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()))
  EDG 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()))
   graph:
    g#(s(x),y) -> g#(f(x,y),0()) -> g#(f(x,y),0()) -> g#(y,0())
    g#(s(x),y) -> g#(f(x,y),0()) -> g#(f(x,y),0()) -> g#(x,0())
    g#(f(x,y),0()) -> g#(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#(y,0())
    g#(f(x,y),0()) -> g#(y,0()) -> g#(f(x,y),0()) -> g#(x,0())
    g#(f(x,y),0()) -> g#(x,0()) -> g#(s(x),y) -> g#(f(x,y),0())
    g#(f(x,y),0()) -> g#(x,0()) -> g#(f(x,y),0()) -> g#(y,0())
    g#(f(x,y),0()) -> g#(x,0()) -> g#(f(x,y),0()) -> g#(x,0())
    g#(x,s(y)) -> g#(f(x,y),0()) -> g#(f(x,y),0()) -> g#(y,0())
    g#(x,s(y)) -> g#(f(x,y),0()) -> g#(f(x,y),0()) -> g#(x,0())
   CDG 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()))
    graph:
     g#(s(x),y) -> g#(f(x,y),0()) -> g#(f(x,y),0()) -> g#(x,0())
     g#(s(x),y) -> g#(f(x,y),0()) -> g#(f(x,y),0()) -> g#(y,0())
     g#(f(x,y),0()) -> g#(y,0()) -> g#(f(x,y),0()) -> g#(x,0())
     g#(f(x,y),0()) -> g#(y,0()) -> g#(f(x,y),0()) -> g#(y,0())
     g#(f(x,y),0()) -> g#(x,0()) -> g#(f(x,y),0()) -> g#(x,0())
     g#(f(x,y),0()) -> g#(x,0()) -> g#(f(x,y),0()) -> g#(y,0())
     g#(x,s(y)) -> g#(f(x,y),0()) -> g#(f(x,y),0()) -> g#(x,0())
     g#(x,s(y)) -> g#(f(x,y),0()) -> g#(f(x,y),0()) -> g#(y,0())
    SCC Processor:
     #sccs: 1
     #rules: 2
     #arcs: 8/16
     DPs:
      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,
       
       [s](x0) = x0 + 1,
       
       [g](x0, x1) = 2x0 + 4x1,
       
       [f](x0, x1) = x0 + 2x1 + 1,
       
       [0] = 0
      orientation:
       g#(f(x,y),0()) = x + 2y + 1 >= y = g#(y,0())
       
       g#(f(x,y),0()) = x + 2y + 1 >= x = g#(x,0())
       
       g(0(),f(x,x)) = 12x + 4 >= x = x
       
       g(x,s(y)) = 2x + 4y + 4 >= 2x + 4y + 2 = g(f(x,y),0())
       
       g(s(x),y) = 2x + 4y + 2 >= 2x + 4y + 2 = g(f(x,y),0())
       
       g(f(x,y),0()) = 2x + 4y + 2 >= 2x + 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