Remark two_power_nat_infinity:
  forall x, 0 <= x -> exists n, x < two_power_nat n.

Lemma Zshiftin_pos_ind: forall (P: Z -> Prop), P 1 -> (forall b x, 0 < x -> P x -> P (Zshiftin b x)) -> forall x, 0 < x -> P x.