Corollary Ztestbit_two_p_m1:
  forall n i, 0 <= n -> 0 <= i ->
  Z.testbit (two_p n - 1) i = if zlt i n then true else false.

Lemma bits_zero_ext: forall n x i, 0 <= i -> testbit (zero_ext n x) i = if zlt i n then testbit x i else false.