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diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lsx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lsx.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/sn_6.ma".
-include "basic_2/multiple/lleq.ma".
-include "basic_2/reduction/lpx.ma".
-
-(* SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS **********************)
-
-definition lsx: ∀h. sd h → relation4 ynat term genv lenv ≝
-                λh,o,l,T,G. SN … (lpx h o G) (lleq l T).
-
-interpretation
-   "extended strong normalization (local environment)"
-   'SN h o l T G L = (lsx h o T l G L).
-
-(* Basic eliminators ********************************************************)
-
-lemma lsx_ind: ∀h,o,G,T,l. ∀R:predicate lenv.
-               (∀L1. G ⊢ ⬊*[h, o, T, l] L1 →
-                     (∀L2. ⦃G, L1⦄ ⊢ ➡[h, o] L2 → (L1 ≡[T, l] L2 → ⊥) → R L2) →
-                     R L1
-               ) →
-               ∀L. G ⊢ ⬊*[h, o, T, l] L → R L.
-#h #o #G #T #l #R #H0 #L1 #H elim H -L1
-/5 width=1 by lleq_sym, SN_intro/
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma lsx_intro: ∀h,o,G,L1,T,l.
-                 (∀L2. ⦃G, L1⦄ ⊢ ➡[h, o] L2 → (L1 ≡[T, l] L2 → ⊥) → G ⊢ ⬊*[h, o, T, l] L2) →
-                 G ⊢ ⬊*[h, o, T, l] L1.
-/5 width=1 by lleq_sym, SN_intro/ qed.
-
-lemma lsx_atom: ∀h,o,G,T,l. G ⊢ ⬊*[h, o, T, l] ⋆.
-#h #o #G #T #l @lsx_intro
-#X #H #HT lapply (lpx_inv_atom1 … H) -H
-#H destruct elim HT -HT //
-qed.
-
-lemma lsx_sort: ∀h,o,G,L,l,s. G ⊢ ⬊*[h, o, ⋆s, l] L.
-#h #o #G #L1 #l #s @lsx_intro
-#L2 #HL12 #H elim H -H
-/3 width=4 by lpx_fwd_length, lleq_sort/
-qed.
-
-lemma lsx_gref: ∀h,o,G,L,l,p. G ⊢ ⬊*[h, o, §p, l] L.
-#h #o #G #L1 #l #p @lsx_intro
-#L2 #HL12 #H elim H -H
-/3 width=4 by lpx_fwd_length, lleq_gref/
-qed.
-
-lemma lsx_ge_up: ∀h,o,G,L,T,U,lt,l,k. lt ≤ yinj l + yinj k →
-                 ⬆[l, k] T ≡ U → G ⊢ ⬊*[h, o, U, lt] L → G ⊢ ⬊*[h, o, U, l] L.
-#h #o #G #L #T #U #lt #l #k #Hltlm #HTU #H @(lsx_ind … H) -L
-/5 width=7 by lsx_intro, lleq_ge_up/
-qed-.
-
-lemma lsx_ge: ∀h,o,G,L,T,l1,l2. l1 ≤ l2 →
-              G ⊢ ⬊*[h, o, T, l1] L → G ⊢ ⬊*[h, o, T, l2] L.
-#h #o #G #L #T #l1 #l2 #Hl12 #H @(lsx_ind … H) -L
-/5 width=7 by lsx_intro, lleq_ge/
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lsx_fwd_bind_sn: ∀h,o,a,I,G,L,V,T,l. G ⊢ ⬊*[h, o, ⓑ{a,I}V.T, l] L →
-                       G ⊢ ⬊*[h, o, V, l] L.
-#h #o #a #I #G #L #V #T #l #H @(lsx_ind … H) -L
-#L1 #_ #IHL1 @lsx_intro
-#L2 #HL12 #HV @IHL1 /3 width=4 by lleq_fwd_bind_sn/
-qed-.
-
-lemma lsx_fwd_flat_sn: ∀h,o,I,G,L,V,T,l. G ⊢ ⬊*[h, o, ⓕ{I}V.T, l] L →
-                       G ⊢ ⬊*[h, o, V, l] L.
-#h #o #I #G #L #V #T #l #H @(lsx_ind … H) -L
-#L1 #_ #IHL1 @lsx_intro
-#L2 #HL12 #HV @IHL1 /3 width=3 by lleq_fwd_flat_sn/
-qed-.
-
-lemma lsx_fwd_flat_dx: ∀h,o,I,G,L,V,T,l. G ⊢ ⬊*[h, o, ⓕ{I}V.T, l] L →
-                       G ⊢ ⬊*[h, o, T, l] L.
-#h #o #I #G #L #V #T #l #H @(lsx_ind … H) -L
-#L1 #_ #IHL1 @lsx_intro
-#L2 #HL12 #HV @IHL1 /3 width=3 by lleq_fwd_flat_dx/
-qed-.
-
-lemma lsx_fwd_pair_sn: ∀h,o,I,G,L,V,T,l. G ⊢ ⬊*[h, o, ②{I}V.T, l] L →
-                       G ⊢ ⬊*[h, o, V, l] L.
-#h #o * /2 width=4 by lsx_fwd_bind_sn, lsx_fwd_flat_sn/
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma lsx_inv_flat: ∀h,o,I,G,L,V,T,l. G ⊢ ⬊*[h, o, ⓕ{I}V.T, l] L →
-                    G ⊢ ⬊*[h, o, V, l] L ∧ G ⊢ ⬊*[h, o, T, l] L.
-/3 width=3 by lsx_fwd_flat_sn, lsx_fwd_flat_dx, conj/ qed-.