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:
 DP Processor:
  DPs:
   g#(x,s(y)) -> g#(f(x,y),0())
   g#(s(x),y) -> g#(f(x,y),0())
   g#(f(x,y),0()) -> g#(y,0())
   g#(f(x,y),0()) -> g#(x,0())
  TRS:
   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()))
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [g#](x0, x1) = x0 + 4x1,
    
    [s](x0) = 2x0 + 2,
    
    [g](x0, x1) = x0 + 4x1,
    
    [f](x0, x1) = x0 + 4x1 + 1,
    
    [0] = 0
   orientation:
    g#(x,s(y)) = x + 8y + 8 >= x + 4y + 1 = g#(f(x,y),0())
    
    g#(s(x),y) = 2x + 4y + 2 >= x + 4y + 1 = g#(f(x,y),0())
    
    g#(f(x,y),0()) = x + 4y + 1 >= y = g#(y,0())
    
    g#(f(x,y),0()) = x + 4y + 1 >= x = g#(x,0())
    
    g(0(),f(x,x)) = 20x + 4 >= x = x
    
    g(x,s(y)) = x + 8y + 8 >= x + 4y + 1 = g(f(x,y),0())
    
    g(s(x),y) = 2x + 4y + 2 >= x + 4y + 1 = g(f(x,y),0())
    
    g(f(x,y),0()) = x + 4y + 1 >= x + 4y + 1 = f(g(x,0()),g(y,0()))
   problem:
    DPs:
     
    TRS:
     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()))
   Qed