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(*       ___                                                              *)
(*      ||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/reduction/cpx_lleq.ma".
include "basic_2/reduction/llpx.ma".

(* LAZY SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ***************)

(* Properties on lazy equivalence for local environments ********************)

lemma llpx_lrefl: ∀h,g,G,L1,L2,T,d. L1 ⋕[T, d] L2 → ⦃G, L1⦄ ⊢ ➡[h, g, T, d] L2.
/2 width=1 by llpx_sn_lrefl/ qed-.

lemma lleq_llpx_trans: ∀h,g,G,L1,L2,T,d. L1 ⋕[T, d] L2 →
                       ∀L.⦃G, L2⦄ ⊢ ➡[h, g, T, d] L → ⦃G, L1⦄ ⊢ ➡[h, g, T, d] L.
/3 width=3 by lleq_cpx_trans, lleq_llpx_sn_trans/ qed-.

lemma lleq_llpx_conf: ∀h,g,G,L1,L2,T,d. L1 ⋕[T, d] L2 →
                      ∀L.⦃G, L1⦄ ⊢ ➡[h, g, T, d] L → ⦃G, L2⦄ ⊢ ➡[h, g, T, d] L.
/3 width=3 by lleq_cpx_trans, lleq_llpx_sn_conf/ qed-.