YES

TRS:

  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

linear polynomial interpretations on N:

  f_A(x1,x2) = x1 + x2
  f#_A(x1,x2) = x1 + x2 + 2
  g1_A(x1,x2,x3) = x2
  g1#_A(x1,x2,x3) = x1 + x3 + 1
  g2_A(x1,x2,x3) = x1 + x3
  g2#_A(x1,x2,x3) = x1 + x3 + 1
  h_A(x1,x2) = x1
  h#_A(x1,x2) = 0

precedence:

  f > g1 > h > g2