(**************************************************************************)
(*       ___                                                              *)
(*      ||M||                                                             *)
(*      ||A||       A project by Andrea Asperti                           *)
(*      ||T||                                                             *)
(*      ||I||       Developers:                                           *)
(*      ||T||         The HELM team.                                      *)
(*      ||A||         http://helm.cs.unibo.it                             *)
(*      \   /                                                             *)
(*       \ /        This file is distributed under the terms of the       *)
(*        v         GNU General Public License Version 2                  *)
(*                                                                        *)
(**************************************************************************)

include "basic_2/grammar/fsups.ma".
include "basic_2/substitution/ldrop_fsup.ma".

(* LOCAL ENVIRONMENT SLICING ************************************************)

(* Inversion lemmas on plus-iterated supclosure  ****************************)

lemma fsupp_inv_atom1_fsups: ∀J,L1,L2,T2. ⦃L1, ⓪{J}⦄ ⊃+ ⦃L2, T2⦄ →
                             ∃∃I,K,V,i. ⇩[0, i] L1 ≡ K.ⓑ{I}V &
                             ⦃K, V⦄ ⊃* ⦃L2, T2⦄ & J = LRef i.
#J #L1 #L2 #T2 #H @(fsupp_ind … H) -L2 -T2
[ #L2 #T2 #H
  elim (fsup_inv_atom1_ldrop … H) -H * #i #HL12 #H destruct /2 width=7/
| #L #T #L2 #T2 #_ #HT2 * #I #K #V #i #HLK #HVT #H destruct /3 width=8/
]
qed-.