Canonical bin_nat_eqMixin := EqMixin eq_binP.Canonical bin_nat_eqType := Eval hnf in EqType N bin_nat_eqMixin.Fixpoint nat_of_pos p0 :=
  match p0 with
  | xO p => (nat_of_pos p).*2
  | xI p => (nat_of_pos p).*2.+1
  | xH   => 1
  end.Coercion nat_of_bin b := if b is Npos p then p : nat else 0.Fixpoint pos_of_nat n0 m0 :=
  match n0, m0 with
  | n.+1, m.+2 => pos_of_nat n m
  | n.+1,    1 => xO (pos_of_nat n n)
  | n.+1,    0 => xI (pos_of_nat n n)
  |    0,    _ => xH
  end.Definition bin_of_nat n0 := if n0 is n.+1 then Npos (pos_of_nat n n) else 0%num.

Lemma nat_of_exp_bin n (b : N) : n ^ b = pow_N 1 muln n b.