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diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/aaa_lleq.ma b/matita/matita/contribs/lambdadelta/basic_2/static/aaa_lleq.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/substitution/lleq_ldrop.ma".
+include "basic_2/static/aaa.ma".
+
+(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
+
+(* Properties on lazy equivalence for local environments ********************)
+
+lemma lleq_aaa_trans: ∀G,L2,T,A. ⦃G, L2⦄ ⊢ T ⁝ A →
+                      ∀L1. L1 ⋕[T, 0] L2 → ⦃G, L1⦄ ⊢ T ⁝ A.
+#G #L2 #T #A #H elim H -G -L2 -T -A /2 width=1 by aaa_sort/
+[ #I #G #L2 #K2 #V2 #A #i #HLK2 #_ #IHV2 #L1 #H elim (lleq_fwd_lref_dx … H … HLK2) -L2
+  [ #H elim (ylt_yle_false … H) //
+  | * /3 width=5 by aaa_lref/
+  ]
+| #a #G #L2 #V #T #B #A #_ #_ #IHV #IHT #L1 #H elim (lleq_inv_bind_O … H) -H
+  /3 width=2 by aaa_abbr/
+| #a #G #L2 #V #T #B #A #_ #_ #IHV #IHT #L1 #H elim (lleq_inv_bind_O … H) -H
+  /3 width=1 by aaa_abst/
+| #G #L2 #V #T #B #A #_ #_ #IHV #IHT #L1 #H elim (lleq_inv_flat … H) -H
+  /3 width=3 by aaa_appl/
+| #G #L2 #V #T #A #_ #_ #IHV #IHT #L1 #H elim (lleq_inv_flat … H) -H
+  /3 width=1 by aaa_cast/
+]
+qed-.
+
+lemma aaa_lleq_conf: ∀G,L2,T,A. ⦃G, L2⦄ ⊢ T ⁝ A →
+                     ∀L1. L2 ⋕[T, 0] L1 → ⦃G, L1⦄ ⊢ T ⁝ A.
+/3 width=3 by lleq_aaa_trans, lleq_sym/ qed-.