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diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_lleq.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_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/cpys_cpys.ma".
+include "basic_2/substitution/lleq.ma".
+
+(* Advanced forward lemmas **************************************************)
+
+lemma lleq_fwd_lref: ∀L1,L2,d,i. L1 ⋕[#i, d] L2 →
+                     ∨∨ |L1| ≤ i ∧ |L2| ≤ i
+                      | i < d
+                      | ∃∃I1,I2,K1,K2,V. ⇩[0, i] L1 ≡ K1.ⓑ{I1}V &
+                                         ⇩[0, i] L2 ≡ K2.ⓑ{I2}V &
+                                         K1 ⋕[V, 0] K2 & d ≤ i.
+#L1 #L2 #d #i * #HL12 #IH elim (lt_or_ge i (|L1|)) /3 width=3 by or3_intro0, conj/
+elim (lt_or_ge i d) /2 width=1 by or3_intro1/ #Hdi #Hi
+elim (ldrop_O1_lt … Hi) #I1 #K1 #V1 #HLK1
+elim (ldrop_O1_lt L2 i) // -Hi #I2 #K2 #V2 #HLK2
+lapply (ldrop_fwd_length_minus2 … HLK2) #H
+lapply (ldrop_fwd_length_minus2 … HLK1) >HL12 <H -HL12 -H
+#H lapply (injective_plus_l … H) -H #HK12
+elim (lift_total V1 0 (i+1)) #W1 #HVW1
+elim (lift_total V2 0 (i+1)) #W2 #HVW2
+elim (IH W1) elim (IH W2) #_ #H2 #H1 #_
+elim (cpys_inv_lref1_ldrop … (H1 ?) … HLK2 … HVW1) -H1
+[ elim (cpys_inv_lref1_ldrop … (H2 ?) … HLK1 … HVW2) -H2 ]
+/3 width=7 by cpys_subst, yle_inj/ -W1 -W2 #H12 #_ #_ #H21 #_ #_
+lapply (cpys_antisym_eq … H12 … H21) -H12 -H21 #H destruct
+@or3_intro2 @(ex4_5_intro … HLK1 HLK2) // @conj // -HK12
+#V elim (lift_total V 0 (i+1))
+#W #HVW elim (IH W) -IH #H12 #H21 @conj #H
+[ elim (cpys_inv_lref1_ldrop … (H12 ?) … HLK2 … HVW) -H12 -H21
+| elim (cpys_inv_lref1_ldrop … (H21 ?) … HLK1 … HVW) -H21 -H12
+] /3 width=7 by cpys_subst, yle_inj/
+qed-.