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diff --git a/matita/matita/contribs/lambda_delta/basic_2/grammar/lsubs.ma b/matita/matita/contribs/lambda_delta/basic_2/grammar/lsubs.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/grammar/lenv_length.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR SUBSTITUTION ****************************)
-
-inductive lsubs: nat → nat → relation lenv ≝
-| lsubs_sort: ∀d,e. lsubs d e (⋆) (⋆)
-| lsubs_OO:   ∀L1,L2. lsubs 0 0 L1 L2
-| lsubs_abbr: ∀L1,L2,V,e. lsubs 0 e L1 L2 →
-              lsubs 0 (e + 1) (L1. ⓓV) (L2.ⓓV)
-| lsubs_abst: ∀L1,L2,I,V1,V2,e. lsubs 0 e L1 L2 →
-              lsubs 0 (e + 1) (L1. ⓛV1) (L2.ⓑ{I} V2)
-| lsubs_skip: ∀L1,L2,I1,I2,V1,V2,d,e.
-              lsubs d e L1 L2 → lsubs (d + 1) e (L1. ⓑ{I1} V1) (L2. ⓑ{I2} V2)
-.
-
-interpretation
-  "local environment refinement (substitution)"
-  'SubEq L1 d e L2 = (lsubs d e L1 L2).
-
-definition lsubs_conf: ∀S. (lenv → relation S) → Prop ≝ λS,R.
-                       ∀L1,s1,s2. R L1 s1 s2 →
-                       ∀L2,d,e. L1 [d, e] ≼ L2 → R L2 s1 s2.
-
-(* Basic properties *********************************************************)
-
-lemma TC_lsubs_conf: ∀S,R. lsubs_conf S R → lsubs_conf S (λL. (TC … (R L))).
-#S #R #HR #L1 #s1 #s2 #H elim H -s2
-[ /3 width=5/
-| #s #s2 #_ #Hs2 #IHs1 #L2 #d #e #HL12
-  lapply (HR … Hs2 … HL12) -HR -Hs2 -HL12 /3 width=3/
-]
-qed.
-
-lemma lsubs_bind_eq: ∀L1,L2,e. L1 [0, e] ≼ L2 → ∀I,V.
-                     L1. ⓑ{I} V [0, e + 1] ≼ L2.ⓑ{I} V.
-#L1 #L2 #e #HL12 #I #V elim I -I /2 width=1/
-qed.
-
-lemma lsubs_refl: ∀d,e,L. L [d, e] ≼ L.
-#d elim d -d
-[ #e elim e -e // #e #IHe #L elim L -L // /2 width=1/
-| #d #IHd #e #L elim L -L // /2 width=1/
-]
-qed.
-
-lemma lsubs_skip_lt: ∀L1,L2,d,e. L1 [d - 1, e] ≼ L2 → 0 < d →
-                     ∀I1,I2,V1,V2. L1. ⓑ{I1} V1 [d, e] ≼ L2. ⓑ{I2} V2.
-
-#L1 #L2 #d #e #HL12 #Hd >(plus_minus_m_m d 1) // /2 width=1/
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-(* Basic forward lemmas ***************************************************)
-
-fact lsubs_fwd_length_full1_aux: ∀L1,L2,d,e. L1 [d, e] ≼ L2 →
-                                 d = 0 → e = |L1| → |L1| ≤ |L2|.
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e normalize
-[ //
-| /2 width=1/
-| /3 width=1/
-| /3 width=1/
-| #L1 #L2 #_ #_ #_ #_ #d #e #_ #_ >commutative_plus normalize #H destruct
-]
-qed.
-
-lemma lsubs_fwd_length_full1: ∀L1,L2. L1 [0, |L1|] ≼ L2 → |L1| ≤ |L2|.
-/2 width=5/ qed-.
-
-fact lsubs_fwd_length_full2_aux: ∀L1,L2,d,e. L1 [d, e] ≼ L2 →
-                                 d = 0 → e = |L2| → |L2| ≤ |L1|.
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e normalize
-[ //
-| /2 width=1/
-| /3 width=1/
-| /3 width=1/
-| #L1 #L2 #_ #_ #_ #_ #d #e #_ #_ >commutative_plus normalize #H destruct
-]
-qed.
-
-lemma lsubs_fwd_length_full2: ∀L1,L2. L1 [0, |L2|] ≼ L2 → |L2| ≤ |L1|.
-/2 width=5/ qed-.