intros P base step_nn step_rn step_rr step_nr ni1.
elim ni1; clear ni1.
intros ni2; case ni2; clear ni2.
intros; apply base.
intros a ni2 eqv; elimtype False; inversion_clear eqv.
intros a; case a; clear a.
intros x ni1 ih ni2.
case ni2; clear ni2.
intros eqv; elimtype False; inversion_clear eqv.
intros a; case a; clear a.
intros x' ni2 eqv.
cut (x = x').
intro eq_x;  rewrite eq_x; clear eq_x.
apply step_nn.
inversion_clear eqv; assumption.
apply ih.
inversion_clear eqv; assumption.
inversion_clear eqv; trivial.
intros x' k' ni2 eqv.
cut (x = x').
intro eq_x;  rewrite eq_x; clear eq_x.
apply step_nr.
inversion_clear eqv; assumption.
apply ih.
inversion_clear eqv; assumption.
inversion_clear eqv; trivial.
intros x k ni1 ih ni2.
case ni2; clear ni2.
intros eqv; elimtype False; inversion_clear eqv.
intros a; case a; clear a.
intros x' ni2 eqv.
cut (x = x').
intro eq_x;  rewrite eq_x; clear eq_x.
apply step_rn.
inversion_clear eqv; assumption.
apply ih.
inversion_clear eqv; assumption.
inversion_clear eqv; trivial.
intros x' k' ni2 eqv.
cut (x = x').
intro eq_x;  rewrite eq_x; clear eq_x.
apply step_rr.
inversion_clear eqv; assumption.
apply ih.
inversion_clear eqv; assumption.
inversion_clear eqv; trivial.
Qed.
(*****************************************************************)
Inductive inf_deco_spec (ni1 ni2 : nested_imps) : Set :=
Inf_Deco_Spec_Intro :
forall ni : nested_imps,
le_ni ni ni1 ->
eqv_ni ni ni2 ->
(forall (x : nimp) (k : kripke_tree),
In (Decorated x k) ni ->
In (Decorated x k) ni1 \/ In (Decorated x k) ni2) ->
inf_deco_spec ni1 ni2.

Lemma inf_deco :
forall ni1 ni2 : nested_imps, eqv_ni ni1 ni2 -> inf_deco_spec ni1 ni2.