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diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpx.ma
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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                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/predsn_5.ma".
+include "basic_2/reduction/lpr.ma".
+include "basic_2/reduction/cpx.ma".
+
+(* SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ********************)
+
+definition lpx: ∀h. sd h → relation3 genv lenv lenv ≝
+                λh,o,G. lpx_sn (cpx h o G).
+
+interpretation "extended parallel reduction (local environment, sn variant)"
+   'PRedSn h o G L1 L2 = (lpx h o G L1 L2).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma lpx_inv_atom1: ∀h,o,G,L2. ⦃G, ⋆⦄ ⊢ ➡[h, o] L2 → L2 = ⋆.
+/2 width=4 by lpx_sn_inv_atom1_aux/ qed-.
+
+lemma lpx_inv_pair1: ∀h,o,I,G,K1,V1,L2. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡[h, o] L2 →
+                     ∃∃K2,V2. ⦃G, K1⦄ ⊢ ➡[h, o] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h, o] V2 &
+                              L2 = K2. ⓑ{I} V2.
+/2 width=3 by lpx_sn_inv_pair1_aux/ qed-.
+
+lemma lpx_inv_atom2: ∀h,o,G,L1.  ⦃G, L1⦄ ⊢ ➡[h, o] ⋆ → L1 = ⋆.
+/2 width=4 by lpx_sn_inv_atom2_aux/ qed-.
+
+lemma lpx_inv_pair2: ∀h,o,I,G,L1,K2,V2.  ⦃G, L1⦄ ⊢ ➡[h, o] K2.ⓑ{I}V2 →
+                     ∃∃K1,V1. ⦃G, K1⦄ ⊢ ➡[h, o] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h, o] V2 &
+                             L1 = K1. ⓑ{I} V1.
+/2 width=3 by lpx_sn_inv_pair2_aux/ qed-.
+
+lemma lpx_inv_pair: ∀h,o,I1,I2,G,L1,L2,V1,V2.  ⦃G, L1.ⓑ{I1}V1⦄ ⊢ ➡[h, o] L2.ⓑ{I2}V2 →
+                    ∧∧ ⦃G, L1⦄ ⊢ ➡[h, o] L2 & ⦃G, L1⦄ ⊢ V1 ➡[h, o] V2 & I1 = I2.
+/2 width=1 by lpx_sn_inv_pair/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lpx_refl: ∀h,o,G,L.  ⦃G, L⦄ ⊢ ➡[h, o] L.
+/2 width=1 by lpx_sn_refl/ qed.
+
+lemma lpx_pair: ∀h,o,I,G,K1,K2,V1,V2. ⦃G, K1⦄ ⊢ ➡[h, o] K2 → ⦃G, K1⦄ ⊢ V1 ➡[h, o] V2 →
+                ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡[h, o] K2.ⓑ{I}V2.
+/2 width=1 by lpx_sn_pair/ qed.
+
+lemma lpr_lpx: ∀h,o,G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L1⦄ ⊢ ➡[h, o] L2.
+#h #o #G #L1 #L2 #H elim H -L1 -L2 /3 width=1 by lpx_pair, cpr_cpx/
+qed.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lpx_fwd_length: ∀h,o,G,L1,L2. ⦃G, L1⦄ ⊢ ➡[h, o] L2 → |L1| = |L2|.
+/2 width=2 by lpx_sn_fwd_length/ qed-.