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diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lpxs.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lpxs.ma
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--- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lpxs.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/predsnstar_5.ma".
-include "basic_2/reduction/lpx.ma".
-include "basic_2/computation/lprs.ma".
-
-(* SN EXTENDED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *******************)
-
-definition lpxs: ∀h. sd h → relation3 genv lenv lenv ≝
-                 λh,o,G. TC … (lpx h o G).
-
-interpretation "extended parallel computation (local environment, sn variant)"
-   'PRedSnStar h o G L1 L2 = (lpxs h o G L1 L2).
-
-(* Basic eliminators ********************************************************)
-
-lemma lpxs_ind: ∀h,o,G,L1. ∀R:predicate lenv. R L1 →
-                (∀L,L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L → ⦃G, L⦄ ⊢ ➡[h, o] L2 → R L → R L2) →
-                ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 → R L2.
-#h #o #G #L1 #R #HL1 #IHL1 #L2 #HL12
-@(TC_star_ind … HL1 IHL1 … HL12) //
-qed-.
-
-lemma lpxs_ind_dx: ∀h,o,G,L2. ∀R:predicate lenv. R L2 →
-                   (∀L1,L. ⦃G, L1⦄ ⊢ ➡[h, o] L → ⦃G, L⦄ ⊢ ➡*[h, o] L2 → R L → R L1) →
-                   ∀L1. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 → R L1.
-#h #o #G #L2 #R #HL2 #IHL2 #L1 #HL12
-@(TC_star_ind_dx … HL2 IHL2 … HL12) //
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma lprs_lpxs: ∀h,o,G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 → ⦃G, L1⦄ ⊢ ➡*[h, o] L2.
-/3 width=3 by lpr_lpx, monotonic_TC/ qed.
-
-lemma lpx_lpxs: ∀h,o,G,L1,L2. ⦃G, L1⦄ ⊢ ➡[h, o] L2 → ⦃G, L1⦄ ⊢ ➡*[h, o] L2.
-/2 width=1 by inj/ qed.
-
-lemma lpxs_refl: ∀h,o,G,L. ⦃G, L⦄ ⊢ ➡*[h, o] L.
-/2 width=1 by lprs_lpxs/ qed.
-
-lemma lpxs_strap1: ∀h,o,G,L1,L,L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L → ⦃G, L⦄ ⊢ ➡[h, o] L2 → ⦃G, L1⦄ ⊢ ➡*[h, o] L2.
-/2 width=3 by step/ qed.
-
-lemma lpxs_strap2: ∀h,o,G,L1,L,L2. ⦃G, L1⦄ ⊢ ➡[h, o] L → ⦃G, L⦄ ⊢ ➡*[h, o] L2 → ⦃G, L1⦄ ⊢ ➡*[h, o] L2.
-/2 width=3 by TC_strap/ qed.
-
-lemma lpxs_pair_refl: ∀h,o,G,L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 → ∀I,V. ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡*[h, o] L2.ⓑ{I}V.
-/2 width=1 by TC_lpx_sn_pair_refl/ qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma lpxs_inv_atom1: ∀h,o,G,L2. ⦃G, ⋆⦄ ⊢ ➡*[h, o] L2 → L2 = ⋆.
-/2 width=2 by TC_lpx_sn_inv_atom1/ qed-.
-
-lemma lpxs_inv_atom2: ∀h,o,G,L1. ⦃G, L1⦄ ⊢ ➡*[h, o] ⋆ → L1 = ⋆.
-/2 width=2 by TC_lpx_sn_inv_atom2/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lpxs_fwd_length: ∀h,o,G,L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 → |L1| = |L2|.
-/2 width=2 by TC_lpx_sn_fwd_length/ qed-.