rewrite ZL6.
destruct (ZL4 p) as [x Heq]; rewrite Heq; simpl.
replace (x + S x) with (S (2 * x)) by omega.
symmetry; apply mod2_S_double.
rewrite IHp.
rewrite ZL6.
destruct (nat_of_P p); simpl; intuition.
replace (n + S n) with (S (2 * n)) by omega.
rewrite div2_S_double; auto.
unfold nat_of_P; simpl.
rewrite ZL6.
replace (nat_of_P p + nat_of_P p) with (2 * nat_of_P p) by omega.
symmetry; apply mod2_double.
rewrite IHp.
unfold nat_of_P; simpl.
rewrite ZL6.
replace (nat_of_P p + nat_of_P p) with (2 * nat_of_P p) by omega.
rewrite div2_double.
auto.
auto.
Qed.

Lemma posToWord_sz0: forall p, posToWord 0 p = $0.