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diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/fsle_fqup.ma b/matita/matita/contribs/lambdadelta/basic_2/static/fsle_fqup.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/frees_fqup.ma".
+include "basic_2/static/fsle_length.ma".
+
+(* FREE VARIABLES INCLUSION FOR RESTRICTED CLOSURES *************************)
+
+(* Advanced properties ******************************************************)
+
+lemma fsle_refl: bi_reflexive … fsle.
+#L #T
+elim (frees_total L T) #f #Hf
+/2 width=8 by sle_refl, ex4_4_intro/
+qed.
+
+lemma fsle_shift: ∀L1,L2. |L1| = |L2| →
+                  ∀I,T1,T2,V.  ⦃L1.ⓧ, T1⦄ ⊆ ⦃L2.ⓑ{I}V, T2⦄ →
+                  ∀p. ⦃L1.ⓧ, T1⦄ ⊆ ⦃L2, ⓑ{p,I}V.T2⦄.
+#L1 #L2 #H1L #I #T1 #T2 #V
+* #n #m #f2 #g2 #Hf2 #Hg2 #H2L #Hfg2 #p
+elim (lveq_inj_length … H2L) // -H1L #H1 #H2 destruct
+lapply (lveq_inv_bind … H2L) -H2L #HL
+elim (frees_total L2 V) #g1 #Hg1
+elim (sor_isfin_ex g1 (⫱g2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #g #Hg #_
+lapply (sor_inv_sle_dx … Hg) #H0g
+/4 width=10 by frees_bind, lveq_void_sn, sle_tl, sle_trans, ex4_4_intro/
+qed.
+
+lemma fsle_bind_dx_sn: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ⊆ ⦃L2, V2⦄ →
+                       ∀p,I,T2. ⦃L1, V1⦄ ⊆ ⦃L2, ⓑ{p,I}V2.T2⦄.
+#L1 #L2 #V1 #V2 * #n1 #m1 #f1 #g1 #Hf1 #Hg1 #HL12 #Hfg1 #p #I #T2
+elim (frees_total (L2.ⓧ) T2) #g2 #Hg2
+elim (sor_isfin_ex g1 (⫱g2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #g #Hg #_
+@(ex4_4_intro … g Hf1 … HL12) (**) (* full auto too slow *)
+/4 width=5 by frees_bind_void, sor_inv_sle_sn, sor_tls, sle_trans/
+qed.
+
+lemma fsle_bind_dx_dx: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2.ⓧ, T2⦄ → |L1| ≤ |L2| →
+                       ∀p,I,V2. ⦃L1, T1⦄ ⊆ ⦃L2, ⓑ{p,I}V2.T2⦄.
+#L1 #L2 #T1 #T2 * #n1 #x1 #f2 #g2 #Hf2 #Hg2 #H #Hfg2 #HL12 #p #I #V2
+elim (lveq_inv_void_dx_length … H HL12) -H -HL12 #m1 #HL12 #H1 #H2 destruct
+<tls_xn in Hfg2; #Hfg2
+elim (frees_total L2 V2) #g1 #Hg1
+elim (sor_isfin_ex g1 (⫱g2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #g #Hg #_
+@(ex4_4_intro … g Hf2 … HL12) (**) (* full auto too slow *)
+/4 width=5 by frees_bind_void, sor_inv_sle_dx, sor_tls, sle_trans/
+qed.
+
+lemma fsle_flat_dx_sn: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ⊆ ⦃L2, V2⦄ →
+                       ∀I,T2. ⦃L1, V1⦄ ⊆ ⦃L2, ⓕ{I}V2.T2⦄.
+#L1 #L2 #V1 #V2 * #n1 #m1 #f1 #g1 #Hf1 #Hg1 #HL12 #Hfg1 #I #T2
+elim (frees_total L2 T2) #g2 #Hg2
+elim (sor_isfin_ex g1 g2) /2 width=3 by frees_fwd_isfin/ #g #Hg #_
+@(ex4_4_intro … g Hf1 … HL12) (**) (* full auto too slow *)
+/4 width=5 by frees_flat, sor_inv_sle_sn, sor_tls, sle_trans/
+qed.
+
+lemma fsle_flat_dx_dx: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ →
+                       ∀I,V2. ⦃L1, T1⦄ ⊆ ⦃L2, ⓕ{I}V2.T2⦄.
+#L1 #L2 #T1 #T2 * #n1 #m1 #f2 #g2 #Hf2 #Hg2 #HL12 #Hfg2 #I #V2
+elim (frees_total L2 V2) #g1 #Hg1
+elim (sor_isfin_ex g1 g2) /2 width=3 by frees_fwd_isfin/ #g #Hg #_
+@(ex4_4_intro … g Hf2 … HL12) (**) (* full auto too slow *)
+/4 width=5 by frees_flat, sor_inv_sle_dx, sor_tls, sle_trans/
+qed.
+
+(* Advanced forward lemmas ***************************************************)
+
+lemma fsle_fwd_pair_sn: ∀I1,I2,L1,L2,V1,V2,T1,T2. ⦃L1.ⓑ{I1}V1, T1⦄ ⊆ ⦃L2.ⓑ{I2}V2, T2⦄ →
+                        ⦃L1.ⓧ, T1⦄ ⊆ ⦃L2.ⓑ{I2}V2, T2⦄.
+#I1 #I2 #L1 #L2 #V1 #V2 #T1 #T2 *
+#n1 #n2 #f1 #f2 #Hf1 #Hf2 #HL12 #Hf12
+elim (lveq_inv_pair_pair … HL12) -HL12 #HL12 #H1 #H2 destruct
+elim (frees_total (L1.ⓧ) T1) #g1 #Hg1
+lapply (lsubr_lsubf … Hg1 … Hf1) -Hf1 /2 width=1 by lsubr_unit/ #Hfg1
+/5 width=10 by lsubf_fwd_sle, lveq_bind, sle_trans, ex4_4_intro/ (**) (* full auto too slow *)
+qed-.