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 Znatlike_ind: forall (P: Z -> Prop), (forall n, n <= 0 -> P n) -> (forall n, 0 <= n -> P n -> P (Z.succ n)) -> forall n, P n.