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
  Usable Rule 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:
    activate(n__f(X)) -> f(X)
    activate(n__d(X)) -> d(X)
    activate(X) -> X
    f(f(X)) -> c(n__f(g(n__f(X))))
    f(X) -> n__f(X)
    c(X) -> d(activate(X))
    d(X) -> n__d(X)
   Arctic Interpretation Processor:
    dimension: 1
    usable rules:
     activate(n__f(X)) -> f(X)
     activate(n__d(X)) -> d(X)
     activate(X) -> X
     f(f(X)) -> c(n__f(g(n__f(X))))
     f(X) -> n__f(X)
     c(X) -> d(activate(X))
     d(X) -> n__d(X)
    interpretation:
     [h#](x0) = 1x0 + 12,
     
     [d#](x0) = -13x0 + 0,
     
     [activate#](x0) = 2x0 + -16,
     
     [c#](x0) = 3x0 + 1,
     
     [f#](x0) = x0,
     
     [n__d](x0) = -11x0 + 8,
     
     [d](x0) = -7x0 + 9,
     
     [activate](x0) = 13x0 + 3,
     
     [c](x0) = 8x0 + 10,
     
     [g](x0) = x0 + -16,
     
     [n__f](x0) = x0,
     
     [f](x0) = 10x0 + 2
    orientation:
     f#(f(X)) = 10X + 2 >= 3X + 1 = c#(n__f(g(n__f(X))))
     
     c#(X) = 3X + 1 >= 2X + -16 = activate#(X)
     
     c#(X) = 3X + 1 >= X + 0 = d#(activate(X))
     
     h#(X) = 1X + 12 >= -8X + 11 = c#(n__d(X))
     
     activate#(n__f(X)) = 2X + -16 >= X = f#(X)
     
     activate#(n__d(X)) = -9X + 10 >= -13X + 0 = d#(X)
     
     activate(n__f(X)) = 13X + 3 >= 10X + 2 = f(X)
     
     activate(n__d(X)) = 2X + 21 >= -7X + 9 = d(X)
     
     activate(X) = 13X + 3 >= X = X
     
     f(f(X)) = 20X + 12 >= 8X + 10 = c(n__f(g(n__f(X))))
     
     f(X) = 10X + 2 >= X = n__f(X)
     
     c(X) = 8X + 10 >= 6X + 9 = d(activate(X))
     
     d(X) = -7X + 9 >= -11X + 8 = n__d(X)
    problem:
     DPs:
      
     TRS:
      activate(n__f(X)) -> f(X)
      activate(n__d(X)) -> d(X)
      activate(X) -> X
      f(f(X)) -> c(n__f(g(n__f(X))))
      f(X) -> n__f(X)
      c(X) -> d(activate(X))
      d(X) -> n__d(X)
    Qed