YES

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

Proof:
 DP Processor:
  DPs:
   f#(g(x,y),f(y,y)) -> f#(g(y,x),y)
  TRS:
   f(g(x,y),f(y,y)) -> f(g(y,x),y)
  Arctic Interpretation Processor:
   dimension: 1
   interpretation:
    [f#](x0, x1) = x0 + 1x1 + 2,
    
    [f](x0, x1) = 2x0 + 3x1 + 3,
    
    [g](x0, x1) = 2x0 + 2
   orientation:
    f#(g(x,y),f(y,y)) = 2x + 4y + 4 >= 2y + 2 = f#(g(y,x),y)
    
    f(g(x,y),f(y,y)) = 4x + 6y + 6 >= 4y + 4 = f(g(y,x),y)
   problem:
    DPs:
     
    TRS:
     f(g(x,y),f(y,y)) -> f(g(y,x),y)
   Qed