(**************************************************************************)
(*       ___                                                              *)
(*      ||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                  *)
(*                                                                        *)
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include "basic_2/rt_conversion/cpce.ma".
include "basic_2/rt_equivalence/cpcs.ma".
include "basic_2/dynamic/lsubv.ma".

(* LOCAL ENVIRONMENT REFINEMENT FOR NATIVE VALIDITY *************************)

lemma lsubv_inv_unit_dx (h) (a) (G):
      ∀I,L1,K2. G ⊢ L1 ⫃![h,a] K2.ⓤ{I} →
      ∃∃K1. G ⊢ K1 ⫃![h,a] K2 & L1 = K1.ⓤ{I}.
#h #a #G #I #L1 #K2 #H
elim (lsubv_inv_bind_dx … H) -H // *
#K1 #XW #XV #_ #_ #H1 #H2 destruct
qed-.

lemma lsubv_inv_abbr_dx (h) (a) (G):
      ∀L1,K2,V. G ⊢ L1 ⫃![h,a] K2.ⓓV →
      ∃∃K1. G ⊢ K1 ⫃![h,a] K2 & L1 = K1.ⓓV.
#h #a #G #L1 #K2 #V #H
elim (lsubv_inv_bind_dx … H) -H // *
#K1 #XW #XV #_ #_ #H1 #H2 destruct
qed-.

lemma lsubv_cpce_trans_cpcs (h) (a) (G) (T0):
      ∀L2,T2. ⦃G,L2⦄ ⊢ T0 ⬌η[h] T2 → ∀L1. G ⊢ L1 ⫃![h,a] L2 →
      ∃∃T1. ⦃G,L1⦄ ⊢ T0 ⬌η[h] T1 & ⦃G,L1⦄ ⊢ T1 ⬌*[h] T2.
#h #a #G #T0 #L2 #T2 #H elim H -G -L2 -T0 -T2
[ #G #L2 #s #L1 #HL12
  /2 width=3 by cpce_sort, ex2_intro/
| #G #i #Y1 #HY1
  lapply (lsubv_inv_atom2 … HY1) -HY1 #H destruct
   /2 width=3 by cpce_atom, ex2_intro/
| #I #G #K2 #Y1 #HY1
  elim (lsubv_inv_unit_dx … HY1) -HY1 #K2 #_ #H destruct
  /2 width=3 by cpce_unit, ex2_intro/
| #G #K2 #V2 #Y1 #HY1
  elim (lsubv_inv_abbr_dx … HY1) -HY1 #K2 #_ #H destruct
  /2 width=3 by cpce_ldef, ex2_intro/
| #G #K2 #W2 #HW2 #Y1 #HY1
  elim (lsubv_inv_bind_dx … HY1) -HY1 *
  [ #K1 #HK12 #H destruct
    @(ex2_intro … (#0)) [| // ]
    @cpce_ldec #n #p #V2 #U2 #HWU2