--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/computation/lsubc_ldrop.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR ABSTRACT CANDIDATES OF REDUCIBILITY *****)
+
+(* Properties concerning generic local environment slicing ******************)
+
+(* Basic_1: was: csubc_drop1_conf_rev *)
+lemma ldrops_lsubc_trans: ∀RR,RS,RP.
+ acp RR RS RP → acr RR RS RP (λL,T. RP L T) →
+ ∀L1,K1,des. ⇩*[des] L1 ≡ K1 → ∀K2. K1 [RP] ⊑ K2 →
+ ∃∃L2. L1 [RP] ⊑ L2 & ⇩*[des] L2 ≡ K2.
+#RR #RS #RP #Hacp #Hacr #L1 #K1 #des #H elim H -L1 -K1 -des
+[ /2 width=3/
+| #L1 #L #K1 #des #d #e #_ #HLK1 #IHL #K2 #HK12
+ elim (ldrop_lsubc_trans … Hacp Hacr … HLK1 … HK12) -Hacp -Hacr -K1 #K #HLK #HK2
+ elim (IHL … HLK) -IHL -HLK /3 width=5/
+]
+qed-.