intros ni ni1 dni dni1 x k le.
apply lt_ks_count_undecs.
apply le_ni_trans with (rev_app ((x, k) :: dni) ni); try assumption.
simpl in |- *.
rewrite (rev_app_app dni (Decorated x k :: ni)).
rewrite (rev_app_app dni (Undecorated x :: ni)).
apply le_ni_app_dn.
trivial.
apply le_ni_refl.
apply le_lt_trans with (count_undecs ni).
apply le_ks_le_count_undecs with dni1 ((x, k) :: dni); assumption.
simpl in |- *; apply lt_n_Sn.
Qed.

(* Specifications: *)
Lemma lt_ks_shift_dd :
forall (ni : nested_imps) (dni : decorated_nested_imps)
(x : nimp) (k : kripke_tree),
Lt_Ks ni ((x, k) :: dni) (Decorated x k :: ni) dni.