YES

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

Proof:
 DP Processor:
  DPs:
   f#(s(x),y) -> g#(x,y)
   f#(s(x),y) -> f#(x,g(x,y))
  TRS:
   f(s(x),y) -> f(x,g(x,y))
   f(0(),y) -> y
   g(x,y) -> y
  Usable Rule Processor:
   DPs:
    f#(s(x),y) -> g#(x,y)
    f#(s(x),y) -> f#(x,g(x,y))
   TRS:
    f6(x,y) -> x
    f6(x,y) -> y
    g(x,y) -> y
   TDG Processor:
    DPs:
     f#(s(x),y) -> g#(x,y)
     f#(s(x),y) -> f#(x,g(x,y))
    TRS:
     f6(x,y) -> x
     f6(x,y) -> y
     g(x,y) -> y
    graph:
     f#(s(x),y) -> f#(x,g(x,y)) -> f#(s(x),y) -> f#(x,g(x,y))
     f#(s(x),y) -> f#(x,g(x,y)) -> f#(s(x),y) -> g#(x,y)
    Restore Modifier:
     DPs:
      f#(s(x),y) -> g#(x,y)
      f#(s(x),y) -> f#(x,g(x,y))
     TRS:
      f(s(x),y) -> f(x,g(x,y))
      f(0(),y) -> y
      g(x,y) -> y
     SCC Processor:
      #sccs: 1
      #rules: 1
      #arcs: 2/4
      DPs:
       f#(s(x),y) -> f#(x,g(x,y))
      TRS:
       f(s(x),y) -> f(x,g(x,y))
       f(0(),y) -> y
       g(x,y) -> y
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [f#](x0, x1) = x0 + 1,
        
        [0] = 0,
        
        [g](x0, x1) = x1,
        
        [f](x0, x1) = x0 + x1 + 1,
        
        [s](x0) = x0 + 1
       orientation:
        f#(s(x),y) = x + 2 >= x + 1 = f#(x,g(x,y))
        
        f(s(x),y) = x + y + 2 >= x + y + 1 = f(x,g(x,y))
        
        f(0(),y) = y + 1 >= y = y
        
        g(x,y) = y >= y = y
       problem:
        DPs:
         
        TRS:
         f(s(x),y) -> f(x,g(x,y))
         f(0(),y) -> y
         g(x,y) -> y
       Qed