(**************************************************************************)
(*       ___                                                              *)
(*      ||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/dynamic/snta_snta.ma".
include "basic_2/dynamic/lsubsn_ldrop.ma".
include "basic_2/dynamic/lsubsn_cpcs.ma".

(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE TYPE ASSIGNMENT *******)

(* Properties concerning stratified native type assignment ******************)

(* Note: the corresponding confluence property does not hold *)
lemma lsubsn_snta_trans: ∀h,L2,T,U,l. ⦃h, L2⦄ ⊢ T :[l] U →
                         ∀L1. h ⊢ L1 :⊑[] L2 → ⦃h, L1⦄ ⊢ T :[l] U.
#h #L2 #T #U #l #H elim H -L2 -T -U -l
[ //
| #L2 #K2 #V2 #W2 #U2 #i #l #HLK2 #_ #WU2 #IHVW2 #L1 #HL12
  elim (lsubsn_ldrop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1
  elim (lsubsn_inv_pair2 … H) -H * #K1
  [ #HK12 #H destruct /3 width=6/
  | #V1 #l0 #_ #_ #_ #_ #H destruct
  ]
| #L2 #K2 #W2 #V2 #U2 #i #l #HLK2 #HWV2 #HWU2 #IHWV2 #L1 #HL12
  elim (lsubsn_ldrop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1
  elim (lsubsn_inv_pair2 … H) -H * #K1 [ -HWV2 | -IHWV2 ]
  [ #HK12 #H destruct /3 width=6/
  | #V1 #l0 #H1 #H2 #_ #H #_ destruct
    elim (snta_fwd_correct … H2) -H2 #V #H
    elim (snta_mono … HWV2 … H) -HWV2 -H /2 width=6/
  ]
| /4 width=3/
| /3 width=2/
| /3 width=2/
| /3 width=1/
| #L2 #T #U1 #U2 #V2 #l #_ #HU12 #_ #IHTU1 #IHUV2 #L1 #HL12
  lapply (cpcs_lsubsn_trans … HL12 … HU12) -HU12 /3 width=3/
]
qed.