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