YES

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

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