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:
   [s](x0) = 2x0 + 6,
   
   [g](x0, x1) = 2x0 + 4x1,
   
   [f](x0, x1) = x0 + 2x1,
   
   [0] = 0
  orientation:
   g(0(),f(x,x)) = 12x >= x = x
   
   g(x,s(y)) = 2x + 8y + 24 >= 2x + 4y = g(f(x,y),0())
   
   g(s(x),y) = 4x + 4y + 12 >= 2x + 4y = g(f(x,y),0())
   
   g(f(x,y),0()) = 2x + 4y >= 2x + 4y = f(g(x,0()),g(y,0()))
  problem:
   g(0(),f(x,x)) -> x
   g(f(x,y),0()) -> f(g(x,0()),g(y,0()))
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [g](x0, x1) = 3x0 + 4x1 + 1,
    
    [f](x0, x1) = 3x0 + 2x1 + 2,
    
    [0] = 0
   orientation:
    g(0(),f(x,x)) = 20x + 9 >= x = x
    
    g(f(x,y),0()) = 9x + 6y + 7 >= 9x + 6y + 7 = f(g(x,0()),g(y,0()))
   problem:
    g(f(x,y),0()) -> f(g(x,0()),g(y,0()))
   Matrix Interpretation Processor: dim=1
    
    interpretation:
     [g](x0, x1) = 2x0 + 2x1 + 1,
     
     [f](x0, x1) = 2x0 + 2x1 + 4,
     
     [0] = 0
    orientation:
     g(f(x,y),0()) = 4x + 4y + 9 >= 4x + 4y + 8 = f(g(x,0()),g(y,0()))
    problem:
     
    Qed