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diff --git a/matita/matita/contribs/lambdadelta/static_2/static/frees_drops.ma b/matita/matita/contribs/lambdadelta/static_2/static/frees_drops.ma
index d936eda81..5cbeaea87 100644
--- a/matita/matita/contribs/lambdadelta/static_2/static/frees_drops.ma
+++ b/matita/matita/contribs/lambdadelta/static_2/static/frees_drops.ma
@@ -21,7 +21,7 @@ include "static_2/static/frees_fqup.ma".
 (* Advanced properties ******************************************************)
 
 lemma frees_atom_drops: 
-      ∀b,L,i. ⬇*[b,𝐔❴i❵] L ≘ ⋆ →
+      ∀b,L,i. ⇩*[b,𝐔❴i❵] L ≘ ⋆ →
       ∀f. 𝐈⦃f⦄ → L ⊢ 𝐅+⦃#i⦄ ≘ ⫯*[i]↑f.
 #b #L elim L -L /2 width=1 by frees_atom/
 #L #I #IH *
@@ -32,7 +32,7 @@ qed.
 
 lemma frees_pair_drops:
       ∀f,K,V. K ⊢ 𝐅+⦃V⦄ ≘ f → 
-      ∀i,I,L. ⬇*[i] L ≘ K.ⓑ{I}V → L ⊢ 𝐅+⦃#i⦄ ≘ ⫯*[i] ↑f.
+      ∀i,I,L. ⇩*[i] L ≘ K.ⓑ{I}V → L ⊢ 𝐅+⦃#i⦄ ≘ ⫯*[i] ↑f.
 #f #K #V #Hf #i elim i -i
 [ #I #L #H lapply (drops_fwd_isid … H ?) -H /2 width=1 by frees_pair/
 | #i #IH #I #L #H elim (drops_inv_succ … H) -H /3 width=2 by frees_lref/
@@ -40,7 +40,7 @@ lemma frees_pair_drops:
 qed.
 
 lemma frees_unit_drops:
-      ∀f.  𝐈⦃f⦄ → ∀I,K,i,L. ⬇*[i] L ≘ K.ⓤ{I} →
+      ∀f.  𝐈⦃f⦄ → ∀I,K,i,L. ⇩*[i] L ≘ K.ⓤ{I} →
       L ⊢ 𝐅+⦃#i⦄ ≘ ⫯*[i] ↑f.
 #f #Hf #I #K #i elim i -i
 [ #L #H lapply (drops_fwd_isid … H ?) -H /2 width=1 by frees_unit/
@@ -51,7 +51,7 @@ qed.
 (*
 lemma frees_sort_pushs:
       ∀f,K,s. K ⊢ 𝐅+⦃⋆s⦄ ≘ f →
-      ∀i,L. ⬇*[i] L ≘ K → L ⊢ 𝐅+⦃⋆s⦄ ≘ ⫯*[i] f.
+      ∀i,L. ⇩*[i] L ≘ K → L ⊢ 𝐅+⦃⋆s⦄ ≘ ⫯*[i] f.
 #f #K #s #Hf #i elim i -i
 [ #L #H lapply (drops_fwd_isid … H ?) -H //
 | #i #IH #L #H elim (drops_inv_succ … H) -H /3 width=1 by frees_sort/
@@ -60,7 +60,7 @@ qed.
 *)
 lemma frees_lref_pushs:
       ∀f,K,j. K ⊢ 𝐅+⦃#j⦄ ≘ f →
-      ∀i,L. ⬇*[i] L ≘ K → L ⊢ 𝐅+⦃#(i+j)⦄ ≘ ⫯*[i] f.
+      ∀i,L. ⇩*[i] L ≘ K → L ⊢ 𝐅+⦃#(i+j)⦄ ≘ ⫯*[i] f.
 #f #K #j #Hf #i elim i -i
 [ #L #H lapply (drops_fwd_isid … H ?) -H //
 | #i #IH #L #H elim (drops_inv_succ … H) -H
@@ -70,7 +70,7 @@ qed.
 (*
 lemma frees_gref_pushs:
       ∀f,K,l. K ⊢ 𝐅+⦃§l⦄ ≘ f →
-      ∀i,L. ⬇*[i] L ≘ K → L ⊢ 𝐅+⦃§l⦄ ≘ ⫯*[i] f.
+      ∀i,L. ⇩*[i] L ≘ K → L ⊢ 𝐅+⦃§l⦄ ≘ ⫯*[i] f.
 #f #K #l #Hf #i elim i -i
 [ #L #H lapply (drops_fwd_isid … H ?) -H //
 | #i #IH #L #H elim (drops_inv_succ … H) -H /3 width=1 by frees_gref/
@@ -81,9 +81,9 @@ qed.
 
 lemma frees_inv_lref_drops:
       ∀L,i,f. L ⊢ 𝐅+⦃#i⦄ ≘ f →
-      ∨∨ ∃∃g. ⬇*[Ⓕ,𝐔❴i❵] L ≘ ⋆ & 𝐈⦃g⦄ & f = ⫯*[i] ↑g
-       | ∃∃g,I,K,V. K ⊢ 𝐅+⦃V⦄ ≘ g & ⬇*[i] L ≘ K.ⓑ{I}V & f = ⫯*[i] ↑g
-       | ∃∃g,I,K. ⬇*[i] L ≘ K.ⓤ{I} & 𝐈⦃g⦄ & f = ⫯*[i] ↑g.
+      ∨∨ ∃∃g. ⇩*[Ⓕ,𝐔❴i❵] L ≘ ⋆ & 𝐈⦃g⦄ & f = ⫯*[i] ↑g
+       | ∃∃g,I,K,V. K ⊢ 𝐅+⦃V⦄ ≘ g & ⇩*[i] L ≘ K.ⓑ{I}V & f = ⫯*[i] ↑g
+       | ∃∃g,I,K. ⇩*[i] L ≘ K.ⓤ{I} & 𝐈⦃g⦄ & f = ⫯*[i] ↑g.
 #L elim L -L
 [ #i #g | #L #I #IH * [ #g cases I -I [ #I | #I #V ] -IH | #i #g ] ] #H
 [ elim (frees_inv_atom … H) -H #f #Hf #H destruct
@@ -105,7 +105,7 @@ qed-.
 
 lemma frees_lifts:
       ∀b,f1,K,T. K ⊢ 𝐅+⦃T⦄ ≘ f1 →
-      ∀f,L. ⬇*[b,f] L ≘ K → ∀U. ⬆*[f] T ≘ U →
+      ∀f,L. ⇩*[b,f] L ≘ K → ∀U. ⇧*[f] T ≘ U →
       ∀f2. f ~⊚ f1 ≘ f2 → L ⊢ 𝐅+⦃U⦄ ≘ f2.
 #b #f1 #K #T #H lapply (frees_fwd_isfin … H) elim H -f1 -K -T
 [ #f1 #K #s #Hf1 #_ #f #L #HLK #U #H2 #f2 #H3
@@ -162,7 +162,7 @@ lemma frees_lifts:
 qed-.
 
 lemma frees_lifts_SO:
-      ∀b,L,K. ⬇*[b,𝐔❴1❵] L ≘ K → ∀T,U. ⬆*[1] T ≘ U →
+      ∀b,L,K. ⇩*[b,𝐔❴1❵] L ≘ K → ∀T,U. ⇧*[1] T ≘ U →
       ∀f. K ⊢ 𝐅+⦃T⦄ ≘ f → L ⊢ 𝐅+⦃U⦄ ≘ ⫯f.
 #b #L #K #HLK #T #U #HTU #f #Hf
 @(frees_lifts b … Hf … HTU) //  (**) (* auto fails *)
@@ -172,7 +172,7 @@ qed.
 
 lemma frees_fwd_coafter:
       ∀b,f2,L,U. L ⊢ 𝐅+⦃U⦄ ≘ f2 →
-      ∀f,K. ⬇*[b,f] L ≘ K → ∀T. ⬆*[f] T ≘ U →
+      ∀f,K. ⇩*[b,f] L ≘ K → ∀T. ⇧*[f] T ≘ U →
       ∀f1. K ⊢ 𝐅+⦃T⦄ ≘ f1 → f ~⊚ f1 ≘ f2.
 /4 width=11 by frees_lifts, frees_mono, coafter_eq_repl_back0/ qed-.
 
@@ -180,7 +180,7 @@ lemma frees_fwd_coafter:
 
 lemma frees_inv_lifts_ex:
       ∀b,f2,L,U. L ⊢ 𝐅+⦃U⦄ ≘ f2 →
-      ∀f,K. ⬇*[b,f] L ≘ K → ∀T. ⬆*[f] T ≘ U →
+      ∀f,K. ⇩*[b,f] L ≘ K → ∀T. ⇧*[f] T ≘ U →
       ∃∃f1. f ~⊚ f1 ≘ f2 & K ⊢ 𝐅+⦃T⦄ ≘ f1.
 #b #f2 #L #U #Hf2 #f #K #HLK #T elim (frees_total K T)
 /3 width=9 by frees_fwd_coafter, ex2_intro/
@@ -188,7 +188,7 @@ qed-.
 
 lemma frees_inv_lifts_SO:
       ∀b,f,L,U. L ⊢ 𝐅+⦃U⦄ ≘ f →
-      ∀K. ⬇*[b,𝐔❴1❵] L ≘ K → ∀T. ⬆*[1] T ≘ U →
+      ∀K. ⇩*[b,𝐔❴1❵] L ≘ K → ∀T. ⇧*[1] T ≘ U →
       K ⊢ 𝐅+⦃T⦄ ≘ ⫱f.
 #b #f #L #U #H #K #HLK #T #HTU elim(frees_inv_lifts_ex … H … HLK … HTU) -b -L -U
 #f1 #Hf #Hf1 elim (coafter_inv_nxx … Hf) -Hf
@@ -197,7 +197,7 @@ qed-.
 
 lemma frees_inv_lifts:
       ∀b,f2,L,U. L ⊢ 𝐅+⦃U⦄ ≘ f2 →
-      ∀f,K. ⬇*[b,f] L ≘ K → ∀T. ⬆*[f] T ≘ U →
+      ∀f,K. ⇩*[b,f] L ≘ K → ∀T. ⇧*[f] T ≘ U →
       ∀f1. f ~⊚ f1 ≘ f2 → K ⊢ 𝐅+⦃T⦄ ≘ f1.
 #b #f2 #L #U #H #f #K #HLK #T #HTU #f1 #Hf2 elim (frees_inv_lifts_ex … H … HLK … HTU) -b -L -U
 /3 width=7 by frees_eq_repl_back, coafter_inj/
@@ -206,7 +206,7 @@ qed-.
 (* Note: this is used by rex_conf and might be modified *)
 lemma frees_inv_drops_next:
       ∀f1,L1,T1. L1 ⊢ 𝐅+⦃T1⦄ ≘ f1 →
-      ∀I2,L2,V2,n. ⬇*[n] L1 ≘ L2.ⓑ{I2}V2 →
+      ∀I2,L2,V2,n. ⇩*[n] L1 ≘ L2.ⓑ{I2}V2 →
       ∀g1. ↑g1 = ⫱*[n] f1 →
       ∃∃g2. L2 ⊢ 𝐅+⦃V2⦄ ≘ g2 & g2 ⊆ g1.
 #f1 #L1 #T1 #H elim H -f1 -L1 -T1