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
  TDG 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
   graph:
    h#(X) -> c#(n__d(X)) -> c#(X) -> d#(activate(X))
    h#(X) -> c#(n__d(X)) -> c#(X) -> activate#(X)
    activate#(n__f(X)) -> activate#(X) ->
    activate#(n__d(X)) -> d#(X)
    activate#(n__f(X)) -> activate#(X) ->
    activate#(n__g(X)) -> g#(X)
    activate#(n__f(X)) -> activate#(X) ->
    activate#(n__f(X)) -> f#(activate(X))
    activate#(n__f(X)) -> activate#(X) ->
    activate#(n__f(X)) -> activate#(X)
    activate#(n__f(X)) -> f#(activate(X)) ->
    f#(f(X)) -> c#(n__f(n__g(n__f(X))))
    c#(X) -> activate#(X) -> activate#(n__d(X)) -> d#(X)
    c#(X) -> activate#(X) -> activate#(n__g(X)) -> g#(X)
    c#(X) -> activate#(X) -> activate#(n__f(X)) -> f#(activate(X))
    c#(X) -> activate#(X) -> activate#(n__f(X)) -> activate#(X)
    f#(f(X)) -> c#(n__f(n__g(n__f(X)))) ->
    c#(X) -> d#(activate(X))
    f#(f(X)) -> c#(n__f(n__g(n__f(X)))) -> c#(X) -> activate#(X)
   CDG 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
    graph:
     h#(X) -> c#(n__d(X)) -> c#(X) -> activate#(X)
     h#(X) -> c#(n__d(X)) -> c#(X) -> d#(activate(X))
     activate#(n__f(X)) -> activate#(X) ->
     activate#(n__f(X)) -> activate#(X)
     activate#(n__f(X)) -> activate#(X) ->
     activate#(n__f(X)) -> f#(activate(X))
     activate#(n__f(X)) -> activate#(X) ->
     activate#(n__g(X)) -> g#(X)
     activate#(n__f(X)) -> f#(activate(X)) ->
     f#(f(X)) -> c#(n__f(n__g(n__f(X))))
     c#(X) -> activate#(X) -> activate#(n__f(X)) -> activate#(X)
     c#(X) -> activate#(X) -> activate#(n__f(X)) -> f#(activate(X))
     c#(X) -> activate#(X) -> activate#(n__g(X)) -> g#(X)
     f#(f(X)) -> c#(n__f(n__g(n__f(X)))) ->
     c#(X) -> activate#(X)
     f#(f(X)) -> c#(n__f(n__g(n__f(X)))) -> c#(X) -> d#(activate(X))
    SCC Processor:
     #sccs: 1
     #rules: 4
     #arcs: 11/64
     DPs:
      c#(X) -> activate#(X)
      activate#(n__f(X)) -> f#(activate(X))
      f#(f(X)) -> c#(n__f(n__g(n__f(X))))
      activate#(n__f(X)) -> activate#(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
     Arctic Interpretation Processor:
      dimension: 1
      interpretation:
       [activate#](x0) = x0,
       
       [c#](x0) = x0,
       
       [f#](x0) = 4x0,
       
       [g](x0) = x0,
       
       [n__d](x0) = x0,
       
       [h](x0) = 3x0 + 2,
       
       [d](x0) = x0,
       
       [activate](x0) = x0,
       
       [c](x0) = x0,
       
       [n__g](x0) = x0,
       
       [n__f](x0) = 4x0,
       
       [f](x0) = 4x0
      orientation:
       c#(X) = X >= X = activate#(X)
       
       activate#(n__f(X)) = 4X >= 4X = f#(activate(X))
       
       f#(f(X)) = 8X >= 8X = c#(n__f(n__g(n__f(X))))
       
       activate#(n__f(X)) = 4X >= X = activate#(X)
       
       f(f(X)) = 8X >= 8X = c(n__f(n__g(n__f(X))))
       
       c(X) = X >= X = d(activate(X))
       
       h(X) = 3X + 2 >= X = c(n__d(X))
       
       f(X) = 4X >= 4X = n__f(X)
       
       g(X) = X >= X = n__g(X)
       
       d(X) = X >= X = n__d(X)
       
       activate(n__f(X)) = 4X >= 4X = f(activate(X))
       
       activate(n__g(X)) = X >= X = g(X)
       
       activate(n__d(X)) = X >= X = d(X)
       
       activate(X) = X >= X = X
      problem:
       DPs:
        c#(X) -> activate#(X)
        activate#(n__f(X)) -> f#(activate(X))
        f#(f(X)) -> c#(n__f(n__g(n__f(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
      EDG Processor:
       DPs:
        c#(X) -> activate#(X)
        activate#(n__f(X)) -> f#(activate(X))
        f#(f(X)) -> c#(n__f(n__g(n__f(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
       graph:
        activate#(n__f(X)) -> f#(activate(X)) ->
        f#(f(X)) -> c#(n__f(n__g(n__f(X))))
        c#(X) -> activate#(X) ->
        activate#(n__f(X)) -> f#(activate(X))
        f#(f(X)) -> c#(n__f(n__g(n__f(X)))) -> c#(X) -> activate#(X)
       Arctic Interpretation Processor:
        dimension: 1
        interpretation:
         [activate#](x0) = 6x0 + 0,
         
         [c#](x0) = 6x0 + 0,
         
         [f#](x0) = 8x0 + 0,
         
         [g](x0) = 1,
         
         [n__d](x0) = x0,
         
         [h](x0) = 10x0 + 0,
         
         [d](x0) = x0 + 0,
         
         [activate](x0) = x0 + 0,
         
         [c](x0) = 1x0 + 0,
         
         [n__g](x0) = 1,
         
         [n__f](x0) = 2x0 + 2,
         
         [f](x0) = 2x0 + 2
        orientation:
         c#(X) = 6X + 0 >= 6X + 0 = activate#(X)
         
         activate#(n__f(X)) = 8X + 8 >= 8X + 8 = f#(activate(X))
         
         f#(f(X)) = 10X + 10 >= 9 = c#(n__f(n__g(n__f(X))))
         
         f(f(X)) = 4X + 4 >= 4 = c(n__f(n__g(n__f(X))))
         
         c(X) = 1X + 0 >= X + 0 = d(activate(X))
         
         h(X) = 10X + 0 >= 1X + 0 = c(n__d(X))
         
         f(X) = 2X + 2 >= 2X + 2 = n__f(X)
         
         g(X) = 1 >= 1 = n__g(X)
         
         d(X) = X + 0 >= X = n__d(X)
         
         activate(n__f(X)) = 2X + 2 >= 2X + 2 = f(activate(X))
         
         activate(n__g(X)) = 1 >= 1 = g(X)
         
         activate(n__d(X)) = X + 0 >= X + 0 = d(X)
         
         activate(X) = X + 0 >= X = X
        problem:
         DPs:
          c#(X) -> activate#(X)
          activate#(n__f(X)) -> f#(activate(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
        EDG Processor:
         DPs:
          c#(X) -> activate#(X)
          activate#(n__f(X)) -> f#(activate(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
         graph:
          c#(X) -> activate#(X) -> activate#(n__f(X)) -> f#(activate(X))
         CDG Processor:
          DPs:
           c#(X) -> activate#(X)
           activate#(n__f(X)) -> f#(activate(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
          graph:
           
          Qed