apply Eqv_NI_Nil.
intros x ni1 ni2 le12 ih; apply Eqv_NI_Cons_NN; assumption.
intros x k ni1 ni2 le12 ih; apply Eqv_NI_Cons_DN; assumption.
intros x k ni1 ni2 le12 ih; apply Eqv_NI_Cons_DD; assumption.
Qed.

Lemma eqv_sym :
forall ni1 ni2 : nested_imps, eqv_ni ni1 ni2 -> eqv_ni ni2 ni1.