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 sign_ext_widen: forall x n n', 0 < n <= n' -> sign_ext n' (sign_ext n x) = sign_ext n x.