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:
 Arctic Interpretation Processor:
  dimension: 1
  interpretation:
   [g](x0) = x0,
   
   [n__d](x0) = x0,
   
   [h](x0) = 8x0,
   
   [d](x0) = x0,
   
   [activate](x0) = x0,
   
   [c](x0) = x0,
   
   [n__g](x0) = x0,
   
   [n__f](x0) = x0,
   
   [f](x0) = x0
  orientation:
   f(f(X)) = X >= X = c(n__f(n__g(n__f(X))))
   
   c(X) = X >= X = d(activate(X))
   
   h(X) = 8X >= X = c(n__d(X))
   
   f(X) = X >= X = n__f(X)
   
   g(X) = X >= X = n__g(X)
   
   d(X) = X >= X = n__d(X)
   
   activate(n__f(X)) = X >= X = 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:
   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
  Arctic Interpretation Processor:
   dimension: 2
   interpretation:
              [0  -&]  
    [g](x0) = [0  -&]x0,
    
                 [0  -&]  
    [n__d](x0) = [1  0 ]x0,
    
              [0  -&]  
    [d](x0) = [1  0 ]x0,
    
                     [0 0]  
    [activate](x0) = [2 2]x0,
    
              [0 0]  
    [c](x0) = [2 2]x0,
    
                 [0  -&]  
    [n__g](x0) = [0  -&]x0,
    
                   
    [n__f](x0) = x0,
    
              [0  -&]  
    [f](x0) = [2  0 ]x0
   orientation:
              [0  -&]     [0  -&]                          
    f(f(X)) = [2  0 ]X >= [2  -&]X = c(n__f(n__g(n__f(X))))
    
           [0 0]     [0 0]                  
    c(X) = [2 2]X >= [2 2]X = d(activate(X))
    
           [0  -&]                
    f(X) = [2  0 ]X >= X = n__f(X)
    
           [0  -&]     [0  -&]           
    g(X) = [0  -&]X >= [0  -&]X = n__g(X)
    
           [0  -&]     [0  -&]           
    d(X) = [1  0 ]X >= [1  0 ]X = n__d(X)
    
                        [0 0]     [0 0]                  
    activate(n__f(X)) = [2 2]X >= [2 2]X = f(activate(X))
    
                        [0  -&]     [0  -&]        
    activate(n__g(X)) = [2  -&]X >= [0  -&]X = g(X)
    
                        [1 0]     [0  -&]        
    activate(n__d(X)) = [3 2]X >= [1  0 ]X = d(X)
    
                  [0 0]          
    activate(X) = [2 2]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(X) -> X
   Arctic Interpretation Processor:
    dimension: 2
    interpretation:
               [0  -&]  
     [g](x0) = [0  -&]x0,
     
                  [0  -&]  
     [n__d](x0) = [0  0 ]x0,
     
               [1  -&]  
     [d](x0) = [1  1 ]x0,
     
                      [0  -&]  
     [activate](x0) = [0  0 ]x0,
     
               [1  -&]  
     [c](x0) = [2  1 ]x0,
     
                  [0  -&]  
     [n__g](x0) = [0  -&]x0,
     
                  [0 0]  
     [n__f](x0) = [1 1]x0,
     
               [0 0]  
     [f](x0) = [1 1]x0
    orientation:
               [1 1]     [1 1]                          
     f(f(X)) = [2 2]X >= [2 2]X = c(n__f(n__g(n__f(X))))
     
            [1  -&]     [1  -&]                  
     c(X) = [2  1 ]X >= [1  1 ]X = d(activate(X))
     
            [0 0]     [0 0]           
     f(X) = [1 1]X >= [1 1]X = n__f(X)
     
            [0  -&]     [0  -&]           
     g(X) = [0  -&]X >= [0  -&]X = n__g(X)
     
            [1  -&]     [0  -&]           
     d(X) = [1  1 ]X >= [0  0 ]X = n__d(X)
     
                         [0 0]     [0 0]                  
     activate(n__f(X)) = [1 1]X >= [1 1]X = f(activate(X))
     
                         [0  -&]     [0  -&]        
     activate(n__g(X)) = [0  -&]X >= [0  -&]X = g(X)
     
                   [0  -&]          
     activate(X) = [0  0 ]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)
     activate(n__f(X)) -> f(activate(X))
     activate(n__g(X)) -> g(X)
     activate(X) -> X
    Arctic Interpretation Processor:
     dimension: 2
     interpretation:
                [1  -&]  
      [g](x0) = [1  -&]x0,
      
                [0  -&]  
      [d](x0) = [1  0 ]x0,
      
                       [1  -&]  
      [activate](x0) = [0  1 ]x0,
      
                [1  -&]  
      [c](x0) = [2  1 ]x0,
      
                   [0  -&]  
      [n__g](x0) = [1  -&]x0,
      
                   [0 0]  
      [n__f](x0) = [2 3]x0,
      
                [0 0]  
      [f](x0) = [2 3]x0
     orientation:
                [2 3]     [2 2]                          
      f(f(X)) = [5 6]X >= [5 5]X = c(n__f(n__g(n__f(X))))
      
             [1  -&]     [1  -&]                  
      c(X) = [2  1 ]X >= [2  1 ]X = d(activate(X))
      
             [0 0]     [0 0]           
      f(X) = [2 3]X >= [2 3]X = n__f(X)
      
             [1  -&]     [0  -&]           
      g(X) = [1  -&]X >= [1  -&]X = n__g(X)
      
                          [1 1]     [1 1]                  
      activate(n__f(X)) = [3 4]X >= [3 4]X = f(activate(X))
      
                          [1  -&]     [1  -&]        
      activate(n__g(X)) = [2  -&]X >= [1  -&]X = g(X)
      
                    [1  -&]          
      activate(X) = [0  1 ]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)
      activate(n__f(X)) -> f(activate(X))
      activate(n__g(X)) -> g(X)
     Arctic Interpretation Processor:
      dimension: 2
      interpretation:
                 [0  -&]  
       [g](x0) = [0  -&]x0,
       
                 [0  -&]  
       [d](x0) = [0  0 ]x0,
       
                        [0  -&]  
       [activate](x0) = [0  0 ]x0,
       
                 [1  -&]  
       [c](x0) = [2  1 ]x0,
       
                    [0  -&]  
       [n__g](x0) = [0  -&]x0,
       
                    [0 0]  
       [n__f](x0) = [1 1]x0,
       
                 [0 0]  
       [f](x0) = [1 1]x0
      orientation:
                 [1 1]     [1 1]                          
       f(f(X)) = [2 2]X >= [2 2]X = c(n__f(n__g(n__f(X))))
       
              [1  -&]     [0  -&]                  
       c(X) = [2  1 ]X >= [0  0 ]X = d(activate(X))
       
              [0 0]     [0 0]           
       f(X) = [1 1]X >= [1 1]X = n__f(X)
       
              [0  -&]     [0  -&]           
       g(X) = [0  -&]X >= [0  -&]X = n__g(X)
       
                           [0 0]     [0 0]                  
       activate(n__f(X)) = [1 1]X >= [1 1]X = f(activate(X))
       
                           [0  -&]     [0  -&]        
       activate(n__g(X)) = [0  -&]X >= [0  -&]X = g(X)
      problem:
       f(f(X)) -> c(n__f(n__g(n__f(X))))
       f(X) -> n__f(X)
       g(X) -> n__g(X)
       activate(n__f(X)) -> f(activate(X))
       activate(n__g(X)) -> g(X)
      Arctic Interpretation Processor:
       dimension: 2
       interpretation:
                  [0  -&]  
        [g](x0) = [0  -&]x0,
        
                         [0 0]  
        [activate](x0) = [0 2]x0,
        
                  [1  0 ]  
        [c](x0) = [-& -&]x0,
        
                     [0  -&]  
        [n__g](x0) = [-& -&]x0,
        
                     [0 0]  
        [n__f](x0) = [0 2]x0,
        
                  [0 0]  
        [f](x0) = [2 2]x0
       orientation:
                  [2 2]     [1  1 ]                          
        f(f(X)) = [4 4]X >= [-& -&]X = c(n__f(n__g(n__f(X))))
        
               [0 0]     [0 0]           
        f(X) = [2 2]X >= [0 2]X = n__f(X)
        
               [0  -&]     [0  -&]           
        g(X) = [0  -&]X >= [-& -&]X = n__g(X)
        
                            [0 2]     [0 2]                  
        activate(n__f(X)) = [2 4]X >= [2 4]X = f(activate(X))
        
                            [0  -&]     [0  -&]        
        activate(n__g(X)) = [0  -&]X >= [0  -&]X = g(X)
       problem:
        f(X) -> n__f(X)
        g(X) -> n__g(X)
        activate(n__f(X)) -> f(activate(X))
        activate(n__g(X)) -> g(X)
       Arctic Interpretation Processor:
        dimension: 3
        interpretation:
                   [0 0 0]  
         [g](x0) = [1 1 0]x0
                   [0 1 0]  ,
         
                          [0  1  0 ]  
         [activate](x0) = [-& 1  1 ]x0
                          [1  1  1 ]  ,
         
                      [0  0  -&]  
         [n__g](x0) = [1  0  0 ]x0
                      [0  1  0 ]  ,
         
                      [0  0  -&]  
         [n__f](x0) = [1  1  1 ]x0
                      [-& 0  0 ]  ,
         
                   [0  0  1 ]  
         [f](x0) = [1  1  1 ]x0
                   [-& 1  1 ]  
        orientation:
                [0  0  1 ]     [0  0  -&]           
         f(X) = [1  1  1 ]X >= [1  1  1 ]X = n__f(X)
                [-& 1  1 ]     [-& 0  0 ]           
         
                [0 0 0]     [0  0  -&]           
         g(X) = [1 1 0]X >= [1  0  0 ]X = n__g(X)
                [0 1 0]     [0  1  0 ]           
         
                             [2 2 2]     [2 2 2]                  
         activate(n__f(X)) = [2 2 2]X >= [2 2 2]X = f(activate(X))
                             [2 2 2]     [2 2 2]                  
         
                             [2 1 1]     [0 0 0]        
         activate(n__g(X)) = [2 2 1]X >= [1 1 0]X = g(X)
                             [2 2 1]     [0 1 0]        
        problem:
         f(X) -> n__f(X)
         g(X) -> n__g(X)
         activate(n__f(X)) -> f(activate(X))
        Arctic Interpretation Processor:
         dimension: 3
         interpretation:
                    [2 0 0]  
          [g](x0) = [2 0 0]x0
                    [0 0 0]  ,
          
                           [3 0 2]  
          [activate](x0) = [0 0 0]x0
                           [0 0 0]  ,
          
                       [0  -& -&]  
          [n__g](x0) = [0  -& -&]x0
                       [-& -& -&]  ,
          
                       [0  -& 0 ]  
          [n__f](x0) = [-& -& 0 ]x0
                       [-& 0  0 ]  ,
          
                    [0  0  0 ]  
          [f](x0) = [-& 0  0 ]x0
                    [-& 0  0 ]  
         orientation:
                 [0  0  0 ]     [0  -& 0 ]           
          f(X) = [-& 0  0 ]X >= [-& -& 0 ]X = n__f(X)
                 [-& 0  0 ]     [-& 0  0 ]           
          
                 [2 0 0]     [0  -& -&]           
          g(X) = [2 0 0]X >= [0  -& -&]X = n__g(X)
                 [0 0 0]     [-& -& -&]           
          
                              [3 2 3]     [3 0 2]                  
          activate(n__f(X)) = [0 0 0]X >= [0 0 0]X = f(activate(X))
                              [0 0 0]     [0 0 0]                  
         problem:
          f(X) -> n__f(X)
          activate(n__f(X)) -> f(activate(X))
         KBO Processor:
          weight function:
           w0 = 1
           w(activate) = w(n__f) = w(f) = 1
          precedence:
           activate > f > n__f
          problem:
           
          Qed