intuition.
unfold N.of_nat, posnatToNat.
destruct p.
destruct x.
omega.
destruct x; intuition; simpl in *.
case_eq (N.compare 0 1)%N; intuition.
inversion H.
inversion H.
case_eq (N.compare 0 (N.pos (Pos.succ (Pos.of_succ_nat x))))%N; intuition.
inversion H.
inversion H.
Qed.

Lemma modNat_lt : forall x p, (modNat x p < p)%nat.
intuition.