YES

Problem:
 f(f(x)) -> f(g(f(x),x))
 f(f(x)) -> f(h(f(x),f(x)))
 g(x,y) -> y
 h(x,x) -> g(x,0())

Proof:
 Matrix Interpretation Processor: dim=3
  
  interpretation:
         [0]
   [0] = [1]
         [0],
   
                 [1 0 0]     [1 0 0]     [0]
   [h](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1]
                 [0 0 0]     [0 1 0]     [0],
   
                 [1 0 0]       
   [g](x0, x1) = [0 0 0]x0 + x1
                 [0 0 0]       ,
   
             [1 0 1]     [1]
   [f](x0) = [0 0 0]x0 + [0]
             [1 0 1]     [1]
  orientation:
             [2 0 2]    [3]    [2 0 2]    [2]               
   f(f(x)) = [0 0 0]x + [0] >= [0 0 0]x + [0] = f(g(f(x),x))
             [2 0 2]    [3]    [2 0 2]    [2]               
   
             [2 0 2]    [3]    [2 0 2]    [3]                  
   f(f(x)) = [0 0 0]x + [0] >= [0 0 0]x + [0] = f(h(f(x),f(x)))
             [2 0 2]    [3]    [2 0 2]    [3]                  
   
            [1 0 0]              
   g(x,y) = [0 0 0]x + y >= y = y
            [0 0 0]              
   
            [2 0 0]    [0]    [1 0 0]    [0]           
   h(x,x) = [0 0 0]x + [1] >= [0 0 0]x + [1] = g(x,0())
            [0 1 0]    [0]    [0 0 0]    [0]           
  problem:
   f(f(x)) -> f(h(f(x),f(x)))
   g(x,y) -> y
   h(x,x) -> g(x,0())
  Matrix Interpretation Processor: dim=2
   
   interpretation:
          [0]
    [0] = [1],
    
                  [1 0]     [2 0]     [0]
    [h](x0, x1) = [0 0]x0 + [0 0]x1 + [1],
    
                  [1 0]       
    [g](x0, x1) = [0 0]x0 + x1,
    
              [1 1]     [0]
    [f](x0) = [2 2]x0 + [2]
   orientation:
              [3 3]    [2]    [3 3]    [1]                  
    f(f(x)) = [6 6]x + [6] >= [6 6]x + [4] = f(h(f(x),f(x)))
    
             [1 0]              
    g(x,y) = [0 0]x + y >= y = y
    
             [3 0]    [0]    [1 0]    [0]           
    h(x,x) = [0 0]x + [1] >= [0 0]x + [1] = g(x,0())
   problem:
    g(x,y) -> y
    h(x,x) -> g(x,0())
   Matrix Interpretation Processor: dim=1
    
    interpretation:
     [0] = 0,
     
     [h](x0, x1) = 6x0 + x1 + 4,
     
     [g](x0, x1) = 4x0 + 2x1 + 4
    orientation:
     g(x,y) = 4x + 2y + 4 >= y = y
     
     h(x,x) = 7x + 4 >= 4x + 4 = g(x,0())
    problem:
     h(x,x) -> g(x,0())
    Matrix Interpretation Processor: dim=3
     
     interpretation:
            [0]
      [0] = [0]
            [0],
      
                    [1 0 0]     [1 0 0]     [1]
      [h](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0]
                    [0 0 0]     [0 0 0]     [0],
      
                    [1 0 0]     [1 0 0]  
      [g](x0, x1) = [0 0 0]x0 + [0 0 0]x1
                    [0 0 0]     [0 0 0]  
     orientation:
               [2 0 0]    [1]    [1 0 0]            
      h(x,x) = [0 0 0]x + [0] >= [0 0 0]x = g(x,0())
               [0 0 0]    [0]    [0 0 0]            
     problem:
      
     Qed