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diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_ldrop.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_ldrop.ma
index 61466ca35..476680a16 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_ldrop.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_ldrop.ma
@@ -12,71 +12,85 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/substitution/lleq_ldrop.ma".
-include "basic_2/computation/lpxs_ldrop.ma".
+include "basic_2/multiple/lleq_ldrop.ma".
+include "basic_2/reduction/lpx_ldrop.ma".
 include "basic_2/computation/lsx.ma".
 
 (* SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS **********************)
 
 (* Advanced properties ******************************************************)
 
-lemma lsx_lref_free: ∀h,g,G,L,d,i. |L| ≤ i → G ⊢ ⋕⬊*[h, g, #i, d] L.
+lemma lsx_lref_free: ∀h,g,G,L,d,i. |L| ≤ i → G ⊢ ⬊*[h, g, #i, d] L.
 #h #g #G #L1 #d #i #HL1 @lsx_intro
 #L2 #HL12 #H elim H -H
-/4 width=6 by lpxs_fwd_length, lleq_free, le_repl_sn_conf_aux/
+/4 width=6 by lpx_fwd_length, lleq_free, le_repl_sn_conf_aux/
 qed.
 
-lemma lsx_lref_skip: ∀h,g,G,L,d,i. yinj i < d → G ⊢ ⋕⬊*[h, g, #i, d] L.
+lemma lsx_lref_skip: ∀h,g,G,L,d,i. yinj i < d → G ⊢ ⬊*[h, g, #i, d] L.
 #h #g #G #L1 #d #i #HL1 @lsx_intro
 #L2 #HL12 #H elim H -H
-/3 width=4 by lpxs_fwd_length, lleq_skip/
+/3 width=4 by lpx_fwd_length, lleq_skip/
 qed.
 
+(* Advanced forward lemmas **************************************************)
+
+lemma lsx_fwd_lref_be: ∀h,g,I,G,L,d,i. d ≤ yinj i → G ⊢ ⬊*[h, g, #i, d] L →
+                       ∀K,V. ⇩[i] L ≡ K.ⓑ{I}V → G ⊢ ⬊*[h, g, V, 0] K.
+#h #g #I #G #L #d #i #Hdi #H @(lsx_ind … H) -L
+#L1 #_ #IHL1 #K1 #V #HLK1 @lsx_intro
+#K2 #HK12 #HnK12 lapply (ldrop_fwd_drop2 … HLK1)
+#H2LK1 elim (ldrop_lpx_trans … H2LK1 … HK12) -H2LK1 -HK12
+#L2 #HL12 #H2LK2 #H elim (leq_ldrop_conf_be … H … HLK1) -H /2 width=1 by ylt_inj/
+#Y #_ #HLK2 lapply (ldrop_fwd_drop2 … HLK2)
+#HY lapply (ldrop_mono … HY … H2LK2) -HY -H2LK2 #H destruct
+/4 width=10 by lleq_inv_lref_ge/
+qed-.
+
 (* Properties on relocation *************************************************)
 
 lemma lsx_lift_le: ∀h,g,G,K,T,U,dt,d,e. dt ≤ yinj d →
-                   ⇧[d, e] T ≡ U → G ⊢ ⋕⬊*[h, g, T, dt] K →
-                   ∀L. ⇩[Ⓕ, d, e] L ≡ K → G ⊢ ⋕⬊*[h, g, U, dt] L.
+                   ⇧[d, e] T ≡ U → G ⊢ ⬊*[h, g, T, dt] K →
+                   ∀L. ⇩[Ⓕ, d, e] L ≡ K → G ⊢ ⬊*[h, g, U, dt] L.
 #h #g #G #K #T #U #dt #d #e #Hdtd #HTU #H @(lsx_ind … H) -K
 #K1 #_ #IHK1 #L1 #HLK1 @lsx_intro
-#L2 #HL12 #HnU elim (lpxs_ldrop_conf … HLK1 … HL12) -HL12
+#L2 #HL12 #HnU elim (lpx_ldrop_conf … HLK1 … HL12) -HL12
 /4 width=10 by lleq_lift_le/
 qed-.
 
 lemma lsx_lift_ge: ∀h,g,G,K,T,U,dt,d,e. yinj d ≤ dt →
-                   ⇧[d, e] T ≡ U → G ⊢ ⋕⬊*[h, g, T, dt] K →
-                   ∀L. ⇩[Ⓕ, d, e] L ≡ K → G ⊢ ⋕⬊*[h, g, U, dt + e] L.
+                   ⇧[d, e] T ≡ U → G ⊢ ⬊*[h, g, T, dt] K →
+                   ∀L. ⇩[Ⓕ, d, e] L ≡ K → G ⊢ ⬊*[h, g, U, dt + e] L.
 #h #g #G #K #T #U #dt #d #e #Hddt #HTU #H @(lsx_ind … H) -K
 #K1 #_ #IHK1 #L1 #HLK1 @lsx_intro
-#L2 #HL12 #HnU elim (lpxs_ldrop_conf … HLK1 … HL12) -HL12
+#L2 #HL12 #HnU elim (lpx_ldrop_conf … HLK1 … HL12) -HL12
 /4 width=9 by lleq_lift_ge/
 qed-.
 
 (* Inversion lemmas on relocation *******************************************)
 
 lemma lsx_inv_lift_le: ∀h,g,G,L,T,U,dt,d,e. dt ≤ yinj d →
-                       ⇧[d, e] T ≡ U → G ⊢ ⋕⬊*[h, g, U, dt] L →
-                       ∀K. ⇩[Ⓕ, d, e] L ≡ K → G ⊢ ⋕⬊*[h, g, T, dt] K.
+                       ⇧[d, e] T ≡ U → G ⊢ ⬊*[h, g, U, dt] L →
+                       ∀K. ⇩[Ⓕ, d, e] L ≡ K → G ⊢ ⬊*[h, g, T, dt] K.
 #h #g #G #L #T #U #dt #d #e #Hdtd #HTU #H @(lsx_ind … H) -L
 #L1 #_ #IHL1 #K1 #HLK1 @lsx_intro
-#K2 #HK12 #HnT elim (ldrop_lpxs_trans … HLK1 … HK12) -HK12
+#K2 #HK12 #HnT elim (ldrop_lpx_trans … HLK1 … HK12) -HK12
 /4 width=10 by lleq_inv_lift_le/
 qed-.
 
 lemma lsx_inv_lift_be: ∀h,g,G,L,T,U,dt,d,e. yinj d ≤ dt → dt ≤ d + e →
-                       ⇧[d, e] T ≡ U → G ⊢ ⋕⬊*[h, g, U, dt] L →
-                       ∀K. ⇩[Ⓕ, d, e] L ≡ K → G ⊢ ⋕⬊*[h, g, T, d] K.
+                       ⇧[d, e] T ≡ U → G ⊢ ⬊*[h, g, U, dt] L →
+                       ∀K. ⇩[Ⓕ, d, e] L ≡ K → G ⊢ ⬊*[h, g, T, d] K.
 #h #g #G #L #T #U #dt #d #e #Hddt #Hdtde #HTU #H @(lsx_ind … H) -L
 #L1 #_ #IHL1 #K1 #HLK1 @lsx_intro
-#K2 #HK12 #HnT elim (ldrop_lpxs_trans … HLK1 … HK12) -HK12
+#K2 #HK12 #HnT elim (ldrop_lpx_trans … HLK1 … HK12) -HK12
 /4 width=11 by lleq_inv_lift_be/
 qed-.
 
 lemma lsx_inv_lift_ge: ∀h,g,G,L,T,U,dt,d,e. yinj d + yinj e ≤ dt →
-                       ⇧[d, e] T ≡ U → G ⊢ ⋕⬊*[h, g, U, dt] L →
-                       ∀K. ⇩[Ⓕ, d, e] L ≡ K → G ⊢ ⋕⬊*[h, g, T, dt-e] K.
+                       ⇧[d, e] T ≡ U → G ⊢ ⬊*[h, g, U, dt] L →
+                       ∀K. ⇩[Ⓕ, d, e] L ≡ K → G ⊢ ⬊*[h, g, T, dt-e] K.
 #h #g #G #L #T #U #dt #d #e #Hdedt #HTU #H @(lsx_ind … H) -L
 #L1 #_ #IHL1 #K1 #HLK1 @lsx_intro
-#K2 #HK12 #HnT elim (ldrop_lpxs_trans … HLK1 … HK12) -HK12
+#K2 #HK12 #HnT elim (ldrop_lpx_trans … HLK1 … HK12) -HK12
 /4 width=9 by lleq_inv_lift_ge/
 qed-.