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.

Theorem zero_ext_and: forall n x, 0 <= n -> zero_ext n x = and x (repr (two_p n - 1)).