YES

Problem:
 f(X) -> g(n__h(n__f(X)))
 h(X) -> n__h(X)
 f(X) -> n__f(X)
 activate(n__h(X)) -> h(activate(X))
 activate(n__f(X)) -> f(activate(X))
 activate(X) -> X

Proof:
 Arctic Interpretation Processor:
  dimension: 1
  interpretation:
   [activate](x0) = 1x0,
   
   [h](x0) = x0,
   
   [g](x0) = x0,
   
   [n__h](x0) = x0,
   
   [n__f](x0) = x0,
   
   [f](x0) = x0
  orientation:
   f(X) = X >= X = g(n__h(n__f(X)))
   
   h(X) = X >= X = n__h(X)
   
   f(X) = X >= X = n__f(X)
   
   activate(n__h(X)) = 1X >= 1X = h(activate(X))
   
   activate(n__f(X)) = 1X >= 1X = f(activate(X))
   
   activate(X) = 1X >= X = X
  problem:
   f(X) -> g(n__h(n__f(X)))
   h(X) -> n__h(X)
   f(X) -> n__f(X)
   activate(n__h(X)) -> h(activate(X))
   activate(n__f(X)) -> f(activate(X))
  String Reversal Processor:
   f(X) -> n__f(n__h(g(X)))
   h(X) -> n__h(X)
   f(X) -> n__f(X)
   n__h(activate(X)) -> activate(h(X))
   n__f(activate(X)) -> activate(f(X))
   Matrix Interpretation Processor: dim=3
    
    interpretation:
                      [1 1 0]     [0]
     [activate](x0) = [0 0 0]x0 + [0]
                      [0 0 1]     [1],
     
               [1 1 1]     [1]
     [h](x0) = [0 0 0]x0 + [0]
               [0 0 1]     [0],
     
               [1 0 0]  
     [g](x0) = [0 0 0]x0
               [0 0 0]  ,
     
                  [1 0 1]  
     [n__h](x0) = [0 0 0]x0
                  [0 0 1]  ,
     
                  [1 0 0]  
     [n__f](x0) = [0 0 0]x0
                  [0 0 1]  ,
     
                 
     [f](x0) = x0
                 
    orientation:
                 [1 0 0]                    
     f(X) = X >= [0 0 0]X = n__f(n__h(g(X)))
                 [0 0 0]                    
     
            [1 1 1]    [1]    [1 0 1]           
     h(X) = [0 0 0]X + [0] >= [0 0 0]X = n__h(X)
            [0 0 1]    [0]    [0 0 1]           
     
                 [1 0 0]           
     f(X) = X >= [0 0 0]X = n__f(X)
                 [0 0 1]           
     
                         [1 1 1]    [1]    [1 1 1]    [1]                 
     n__h(activate(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = activate(h(X))
                         [0 0 1]    [1]    [0 0 1]    [1]                 
     
                         [1 1 0]    [0]    [1 1 0]    [0]                 
     n__f(activate(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = activate(f(X))
                         [0 0 1]    [1]    [0 0 1]    [1]                 
    problem:
     f(X) -> n__f(n__h(g(X)))
     f(X) -> n__f(X)
     n__h(activate(X)) -> activate(h(X))
     n__f(activate(X)) -> activate(f(X))
    Arctic Interpretation Processor:
     dimension: 2
     interpretation:
                       [0  -&]  
      [activate](x0) = [0  0 ]x0,
      
                [0 0]  
      [h](x0) = [0 0]x0,
      
                [0  0 ]  
      [g](x0) = [-& -&]x0,
      
                   [0 1]  
      [n__h](x0) = [0 1]x0,
      
                   [0  0 ]  
      [n__f](x0) = [-& 2 ]x0,
      
                [0 0]  
      [f](x0) = [2 2]x0
     orientation:
             [0 0]     [0 0]                    
      f(X) = [2 2]X >= [2 2]X = n__f(n__h(g(X)))
      
             [0 0]     [0  0 ]           
      f(X) = [2 2]X >= [-& 2 ]X = n__f(X)
      
                          [1 1]     [0 0]                  
      n__h(activate(X)) = [1 1]X >= [0 0]X = activate(h(X))
      
                          [0 0]     [0 0]                  
      n__f(activate(X)) = [2 2]X >= [2 2]X = activate(f(X))
     problem:
      f(X) -> n__f(n__h(g(X)))
      f(X) -> n__f(X)
      n__f(activate(X)) -> activate(f(X))
     Arctic Interpretation Processor:
      dimension: 2
      interpretation:
                        [0 0]  
       [activate](x0) = [1 2]x0,
       
                 [0  -&]  
       [g](x0) = [2  2 ]x0,
       
                    [0  -&]  
       [n__h](x0) = [-& -&]x0,
       
                    [0  0 ]  
       [n__f](x0) = [-& 2 ]x0,
       
                 [1 0]  
       [f](x0) = [0 2]x0
      orientation:
              [1 0]     [0  -&]                    
       f(X) = [0 2]X >= [-& -&]X = n__f(n__h(g(X)))
       
              [1 0]     [0  0 ]           
       f(X) = [0 2]X >= [-& 2 ]X = n__f(X)
       
                           [1 2]     [1 2]                  
       n__f(activate(X)) = [3 4]X >= [2 4]X = activate(f(X))
      problem:
       f(X) -> n__f(X)
       n__f(activate(X)) -> activate(f(X))
      KBO Processor:
       weight function:
        w0 = 1
        w(activate) = w(n__f) = w(f) = 1
       precedence:
        f > n__f > activate
       problem:
        
       Qed