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())
   SCC Processor:
    #sccs: 1
    #rules: 3
    #arcs: 10/16
    DPs:
     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())
    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:
     dimension: 1
     interpretation:
      [g#](x0, x1) = x0 + x1 + 1,
      
      [s](x0) = x0 + 1,
      
      [g](x0, x1) = x0 + x1,
      
      [f](x0, x1) = x0 + x1 + 1,
      
      [0] = 0
     orientation:
      g#(s(x),y) = x + y + 2 >= x + y + 2 = g#(f(x,y),0())
      
      g#(f(x,y),0()) = x + y + 2 >= x + 1 = g#(x,0())
      
      g#(f(x,y),0()) = x + y + 2 >= y + 1 = g#(y,0())
      
      g(0(),f(x,x)) = 2x + 1 >= x = x
      
      g(x,s(y)) = x + y + 1 >= x + y + 1 = g(f(x,y),0())
      
      g(s(x),y) = x + y + 1 >= x + y + 1 = g(f(x,y),0())
      
      g(f(x,y),0()) = x + y + 1 >= x + y + 1 = f(g(x,0()),g(y,0()))
     problem:
      DPs:
       g#(s(x),y) -> g#(f(x,y),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:
      dimension: 1
      interpretation:
       [g#](x0, x1) = x0,
       
       [s](x0) = x0 + 1,
       
       [g](x0, x1) = x1,
       
       [f](x0, x1) = x0,
       
       [0] = 0
      orientation:
       g#(s(x),y) = x + 1 >= x = g#(f(x,y),0())
       
       g(0(),f(x,x)) = x >= x = x
       
       g(x,s(y)) = y + 1 >= 0 = g(f(x,y),0())
       
       g(s(x),y) = y >= 0 = g(f(x,y),0())
       
       g(f(x,y),0()) = 0 >= 0 = 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