intuition.
destruct (Even.even_odd_dec n).
rewrite (even_double n) at 1.
unfold double.
omega.
trivial.
rewrite (odd_double n) at 1.
unfold double.
omega.
trivial.
Qed.
Local Open Scope N_scope.
Definition modNat (n : nat)(p : posnat) : nat :=
N.to_nat ((N.of_nat n) mod (N.of_nat p)).

Lemma Npos_nz : forall p,
Npos p <> N0.