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
     Matrix Interpretation Processor: dim=3
      
      interpretation:
       [activate#](x0) = [1 0 1]x0,
       
       [c#](x0) = [1 0 1]x0 + [1],
       
       [f#](x0) = [1 1 1]x0,
       
                 [1 0 0]  
       [g](x0) = [0 0 0]x0
                 [0 0 0]  ,
       
                    [0]
       [n__d](x0) = [0]
                    [0],
       
                 [0 1 1]     [1]
       [h](x0) = [0 0 1]x0 + [0]
                 [0 1 0]     [1],
       
                 [0]
       [d](x0) = [0]
                 [0],
       
                        [1 0 1]  
       [activate](x0) = [0 1 0]x0
                        [0 0 1]  ,
       
                 [1]
       [c](x0) = [0]
                 [1],
       
                    [1 0 0]  
       [n__g](x0) = [0 0 0]x0
                    [0 0 0]  ,
       
                    [1 0 1]     [0]
       [n__f](x0) = [0 0 0]x0 + [1]
                    [0 1 1]     [1],
       
                 [1 0 1]     [1]
       [f](x0) = [0 0 0]x0 + [1]
                 [0 1 1]     [1]
      orientation:
       c#(X) = [1 0 1]X + [1] >= [1 0 1]X = activate#(X)
       
       activate#(n__f(X)) = [1 1 2]X + [1] >= [1 1 2]X = f#(activate(X))
       
       f#(f(X)) = [1 1 2]X + [3] >= [1 0 1]X + [2] = c#(n__f(n__g(n__f(X))))
       
       activate#(n__f(X)) = [1 1 2]X + [1] >= [1 0 1]X = activate#(X)
       
                 [1 1 2]    [3]    [1]                         
       f(f(X)) = [0 0 0]X + [1] >= [0] = c(n__f(n__g(n__f(X))))
                 [0 1 1]    [3]    [1]                         
       
              [1]    [0]                 
       c(X) = [0] >= [0] = d(activate(X))
              [1]    [0]                 
       
              [0 1 1]    [1]    [1]             
       h(X) = [0 0 1]X + [0] >= [0] = c(n__d(X))
              [0 1 0]    [1]    [1]             
       
              [1 0 1]    [1]    [1 0 1]    [0]          
       f(X) = [0 0 0]X + [1] >= [0 0 0]X + [1] = n__f(X)
              [0 1 1]    [1]    [0 1 1]    [1]          
       
              [1 0 0]     [1 0 0]           
       g(X) = [0 0 0]X >= [0 0 0]X = n__g(X)
              [0 0 0]     [0 0 0]           
       
              [0]    [0]          
       d(X) = [0] >= [0] = n__d(X)
              [0]    [0]          
       
                           [1 1 2]    [1]    [1 0 2]    [1]                 
       activate(n__f(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = f(activate(X))
                           [0 1 1]    [1]    [0 1 1]    [1]                 
       
                           [1 0 0]     [1 0 0]        
       activate(n__g(X)) = [0 0 0]X >= [0 0 0]X = g(X)
                           [0 0 0]     [0 0 0]        
       
                           [0]    [0]       
       activate(n__d(X)) = [0] >= [0] = d(X)
                           [0]    [0]       
       
                     [1 0 1]          
       activate(X) = [0 1 0]X >= X = X
                     [0 0 1]          
      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