YES

Problem:
 f(x,y) -> g1(x,x,y)
 f(x,y) -> g1(y,x,x)
 f(x,y) -> g2(x,y,y)
 f(x,y) -> g2(y,y,x)
 g1(x,x,y) -> h(x,y)
 g1(y,x,x) -> h(x,y)
 g2(x,y,y) -> h(x,y)
 g2(y,y,x) -> h(x,y)
 h(x,x) -> x

Proof:
 Bounds Processor:
  bound: 2
  enrichment: top
  automaton:
   final states: {5}
   transitions:
    f0(5,5) -> 5*
    g10(5,5,5) -> 5*
    g20(5,5,5) -> 5*
    h0(5,5) -> 5*
    h1(5,5) -> 5*
    g21(5,5,5) -> 5*
    g11(5,5,5) -> 5*
    h2(5,5) -> 5*
  problem:
   
  Qed