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diff --git a/matita/matita/contribs/lambdadelta/apps_2/functional/mf_vpush_exteq.ma b/matita/matita/contribs/lambdadelta/apps_2/functional/mf_vpush_exteq.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 "ground_2/lib/functions.ma".
+include "ground_2/lib/exteq.ma".
+include "apps_2/functional/flifts_flifts_basic.ma".
+include "apps_2/functional/mf_vpush.ma".
+
+(* MULTIPLE FILLING PUSH ****************************************************)
+
+(* Properties with extensional equivalence **********************************)
+
+lemma mf_lc_id: ⇡[0←#0]mf_li ≐ mf_li.
+#i elim (eq_or_gt i) #Hi destruct //
+>mf_vpush_gt // >(flifts_lref_uni 1) <(S_pred … Hi) in ⊢ (???%); -Hi //
+qed.
+
+lemma mf_vpush_comp (i): compatible_3 … (mf_vpush i) (eq …) (exteq …) (exteq …).
+#i #T1 #T2 #HT12 #lv1 #lv2 #Hlv #j
+elim (lt_or_eq_or_gt j i) #Hij destruct
+[ >mf_vpush_lt // >mf_vpush_lt //
+| >mf_vpush_eq >mf_vpush_eq //
+| >mf_vpush_gt // >mf_vpush_gt //
+]
+qed-.
+
+(* Main properties with extensional equivalence *****************************)
+
+theorem mf_vpush_swap: ∀l1,l2. l2 ≤ l1 →
+                       ∀v,T1,T2. ⇡[l2←T2]⇡[l1←T1]v ≐ ⇡[↑l1←↑[l2,1]T1]⇡[l2←T2]v.
+#l1 #l2 #Hl21 #v #T1 #T2 #i
+elim (lt_or_eq_or_gt i l2) #Hl2 destruct
+[ lapply (lt_to_le_to_lt … Hl2 Hl21) #Hl1
+  >mf_vpush_lt // >mf_vpush_lt // >mf_vpush_lt /2 width=1 by lt_S/ >mf_vpush_lt //
+| >mf_vpush_eq >mf_vpush_lt /2 width=1 by monotonic_le_plus_l/
+| >mf_vpush_gt // elim (lt_or_eq_or_gt (↓i) l1) #Hl1 destruct
+  [ >mf_vpush_lt // >mf_vpush_lt /2 width=1 by lt_minus_to_plus/ >mf_vpush_gt //
+  | >mf_vpush_eq <(lt_succ_pred … Hl2) >mf_vpush_eq //
+  | lapply (le_to_lt_to_lt … Hl21 Hl1) -Hl2 #Hl2
+    >mf_vpush_gt // >mf_vpush_gt /2 width=1 by lt_minus_to_plus_r/ >mf_vpush_gt //
+    /2 width=1 by flifts_basic_swap/
+  ]
+]
+qed-.