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diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/frees_append.ma b/matita/matita/contribs/lambdadelta/basic_2/static/frees_append.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/syntax/append.ma".
+include "basic_2/static/frees.ma".
+
+(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
+
+(* Properties with append for local environments ****************************)
+
+lemma frees_append_void: ∀f,K,T. K ⊢ 𝐅*⦃T⦄ ≡ f → ⓧ.K ⊢ 𝐅*⦃T⦄ ≡ f.
+#f #K #T #H elim H -f -K -T
+[ /2 width=1 by frees_sort/
+| #f * /3 width=1 by frees_atom, frees_unit, frees_lref/
+| /2 width=1 by frees_pair/
+| /2 width=1 by frees_unit/
+| /2 width=1 by frees_lref/
+| /2 width=1 by frees_gref/
+| /3 width=5 by frees_bind/
+| /3 width=5 by frees_flat/
+]
+qed.
+
+(* Inversion lemmas with append for local environments **********************)
+
+fact frees_inv_append_void_aux: ∀f,L,T. L ⊢ 𝐅*⦃T⦄ ≡ f →
+                                ∀K. L = ⓧ.K → K ⊢ 𝐅*⦃T⦄ ≡ f.
+#f #L #T #H elim H -f -L -T
+[ /2 width=1 by frees_sort/
+| #f #i #_ #K #H 
+  elim (append_inv_atom3_sn … H) -H #H1 #H2 destruct
+| #f #I #L #V #_ #IH #K #H
+  elim (append_inv_bind3_sn … H) -H * [ | #Y ] #H1 #H2 destruct
+  /3 width=1 by frees_pair/
+| #f #I #L #Hf #K #H
+  elim (append_inv_bind3_sn … H) -H * [ | #Y ] #H1 #H2 destruct
+  /2 width=1 by frees_atom, frees_unit/
+| #f #I #L #i #Hf #IH #K #H
+  elim (append_inv_bind3_sn … H) -H * [ | #Y ] #H1 #H2 destruct
+  /3 width=1 by frees_lref, frees_lref_push/
+| /2 width=1 by frees_gref/
+| /3 width=5 by frees_bind/
+| /3 width=5 by frees_flat/
+]
+qed-.
+
+lemma frees_inv_append_void: ∀f,K,T. ⓧ.K  ⊢ 𝐅*⦃T⦄ ≡ f → K ⊢ 𝐅*⦃T⦄ ≡ f.
+/2 width=3 by frees_inv_append_void_aux/ qed-.