(**************************************************************************)
(*       ___                                                              *)
(*      ||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/relocation/llor.ma".
include "basic_2/substitution/lleq_alt.ma".

(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)

(* Properties on poinwise union for local environments **********************)

lemma llor_lleq_O: ∀T,L1,L2,L. L1 ⩖[T] L2 ≡ L → |L1| ≤ |L2| → |L2| = |L| → L2 ⋕[T, yinj 0] L.
#T #L1 @(f2_ind … rfw … L1 T) -L1 -T
#n #IH #L1 #T #Hn #L2 #L #H #HL12 #HL2 elim (llor_inv_alt … H) // destruct
#HL1 #IHT @lleq_intro_alt //
#I2 #I #K2 #K #V2 #V #i #_ #HnT #HLK2 #HLK lapply (ldrop_fwd_length_lt2 … HLK)
#Hi elim (ldrop_O1_lt L1 i) //
#I1 #K1 #V1 #HLK1 elim (IHT … HLK1 HLK) >HL1 >HL2 <minus_n_n -IHT -HLK1 -HLK
[
| * #I0 #K0 #V0 #_ #HLK0 lapply (ldrop_mono … HLK0 … HLK2) -HLK0 -HLK2
  #H1 #H2 #H3 #_ destruct