)
     activate(n__d(X)) -> d(X)
     activate(X) -> X
   Qed