apply Le_NI_Nil.
intros a; case a; clear a.
intros x N ih.
apply Le_NI_Cons_NN; assumption.
intros x k ni ih.
apply Le_NI_Cons_DD; assumption.
Qed.

Lemma le_ni_trans :
forall ni1 ni2 ni3 : nested_imps,
le_ni ni1 ni2 -> le_ni ni2 ni3 -> le_ni ni1 ni3.