YES

TRS:

  f(f(x)) -> f(c(f(x)))
  f(f(x)) -> f(d(f(x)))
  g(c(x)) -> x
  g(d(x)) -> x
  g(c(0())) -> g(d(1()))
  g(c(1())) -> g(d(0()))

max/plus interpretations on N:

  f_A(x1) = max{16, 11 + x1}
  f#_A(x1) = max{0, 5 + x1}
  c_A(x1) = max{8, -17 + x1}
  c#_A(x1) = max{20, 6}
  d_A(x1) = max{6, -20 + x1}
  d#_A(x1) = max{20, 7}
  g_A(x1) = max{26, 20 + x1}
  g#_A(x1) = max{8, 13 + x1}
  0_A = 16
  0#_A = 11
  1_A = 14
  1#_A = 20

precedence:

  f > c > 1 > d > g > 0