YES

Problem:
 f(f(X)) -> c(n__f(g(n__f(X))))
 c(X) -> d(activate(X))
 h(X) -> c(n__d(X))
 f(X) -> n__f(X)
 d(X) -> n__d(X)
 activate(n__f(X)) -> f(X)
 activate(n__d(X)) -> d(X)
 activate(X) -> X

Proof:
 Arctic Interpretation Processor:
  dimension: 2
  interpretation:
                [0  1 ]  
   [n__d](x0) = [-& 0 ]x0,
   
             [3 3]  
   [h](x0) = [2 1]x0,
   
             [0  2 ]  
   [d](x0) = [-& 0 ]x0,
   
                    [2 2]  
   [activate](x0) = [0 0]x0,
   
             [2 3]  
   [c](x0) = [0 0]x0,
   
             [0 0]  
   [g](x0) = [0 1]x0,
   
                [1  0 ]  
   [n__f](x0) = [-& 0 ]x0,
   
             [3 1]  
   [f](x0) = [0 0]x0
  orientation:
             [6 4]     [4 4]                       
   f(f(X)) = [3 1]X >= [2 1]X = c(n__f(g(n__f(X))))
   
          [2 3]     [2 2]                  
   c(X) = [0 0]X >= [0 0]X = d(activate(X))
   
          [3 3]     [2 3]              
   h(X) = [2 1]X >= [0 1]X = c(n__d(X))
   
          [3 1]     [1  0 ]           
   f(X) = [0 0]X >= [-& 0 ]X = n__f(X)
   
          [0  2 ]     [0  1 ]           
   d(X) = [-& 0 ]X >= [-& 0 ]X = n__d(X)
   
                       [3 2]     [3 1]        
   activate(n__f(X)) = [1 0]X >= [0 0]X = f(X)
   
                       [2 3]     [0  2 ]        
   activate(n__d(X)) = [0 1]X >= [-& 0 ]X = d(X)
   
                 [2 2]          
   activate(X) = [0 0]X >= X = X
  problem:
   f(f(X)) -> c(n__f(g(n__f(X))))
   c(X) -> d(activate(X))
   h(X) -> c(n__d(X))
   f(X) -> n__f(X)
   d(X) -> n__d(X)
   activate(n__f(X)) -> f(X)
   activate(X) -> X
  Arctic Interpretation Processor:
   dimension: 2
   interpretation:
                 [0  0 ]  
    [n__d](x0) = [-& 0 ]x0,
    
              [2 2]  
    [h](x0) = [2 2]x0,
    
              [0 0]  
    [d](x0) = [0 0]x0,
    
                     [0  -&]  
    [activate](x0) = [0  0 ]x0,
    
              [0 0]  
    [c](x0) = [0 0]x0,
    
              [0 0]  
    [g](x0) = [0 2]x0,
    
                 [0  0 ]  
    [n__f](x0) = [-& -&]x0,
    
              [0  0 ]  
    [f](x0) = [0  -&]x0
   orientation:
              [0 0]     [0 0]                       
    f(f(X)) = [0 0]X >= [0 0]X = c(n__f(g(n__f(X))))
    
           [0 0]     [0 0]                  
    c(X) = [0 0]X >= [0 0]X = d(activate(X))
    
           [2 2]     [0 0]              
    h(X) = [2 2]X >= [0 0]X = c(n__d(X))
    
           [0  0 ]     [0  0 ]           
    f(X) = [0  -&]X >= [-& -&]X = n__f(X)
    
           [0 0]     [0  0 ]           
    d(X) = [0 0]X >= [-& 0 ]X = n__d(X)
    
                        [0 0]     [0  0 ]        
    activate(n__f(X)) = [0 0]X >= [0  -&]X = f(X)
    
                  [0  -&]          
    activate(X) = [0  0 ]X >= X = X
   problem:
    f(f(X)) -> c(n__f(g(n__f(X))))
    c(X) -> d(activate(X))
    f(X) -> n__f(X)
    d(X) -> n__d(X)
    activate(n__f(X)) -> f(X)
    activate(X) -> X
   Arctic Interpretation Processor:
    dimension: 2
    interpretation:
                  [0  -&]  
     [n__d](x0) = [-& -&]x0,
     
               [0  0 ]  
     [d](x0) = [-& -&]x0,
     
                      [2 1]  
     [activate](x0) = [0 2]x0,
     
               [2 2]  
     [c](x0) = [1 0]x0,
     
               [0  -&]  
     [g](x0) = [-& -&]x0,
     
                  [0  -&]  
     [n__f](x0) = [0  3 ]x0,
     
               [1  -&]  
     [f](x0) = [0  3 ]x0
    orientation:
               [2  -&]     [2  -&]                       
     f(f(X)) = [3  6 ]X >= [1  -&]X = c(n__f(g(n__f(X))))
     
            [2 2]     [2  2 ]                  
     c(X) = [1 0]X >= [-& -&]X = d(activate(X))
     
            [1  -&]     [0  -&]           
     f(X) = [0  3 ]X >= [0  3 ]X = n__f(X)
     
            [0  0 ]     [0  -&]           
     d(X) = [-& -&]X >= [-& -&]X = n__d(X)
     
                         [2 4]     [1  -&]        
     activate(n__f(X)) = [2 5]X >= [0  3 ]X = f(X)
     
                   [2 1]          
     activate(X) = [0 2]X >= X = X
    problem:
     f(f(X)) -> c(n__f(g(n__f(X))))
     c(X) -> d(activate(X))
     f(X) -> n__f(X)
     d(X) -> n__d(X)
    Arctic Interpretation Processor:
     dimension: 2
     interpretation:
                   [0  0 ]  
      [n__d](x0) = [-& -&]x0,
      
                [0  3 ]  
      [d](x0) = [-& 2 ]x0,
      
                       [1  -&]  
      [activate](x0) = [0  0 ]x0,
      
                [3 3]  
      [c](x0) = [2 2]x0,
      
                [0 1]  
      [g](x0) = [2 1]x0,
      
                   [0 0]  
      [n__f](x0) = [0 1]x0,
      
                [3 3]  
      [f](x0) = [3 2]x0
     orientation:
                [6 6]     [6 6]                       
      f(f(X)) = [6 6]X >= [5 5]X = c(n__f(g(n__f(X))))
      
             [3 3]     [3 3]                  
      c(X) = [2 2]X >= [2 2]X = d(activate(X))
      
             [3 3]     [0 0]           
      f(X) = [3 2]X >= [0 1]X = n__f(X)
      
             [0  3 ]     [0  0 ]           
      d(X) = [-& 2 ]X >= [-& -&]X = n__d(X)
     problem:
      f(f(X)) -> c(n__f(g(n__f(X))))
      c(X) -> d(activate(X))
      d(X) -> n__d(X)
     Arctic Interpretation Processor:
      dimension: 2
      interpretation:
                    [0  -&]  
       [n__d](x0) = [-& -&]x0,
       
                 [0  0 ]  
       [d](x0) = [-& -&]x0,
       
                        [0 0]  
       [activate](x0) = [0 2]x0,
       
                 [2 3]  
       [c](x0) = [0 1]x0,
       
                 [0  1 ]  
       [g](x0) = [-& 1 ]x0,
       
                    [0 2]  
       [n__f](x0) = [0 1]x0,
       
                 [0 3]  
       [f](x0) = [2 3]x0
      orientation:
                 [5 6]     [5 6]                       
       f(f(X)) = [5 6]X >= [3 4]X = c(n__f(g(n__f(X))))
       
              [2 3]     [0  2 ]                  
       c(X) = [0 1]X >= [-& -&]X = d(activate(X))
       
              [0  0 ]     [0  -&]           
       d(X) = [-& -&]X >= [-& -&]X = n__d(X)
      problem:
       f(f(X)) -> c(n__f(g(n__f(X))))
       d(X) -> n__d(X)
      Arctic Interpretation Processor:
       dimension: 2
       interpretation:
                     [0  -&]  
        [n__d](x0) = [-& -&]x0,
        
                  [3 0]  
        [d](x0) = [0 0]x0,
        
                  [0  2 ]  
        [c](x0) = [0  -&]x0,
        
                  [0 2]  
        [g](x0) = [0 1]x0,
        
                     [0  1 ]  
        [n__f](x0) = [-& 0 ]x0,
        
                  [0 2]  
        [f](x0) = [0 1]x0
       orientation:
                  [2 3]     [2 3]                       
        f(f(X)) = [1 2]X >= [1 2]X = c(n__f(g(n__f(X))))
        
               [3 0]     [0  -&]           
        d(X) = [0 0]X >= [-& -&]X = n__d(X)
       problem:
        f(f(X)) -> c(n__f(g(n__f(X))))
       Arctic Interpretation Processor:
        dimension: 2
        interpretation:
                   [0 1]  
         [c](x0) = [0 0]x0,
         
                   [0  -&]  
         [g](x0) = [0  0 ]x0,
         
                      [0  1 ]  
         [n__f](x0) = [3  -&]x0,
         
                   [0 3]  
         [f](x0) = [3 3]x0
        orientation:
                   [6 6]     [4 5]                       
         f(f(X)) = [6 6]X >= [4 4]X = c(n__f(g(n__f(X))))
        problem:
         
        Qed