intros ni1 ni2 ni3 le12.
generalize ni3; clear ni3.
elim le12; clear le12 ni1 ni2.
intros; assumption.
intros x ni1 ni2 le12 ih ni3 le23.
inversion_clear le23.
apply Le_NI_Cons_NN.
apply ih; assumption.
intros x k ni1 ni2 le12 ih ni3 le23.
inversion_clear le23.
apply Le_NI_Cons_DN.
apply ih; assumption.
intros x k ni1 ni2 le12 ih ni3 le23.
inversion_clear le23.
apply Le_NI_Cons_DN.
apply ih; assumption.
apply Le_NI_Cons_DD.
apply ih; assumption.
Qed.
(*****************************************************************)
Inductive filter_deco_spec (i : Int) (N : nested_imps) : Set :=
Filter_Deco_Spec_Intro :
forall ni1 : nested_imps,
le_ni N ni1 ->
(forall (x : nimp) (k : kripke_tree),
In (Decorated x k) ni1 -> forces_t k (Atom i)) ->
filter_deco_spec i N.

Lemma filter_deco :
forall (i : Int) (ni : nested_imps), filter_deco_spec i ni.