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: 1
   interpretation:
    [n__d](x0) = x0,
    
    [h](x0) = 8x0,
    
    [d](x0) = x0,
    
    [activate](x0) = 1x0,
    
    [c](x0) = 1x0,
    
    [g](x0) = x0,
    
    [n__f](x0) = x0,
    
    [f](x0) = 1x0
   orientation:
    f(f(X)) = 2X >= 1X = c(n__f(g(n__f(X))))
    
    c(X) = 1X >= 1X = d(activate(X))
    
    h(X) = 8X >= 1X = c(n__d(X))
    
    f(X) = 1X >= X = n__f(X)
    
    d(X) = X >= X = n__d(X)
    
    activate(n__f(X)) = 1X >= 1X = f(X)
    
    activate(X) = 1X >= X = X
   problem:
    c(X) -> d(activate(X))
    d(X) -> n__d(X)
    activate(n__f(X)) -> f(X)
   Arctic Interpretation Processor:
    dimension: 3
    interpretation:
                  [0  -& -&]  
     [n__d](x0) = [0  -& -&]x0
                  [-& 2  0 ]  ,
     
               [0 0 2]  
     [d](x0) = [2 1 1]x0
               [0 2 1]  ,
     
                      [0  -& 0 ]  
     [activate](x0) = [0  0  1 ]x0
                      [1  1  -&]  ,
     
               [3 3 1]  
     [c](x0) = [2 2 2]x0
               [2 2 3]  ,
     
                  [1 0 0]  
     [n__f](x0) = [0 0 1]x0
                  [0 2 1]  ,
     
               [0  0  -&]  
     [f](x0) = [-& 0  0 ]x0
               [0  -& -&]  
    orientation:
            [3 3 1]     [3 3 1]                  
     c(X) = [2 2 2]X >= [2 2 2]X = d(activate(X))
            [2 2 3]     [2 2 3]                  
     
            [0 0 2]     [0  -& -&]           
     d(X) = [2 1 1]X >= [0  -& -&]X = n__d(X)
            [0 2 1]     [-& 2  0 ]           
     
                         [1 2 1]     [0  0  -&]        
     activate(n__f(X)) = [1 3 2]X >= [-& 0  0 ]X = f(X)
                         [2 1 2]     [0  -& -&]        
    problem:
     c(X) -> d(activate(X))
     d(X) -> n__d(X)
    Arctic Interpretation Processor:
     dimension: 3
     interpretation:
                   [0  -& -&]  
      [n__d](x0) = [-& -& -&]x0
                   [-& -& -&]  ,
      
                [1  -& -&]  
      [d](x0) = [2  -& 0 ]x0
                [-& 0  2 ]  ,
      
                       [1  0  -&]  
      [activate](x0) = [0  2  -&]x0
                       [0  1  0 ]  ,
      
                [2 1 0]  
      [c](x0) = [3 2 2]x0
                [2 3 2]  
     orientation:
             [2 1 0]     [2  1  -&]                  
      c(X) = [3 2 2]X >= [3  2  0 ]X = d(activate(X))
             [2 3 2]     [2  3  2 ]                  
      
             [1  -& -&]     [0  -& -&]           
      d(X) = [2  -& 0 ]X >= [-& -& -&]X = n__d(X)
             [-& 0  2 ]     [-& -& -&]           
     problem:
      c(X) -> d(activate(X))
     Arctic Interpretation Processor:
      dimension: 3
      interpretation:
                 [0  0  1 ]  
       [d](x0) = [-& -& 0 ]x0
                 [0  2  1 ]  ,
       
                        [0  2  0 ]  
       [activate](x0) = [0  0  -&]x0
                        [0  1  0 ]  ,
       
                 [2 3 2]  
       [c](x0) = [2 2 2]x0
                 [3 3 2]  
      orientation:
              [2 3 2]     [1 2 1]                  
       c(X) = [2 2 2]X >= [0 1 0]X = d(activate(X))
              [3 3 2]     [2 2 1]                  
      problem:
       
      Qed