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)
  TDG 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)
   graph:
    g#(0(),x) -> g#(f(x,x),x) -> g#(0(),x) -> g#(f(x,x),x)
    g#(0(),x) -> g#(f(x,x),x) -> g#(0(),x) -> f#(x,x)
    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)
   EDG 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)
    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)
    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)
     KBO Processor:
      argument filtering:
       pi(0) = []
       pi(f) = [0,1]
       pi(s) = [0]
       pi(g) = 1
       pi(f#) = 1
      weight function:
       w0 = 1
       w(f#) = w(g) = w(s) = w(f) = w(0) = 1
      precedence:
       f# ~ g ~ s ~ f ~ 0
      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