YES

TRS:

  half(0()) -> 0()
  half(s(s(x))) -> s(half(x))
  log(s(0())) -> 0()
  log(s(s(x))) -> s(log(s(half(x))))

max/plus interpretations on N:

  half_A(x1) = max{0, x1}
  half#_A(x1) = max{0, x1}
  0_A = 0
  0#_A = 0
  s_A(x1) = max{0, x1}
  s#_A(x1) = max{0, x1}
  log_A(x1) = max{0, x1}
  log#_A(x1) = max{0, x1}

precedence:

  log > s > half = 0