YES

TRS:

  f(a()) -> b()
  f(c()) -> d()
  f(g(x,y)) -> g(f(x),f(y))
  f(h(x,y)) -> g(h(y,f(x)),h(x,f(y)))
  g(x,x) -> h(e(),x)

linear polynomial interpretations on N:

  f_A(x1) = x1 + 1
  f#_A(x1) = 2
  a_A = 1
  a#_A = 1
  b_A = 0
  b#_A = 0
  c_A = 1
  c#_A = 1
  d_A = 0
  d#_A = 0
  g_A(x1,x2) = 1
  g#_A(x1,x2) = 1
  h_A(x1,x2) = 1
  h#_A(x1,x2) = 0
  e_A = 1
  e#_A = 0

precedence:

  g > f = c = e > b = d = h > a