apply eqv_sym.
apply le_eqv; assumption.
Qed.
(*****************************************************************)

Lemma eqv_ni_rec :
forall P : nested_imps -> nested_imps -> Set,
P NNil NNil ->
(forall (x : nimp) (ni1 ni2 : nested_imps),
eqv_ni ni1 ni2 ->
P ni1 ni2 -> P (Undecorated x :: ni1) (Undecorated x :: ni2)) ->
(forall (x : nimp) (k : kripke_tree) (ni1 ni2 : nested_imps),
eqv_ni ni1 ni2 ->
P ni1 ni2 -> P (Decorated x k :: ni1) (Undecorated x :: ni2)) ->
(forall (x : nimp) (k k' : kripke_tree) (ni1 ni2 : nested_imps),
eqv_ni ni1 ni2 ->
P ni1 ni2 -> P (Decorated x k :: ni1) (Decorated x k' :: ni2)) ->
(forall (x : nimp) (k : kripke_tree) (ni1 ni2 : nested_imps),
eqv_ni ni1 ni2 ->
P ni1 ni2 -> P (Undecorated x :: ni1) (Decorated x k :: ni2)) ->
forall ni1 ni2 : nested_imps, eqv_ni ni1 ni2 -> P ni1 ni2.