YES

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

Proof:
 Matrix Interpretation Processor: dim=1
  
  interpretation:
   [f](x0, x1) = x0 + 2x1 + 4,
   
   [s](x0) = 4x0 + 4,
   
   [0] = 0,
   
   [g](x0, x1) = x0 + 4x1
  orientation:
   g(0(),f(x,x)) = 12x + 16 >= x = x
   
   g(x,s(y)) = x + 16y + 16 >= x + 2y + 4 = g(f(x,y),0())
   
   g(s(x),y) = 4x + 4y + 4 >= x + 2y + 4 = g(f(x,y),0())
   
   g(f(x,y),0()) = x + 2y + 4 >= x + 2y + 4 = f(g(x,0()),g(y,0()))
  problem:
   g(s(x),y) -> g(f(x,y),0())
   g(f(x,y),0()) -> f(g(x,0()),g(y,0()))
  Matrix Interpretation Processor: dim=3
   
   interpretation:
                  [1 0 0]     [1 0 0]     [0]
    [f](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0]
                  [0 0 0]     [0 0 0]     [1],
    
              [1 0 0]     [1]
    [s](x0) = [0 0 0]x0 + [0]
              [0 0 0]     [0],
    
          [0]
    [0] = [0]
          [0],
    
                  [1 0 0]     [1 0 0]     [0]
    [g](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0]
                  [0 0 0]     [0 0 0]     [1]
   orientation:
                [1 0 0]    [1 0 0]    [1]    [1 0 0]    [1 0 0]    [0]                
    g(s(x),y) = [0 0 0]x + [0 0 0]y + [0] >= [0 0 0]x + [0 0 0]y + [0] = g(f(x,y),0())
                [0 0 0]    [0 0 0]    [1]    [0 0 0]    [0 0 0]    [1]                
    
                    [1 0 0]    [1 0 0]    [0]    [1 0 0]    [1 0 0]    [0]                       
    g(f(x,y),0()) = [0 0 0]x + [0 0 0]y + [0] >= [0 0 0]x + [0 0 0]y + [0] = f(g(x,0()),g(y,0()))
                    [0 0 0]    [0 0 0]    [1]    [0 0 0]    [0 0 0]    [1]                       
   problem:
    g(f(x,y),0()) -> f(g(x,0()),g(y,0()))
   Matrix Interpretation Processor: dim=3
    
    interpretation:
                   [1 1 0]          [0]
     [f](x0, x1) = [0 0 0]x0 + x1 + [0]
                   [0 0 1]          [1],
     
           [0]
     [0] = [0]
           [0],
     
                   [1 0 1]     [1 0 0]  
     [g](x0, x1) = [0 0 0]x0 + [0 0 0]x1
                   [1 1 1]     [0 0 0]  
    orientation:
                     [1 1 1]    [1 0 1]    [1]    [1 0 1]    [1 0 1]    [0]                       
     g(f(x,y),0()) = [0 0 0]x + [0 0 0]y + [0] >= [0 0 0]x + [0 0 0]y + [0] = f(g(x,0()),g(y,0()))
                     [1 1 1]    [1 1 1]    [1]    [1 1 1]    [1 1 1]    [1]                       
    problem:
     
    Qed