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)

max/plus interpretations on N:

  f_A(x1) = max{7, 2 + x1}
  f#_A(x1) = max{7, 2 + x1}
  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) = max{4, x1, x2}
  g#_A(x1,x2) = max{4, x1, x2}
  h_A(x1,x2) = max{1, 1 + x1, x2}
  h#_A(x1,x2) = max{1, 1 + x1, x2}
  e_A = 1
  e#_A = 1

precedence:

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