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:
 Matrix Interpretation Processor: dim=3
  
  interpretation:
             [1 0 0]     [1]
   [g](x0) = [1 0 1]x0 + [1]
             [0 0 0]     [0],
   
                [1 0 0]  
   [n__d](x0) = [0 0 0]x0
                [0 0 0]  ,
   
             [1 1 1]     [1]
   [h](x0) = [0 1 0]x0 + [1]
             [0 0 1]     [1],
   
             [1 0 0]  
   [d](x0) = [0 0 0]x0
             [0 0 0]  ,
   
                      
   [activate](x0) = x0
                      ,
   
             [1 0 0]     [0]
   [c](x0) = [0 0 0]x0 + [0]
             [0 0 0]     [1],
   
                [1 0 0]     [1]
   [n__g](x0) = [1 0 1]x0 + [1]
                [0 0 0]     [0],
   
                [1 0 1]     [0]
   [n__f](x0) = [1 1 0]x0 + [0]
                [0 0 1]     [1],
   
             [1 0 1]     [0]
   [f](x0) = [1 1 0]x0 + [0]
             [0 0 1]     [1]
  orientation:
             [1 0 2]    [1]    [1 0 1]    [1]                         
   f(f(X)) = [2 1 1]X + [0] >= [0 0 0]X + [0] = c(n__f(n__g(n__f(X))))
             [0 0 1]    [2]    [0 0 0]    [1]                         
   
          [1 0 0]    [0]    [1 0 0]                  
   c(X) = [0 0 0]X + [0] >= [0 0 0]X = d(activate(X))
          [0 0 0]    [1]    [0 0 0]                  
   
          [1 1 1]    [1]    [1 0 0]    [0]             
   h(X) = [0 1 0]X + [1] >= [0 0 0]X + [0] = c(n__d(X))
          [0 0 1]    [1]    [0 0 0]    [1]             
   
          [1 0 1]    [0]    [1 0 1]    [0]          
   f(X) = [1 1 0]X + [0] >= [1 1 0]X + [0] = n__f(X)
          [0 0 1]    [1]    [0 0 1]    [1]          
   
          [1 0 0]    [1]    [1 0 0]    [1]          
   g(X) = [1 0 1]X + [1] >= [1 0 1]X + [1] = n__g(X)
          [0 0 0]    [0]    [0 0 0]    [0]          
   
          [1 0 0]     [1 0 0]           
   d(X) = [0 0 0]X >= [0 0 0]X = n__d(X)
          [0 0 0]     [0 0 0]           
   
                       [1 0 1]    [0]    [1 0 1]    [0]                 
   activate(n__f(X)) = [1 1 0]X + [0] >= [1 1 0]X + [0] = f(activate(X))
                       [0 0 1]    [1]    [0 0 1]    [1]                 
   
                       [1 0 0]    [1]    [1 0 0]    [1]       
   activate(n__g(X)) = [1 0 1]X + [1] >= [1 0 1]X + [1] = g(X)
                       [0 0 0]    [0]    [0 0 0]    [0]       
   
                       [1 0 0]     [1 0 0]        
   activate(n__d(X)) = [0 0 0]X >= [0 0 0]X = d(X)
                       [0 0 0]     [0 0 0]        
   
                           
   activate(X) = X >= X = X
                           
  problem:
   f(f(X)) -> c(n__f(n__g(n__f(X))))
   c(X) -> d(activate(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:
              [1 0 0]  
    [g](x0) = [1 0 0]x0
              [0 0 0]  ,
    
                 [1 0 0]  
    [n__d](x0) = [0 0 0]x0
                 [0 0 0]  ,
    
              [1 0 0]  
    [d](x0) = [0 0 0]x0
              [0 0 0]  ,
    
                     [1 1 0]     [0]
    [activate](x0) = [1 1 0]x0 + [0]
                     [0 0 1]     [1],
    
              [1 1 0]     [1]
    [c](x0) = [0 0 0]x0 + [0]
              [0 0 0]     [1],
    
                 [1 0 0]  
    [n__g](x0) = [0 0 0]x0
                 [0 0 0]  ,
    
                 [1 1 0]     [0]
    [n__f](x0) = [1 1 0]x0 + [1]
                 [0 0 0]     [0],
    
              [1 1 0]     [1]
    [f](x0) = [1 1 0]x0 + [1]
              [0 0 0]     [1]
   orientation:
              [2 2 0]    [3]    [2 2 0]    [2]                         
    f(f(X)) = [2 2 0]X + [3] >= [0 0 0]X + [0] = c(n__f(n__g(n__f(X))))
              [0 0 0]    [1]    [0 0 0]    [1]                         
    
           [1 1 0]    [1]    [1 1 0]                  
    c(X) = [0 0 0]X + [0] >= [0 0 0]X = d(activate(X))
           [0 0 0]    [1]    [0 0 0]                  
    
           [1 1 0]    [1]    [1 1 0]    [0]          
    f(X) = [1 1 0]X + [1] >= [1 1 0]X + [1] = n__f(X)
           [0 0 0]    [1]    [0 0 0]    [0]          
    
           [1 0 0]     [1 0 0]           
    g(X) = [1 0 0]X >= [0 0 0]X = n__g(X)
           [0 0 0]     [0 0 0]           
    
           [1 0 0]     [1 0 0]           
    d(X) = [0 0 0]X >= [0 0 0]X = n__d(X)
           [0 0 0]     [0 0 0]           
    
                        [2 2 0]    [1]    [2 2 0]    [1]                 
    activate(n__f(X)) = [2 2 0]X + [1] >= [2 2 0]X + [1] = f(activate(X))
                        [0 0 0]    [1]    [0 0 0]    [1]                 
    
                        [1 0 0]    [0]    [1 0 0]        
    activate(n__g(X)) = [1 0 0]X + [0] >= [1 0 0]X = g(X)
                        [0 0 0]    [1]    [0 0 0]        
    
                        [1 0 0]    [0]    [1 0 0]        
    activate(n__d(X)) = [1 0 0]X + [0] >= [0 0 0]X = d(X)
                        [0 0 0]    [1]    [0 0 0]        
    
                  [1 1 0]    [0]         
    activate(X) = [1 1 0]X + [0] >= X = X
                  [0 0 1]    [1]         
   problem:
    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
   String Reversal Processor:
    g(X) -> n__g(X)
    d(X) -> n__d(X)
    n__f(activate(X)) -> activate(f(X))
    n__g(activate(X)) -> g(X)
    n__d(activate(X)) -> d(X)
    activate(X) -> X
    KBO Processor:
     weight function:
      w0 = 1
      w(g) = w(n__d) = w(d) = w(activate) = w(n__f) = w(f) = 1
      w(n__g) = 0
     precedence:
      n__g > d > n__d > g > n__f > activate > f
     problem:
      
     Qed