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diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/lsubc.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/lsubc.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/static/aaa.ma".
+include "Basic_2/computation/acp_cr.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR ABSTRACT CANDIDATES OF REDUCIBILITY *****)
+
+inductive lsubc (RP:lenv→predicate term): relation lenv ≝
+| lsubc_atom: lsubc RP (⋆) (⋆)
+| lsubc_pair: ∀I,L1,L2,V. lsubc RP L1 L2 → lsubc RP (L1. ⓑ{I} V) (L2. ⓑ{I} V)
+| lsubc_abbr: ∀L1,L2,V,W,A. ⦃L1, V⦄ [RP] ϵ 〚A〛 → L2 ⊢ W ÷ A →
+              lsubc RP L1 L2 → lsubc RP (L1. ⓓV) (L2. ⓛW)
+.
+
+interpretation
+  "local environment refinement (abstract candidates of reducibility)"
+  'CrSubEq L1 RP L2 = (lsubc RP L1 L2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lsubc_inv_atom1_aux: ∀RP,L1,L2. L1 [RP] ⊑ L2 → L1 = ⋆ → L2 = ⋆.
+#RP #L1 #L2 * -L1 -L2
+[ //
+| #I #L1 #L2 #V #_ #H destruct
+| #L1 #L2 #V #W #A #_ #_ #_ #H destruct
+]
+qed.
+
+(* Basic_1: was: csubc_gen_sort_r *)
+lemma lsubc_inv_atom1: ∀RP,L2. ⋆ [RP] ⊑ L2 → L2 = ⋆.
+/2 width=4/ qed-.
+
+fact lsubc_inv_pair1_aux: ∀RP,L1,L2. L1 [RP] ⊑ L2 → ∀I,K1,V. L1 = K1. ⓑ{I} V →
+                          (∃∃K2. K1 [RP] ⊑ K2 & L2 = K2. ⓑ{I} V) ∨
+                          ∃∃K2,W,A. ⦃K1, V⦄ [RP] ϵ 〚A〛 & K2 ⊢ W ÷ A &
+                                    K1 [RP] ⊑ K2 &
+                                    L2 = K2. ⓛW & I = Abbr.
+#RP #L1 #L2 * -L1 -L2
+[ #I #K1 #V #H destruct
+| #J #L1 #L2 #V #HL12 #I #K1 #W #H destruct /3 width=3/
+| #L1 #L2 #V1 #W2 #A #HV1 #HW2 #HL12 #I #K1 #V #H destruct /3 width=7/
+]
+qed.
+
+(* Basic_1: was: csubc_gen_head_r *)
+lemma lsubc_inv_pair1: ∀RP,I,K1,L2,V. K1. ⓑ{I} V [RP] ⊑ L2 →
+                       (∃∃K2. K1 [RP] ⊑ K2 & L2 = K2. ⓑ{I} V) ∨
+                       ∃∃K2,W,A. ⦃K1, V⦄ [RP] ϵ 〚A〛 & K2 ⊢ W ÷ A &
+                                 K1 [RP] ⊑ K2 &
+                                 L2 = K2. ⓛW & I = Abbr.
+/2 width=3/ qed-.
+
+fact lsubc_inv_atom2_aux: ∀RP,L1,L2. L1 [RP] ⊑ L2 → L2 = ⋆ → L1 = ⋆.
+#RP #L1 #L2 * -L1 -L2
+[ //
+| #I #L1 #L2 #V #_ #H destruct
+| #L1 #L2 #V #W #A #_ #_ #_ #H destruct
+]
+qed.
+
+(* Basic_1: was: csubc_gen_sort_l *)
+lemma lsubc_inv_atom2: ∀RP,L1. L1 [RP] ⊑ ⋆ → L1 = ⋆.
+/2 width=4/ qed-.
+
+fact lsubc_inv_pair2_aux: ∀RP,L1,L2. L1 [RP] ⊑ L2 → ∀I,K2,W. L2 = K2. ⓑ{I} W →
+                          (∃∃K1. K1 [RP] ⊑ K2 & L1 = K1. ⓑ{I} W) ∨
+                          ∃∃K1,V,A. ⦃K1, V⦄ [RP] ϵ 〚A〛 & K2 ⊢ W ÷ A &
+                                    K1 [RP] ⊑ K2 &
+                                    L1 = K1. ⓓV & I = Abst.
+#RP #L1 #L2 * -L1 -L2
+[ #I #K2 #W #H destruct
+| #J #L1 #L2 #V #HL12 #I #K2 #W #H destruct /3 width=3/
+| #L1 #L2 #V1 #W2 #A #HV1 #HW2 #HL12 #I #K2 #W #H destruct /3 width=7/
+]
+qed.
+
+(* Basic_1: was: csubc_gen_head_l *)
+lemma lsubc_inv_pair2: ∀RP,I,L1,K2,W. L1 [RP] ⊑ K2. ⓑ{I} W →
+                       (∃∃K1. K1 [RP] ⊑ K2 & L1 = K1. ⓑ{I} W) ∨
+                       ∃∃K1,V,A. ⦃K1, V⦄ [RP] ϵ 〚A〛 & K2 ⊢ W ÷ A &
+                                 K1 [RP] ⊑ K2 &
+                                 L1 = K1. ⓓV & I = Abst.
+/2 width=3/ qed-.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was: csubc_refl *)
+lemma lsubc_refl: ∀RP,L. L [RP] ⊑ L.
+#RP #L elim L -L // /2 width=1/
+qed.
+
+(* Basic_1: removed theorems 2: csubc_clear_conf csubc_getl_conf *)