YES

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

Proof:
 DP Processor:
  DPs:
   f#(s(x),s(y)) -> f#(x,y)
   g#(0(),x) -> f#(x,x)
   g#(0(),x) -> g#(f(x,x),x)
  TRS:
   f(x,0()) -> s(0())
   f(s(x),s(y)) -> s(f(x,y))
   g(0(),x) -> g(f(x,x),x)
  Usable Rule Processor:
   DPs:
    f#(s(x),s(y)) -> f#(x,y)
    g#(0(),x) -> f#(x,x)
    g#(0(),x) -> g#(f(x,x),x)
   TRS:
    f6(x,y) -> x
    f6(x,y) -> y
    f(x,0()) -> s(0())
    f(s(x),s(y)) -> s(f(x,y))
   ADG Processor:
    DPs:
     f#(s(x),s(y)) -> f#(x,y)
     g#(0(),x) -> f#(x,x)
     g#(0(),x) -> g#(f(x,x),x)
    TRS:
     f6(x,y) -> x
     f6(x,y) -> y
     f(x,0()) -> s(0())
     f(s(x),s(y)) -> s(f(x,y))
    graph:
     g#(0(),x) -> f#(x,x) -> f#(s(x),s(y)) -> f#(x,y)
     f#(s(x),s(y)) -> f#(x,y) -> f#(s(x),s(y)) -> f#(x,y)
    Restore Modifier:
     DPs:
      f#(s(x),s(y)) -> f#(x,y)
      g#(0(),x) -> f#(x,x)
      g#(0(),x) -> g#(f(x,x),x)
     TRS:
      f(x,0()) -> s(0())
      f(s(x),s(y)) -> s(f(x,y))
      g(0(),x) -> g(f(x,x),x)
     SCC Processor:
      #sccs: 1
      #rules: 1
      #arcs: 2/9
      DPs:
       f#(s(x),s(y)) -> f#(x,y)
      TRS:
       f(x,0()) -> s(0())
       f(s(x),s(y)) -> s(f(x,y))
       g(0(),x) -> g(f(x,x),x)
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [f#](x0, x1) = x1 + 1,
        
        [g](x0, x1) = x1 + 1,
        
        [s](x0) = x0 + 1,
        
        [f](x0, x1) = x0 + x1 + 1,
        
        [0] = 1
       orientation:
        f#(s(x),s(y)) = y + 2 >= y + 1 = f#(x,y)
        
        f(x,0()) = x + 2 >= 2 = s(0())
        
        f(s(x),s(y)) = x + y + 3 >= x + y + 2 = s(f(x,y))
        
        g(0(),x) = x + 1 >= x + 1 = g(f(x,x),x)
       problem:
        DPs:
         
        TRS:
         f(x,0()) -> s(0())
         f(s(x),s(y)) -> s(f(x,y))
         g(0(),x) -> g(f(x,x),x)
       Qed