induction n; simpl in *; intuition.
econstructor.
eapply app_NoDup; intuition.
econstructor.
simpl; intuition.
econstructor.
simpl in *.
intuition; subst.
eapply allNatsLt_lt in H.
omega.
simpl in *.
intuition; subst.
eapply allNatsLt_lt in H0.
omega.
Qed.

Lemma allNatsLt_lt_if :
forall (n i : nat),
i < n ->
In i (allNatsLt n).