From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Sun, 22 May 2016 11:28:36 +0000 (+0000)
Subject: - notational change for cpg and cpx
X-Git-Tag: make_still_working~584
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=f694e3336cbdabdeefd86f85d827edfd26bf3464;p=helm.git

- notational change for cpg and cpx
- versions table updated on site
---

diff --git a/helm/www/lambdadelta/BTM.html b/helm/www/lambdadelta/BTM.html
index 0ef28910c..34a4afa3e 100644
--- a/helm/www/lambdadelta/BTM.html
+++ b/helm/www/lambdadelta/BTM.html
@@ -222,6 +222,6 @@
     <div xmlns:ld="http://lambdadelta.info/" class="spacer">
       <br />
     </div>
-    <div xmlns:ld="http://lambdadelta.info/" class="spacer">Last update: Sat, 21 May 2016 22:22:39 +0200</div>
+    <div xmlns:ld="http://lambdadelta.info/" class="spacer">Last update: Sun, 22 May 2016 11:12:18 +0200</div>
   </body>
 </html>
diff --git a/helm/www/lambdadelta/apps_2.html b/helm/www/lambdadelta/apps_2.html
index d3a930ba1..47d3e3188 100644
--- a/helm/www/lambdadelta/apps_2.html
+++ b/helm/www/lambdadelta/apps_2.html
@@ -258,6 +258,6 @@
     <div xmlns:ld="http://lambdadelta.info/" class="spacer">
       <br />
     </div>
-    <div xmlns:ld="http://lambdadelta.info/" class="spacer">Last update: Sat, 21 May 2016 22:22:39 +0200</div>
+    <div xmlns:ld="http://lambdadelta.info/" class="spacer">Last update: Sun, 22 May 2016 11:12:18 +0200</div>
   </body>
 </html>
diff --git a/helm/www/lambdadelta/basic_1.html b/helm/www/lambdadelta/basic_1.html
index 0b508490b..2a0caeec0 100644
--- a/helm/www/lambdadelta/basic_1.html
+++ b/helm/www/lambdadelta/basic_1.html
@@ -823,6 +823,6 @@
     <div xmlns:ld="http://lambdadelta.info/" class="spacer">
       <br />
     </div>
-    <div xmlns:ld="http://lambdadelta.info/" class="spacer">Last update: Sat, 21 May 2016 22:22:39 +0200</div>
+    <div xmlns:ld="http://lambdadelta.info/" class="spacer">Last update: Sun, 22 May 2016 11:12:18 +0200</div>
   </body>
 </html>
diff --git a/helm/www/lambdadelta/basic_2.html b/helm/www/lambdadelta/basic_2.html
index 5f68c0935..a50882199 100644
--- a/helm/www/lambdadelta/basic_2.html
+++ b/helm/www/lambdadelta/basic_2.html
@@ -144,29 +144,29 @@
           <tr>
             <td class="snns capitalize italic cyan">sizes</td>
             <td class="snns italic cyan">files</td>
-            <td class="snnn right italic cyan">143</td>
+            <td class="snnn right italic cyan">133</td>
             <td class="snns italic cyan">characters</td>
-            <td class="snnn right italic cyan">116151</td>
+            <td class="snnn right italic cyan">92667</td>
             <td class="snns italic cyan">nodes</td>
-            <td class="ssnn right italic cyan">604153</td>
+            <td class="ssnn right italic cyan">334596</td>
           </tr>
           <tr>
             <td class="snns capitalize italic green">propositions</td>
             <td class="snns italic green">theorems</td>
-            <td class="snnn right italic green">45</td>
+            <td class="snnn right italic green">44</td>
             <td class="snns italic green">lemmas</td>
-            <td class="snnn right italic green">428</td>
+            <td class="snnn right italic green">369</td>
             <td class="snns italic green">total</td>
-            <td class="ssnn right italic green">473</td>
+            <td class="ssnn right italic green">413</td>
           </tr>
           <tr>
             <td class="snss capitalize italic yellow">concepts</td>
             <td class="snss italic yellow">declared</td>
-            <td class="snsn right italic yellow">23</td>
+            <td class="snsn right italic yellow">22</td>
             <td class="snss italic yellow">defined</td>
-            <td class="snsn right italic yellow">33</td>
+            <td class="snsn right italic yellow">32</td>
             <td class="snss italic yellow">total</td>
-            <td class="sssn right italic yellow">56</td>
+            <td class="sssn right italic yellow">54</td>
           </tr>
         </tbody>
       </table>
@@ -795,6 +795,6 @@
     <div xmlns:ld="http://lambdadelta.info/" class="spacer">
       <br />
     </div>
-    <div xmlns:ld="http://lambdadelta.info/" class="spacer">Last update: Sat, 21 May 2016 22:22:39 +0200</div>
+    <div xmlns:ld="http://lambdadelta.info/" class="spacer">Last update: Sun, 22 May 2016 11:12:18 +0200</div>
   </body>
 </html>
diff --git a/helm/www/lambdadelta/documentation.html b/helm/www/lambdadelta/documentation.html
index 92b0a113a..6b1211cc0 100644
--- a/helm/www/lambdadelta/documentation.html
+++ b/helm/www/lambdadelta/documentation.html
@@ -389,6 +389,6 @@
     <div xmlns:ld="http://lambdadelta.info/" class="spacer">
       <br />
     </div>
-    <div xmlns:ld="http://lambdadelta.info/" class="spacer">Last update: Sat, 21 May 2016 22:22:38 +0200</div>
+    <div xmlns:ld="http://lambdadelta.info/" class="spacer">Last update: Sun, 22 May 2016 11:12:18 +0200</div>
   </body>
 </html>
diff --git a/helm/www/lambdadelta/ground_1.html b/helm/www/lambdadelta/ground_1.html
index 8f3f5c5fa..a97822a7f 100644
--- a/helm/www/lambdadelta/ground_1.html
+++ b/helm/www/lambdadelta/ground_1.html
@@ -291,6 +291,6 @@
     <div xmlns:ld="http://lambdadelta.info/" class="spacer">
       <br />
     </div>
-    <div xmlns:ld="http://lambdadelta.info/" class="spacer">Last update: Sat, 21 May 2016 22:22:39 +0200</div>
+    <div xmlns:ld="http://lambdadelta.info/" class="spacer">Last update: Sun, 22 May 2016 11:12:18 +0200</div>
   </body>
 </html>
diff --git a/helm/www/lambdadelta/ground_2.html b/helm/www/lambdadelta/ground_2.html
index bb0294819..e5d2f0e37 100644
--- a/helm/www/lambdadelta/ground_2.html
+++ b/helm/www/lambdadelta/ground_2.html
@@ -134,27 +134,27 @@
             <td class="snns italic cyan">files</td>
             <td class="snnn right italic cyan">86</td>
             <td class="snns italic cyan">characters</td>
-            <td class="snnn right italic cyan">109249</td>
+            <td class="snnn right italic cyan">98269</td>
             <td class="snns italic cyan">nodes</td>
-            <td class="ssnn right italic cyan">217872</td>
+            <td class="ssnn right italic cyan">207871</td>
           </tr>
           <tr>
             <td class="snns capitalize italic green">propositions</td>
             <td class="snns italic green">theorems</td>
-            <td class="snnn right italic green">30</td>
+            <td class="snnn right italic green">23</td>
             <td class="snns italic green">lemmas</td>
-            <td class="snnn right italic green">506</td>
+            <td class="snnn right italic green">497</td>
             <td class="snns italic green">total</td>
-            <td class="ssnn right italic green">536</td>
+            <td class="ssnn right italic green">520</td>
           </tr>
           <tr>
             <td class="snss capitalize italic yellow">concepts</td>
             <td class="snss italic yellow">declared</td>
-            <td class="snsn right italic yellow">59</td>
-            <td class="snss italic yellow">defined</td>
             <td class="snsn right italic yellow">55</td>
+            <td class="snss italic yellow">defined</td>
+            <td class="snsn right italic yellow">57</td>
             <td class="snss italic yellow">total</td>
-            <td class="sssn right italic yellow">114</td>
+            <td class="sssn right italic yellow">112</td>
           </tr>
         </tbody>
       </table>
@@ -759,6 +759,6 @@
     <div xmlns:ld="http://lambdadelta.info/" class="spacer">
       <br />
     </div>
-    <div xmlns:ld="http://lambdadelta.info/" class="spacer">Last update: Sat, 21 May 2016 22:22:39 +0200</div>
+    <div xmlns:ld="http://lambdadelta.info/" class="spacer">Last update: Sun, 22 May 2016 11:12:18 +0200</div>
   </body>
 </html>
diff --git a/helm/www/lambdadelta/implementation.html b/helm/www/lambdadelta/implementation.html
index 868f34477..99992380f 100644
--- a/helm/www/lambdadelta/implementation.html
+++ b/helm/www/lambdadelta/implementation.html
@@ -302,6 +302,6 @@
     <div xmlns:ld="http://lambdadelta.info/" class="spacer">
       <br />
     </div>
-    <div xmlns:ld="http://lambdadelta.info/" class="spacer">Last update: Sat, 21 May 2016 22:22:38 +0200</div>
+    <div xmlns:ld="http://lambdadelta.info/" class="spacer">Last update: Sun, 22 May 2016 11:12:18 +0200</div>
   </body>
 </html>
diff --git a/helm/www/lambdadelta/index.html b/helm/www/lambdadelta/index.html
index 93c9c5e9d..c6a88f020 100644
--- a/helm/www/lambdadelta/index.html
+++ b/helm/www/lambdadelta/index.html
@@ -283,6 +283,6 @@
     <div xmlns:ld="http://lambdadelta.info/" class="spacer">
       <br />
     </div>
-    <div xmlns:ld="http://lambdadelta.info/" class="spacer">Last update: Sat, 21 May 2016 22:22:38 +0200</div>
+    <div xmlns:ld="http://lambdadelta.info/" class="spacer">Last update: Sun, 22 May 2016 11:12:18 +0200</div>
   </body>
 </html>
diff --git a/helm/www/lambdadelta/news.html b/helm/www/lambdadelta/news.html
index 25794154e..f53e74441 100644
--- a/helm/www/lambdadelta/news.html
+++ b/helm/www/lambdadelta/news.html
@@ -380,6 +380,6 @@
     <div xmlns:ld="http://lambdadelta.info/" class="spacer">
       <br />
     </div>
-    <div xmlns:ld="http://lambdadelta.info/" class="spacer">Last update: Sat, 21 May 2016 22:22:37 +0200</div>
+    <div xmlns:ld="http://lambdadelta.info/" class="spacer">Last update: Sun, 22 May 2016 11:12:18 +0200</div>
   </body>
 </html>
diff --git a/helm/www/lambdadelta/specification.html b/helm/www/lambdadelta/specification.html
index 9c941d31c..f1c9fb06e 100644
--- a/helm/www/lambdadelta/specification.html
+++ b/helm/www/lambdadelta/specification.html
@@ -127,8 +127,8 @@
           <tr>
             <td class="snns top capitalize italic gray">version</td>
             <td class="snnn top capitalize italic gray">name</td>
-            <td class="snnn top capitalize italic gray">developed with</td>
             <td class="snnn top capitalize italic gray">stage</td>
+            <td class="snnn top capitalize italic gray">developed with</td>
             <td class="snnn top capitalize italic gray">started</td>
             <td class="snnn top capitalize italic gray">announced</td>
             <td class="snnn top capitalize italic gray">released</td>
@@ -155,17 +155,15 @@
               <a href="http://lambdadelta.info/specification.html#v2">Version 2</a>
             </td>
             <td class="snnn top orange">"basic_2"</td>
+            <td class="snns top orange">"A2"</td>
             <td class="snnn top orange">
-              <a href="http://matita.cs.unibo.it/">Matita 0.99.2</a>
+              <a href="http://matita.cs.unibo.it/">Matita 0.99.3</a>
             </td>
-            <td class="snns top orange">"A2"</td>
             <td class="snnn top orange">October 2015</td>
             <td class="snnn top orange" />
             <td class="snnn top orange" />
             <td class="snnn top orange" />
-            <td class="ssnn top orange">
-              <br />
-            </td>
+            <td class="ssnn top orange" />
           </tr>
           <tr>
             <td class="nnns top orange">
@@ -174,10 +172,10 @@
             <td class="nnnn top orange">
               <br />
             </td>
-            <td class="nnnn top orange">
-              <br />
-            </td>
             <td class="snns top orange">"A1"</td>
+            <td class="snnn top orange">
+              <a href="http://matita.cs.unibo.it/">Matita 0.99.2</a>
+            </td>
             <td class="snnn top orange">April 2011</td>
             <td class="snnn top orange">June 2014</td>
             <td class="snnn top orange">October 2014</td>
@@ -190,27 +188,25 @@
           <tr>
             <td class="snns top orange">Abandoned</td>
             <td class="snnn top orange" />
+            <td class="snnn top orange" />
             <td class="snnn top orange">
               <a href="http://coq.inria.fr/">Coq 7.3.1</a>
             </td>
-            <td class="snnn top orange" />
             <td class="snnn top orange">March 2008</td>
             <td class="snnn top orange" />
             <td class="snnn top orange" />
             <td class="snnn top orange">February 2011</td>
-            <td class="ssnn top orange">
-              <br />
-            </td>
+            <td class="ssnn top orange" />
           </tr>
           <tr>
             <td class="snss top red">
               <a href="http://lambdadelta.info/specification.html#v1">Version 1</a>
             </td>
             <td class="snsn top red">"basic_1"</td>
+            <td class="snsn top red" />
             <td class="snsn top red">
               <a href="http://coq.inria.fr/">Coq 7.3.1</a>
             </td>
-            <td class="snsn top red" />
             <td class="snsn top red">May 2004</td>
             <td class="snsn top red">December 2005</td>
             <td class="snsn top red">November 2006</td>
@@ -382,6 +378,6 @@
     <div xmlns:ld="http://lambdadelta.info/" class="spacer">
       <br />
     </div>
-    <div xmlns:ld="http://lambdadelta.info/" class="spacer">Last update: Sat, 21 May 2016 22:22:39 +0200</div>
+    <div xmlns:ld="http://lambdadelta.info/" class="spacer">Last update: Sun, 22 May 2016 11:12:18 +0200</div>
   </body>
 </html>
diff --git a/helm/www/lambdadelta/web/home/versions.tbl b/helm/www/lambdadelta/web/home/versions.tbl
index 1d88064b3..d89ffd340 100644
--- a/helm/www/lambdadelta/web/home/versions.tbl
+++ b/helm/www/lambdadelta/web/home/versions.tbl
@@ -2,35 +2,41 @@ name "versions"
 
 table {
    class "gray" 
-      [ "version" "name" "developed with"
-        "stage" "started" "announced" "released" "concluded"
+      [ "version" "name"
+        "stage" "developed with"
+        "started" "announced" "released" "concluded"
         "references"
       ]
    class "yellow"
-      [ @@("specification#v3" "Version 3") "\"basic_3\"" ""
-        "" "" "" "" ""
+      [ @@("specification#v3" "Version 3") "\"basic_3\""
+        "" ""
+        "" "" "" ""
         @@("documentation#ldJ3a" "J3a")
       ]
    class "orange" {
       [ { @@("specification#v2" "Version 2") * }
         { "\"basic_2\"" * }
-        { @("http://matita.cs.unibo.it/" "Matita 0.99.2") * }
-        { [ "\"A2\"" "October 2015" "" "" ""
-            *
+        { [ "\"A2\"" @("http://matita.cs.unibo.it/" "Matita 0.99.3")
+            "October 2015" "" "" ""
+            ""  *
           ]
-          [ "\"A1\"" "April 2011" "June 2014" "October 2014" "August 2015" 
-            @@("documentation#ldV2a" "V2a") + " " + @@("documentation#ldR2c" "R2c")
+          [ "\"A1\"" @("http://matita.cs.unibo.it/" "Matita 0.99.2")
+            "April 2011" "June 2014" "October 2014" "August 2015" 
+            @@("documentation#ldV2a" "V2a") + " " + @@("documentation#ldR2c" "R2c") *
           ]
         }
       ]
-      [ "Abandoned" "" @("http://coq.inria.fr/" "Coq 7.3.1")
-        "" "March 2008" "" "" "February 2011" *
+      [ "Abandoned" "" 
+        "" @("http://coq.inria.fr/" "Coq 7.3.1")
+        "March 2008" "" "" "February 2011"
+        ""  *
       ]
    }
    class "red"
-      [ @@("specification#v1" "Version 1") "\"basic_1\"" @("http://coq.inria.fr/" "Coq 7.3.1")
-        "" "May 2004" "December 2005" "November 2006" "May 2008"
-        @@("documentation#ldV1a" "V1a") + " " + @@("documentation#ldJ1a" "J1a")
+      [ @@("specification#v1" "Version 1") "\"basic_1\""
+        "" @("http://coq.inria.fr/" "Coq 7.3.1")
+        "May 2004" "December 2005" "November 2006" "May 2008"
+        @@("documentation#ldV1a" "V1a") + " " + @@("documentation#ldJ1a" "J1a") *
       ]
 }
 
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/pred_6.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/pred_6.ma
deleted file mode 100644
index 8832e1fb4..000000000
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/pred_6.ma
+++ /dev/null
@@ -1,19 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||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                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 ➡ break [ term 46 c , break term 46 h ] break term 46 T2 )"
-   non associative with precedence 45
-   for @{ 'PRed $c $h $G $L $T1 $T2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/predty_5.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/predty_5.ma
new file mode 100644
index 000000000..8ffd00516
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/predty_5.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 ⬈ break [ term 46 h ] break term 46 T2 )"
+   non associative with precedence 45
+   for @{ 'PRedTy $h $G $L $T1 $T2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/predty_6.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/predty_6.ma
new file mode 100644
index 000000000..096a26e4c
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/predty_6.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 ⬈ break [ term 46 c , break term 46 h ] break term 46 T2 )"
+   non associative with precedence 45
+   for @{ 'PRedTy $c $h $G $L $T1 $T2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxe.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxe.ma
index 00fd04247..e71eebdc4 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxe.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxe.ma
@@ -19,14 +19,14 @@ include "basic_2/computation/csx.ma".
 (* EVALUATION FOR CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION ON TERMS ****)
 
 definition cpxe: ∀h. sd h → relation4 genv lenv term term ≝
-                 λh,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 ∧ ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T2⦄.
+                 λh,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h, o] T2 ∧ ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃T2⦄.
 
 interpretation "evaluation for context-sensitive extended parallel reduction (term)"
    'PRedEval h o G L T1 T2 = (cpxe h o G L T1 T2).
 
 (* Basic properties *********************************************************)
 
-lemma csx_cpxe: ∀h,o,G,L,T1. ⦃G, L⦄ ⊢ ⬊*[h, o] T1 → ∃T2. ⦃G, L⦄ ⊢ T1 ➡*[h, o] 𝐍⦃T2⦄.
+lemma csx_cpxe: ∀h,o,G,L,T1. ⦃G, L⦄ ⊢ ⬊*[h, o] T1 → ∃T2. ⦃G, L⦄ ⊢ T1 ⬈*[h, o] 𝐍⦃T2⦄.
 #h #o #G #L #T1 #H @(csx_ind … H) -T1
 #T1 #_ #IHT1 elim (cnx_dec h o G L T1) /3 width=3 by ex_intro, conj/
 * #T #H1T1 #H2T1 elim (IHT1 … H1T1 H2T1) -IHT1 -H2T1
diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs.ma
index 037113f24..d21a90427 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs.ma
@@ -27,45 +27,45 @@ interpretation "extended context-sensitive parallel computation (term)"
 (* Basic eliminators ********************************************************)
 
 lemma cpxs_ind: ∀h,o,G,L,T1. ∀R:predicate term. R T1 →
-                (∀T,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T → ⦃G, L⦄ ⊢ T ➡[h, o] T2 → R T → R T2) →
-                ∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 → R T2.
+                (∀T,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h, o] T → ⦃G, L⦄ ⊢ T ⬈[h, o] T2 → R T → R T2) →
+                ∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h, o] T2 → R T2.
 #h #o #L #G #T1 #R #HT1 #IHT1 #T2 #HT12
 @(TC_star_ind … HT1 IHT1 … HT12) //
 qed-.
 
 lemma cpxs_ind_dx: ∀h,o,G,L,T2. ∀R:predicate term. R T2 →
-                   (∀T1,T. ⦃G, L⦄ ⊢ T1 ➡[h, o] T → ⦃G, L⦄ ⊢ T ➡*[h, o] T2 → R T → R T1) →
-                   ∀T1. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 → R T1.
+                   (∀T1,T. ⦃G, L⦄ ⊢ T1 ⬈[h, o] T → ⦃G, L⦄ ⊢ T ⬈*[h, o] T2 → R T → R T1) →
+                   ∀T1. ⦃G, L⦄ ⊢ T1 ⬈*[h, o] T2 → R T1.
 #h #o #G #L #T2 #R #HT2 #IHT2 #T1 #HT12
 @(TC_star_ind_dx … HT2 IHT2 … HT12) //
 qed-.
 
 (* Basic properties *********************************************************)
 
-lemma cpxs_refl: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ T ➡*[h, o] T.
+lemma cpxs_refl: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ T ⬈*[h, o] T.
 /2 width=1 by inj/ qed.
 
-lemma cpx_cpxs: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2.
+lemma cpx_cpxs: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈[h, o] T2 → ⦃G, L⦄ ⊢ T1 ⬈*[h, o] T2.
 /2 width=1 by inj/ qed.
 
-lemma cpxs_strap1: ∀h,o,G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T →
-                   ∀T2. ⦃G, L⦄ ⊢ T ➡[h, o] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2.
+lemma cpxs_strap1: ∀h,o,G,L,T1,T. ⦃G, L⦄ ⊢ T1 ⬈*[h, o] T →
+                   ∀T2. ⦃G, L⦄ ⊢ T ⬈[h, o] T2 → ⦃G, L⦄ ⊢ T1 ⬈*[h, o] T2.
 normalize /2 width=3 by step/ qed.
 
-lemma cpxs_strap2: ∀h,o,G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡[h, o] T →
-                   ∀T2. ⦃G, L⦄ ⊢ T ➡*[h, o] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2.
+lemma cpxs_strap2: ∀h,o,G,L,T1,T. ⦃G, L⦄ ⊢ T1 ⬈[h, o] T →
+                   ∀T2. ⦃G, L⦄ ⊢ T ⬈*[h, o] T2 → ⦃G, L⦄ ⊢ T1 ⬈*[h, o] T2.
 normalize /2 width=3 by TC_strap/ qed.
 
 lemma lsubr_cpxs_trans: ∀h,o,G. lsub_trans … (cpxs h o G) lsubr.
 /3 width=5 by lsubr_cpx_trans, LTC_lsub_trans/
 qed-.
 
-lemma cprs_cpxs: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2.
+lemma cprs_cpxs: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈* T2 → ⦃G, L⦄ ⊢ T1 ⬈*[h, o] T2.
 #h #o #G #L #T1 #T2 #H @(cprs_ind … H) -T2 /3 width=3 by cpxs_strap1, cpr_cpx/
 qed.
 
 lemma cpxs_sort: ∀h,o,G,L,s,d1. deg h o s d1 →
-                 ∀d2. d2 ≤ d1 → ⦃G, L⦄ ⊢ ⋆s ➡*[h, o] ⋆((next h)^d2 s).
+                 ∀d2. d2 ≤ d1 → ⦃G, L⦄ ⊢ ⋆s ⬈*[h, o] ⋆((next h)^d2 s).
 #h #o #G #L #s #d1 #Hkd1 #d2 @(nat_ind_plus … d2) -d2 /2 width=1 by cpx_cpxs/
 #d2 #IHd2 #Hd21 >iter_SO
 @(cpxs_strap1 … (⋆(iter d2 ℕ (next h) s)))
@@ -75,68 +75,68 @@ lemma cpxs_sort: ∀h,o,G,L,s,d1. deg h o s d1 →
 ]
 qed.
 
-lemma cpxs_bind_dx: ∀h,o,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, o] V2 →
-                    ∀I,T1,T2. ⦃G, L. ⓑ{I}V1⦄ ⊢ T1 ➡*[h, o] T2 →
-                    ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[h, o] ⓑ{a,I}V2.T2.
+lemma cpxs_bind_dx: ∀h,o,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h, o] V2 →
+                    ∀I,T1,T2. ⦃G, L. ⓑ{I}V1⦄ ⊢ T1 ⬈*[h, o] T2 →
+                    ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ⬈*[h, o] ⓑ{a,I}V2.T2.
 #h #o #G #L #V1 #V2 #HV12 #I #T1 #T2 #HT12 #a @(cpxs_ind_dx … HT12) -T1
 /3 width=3 by cpxs_strap2, cpx_cpxs, cpx_pair_sn, cpx_bind/
 qed.
 
-lemma cpxs_flat_dx: ∀h,o,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, o] V2 →
-                    ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 →
-                    ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡*[h, o] ⓕ{I}V2.T2.
+lemma cpxs_flat_dx: ∀h,o,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h, o] V2 →
+                    ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h, o] T2 →
+                    ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬈*[h, o] ⓕ{I}V2.T2.
 #h #o #G #L #V1 #V2 #HV12 #T1 #T2 #HT12 @(cpxs_ind … HT12) -T2
 /3 width=5 by cpxs_strap1, cpx_cpxs, cpx_pair_sn, cpx_flat/
 qed.
 
-lemma cpxs_flat_sn: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 →
-                    ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, o] V2 →
-                    ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡*[h, o] ⓕ{I}V2.T2.
+lemma cpxs_flat_sn: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈[h, o] T2 →
+                    ∀V1,V2. ⦃G, L⦄ ⊢ V1 ⬈*[h, o] V2 →
+                    ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬈*[h, o] ⓕ{I}V2.T2.
 #h #o #G #L #T1 #T2 #HT12 #V1 #V2 #H @(cpxs_ind … H) -V2
 /3 width=5 by cpxs_strap1, cpx_cpxs, cpx_pair_sn, cpx_flat/
 qed.
 
-lemma cpxs_pair_sn: ∀h,o,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, o] V2 →
-                    ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ➡*[h, o] ②{I}V2.T.
+lemma cpxs_pair_sn: ∀h,o,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈*[h, o] V2 →
+                    ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ⬈*[h, o] ②{I}V2.T.
 #h #o #I #G #L #V1 #V2 #H @(cpxs_ind … H) -V2
 /3 width=3 by cpxs_strap1, cpx_pair_sn/
 qed.
 
 lemma cpxs_zeta: ∀h,o,G,L,V,T1,T,T2. ⬆[0, 1] T2 ≡ T →
-                 ⦃G, L.ⓓV⦄ ⊢ T1 ➡*[h, o] T → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡*[h, o] T2.
+                 ⦃G, L.ⓓV⦄ ⊢ T1 ⬈*[h, o] T → ⦃G, L⦄ ⊢ +ⓓV.T1 ⬈*[h, o] T2.
 #h #o #G #L #V #T1 #T #T2 #HT2 #H @(cpxs_ind_dx … H) -T1
 /3 width=3 by cpxs_strap2, cpx_cpxs, cpx_bind, cpx_zeta/
 qed.
 
-lemma cpxs_eps: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 →
-                ∀V. ⦃G, L⦄ ⊢ ⓝV.T1 ➡*[h, o] T2.
+lemma cpxs_eps: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h, o] T2 →
+                ∀V. ⦃G, L⦄ ⊢ ⓝV.T1 ⬈*[h, o] T2.
 #h #o #G #L #T1 #T2 #H @(cpxs_ind … H) -T2
 /3 width=3 by cpxs_strap1, cpx_cpxs, cpx_eps/
 qed.
 
-lemma cpxs_ct: ∀h,o,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, o] V2 →
-               ∀T. ⦃G, L⦄ ⊢ ⓝV1.T ➡*[h, o] V2.
+lemma cpxs_ct: ∀h,o,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈*[h, o] V2 →
+               ∀T. ⦃G, L⦄ ⊢ ⓝV1.T ⬈*[h, o] V2.
 #h #o #G #L #V1 #V2 #H @(cpxs_ind … H) -V2
 /3 width=3 by cpxs_strap1, cpx_cpxs, cpx_ct/
 qed.
 
 lemma cpxs_beta_dx: ∀h,o,a,G,L,V1,V2,W1,W2,T1,T2.
-                    ⦃G, L⦄ ⊢ V1 ➡[h, o] V2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡*[h, o] T2 → ⦃G, L⦄ ⊢ W1 ➡[h, o] W2 →
-                    ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡*[h, o] ⓓ{a}ⓝW2.V2.T2.
+                    ⦃G, L⦄ ⊢ V1 ⬈[h, o] V2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈*[h, o] T2 → ⦃G, L⦄ ⊢ W1 ⬈[h, o] W2 →
+                    ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ⬈*[h, o] ⓓ{a}ⓝW2.V2.T2.
 #h #o #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 * -T2
 /4 width=7 by cpx_cpxs, cpxs_strap1, cpxs_bind_dx, cpxs_flat_dx, cpx_beta/
 qed.
 
 lemma cpxs_theta_dx: ∀h,o,a,G,L,V1,V,V2,W1,W2,T1,T2.
-                     ⦃G, L⦄ ⊢ V1 ➡[h, o] V → ⬆[0, 1] V ≡ V2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡*[h, o] T2 →
-                     ⦃G, L⦄ ⊢ W1 ➡[h, o] W2 → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡*[h, o] ⓓ{a}W2.ⓐV2.T2.
+                     ⦃G, L⦄ ⊢ V1 ⬈[h, o] V → ⬆[0, 1] V ≡ V2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈*[h, o] T2 →
+                     ⦃G, L⦄ ⊢ W1 ⬈[h, o] W2 → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ⬈*[h, o] ⓓ{a}W2.ⓐV2.T2.
 #h #o #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 * -T2 
 /4 width=9 by cpx_cpxs, cpxs_strap1, cpxs_bind_dx, cpxs_flat_dx, cpx_theta/
 qed.
 
 (* Basic inversion lemmas ***************************************************)
 
-lemma cpxs_inv_sort1: ∀h,o,G,L,U2,s. ⦃G, L⦄ ⊢ ⋆s ➡*[h, o] U2 →
+lemma cpxs_inv_sort1: ∀h,o,G,L,U2,s. ⦃G, L⦄ ⊢ ⋆s ⬈*[h, o] U2 →
                       ∃∃n,d. deg h o s (n+d) & U2 = ⋆((next h)^n s).
 #h #o #G #L #U2 #s #H @(cpxs_ind … H) -U2
 [ elim (deg_total h o s) #d #Hkd
@@ -150,10 +150,10 @@ lemma cpxs_inv_sort1: ∀h,o,G,L,U2,s. ⦃G, L⦄ ⊢ ⋆s ➡*[h, o] U2 →
 ]
 qed-.
 
-lemma cpxs_inv_cast1: ∀h,o,G,L,W1,T1,U2. ⦃G, L⦄ ⊢ ⓝW1.T1 ➡*[h, o] U2 →
-                      ∨∨ ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ➡*[h, o] W2 & ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 & U2 = ⓝW2.T2
-                       | ⦃G, L⦄ ⊢ T1 ➡*[h, o] U2
-                       | ⦃G, L⦄ ⊢ W1 ➡*[h, o] U2.
+lemma cpxs_inv_cast1: ∀h,o,G,L,W1,T1,U2. ⦃G, L⦄ ⊢ ⓝW1.T1 ⬈*[h, o] U2 →
+                      ∨∨ ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ⬈*[h, o] W2 & ⦃G, L⦄ ⊢ T1 ⬈*[h, o] T2 & U2 = ⓝW2.T2
+                       | ⦃G, L⦄ ⊢ T1 ⬈*[h, o] U2
+                       | ⦃G, L⦄ ⊢ W1 ⬈*[h, o] U2.
 #h #o #G #L #W1 #T1 #U2 #H @(cpxs_ind … H) -U2 /3 width=5 by or3_intro0, ex3_2_intro/
 #U2 #U #_ #HU2 * /3 width=3 by cpxs_strap1, or3_intro1, or3_intro2/ *
 #W #T #HW1 #HT1 #H destruct
@@ -163,14 +163,14 @@ lapply (cpxs_strap1 … HW1 … HW2) -W
 lapply (cpxs_strap1 … HT1 … HT2) -T /3 width=5 by or3_intro0, ex3_2_intro/
 qed-.
 
-lemma cpxs_inv_cnx1: ∀h,o,G,L,T,U. ⦃G, L⦄ ⊢ T ➡*[h, o] U → ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T⦄ → T = U.
+lemma cpxs_inv_cnx1: ∀h,o,G,L,T,U. ⦃G, L⦄ ⊢ T ⬈*[h, o] U → ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃T⦄ → T = U.
 #h #o #G #L #T #U #H @(cpxs_ind_dx … H) -T //
 #T0 #T #H1T0 #_ #IHT #H2T0
 lapply (H2T0 … H1T0) -H1T0 #H destruct /2 width=1 by/
 qed-.
 
-lemma cpxs_neq_inv_step_sn: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 → (T1 = T2 → ⊥) →
-                            ∃∃T. ⦃G, L⦄ ⊢ T1 ➡[h, o] T & T1 = T → ⊥ & ⦃G, L⦄ ⊢ T ➡*[h, o] T2.
+lemma cpxs_neq_inv_step_sn: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h, o] T2 → (T1 = T2 → ⊥) →
+                            ∃∃T. ⦃G, L⦄ ⊢ T1 ⬈[h, o] T & T1 = T → ⊥ & ⦃G, L⦄ ⊢ T ⬈*[h, o] T2.
 #h #o #G #L #T1 #T2 #H @(cpxs_ind_dx … H) -T1
 [ #H elim H -H //
 | #T1 #T #H1 #H2 #IH2 #H12 elim (eq_term_dec T1 T) #H destruct
diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_aaa.ma
index e3d5e8af1..2266edbad 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_aaa.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_aaa.ma
@@ -20,10 +20,10 @@ include "basic_2/computation/cpxs.ma".
 (* Properties about atomic arity assignment on terms ************************)
 
 lemma cpxs_aaa_conf: ∀h,o,G,L,T1,A. ⦃G, L⦄ ⊢ T1 ⁝ A →
-                     ∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 → ⦃G, L⦄ ⊢ T2 ⁝ A.
+                     ∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h, o] T2 → ⦃G, L⦄ ⊢ T2 ⁝ A.
 #h #o #G #L #T1 #A #HT1 #T2 #HT12
 @(TC_Conf3 … HT1 ? HT12) -A -T1 -T2 /2 width=5 by cpx_aaa_conf/
 qed-.
 
-lemma cprs_aaa_conf: ∀G,L,T1,A. ⦃G, L⦄ ⊢ T1 ⁝ A → ∀T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ T2 ⁝ A.
+lemma cprs_aaa_conf: ∀G,L,T1,A. ⦃G, L⦄ ⊢ T1 ⁝ A → ∀T2. ⦃G, L⦄ ⊢ T1 ⬈* T2 → ⦃G, L⦄ ⊢ T2 ⁝ A.
 /3 width=5 by cpxs_aaa_conf, cprs_cpxs/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_cpxs.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_cpxs.ma
index 8f77d9190..4f1d7ceee 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_cpxs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_cpxs.ma
@@ -22,58 +22,58 @@ include "basic_2/computation/cpxs_lift.ma".
 theorem cpxs_trans: ∀h,o,G,L. Transitive … (cpxs h o G L).
 normalize /2 width=3 by trans_TC/ qed-.
 
-theorem cpxs_bind: ∀h,o,a,I,G,L,V1,V2,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡*[h, o] T2 →
-                   ⦃G, L⦄ ⊢ V1 ➡*[h, o] V2 →
-                   ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[h, o] ⓑ{a,I}V2.T2.
+theorem cpxs_bind: ∀h,o,a,I,G,L,V1,V2,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬈*[h, o] T2 →
+                   ⦃G, L⦄ ⊢ V1 ⬈*[h, o] V2 →
+                   ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ⬈*[h, o] ⓑ{a,I}V2.T2.
 #h #o #a #I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cpxs_ind … H) -V2
 /3 width=5 by cpxs_trans, cpxs_bind_dx/
 qed.
 
-theorem cpxs_flat: ∀h,o,I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 →
-                   ⦃G, L⦄ ⊢ V1 ➡*[h, o] V2 →
-                   ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡*[h, o] ⓕ{I}V2.T2.
+theorem cpxs_flat: ∀h,o,I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h, o] T2 →
+                   ⦃G, L⦄ ⊢ V1 ⬈*[h, o] V2 →
+                   ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬈*[h, o] ⓕ{I}V2.T2.
 #h #o #I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cpxs_ind … H) -V2
 /3 width=5 by cpxs_trans, cpxs_flat_dx/
 qed.
 
 theorem cpxs_beta_rc: ∀h,o,a,G,L,V1,V2,W1,W2,T1,T2.
-                      ⦃G, L⦄ ⊢ V1 ➡[h, o] V2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡*[h, o] T2 → ⦃G, L⦄ ⊢ W1 ➡*[h, o] W2 →
-                      ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡*[h, o] ⓓ{a}ⓝW2.V2.T2.
+                      ⦃G, L⦄ ⊢ V1 ⬈[h, o] V2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈*[h, o] T2 → ⦃G, L⦄ ⊢ W1 ⬈*[h, o] W2 →
+                      ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ⬈*[h, o] ⓓ{a}ⓝW2.V2.T2.
 #h #o #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HT12 #H @(cpxs_ind … H) -W2
 /4 width=5 by cpxs_trans, cpxs_beta_dx, cpxs_bind_dx, cpx_pair_sn/
 qed.
 
 theorem cpxs_beta: ∀h,o,a,G,L,V1,V2,W1,W2,T1,T2.
-                   ⦃G, L.ⓛW1⦄ ⊢ T1 ➡*[h, o] T2 → ⦃G, L⦄ ⊢ W1 ➡*[h, o] W2 → ⦃G, L⦄ ⊢ V1 ➡*[h, o] V2 →
-                   ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡*[h, o] ⓓ{a}ⓝW2.V2.T2.
+                   ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈*[h, o] T2 → ⦃G, L⦄ ⊢ W1 ⬈*[h, o] W2 → ⦃G, L⦄ ⊢ V1 ⬈*[h, o] V2 →
+                   ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ⬈*[h, o] ⓓ{a}ⓝW2.V2.T2.
 #h #o #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HT12 #HW12 #H @(cpxs_ind … H) -V2
 /4 width=5 by cpxs_trans, cpxs_beta_rc, cpxs_bind_dx, cpx_flat/
 qed.
 
 theorem cpxs_theta_rc: ∀h,o,a,G,L,V1,V,V2,W1,W2,T1,T2.
-                       ⦃G, L⦄ ⊢ V1 ➡[h, o] V → ⬆[0, 1] V ≡ V2 →
-                       ⦃G, L.ⓓW1⦄ ⊢ T1 ➡*[h, o] T2 → ⦃G, L⦄ ⊢ W1 ➡*[h, o] W2 →
-                       ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡*[h, o] ⓓ{a}W2.ⓐV2.T2.
+                       ⦃G, L⦄ ⊢ V1 ⬈[h, o] V → ⬆[0, 1] V ≡ V2 →
+                       ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈*[h, o] T2 → ⦃G, L⦄ ⊢ W1 ⬈*[h, o] W2 →
+                       ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ⬈*[h, o] ⓓ{a}W2.ⓐV2.T2.
 #h #o #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HT12 #H @(cpxs_ind … H) -W2
 /3 width=5 by cpxs_trans, cpxs_theta_dx, cpxs_bind_dx/
 qed.
 
 theorem cpxs_theta: ∀h,o,a,G,L,V1,V,V2,W1,W2,T1,T2.
-                    ⬆[0, 1] V ≡ V2 → ⦃G, L⦄ ⊢ W1 ➡*[h, o] W2 →
-                    ⦃G, L.ⓓW1⦄ ⊢ T1 ➡*[h, o] T2 → ⦃G, L⦄ ⊢ V1 ➡*[h, o] V →
-                    ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡*[h, o] ⓓ{a}W2.ⓐV2.T2.
+                    ⬆[0, 1] V ≡ V2 → ⦃G, L⦄ ⊢ W1 ⬈*[h, o] W2 →
+                    ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈*[h, o] T2 → ⦃G, L⦄ ⊢ V1 ⬈*[h, o] V →
+                    ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ⬈*[h, o] ⓓ{a}W2.ⓐV2.T2.
 #h #o #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV2 #HW12 #HT12 #H @(TC_ind_dx … V1 H) -V1
 /3 width=5 by cpxs_trans, cpxs_theta_rc, cpxs_flat_dx/
 qed.
 
 (* Advanced inversion lemmas ************************************************)
 
-lemma cpxs_inv_appl1: ∀h,o,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓐV1.T1 ➡*[h, o] U2 →
-                      ∨∨ ∃∃V2,T2.       ⦃G, L⦄ ⊢ V1 ➡*[h, o] V2 & ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 &
+lemma cpxs_inv_appl1: ∀h,o,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓐV1.T1 ⬈*[h, o] U2 →
+                      ∨∨ ∃∃V2,T2.       ⦃G, L⦄ ⊢ V1 ⬈*[h, o] V2 & ⦃G, L⦄ ⊢ T1 ⬈*[h, o] T2 &
                                         U2 = ⓐV2. T2
-                       | ∃∃a,W,T.       ⦃G, L⦄ ⊢ T1 ➡*[h, o] ⓛ{a}W.T & ⦃G, L⦄ ⊢ ⓓ{a}ⓝW.V1.T ➡*[h, o] U2
-                       | ∃∃a,V0,V2,V,T. ⦃G, L⦄ ⊢ V1 ➡*[h, o] V0 & ⬆[0,1] V0 ≡ V2 &
-                                        ⦃G, L⦄ ⊢ T1 ➡*[h, o] ⓓ{a}V.T & ⦃G, L⦄ ⊢ ⓓ{a}V.ⓐV2.T ➡*[h, o] U2.
+                       | ∃∃a,W,T.       ⦃G, L⦄ ⊢ T1 ⬈*[h, o] ⓛ{a}W.T & ⦃G, L⦄ ⊢ ⓓ{a}ⓝW.V1.T ⬈*[h, o] U2
+                       | ∃∃a,V0,V2,V,T. ⦃G, L⦄ ⊢ V1 ⬈*[h, o] V0 & ⬆[0,1] V0 ≡ V2 &
+                                        ⦃G, L⦄ ⊢ T1 ⬈*[h, o] ⓓ{a}V.T & ⦃G, L⦄ ⊢ ⓓ{a}V.ⓐV2.T ⬈*[h, o] U2.
 #h #o #G #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 [ /3 width=5 by or3_intro0, ex3_2_intro/ ]
 #U #U2 #_ #HU2 * *
 [ #V0 #T0 #HV10 #HT10 #H destruct
@@ -108,9 +108,9 @@ lemma lpx_cpx_trans: ∀h,o,G. b_c_transitive … (cpx h o G) (λ_.lpx h o G).
 ]
 qed-.
 
-lemma cpx_bind2: ∀h,o,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, o] V2 →
-                 ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡[h, o] T2 →
-                 ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[h, o] ⓑ{a,I}V2.T2.
+lemma cpx_bind2: ∀h,o,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h, o] V2 →
+                 ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ⬈[h, o] T2 →
+                 ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ⬈*[h, o] ⓑ{a,I}V2.T2.
 /4 width=5 by lpx_cpx_trans, cpxs_bind_dx, lpx_pair/ qed.
 
 (* Advanced properties ******************************************************)
@@ -119,16 +119,16 @@ lemma lpx_cpxs_trans: ∀h,o,G. b_rs_transitive … (cpx h o G) (λ_.lpx h o G).
 #h #o #G @b_c_trans_LTC1 /2 width=3 by lpx_cpx_trans/ (**) (* full auto fails *)
 qed-.
 
-lemma cpxs_bind2_dx: ∀h,o,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, o] V2 →
-                     ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡*[h, o] T2 →
-                     ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[h, o] ⓑ{a,I}V2.T2.
+lemma cpxs_bind2_dx: ∀h,o,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h, o] V2 →
+                     ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ⬈*[h, o] T2 →
+                     ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ⬈*[h, o] ⓑ{a,I}V2.T2.
 /4 width=5 by lpx_cpxs_trans, cpxs_bind_dx, lpx_pair/ qed.
 
 (* Properties on supclosure *************************************************)
 
 lemma fqu_cpxs_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
-                          ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, o] U2 → (T2 = U2 → ⊥) →
-                          ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
+                          ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h, o] U2 → (T2 = U2 → ⊥) →
+                          ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
 #h #o #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
 [ #I #G #L #V1 #V2 #HV12 #_ elim (lift_total V2 0 1)
   #U2 #HVU2 @(ex3_intro … U2)
@@ -156,8 +156,8 @@ lemma fqu_cpxs_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2,
 qed-.
 
 lemma fquq_cpxs_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
-                           ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, o] U2 → (T2 = U2 → ⊥) →
-                           ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
+                           ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h, o] U2 → (T2 = U2 → ⊥) →
+                           ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
 #h #o #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fquq_inv_gen … H12) -H12
 [ #H12 elim (fqu_cpxs_trans_neq … H12 … HTU2 H) -T2
   /3 width=4 by fqu_fquq, ex3_intro/
@@ -166,8 +166,8 @@ lemma fquq_cpxs_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃
 qed-.
 
 lemma fqup_cpxs_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
-                           ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, o] U2 → (T2 = U2 → ⊥) →
-                           ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
+                           ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h, o] U2 → (T2 = U2 → ⊥) →
+                           ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
 #h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1
 [ #G1 #L1 #T1 #H12 #U2 #HTU2 #H elim (fqu_cpxs_trans_neq … H12 … HTU2 H) -T2
   /3 width=4 by fqu_fqup, ex3_intro/
@@ -178,8 +178,8 @@ lemma fqup_cpxs_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2
 qed-.
 
 lemma fqus_cpxs_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
-                           ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, o] U2 → (T2 = U2 → ⊥) →
-                           ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
+                           ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h, o] U2 → (T2 = U2 → ⊥) →
+                           ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
 #h #o #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fqus_inv_gen … H12) -H12
 [ #H12 elim (fqup_cpxs_trans_neq … H12 … HTU2 H) -T2
   /3 width=4 by fqup_fqus, ex3_intro/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_lift.ma
index 9ff2b6f2c..9b4822e58 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_lift.ma
@@ -21,8 +21,8 @@ include "basic_2/computation/cpxs.ma".
 (* Advanced properties ******************************************************)
 
 lemma cpxs_delta: ∀h,o,I,G,L,K,V,V2,i.
-                  ⬇[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ➡*[h, o] V2 →
-                  ∀W2. ⬆[0, i+1] V2 ≡ W2 → ⦃G, L⦄ ⊢ #i ➡*[h, o] W2.
+                  ⬇[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ⬈*[h, o] V2 →
+                  ∀W2. ⬆[0, i+1] V2 ≡ W2 → ⦃G, L⦄ ⊢ #i ⬈*[h, o] W2.
 #h #o #I #G #L #K #V #V2 #i #HLK #H elim H -V2
 [ /3 width=9 by cpx_cpxs, cpx_delta/
 | #V1 lapply (drop_fwd_drop2 … HLK) -HLK
@@ -31,7 +31,7 @@ lemma cpxs_delta: ∀h,o,I,G,L,K,V,V2,i.
 qed.
 
 lemma lstas_cpxs: ∀h,o,G,L,T1,T2,d2. ⦃G, L⦄ ⊢ T1 •*[h, d2] T2 →
-                  ∀d1. ⦃G, L⦄ ⊢ T1 ▪[h, o] d1 → d2 ≤ d1 → ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2.
+                  ∀d1. ⦃G, L⦄ ⊢ T1 ▪[h, o] d1 → d2 ≤ d1 → ⦃G, L⦄ ⊢ T1 ⬈*[h, o] T2.
 #h #o #G #L #T1 #T2 #d2 #H elim H -G -L -T1 -T2 -d2 //
 [ /3 width=3 by cpxs_sort, da_inv_sort/
 | #G #L #K #V1 #V2 #W2 #i #d2 #HLK #_ #HVW2 #IHV12 #d1 #H #Hd21
@@ -50,9 +50,9 @@ qed.
 
 (* Advanced inversion lemmas ************************************************)
 
-lemma cpxs_inv_lref1: ∀h,o,G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡*[h, o] T2 →
+lemma cpxs_inv_lref1: ∀h,o,G,L,T2,i. ⦃G, L⦄ ⊢ #i ⬈*[h, o] T2 →
                       T2 = #i ∨
-                      ∃∃I,K,V1,T1. ⬇[i] L ≡ K.ⓑ{I}V1 & ⦃G, K⦄ ⊢ V1 ➡*[h, o] T1 &
+                      ∃∃I,K,V1,T1. ⬇[i] L ≡ K.ⓑ{I}V1 & ⦃G, K⦄ ⊢ V1 ⬈*[h, o] T1 &
                                    ⬆[0, i+1] T1 ≡ T2.
 #h #o #G #L #T2 #i #H @(cpxs_ind … H) -T2 /2 width=1 by or_introl/
 #T #T2 #_ #HT2 *
@@ -77,17 +77,17 @@ qed-.
 
 (* Properties on supclosure *************************************************)
 
-lemma fqu_cpxs_trans: ∀h,o,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, o] U2 →
+lemma fqu_cpxs_trans: ∀h,o,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h, o] U2 →
                       ∀T1. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
-                      ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, o] U1 & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
+                      ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h, o] U1 & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
 #h #o #G1 #G2 #L1 #L2 #T2 #U2 #H @(cpxs_ind_dx … H) -T2 /2 width=3 by ex2_intro/
 #T #T2 #HT2 #_ #IHTU2 #T1 #HT1 elim (fqu_cpx_trans … HT1 … HT2) -T
 #T #HT1 #HT2 elim (IHTU2 … HT2) -T2 /3 width=3 by cpxs_strap2, ex2_intro/
 qed-.
 
-lemma fquq_cpxs_trans: ∀h,o,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, o] U2 →
+lemma fquq_cpxs_trans: ∀h,o,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h, o] U2 →
                        ∀T1. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
-                       ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, o] U1 & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
+                       ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h, o] U1 & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
 #h #o #G1 #G2 #L1 #L2 #T2 #U2 #HTU2 #T1 #H elim (fquq_inv_gen … H) -H
 [ #HT12 elim (fqu_cpxs_trans … HTU2 … HT12) /3 width=3 by fqu_fquq, ex2_intro/
 | * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
@@ -97,20 +97,20 @@ qed-.
 lemma fquq_lstas_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
                         ∀U2,d1. ⦃G2, L2⦄ ⊢ T2 •*[h, d1] U2 →
                         ∀d2. ⦃G2, L2⦄ ⊢ T2 ▪[h, o] d2 → d1 ≤ d2 →
-                        ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, o] U1 & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
+                        ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h, o] U1 & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
 /3 width=5 by fquq_cpxs_trans, lstas_cpxs/ qed-.
 
-lemma fqup_cpxs_trans: ∀h,o,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, o] U2 →
+lemma fqup_cpxs_trans: ∀h,o,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h, o] U2 →
                        ∀T1. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
-                       ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, o] U1 & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
+                       ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h, o] U1 & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
 #h #o #G1 #G2 #L1 #L2 #T2 #U2 #H @(cpxs_ind_dx … H) -T2 /2 width=3 by ex2_intro/
 #T #T2 #HT2 #_ #IHTU2 #T1 #HT1 elim (fqup_cpx_trans … HT1 … HT2) -T
 #U1 #HTU1 #H2 elim (IHTU2 … H2) -T2 /3 width=3 by cpxs_strap2, ex2_intro/
 qed-.
 
-lemma fqus_cpxs_trans: ∀h,o,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, o] U2 →
+lemma fqus_cpxs_trans: ∀h,o,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h, o] U2 →
                        ∀T1. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
-                       ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, o] U1 & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
+                       ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h, o] U1 & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
 #h #o #G1 #G2 #L1 #L2 #T2 #U2 #HTU2 #T1 #H elim (fqus_inv_gen … H) -H
 [ #HT12 elim (fqup_cpxs_trans … HTU2 … HT12) /3 width=3 by fqup_fqus, ex2_intro/
 | * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
@@ -120,5 +120,5 @@ qed-.
 lemma fqus_lstas_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
                         ∀U2,d1. ⦃G2, L2⦄ ⊢ T2 •*[h, d1] U2 →
                         ∀d2. ⦃G2, L2⦄ ⊢ T2 ▪[h, o] d2 → d1 ≤ d2 →
-                        ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, o] U1 & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
+                        ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h, o] U1 & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
 /3 width=6 by fqus_cpxs_trans, lstas_cpxs/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_lleq.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_lleq.ma
index be3e0f0b5..2765cf1bc 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_lleq.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_lleq.ma
@@ -19,21 +19,21 @@ include "basic_2/computation/cpxs.ma".
 
 (* Properties on lazy equivalence for local environments ********************)
 
-lemma lleq_cpxs_trans: ∀h,o,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡*[h, o] T2 →
-                       ∀L1. L1 ≡[T1, 0] L2 → ⦃G, L1⦄ ⊢ T1 ➡*[h, o] T2.
+lemma lleq_cpxs_trans: ∀h,o,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ⬈*[h, o] T2 →
+                       ∀L1. L1 ≡[T1, 0] L2 → ⦃G, L1⦄ ⊢ T1 ⬈*[h, o] T2.
 #h #o #G #L2 #T1 #T2 #H @(cpxs_ind_dx … H) -T1
 /4 width=6 by cpx_lleq_conf_dx, lleq_cpx_trans, cpxs_strap2/
 qed-.
 
-lemma cpxs_lleq_conf: ∀h,o,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡*[h, o] T2 →
-                      ∀L1. L2 ≡[T1, 0] L1 → ⦃G, L1⦄ ⊢ T1 ➡*[h, o] T2.
+lemma cpxs_lleq_conf: ∀h,o,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ⬈*[h, o] T2 →
+                      ∀L1. L2 ≡[T1, 0] L1 → ⦃G, L1⦄ ⊢ T1 ⬈*[h, o] T2.
 /3 width=3 by lleq_cpxs_trans, lleq_sym/ qed-.
 
-lemma cpxs_lleq_conf_dx: ∀h,o,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡*[h, o] T2 →
+lemma cpxs_lleq_conf_dx: ∀h,o,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ⬈*[h, o] T2 →
                          ∀L1. L1 ≡[T1, 0] L2 → L1 ≡[T2, 0] L2.
 #h #o #G #L2 #T1 #T2 #H @(cpxs_ind … H) -T2 /3 width=6 by cpx_lleq_conf_dx/
 qed-.
 
-lemma cpxs_lleq_conf_sn: ∀h,o,G,L1,T1,T2. ⦃G, L1⦄ ⊢ T1 ➡*[h, o] T2 →
+lemma cpxs_lleq_conf_sn: ∀h,o,G,L1,T1,T2. ⦃G, L1⦄ ⊢ T1 ⬈*[h, o] T2 →
                          ∀L2. L1 ≡[T1, 0] L2 → L1 ≡[T2, 0] L2.
 /4 width=6 by cpxs_lleq_conf_dx, lleq_sym/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_tsts.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_tsts.ma
index ace2532d9..6434ff37f 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_tsts.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_tsts.ma
@@ -19,13 +19,13 @@ include "basic_2/computation/lpxs_cpxs.ma".
 
 (* Forward lemmas involving same top term structure *************************)
 
-lemma cpxs_fwd_cnx: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T⦄ → ∀U. ⦃G, L⦄ ⊢ T ➡*[h, o] U → T ≂ U.
+lemma cpxs_fwd_cnx: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃T⦄ → ∀U. ⦃G, L⦄ ⊢ T ⬈*[h, o] U → T ≂ U.
 #h #o #G #L #T #HT #U #H
 >(cpxs_inv_cnx1 … H HT) -G -L -T //
 qed-.
 
-lemma cpxs_fwd_sort: ∀h,o,G,L,U,s. ⦃G, L⦄ ⊢ ⋆s ➡*[h, o] U →
-                     ⋆s ≂ U ∨ ⦃G, L⦄ ⊢ ⋆(next h s) ➡*[h, o] U.
+lemma cpxs_fwd_sort: ∀h,o,G,L,U,s. ⦃G, L⦄ ⊢ ⋆s ⬈*[h, o] U →
+                     ⋆s ≂ U ∨ ⦃G, L⦄ ⊢ ⋆(next h s) ⬈*[h, o] U.
 #h #o #G #L #U #s #H
 elim (cpxs_inv_sort1 … H) -H #n #d generalize in match s; -s @(nat_ind_plus … n) -n
 [ #s #_ #H -d destruct /2 width=1 by or_introl/
@@ -41,8 +41,8 @@ elim (cpxs_inv_sort1 … H) -H #n #d generalize in match s; -s @(nat_ind_plus 
 qed-.
 
 (* Basic_1: was just: pr3_iso_beta *)
-lemma cpxs_fwd_beta: ∀h,o,a,G,L,V,W,T,U. ⦃G, L⦄ ⊢ ⓐV.ⓛ{a}W.T ➡*[h, o] U →
-                     ⓐV.ⓛ{a}W.T ≂ U ∨ ⦃G, L⦄ ⊢ ⓓ{a}ⓝW.V.T ➡*[h, o] U.
+lemma cpxs_fwd_beta: ∀h,o,a,G,L,V,W,T,U. ⦃G, L⦄ ⊢ ⓐV.ⓛ{a}W.T ⬈*[h, o] U →
+                     ⓐV.ⓛ{a}W.T ≂ U ∨ ⦃G, L⦄ ⊢ ⓓ{a}ⓝW.V.T ⬈*[h, o] U.
 #h #o #a #G #L #V #W #T #U #H
 elim (cpxs_inv_appl1 … H) -H *
 [ #V0 #T0 #_ #_ #H destruct /2 width=1 by tsts_pair, or_introl/
@@ -58,8 +58,8 @@ qed-.
 (* Note: probably this is an inversion lemma *)
 lemma cpxs_fwd_delta: ∀h,o,I,G,L,K,V1,i. ⬇[i] L ≡ K.ⓑ{I}V1 →
                       ∀V2. ⬆[0, i + 1] V1 ≡ V2 →
-                      ∀U. ⦃G, L⦄ ⊢ #i ➡*[h, o] U →
-                      #i ≂ U ∨ ⦃G, L⦄ ⊢ V2 ➡*[h, o] U.
+                      ∀U. ⦃G, L⦄ ⊢ #i ⬈*[h, o] U →
+                      #i ≂ U ∨ ⦃G, L⦄ ⊢ V2 ⬈*[h, o] U.
 #h #o #I #G #L #K #V1 #i #HLK #V2 #HV12 #U #H
 elim (cpxs_inv_lref1 … H) -H /2 width=1 by or_introl/
 * #I0 #K0 #V0 #U0 #HLK0 #HVU0 #HU0
@@ -67,9 +67,9 @@ lapply (drop_mono … HLK0 … HLK) -HLK0 #H destruct
 /4 width=10 by cpxs_lift, drop_fwd_drop2, or_intror/
 qed-.
 
-lemma cpxs_fwd_theta: ∀h,o,a,G,L,V1,V,T,U. ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}V.T ➡*[h, o] U →
+lemma cpxs_fwd_theta: ∀h,o,a,G,L,V1,V,T,U. ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}V.T ⬈*[h, o] U →
                       ∀V2. ⬆[0, 1] V1 ≡ V2 → ⓐV1.ⓓ{a}V.T ≂ U ∨
-                      ⦃G, L⦄ ⊢ ⓓ{a}V.ⓐV2.T ➡*[h, o] U.
+                      ⦃G, L⦄ ⊢ ⓓ{a}V.ⓐV2.T ⬈*[h, o] U.
 #h #o #a #G #L #V1 #V #T #U #H #V2 #HV12
 elim (cpxs_inv_appl1 … H) -H *
 [ -HV12 #V0 #T0 #_ #_ #H destruct /2 width=1 by tsts_pair, or_introl/
@@ -99,8 +99,8 @@ elim (cpxs_inv_appl1 … H) -H *
 ]
 qed-.
 
-lemma cpxs_fwd_cast: ∀h,o,G,L,W,T,U. ⦃G, L⦄ ⊢ ⓝW.T ➡*[h, o] U →
-                     ∨∨ ⓝW. T ≂ U | ⦃G, L⦄ ⊢ T ➡*[h, o] U | ⦃G, L⦄ ⊢ W ➡*[h, o] U.
+lemma cpxs_fwd_cast: ∀h,o,G,L,W,T,U. ⦃G, L⦄ ⊢ ⓝW.T ⬈*[h, o] U →
+                     ∨∨ ⓝW. T ≂ U | ⦃G, L⦄ ⊢ T ⬈*[h, o] U | ⦃G, L⦄ ⊢ W ⬈*[h, o] U.
 #h #o #G #L #W #T #U #H
 elim (cpxs_inv_cast1 … H) -H /2 width=1 by or3_intro1, or3_intro2/ *
 #W0 #T0 #_ #_ #H destruct /2 width=1 by tsts_pair, or3_intro0/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_tsts_vector.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_tsts_vector.ma
index c408aaf67..738d5907a 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_tsts_vector.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_tsts_vector.ma
@@ -21,8 +21,8 @@ include "basic_2/computation/cpxs_tsts.ma".
 (* Vector form of forward lemmas involving same top term structure **********)
 
 (* Basic_1: was just: nf2_iso_appls_lref *)
-lemma cpxs_fwd_cnx_vector: ∀h,o,G,L,T.  𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T⦄ →
-                           ∀Vs,U. ⦃G, L⦄ ⊢ ⒶVs.T ➡*[h, o] U → ⒶVs.T ≂ U.
+lemma cpxs_fwd_cnx_vector: ∀h,o,G,L,T.  𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃T⦄ →
+                           ∀Vs,U. ⦃G, L⦄ ⊢ ⒶVs.T ⬈*[h, o] U → ⒶVs.T ≂ U.
 #h #o #G #L #T #H1T #H2T #Vs elim Vs -Vs [ @(cpxs_fwd_cnx … H2T) ] (**) (* /2 width=3 by cpxs_fwd_cnx/ does not work *)
 #V #Vs #IHVs #U #H
 elim (cpxs_inv_appl1 … H) -H *
@@ -36,8 +36,8 @@ elim (cpxs_inv_appl1 … H) -H *
 ]
 qed-.
 
-lemma cpxs_fwd_sort_vector: ∀h,o,G,L,s,Vs,U. ⦃G, L⦄ ⊢ ⒶVs.⋆s ➡*[h, o] U →
-                            ⒶVs.⋆s ≂ U ∨ ⦃G, L⦄ ⊢ ⒶVs.⋆(next h s) ➡*[h, o] U.
+lemma cpxs_fwd_sort_vector: ∀h,o,G,L,s,Vs,U. ⦃G, L⦄ ⊢ ⒶVs.⋆s ⬈*[h, o] U →
+                            ⒶVs.⋆s ≂ U ∨ ⦃G, L⦄ ⊢ ⒶVs.⋆(next h s) ⬈*[h, o] U.
 #h #o #G #L #s #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_sort/
 #V #Vs #IHVs #U #H
 elim (cpxs_inv_appl1 … H) -H *
@@ -61,8 +61,8 @@ qed-.
 
 
 (* Basic_1: was just: pr3_iso_appls_beta *)
-lemma cpxs_fwd_beta_vector: ∀h,o,a,G,L,Vs,V,W,T,U. ⦃G, L⦄ ⊢ ⒶVs.ⓐV.ⓛ{a}W.T ➡*[h, o] U →
-                            ⒶVs. ⓐV. ⓛ{a}W. T ≂ U ∨ ⦃G, L⦄ ⊢ ⒶVs.ⓓ{a}ⓝW.V.T ➡*[h, o] U.
+lemma cpxs_fwd_beta_vector: ∀h,o,a,G,L,Vs,V,W,T,U. ⦃G, L⦄ ⊢ ⒶVs.ⓐV.ⓛ{a}W.T ⬈*[h, o] U →
+                            ⒶVs. ⓐV. ⓛ{a}W. T ≂ U ∨ ⦃G, L⦄ ⊢ ⒶVs.ⓓ{a}ⓝW.V.T ⬈*[h, o] U.
 #h #o #a #G #L #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_beta/
 #V0 #Vs #IHVs #V #W #T #U #H
 elim (cpxs_inv_appl1 … H) -H *
@@ -86,8 +86,8 @@ qed-.
 
 lemma cpxs_fwd_delta_vector: ∀h,o,I,G,L,K,V1,i. ⬇[i] L ≡ K.ⓑ{I}V1 →
                              ∀V2. ⬆[0, i + 1] V1 ≡ V2 →
-                             ∀Vs,U. ⦃G, L⦄ ⊢ ⒶVs.#i ➡*[h, o] U →
-                             ⒶVs.#i ≂ U ∨ ⦃G, L⦄ ⊢ ⒶVs.V2 ➡*[h, o] U.
+                             ∀Vs,U. ⦃G, L⦄ ⊢ ⒶVs.#i ⬈*[h, o] U →
+                             ⒶVs.#i ≂ U ∨ ⦃G, L⦄ ⊢ ⒶVs.V2 ⬈*[h, o] U.
 #h #o #I #G #L #K #V1 #i #HLK #V2 #HV12 #Vs elim Vs -Vs /2 width=5 by cpxs_fwd_delta/
 #V #Vs #IHVs #U #H -K -V1
 elim (cpxs_inv_appl1 … H) -H *
@@ -111,8 +111,8 @@ qed-.
 
 (* Basic_1: was just: pr3_iso_appls_abbr *)
 lemma cpxs_fwd_theta_vector: ∀h,o,G,L,V1b,V2b. ⬆[0, 1] V1b ≡ V2b →
-                             ∀a,V,T,U. ⦃G, L⦄ ⊢ ⒶV1b.ⓓ{a}V.T ➡*[h, o] U →
-                             ⒶV1b. ⓓ{a}V. T ≂ U ∨ ⦃G, L⦄ ⊢ ⓓ{a}V.ⒶV2b.T ➡*[h, o] U.
+                             ∀a,V,T,U. ⦃G, L⦄ ⊢ ⒶV1b.ⓓ{a}V.T ⬈*[h, o] U →
+                             ⒶV1b. ⓓ{a}V. T ≂ U ∨ ⦃G, L⦄ ⊢ ⓓ{a}V.ⒶV2b.T ⬈*[h, o] U.
 #h #o #G #L #V1b #V2b * -V1b -V2b /3 width=1 by or_intror/
 #V1b #V2b #V1a #V2a #HV12a #HV12b #a
 generalize in match HV12a; -HV12a
@@ -159,10 +159,10 @@ elim (cpxs_inv_appl1 … H) -H *
 qed-.
 
 (* Basic_1: was just: pr3_iso_appls_cast *)
-lemma cpxs_fwd_cast_vector: ∀h,o,G,L,Vs,W,T,U. ⦃G, L⦄ ⊢ ⒶVs.ⓝW.T ➡*[h, o] U →
+lemma cpxs_fwd_cast_vector: ∀h,o,G,L,Vs,W,T,U. ⦃G, L⦄ ⊢ ⒶVs.ⓝW.T ⬈*[h, o] U →
                             ∨∨ ⒶVs. ⓝW. T ≂ U
-                             | ⦃G, L⦄ ⊢ ⒶVs.T ➡*[h, o] U
-                             | ⦃G, L⦄ ⊢ ⒶVs.W ➡*[h, o] U.
+                             | ⦃G, L⦄ ⊢ ⒶVs.T ⬈*[h, o] U
+                             | ⦃G, L⦄ ⊢ ⒶVs.W ⬈*[h, o] U.
 #h #o #G #L #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_cast/
 #V #Vs #IHVs #W #T #U #H
 elim (cpxs_inv_appl1 … H) -H *
diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg.ma
index 5dab2812e..b8742090c 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg.ma
@@ -14,7 +14,7 @@
 
 include "ground_2/steps/rtc_shift.ma".
 include "ground_2/steps/rtc_plus.ma".
-include "basic_2/notation/relations/pred_6.ma".
+include "basic_2/notation/relations/predty_6.ma".
 include "basic_2/grammar/lenv.ma".
 include "basic_2/grammar/genv.ma".
 include "basic_2/relocation/lifts.ma".
@@ -53,30 +53,30 @@ inductive cpg (h): rtc → relation4 genv lenv term term ≝
 
 interpretation
    "counted context-sensitive parallel rt-transition (term)"
-   'PRed c h G L T1 T2 = (cpg h c G L T1 T2).
+   'PRedTy c h G L T1 T2 = (cpg h c G L T1 T2).
 
 (* Basic properties *********************************************************)
 
 (* Note: this is "∀h,g,L. reflexive … (cpg h (𝟘𝟘) L)" *)
-lemma cpg_refl: ∀h,G,T,L. ⦃G, L⦄ ⊢ T ➡[𝟘𝟘, h] T.
+lemma cpg_refl: ∀h,G,T,L. ⦃G, L⦄ ⊢ T ⬈[𝟘𝟘, h] T.
 #h #G #T elim T -T // * /2 width=1 by cpg_bind, cpg_flat/
 qed.
 
-lemma cpg_pair_sn: ∀c,h,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[c, h] V2 →
-                   ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ➡[↓c, h] ②{I}V2.T.
+lemma cpg_pair_sn: ∀c,h,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[c, h] V2 →
+                   ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ⬈[↓c, h] ②{I}V2.T.
 #c #h * /2 width=1 by cpg_bind, cpg_flat/
 qed.
 
 (* Basic inversion lemmas ***************************************************)
 
-fact cpg_inv_atom1_aux: ∀c,h,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[c, h] T2 → ∀J. T1 = ⓪{J} →
+fact cpg_inv_atom1_aux: ∀c,h,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈[c, h] T2 → ∀J. T1 = ⓪{J} →
                         ∨∨ T2 = ⓪{J} ∧ c = 𝟘𝟘 
                          | ∃∃s. J = Sort s & T2 = ⋆(next h s) & c = 𝟘𝟙
-                         | ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[cV, h] V2 & ⬆*[1] V2 ≡ T2 &
+                         | ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[cV, h] V2 & ⬆*[1] V2 ≡ T2 &
                                          L = K.ⓓV1 & J = LRef 0 & c = cV
-                         | ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[cV, h] V2 & ⬆*[1] V2 ≡ T2 &
+                         | ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[cV, h] V2 & ⬆*[1] V2 ≡ T2 &
                                          L = K.ⓛV1 & J = LRef 0 & c = (↓cV)+𝟘𝟙
-                         | ∃∃I,K,V,T,i. ⦃G, K⦄ ⊢ #i ➡[c, h] T & ⬆*[1] T ≡ T2 &
+                         | ∃∃I,K,V,T,i. ⦃G, K⦄ ⊢ #i ⬈[c, h] T & ⬆*[1] T ≡ T2 &
                                         L = K.ⓑ{I}V & J = LRef (⫯i).
 #c #h #G #L #T1 #T2 * -c -G -L -T1 -T2
 [ #I #G #L #J #H destruct /3 width=1 by or5_intro0, conj/
@@ -94,18 +94,18 @@ fact cpg_inv_atom1_aux: ∀c,h,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[c, h] T2 → ∀
 ]
 qed-.
 
-lemma cpg_inv_atom1: ∀c,h,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ➡[c, h] T2 →
+lemma cpg_inv_atom1: ∀c,h,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ⬈[c, h] T2 →
                      ∨∨ T2 = ⓪{J} ∧ c = 𝟘𝟘 
                       | ∃∃s. J = Sort s & T2 = ⋆(next h s) & c = 𝟘𝟙
-                      | ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[cV, h] V2 & ⬆*[1] V2 ≡ T2 &
+                      | ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[cV, h] V2 & ⬆*[1] V2 ≡ T2 &
                                       L = K.ⓓV1 & J = LRef 0 & c = cV
-                      | ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[cV, h] V2 & ⬆*[1] V2 ≡ T2 &
+                      | ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[cV, h] V2 & ⬆*[1] V2 ≡ T2 &
                                       L = K.ⓛV1 & J = LRef 0 & c = (↓cV)+𝟘𝟙
-                      | ∃∃I,K,V,T,i. ⦃G, K⦄ ⊢ #i ➡[c, h] T & ⬆*[1] T ≡ T2 &
+                      | ∃∃I,K,V,T,i. ⦃G, K⦄ ⊢ #i ⬈[c, h] T & ⬆*[1] T ≡ T2 &
                                      L = K.ⓑ{I}V & J = LRef (⫯i).
 /2 width=3 by cpg_inv_atom1_aux/ qed-.
 
-lemma cpg_inv_sort1: ∀c,h,G,L,T2,s. ⦃G, L⦄ ⊢ ⋆s ➡[c, h] T2 →
+lemma cpg_inv_sort1: ∀c,h,G,L,T2,s. ⦃G, L⦄ ⊢ ⋆s ⬈[c, h] T2 →
                      (T2 = ⋆s ∧ c = 𝟘𝟘) ∨ (T2 = ⋆(next h s) ∧ c = 𝟘𝟙).
 #c #h #G #L #T2 #s #H
 elim (cpg_inv_atom1 … H) -H * /3 width=1 by or_introl, conj/
@@ -115,11 +115,11 @@ elim (cpg_inv_atom1 … H) -H * /3 width=1 by or_introl, conj/
 ]
 qed-.
 
-lemma cpg_inv_zero1: ∀c,h,G,L,T2. ⦃G, L⦄ ⊢ #0 ➡[c, h] T2 →
+lemma cpg_inv_zero1: ∀c,h,G,L,T2. ⦃G, L⦄ ⊢ #0 ⬈[c, h] T2 →
                      ∨∨ (T2 = #0 ∧ c = 𝟘𝟘)
-                      | ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[cV, h] V2 & ⬆*[1] V2 ≡ T2 &
+                      | ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[cV, h] V2 & ⬆*[1] V2 ≡ T2 &
                                       L = K.ⓓV1 & c = cV
-                      | ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[cV, h] V2 & ⬆*[1] V2 ≡ T2 &
+                      | ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[cV, h] V2 & ⬆*[1] V2 ≡ T2 &
                                       L = K.ⓛV1 & c = (↓cV)+𝟘𝟙.
 #c #h #G #L #T2 #H
 elim (cpg_inv_atom1 … H) -H * /3 width=1 by or3_intro0, conj/
@@ -129,9 +129,9 @@ elim (cpg_inv_atom1 … H) -H * /3 width=1 by or3_intro0, conj/
 ]
 qed-.
 
-lemma cpg_inv_lref1: ∀c,h,G,L,T2,i. ⦃G, L⦄ ⊢ #⫯i ➡[c, h] T2 →
+lemma cpg_inv_lref1: ∀c,h,G,L,T2,i. ⦃G, L⦄ ⊢ #⫯i ⬈[c, h] T2 →
                      (T2 = #(⫯i) ∧ c = 𝟘𝟘) ∨
-                     ∃∃I,K,V,T. ⦃G, K⦄ ⊢ #i ➡[c, h] T & ⬆*[1] T ≡ T2 & L = K.ⓑ{I}V.
+                     ∃∃I,K,V,T. ⦃G, K⦄ ⊢ #i ⬈[c, h] T & ⬆*[1] T ≡ T2 & L = K.ⓑ{I}V.
 #c #h #G #L #T2 #i #H
 elim (cpg_inv_atom1 … H) -H * /3 width=1 by or_introl, conj/
 [ #s #H destruct
@@ -140,7 +140,7 @@ elim (cpg_inv_atom1 … H) -H * /3 width=1 by or_introl, conj/
 ]
 qed-.
 
-lemma cpg_inv_gref1: ∀c,h,G,L,T2,l. ⦃G, L⦄ ⊢ §l ➡[c, h] T2 → T2 = §l ∧ c = 𝟘𝟘.
+lemma cpg_inv_gref1: ∀c,h,G,L,T2,l. ⦃G, L⦄ ⊢ §l ⬈[c, h] T2 → T2 = §l ∧ c = 𝟘𝟘.
 #c #h #G #L #T2 #l #H
 elim (cpg_inv_atom1 … H) -H * /2 width=1 by conj/
 [ #s #H destruct
@@ -149,12 +149,12 @@ elim (cpg_inv_atom1 … H) -H * /2 width=1 by conj/
 ]
 qed-.
 
-fact cpg_inv_bind1_aux: ∀c,h,G,L,U,U2. ⦃G, L⦄ ⊢ U ➡[c, h] U2 →
+fact cpg_inv_bind1_aux: ∀c,h,G,L,U,U2. ⦃G, L⦄ ⊢ U ⬈[c, h] U2 →
                         ∀p,J,V1,U1. U = ⓑ{p,J}V1.U1 → (
-                        ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ➡[cV, h] V2 & ⦃G, L.ⓑ{J}V1⦄ ⊢ U1 ➡[cT, h] T2 &
+                        ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[cV, h] V2 & ⦃G, L.ⓑ{J}V1⦄ ⊢ U1 ⬈[cT, h] T2 &
                                        U2 = ⓑ{p,J}V2.T2 & c = (↓cV)+cT
                         ) ∨
-                        ∃∃cT,T. ⦃G, L.ⓓV1⦄ ⊢ U1 ➡[cT, h] T & ⬆*[1] U2 ≡ T &
+                        ∃∃cT,T. ⦃G, L.ⓓV1⦄ ⊢ U1 ⬈[cT, h] T & ⬆*[1] U2 ≡ T &
                                 p = true & J = Abbr & c = (↓cT)+𝟙𝟘.
 #c #h #G #L #U #U2 * -c -G -L -U -U2
 [ #I #G #L #q #J #W #U1 #H destruct
@@ -172,26 +172,26 @@ fact cpg_inv_bind1_aux: ∀c,h,G,L,U,U2. ⦃G, L⦄ ⊢ U ➡[c, h] U2 →
 ]
 qed-.
 
-lemma cpg_inv_bind1: ∀c,h,p,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[c, h] U2 → (
-                     ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ➡[cV, h] V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡[cT, h] T2 &
+lemma cpg_inv_bind1: ∀c,h,p,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[c, h] U2 → (
+                     ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[cV, h] V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬈[cT, h] T2 &
                                     U2 = ⓑ{p,I}V2.T2 & c = (↓cV)+cT
                      ) ∨
-                     ∃∃cT,T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[cT, h] T & ⬆*[1] U2 ≡ T &
+                     ∃∃cT,T. ⦃G, L.ⓓV1⦄ ⊢ T1 ⬈[cT, h] T & ⬆*[1] U2 ≡ T &
                              p = true & I = Abbr & c = (↓cT)+𝟙𝟘.
 /2 width=3 by cpg_inv_bind1_aux/ qed-.
 
-lemma cpg_inv_abbr1: ∀c,h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{p}V1.T1 ➡[c, h] U2 → (
-                     ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ➡[cV, h] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[cT, h] T2 &
+lemma cpg_inv_abbr1: ∀c,h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{p}V1.T1 ⬈[c, h] U2 → (
+                     ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[cV, h] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ⬈[cT, h] T2 &
                                     U2 = ⓓ{p}V2.T2 & c = (↓cV)+cT
                      ) ∨
-                     ∃∃cT,T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[cT, h] T & ⬆*[1] U2 ≡ T &
+                     ∃∃cT,T. ⦃G, L.ⓓV1⦄ ⊢ T1 ⬈[cT, h] T & ⬆*[1] U2 ≡ T &
                              p = true & c = (↓cT)+𝟙𝟘.
 #c #h #p #G #L #V1 #T1 #U2 #H elim (cpg_inv_bind1 … H) -H *
 /3 width=8 by ex4_4_intro, ex4_2_intro, or_introl, or_intror/
 qed-.
 
-lemma cpg_inv_abst1: ∀c,h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{p}V1.T1 ➡[c, h] U2 →
-                     ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ➡[cV, h] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ➡[cT, h] T2 &
+lemma cpg_inv_abst1: ∀c,h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{p}V1.T1 ⬈[c, h] U2 →
+                     ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[cV, h] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ⬈[cT, h] T2 &
                                     U2 = ⓛ{p}V2.T2 & c = (↓cV)+cT.
 #c #h #p #G #L #V1 #T1 #U2 #H elim (cpg_inv_bind1 … H) -H * 
 [ /3 width=8 by ex4_4_intro/
@@ -199,15 +199,15 @@ lemma cpg_inv_abst1: ∀c,h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{p}V1.T1 ➡[c, h]
 ]
 qed-.
 
-fact cpg_inv_flat1_aux: ∀c,h,G,L,U,U2. ⦃G, L⦄ ⊢ U ➡[c, h] U2 →
+fact cpg_inv_flat1_aux: ∀c,h,G,L,U,U2. ⦃G, L⦄ ⊢ U ⬈[c, h] U2 →
                         ∀J,V1,U1. U = ⓕ{J}V1.U1 →
-                        ∨∨ ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ➡[cV, h] V2 & ⦃G, L⦄ ⊢ U1 ➡[cT, h] T2 &
+                        ∨∨ ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[cV, h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[cT, h] T2 &
                                           U2 = ⓕ{J}V2.T2 & c = (↓cV)+cT
-                         | ∃∃cT. ⦃G, L⦄ ⊢ U1 ➡[cT, h] U2 & J = Cast & c = (↓cT)+𝟙𝟘
-                         | ∃∃cV. ⦃G, L⦄ ⊢ V1 ➡[cV, h] U2 & J = Cast & c = (↓cV)+𝟘𝟙
-                         | ∃∃cV,cW,cT,p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[cV, h] V2 & ⦃G, L⦄ ⊢ W1 ➡[cW, h] W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[cT, h] T2 &
+                         | ∃∃cT. ⦃G, L⦄ ⊢ U1 ⬈[cT, h] U2 & J = Cast & c = (↓cT)+𝟙𝟘
+                         | ∃∃cV. ⦃G, L⦄ ⊢ V1 ⬈[cV, h] U2 & J = Cast & c = (↓cV)+𝟘𝟙
+                         | ∃∃cV,cW,cT,p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[cV, h] V2 & ⦃G, L⦄ ⊢ W1 ⬈[cW, h] W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈[cT, h] T2 &
                                                         J = Appl & U1 = ⓛ{p}W1.T1 & U2 = ⓓ{p}ⓝW2.V2.T2 & c = (↓cV)+(↓cW)+(↓cT)+𝟙𝟘
-                         | ∃∃cV,cW,cT,p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[cV, h] V & ⬆*[1] V ≡ V2 & ⦃G, L⦄ ⊢ W1 ➡[cW, h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[cT, h] T2 &
+                         | ∃∃cV,cW,cT,p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[cV, h] V & ⬆*[1] V ≡ V2 & ⦃G, L⦄ ⊢ W1 ⬈[cW, h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈[cT, h] T2 &
                                                           J = Appl & U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2 & c = (↓cV)+(↓cW)+(↓cT)+𝟙𝟘.
 #c #h #G #L #U #U2 * -c -G -L -U -U2
 [ #I #G #L #J #W #U1 #H destruct
@@ -225,23 +225,23 @@ fact cpg_inv_flat1_aux: ∀c,h,G,L,U,U2. ⦃G, L⦄ ⊢ U ➡[c, h] U2 →
 ]
 qed-.
 
-lemma cpg_inv_flat1: ∀c,h,I,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ➡[c, h] U2 →
-                     ∨∨ ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ➡[cV, h] V2 & ⦃G, L⦄ ⊢ U1 ➡[cT, h] T2 &
+lemma cpg_inv_flat1: ∀c,h,I,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ⬈[c, h] U2 →
+                     ∨∨ ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[cV, h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[cT, h] T2 &
                                        U2 = ⓕ{I}V2.T2 & c = (↓cV)+cT
-                      | ∃∃cT. ⦃G, L⦄ ⊢ U1 ➡[cT, h] U2 & I = Cast & c = (↓cT)+𝟙𝟘
-                      | ∃∃cV. ⦃G, L⦄ ⊢ V1 ➡[cV, h] U2 & I = Cast & c = (↓cV)+𝟘𝟙
-                      | ∃∃cV,cW,cT,p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[cV, h] V2 & ⦃G, L⦄ ⊢ W1 ➡[cW, h] W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[cT, h] T2 &
+                      | ∃∃cT. ⦃G, L⦄ ⊢ U1 ⬈[cT, h] U2 & I = Cast & c = (↓cT)+𝟙𝟘
+                      | ∃∃cV. ⦃G, L⦄ ⊢ V1 ⬈[cV, h] U2 & I = Cast & c = (↓cV)+𝟘𝟙
+                      | ∃∃cV,cW,cT,p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[cV, h] V2 & ⦃G, L⦄ ⊢ W1 ⬈[cW, h] W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈[cT, h] T2 &
                                                      I = Appl & U1 = ⓛ{p}W1.T1 & U2 = ⓓ{p}ⓝW2.V2.T2 & c = (↓cV)+(↓cW)+(↓cT)+𝟙𝟘
-                      | ∃∃cV,cW,cT,p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[cV, h] V & ⬆*[1] V ≡ V2 & ⦃G, L⦄ ⊢ W1 ➡[cW, h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[cT, h] T2 &
+                      | ∃∃cV,cW,cT,p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[cV, h] V & ⬆*[1] V ≡ V2 & ⦃G, L⦄ ⊢ W1 ⬈[cW, h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈[cT, h] T2 &
                                                        I = Appl & U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2 & c = (↓cV)+(↓cW)+(↓cT)+𝟙𝟘.
 /2 width=3 by cpg_inv_flat1_aux/ qed-.
 
-lemma cpg_inv_appl1: ∀c,h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐV1.U1 ➡[c, h] U2 →
-                     ∨∨ ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ➡[cV, h] V2 & ⦃G, L⦄ ⊢ U1 ➡[cT, h] T2 &
+lemma cpg_inv_appl1: ∀c,h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐV1.U1 ⬈[c, h] U2 →
+                     ∨∨ ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[cV, h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[cT, h] T2 &
                                        U2 = ⓐV2.T2 & c = (↓cV)+cT
-                      | ∃∃cV,cW,cT,p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[cV, h] V2 & ⦃G, L⦄ ⊢ W1 ➡[cW, h] W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[cT, h] T2 &
+                      | ∃∃cV,cW,cT,p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[cV, h] V2 & ⦃G, L⦄ ⊢ W1 ⬈[cW, h] W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈[cT, h] T2 &
                                                      U1 = ⓛ{p}W1.T1 & U2 = ⓓ{p}ⓝW2.V2.T2 & c = (↓cV)+(↓cW)+(↓cT)+𝟙𝟘
-                      | ∃∃cV,cW,cT,p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[cV, h] V & ⬆*[1] V ≡ V2 & ⦃G, L⦄ ⊢ W1 ➡[cW, h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[cT, h] T2 &
+                      | ∃∃cV,cW,cT,p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[cV, h] V & ⬆*[1] V ≡ V2 & ⦃G, L⦄ ⊢ W1 ⬈[cW, h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈[cT, h] T2 &
                                                        U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2 & c = (↓cV)+(↓cW)+(↓cT)+𝟙𝟘.
 #c #h #G #L #V1 #U1 #U2 #H elim (cpg_inv_flat1 … H) -H *
 [ /3 width=8 by or3_intro0, ex4_4_intro/
@@ -251,11 +251,11 @@ lemma cpg_inv_appl1: ∀c,h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐV1.U1 ➡[c, h] U2 
 ]
 qed-.
 
-lemma cpg_inv_cast1: ∀c,h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝV1.U1 ➡[c, h] U2 →
-                     ∨∨ ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ➡[cV, h] V2 & ⦃G, L⦄ ⊢ U1 ➡[cT, h] T2 &
+lemma cpg_inv_cast1: ∀c,h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝV1.U1 ⬈[c, h] U2 →
+                     ∨∨ ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[cV, h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[cT, h] T2 &
                                        U2 = ⓝV2.T2 & c = (↓cV)+cT
-                      | ∃∃cT. ⦃G, L⦄ ⊢ U1 ➡[cT, h] U2 & c = (↓cT)+𝟙𝟘
-                      | ∃∃cV. ⦃G, L⦄ ⊢ V1 ➡[cV, h] U2 & c = (↓cV)+𝟘𝟙.
+                      | ∃∃cT. ⦃G, L⦄ ⊢ U1 ⬈[cT, h] U2 & c = (↓cT)+𝟙𝟘
+                      | ∃∃cV. ⦃G, L⦄ ⊢ V1 ⬈[cV, h] U2 & c = (↓cV)+𝟘𝟙.
 #c #h #G #L #V1 #U1 #U2 #H elim (cpg_inv_flat1 … H) -H *
 [ /3 width=8 by or3_intro0, ex4_4_intro/
 |2,3: /3 width=3 by or3_intro1, or3_intro2, ex2_intro/
@@ -266,8 +266,8 @@ qed-.
 
 (* Basic forward lemmas *****************************************************)
 
-lemma cpg_fwd_bind1_minus: ∀c,h,I,G,L,V1,T1,T. ⦃G, L⦄ ⊢ -ⓑ{I}V1.T1 ➡[c, h] T → ∀p.
-                           ∃∃V2,T2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[c, h] ⓑ{p,I}V2.T2 &
+lemma cpg_fwd_bind1_minus: ∀c,h,I,G,L,V1,T1,T. ⦃G, L⦄ ⊢ -ⓑ{I}V1.T1 ⬈[c, h] T → ∀p.
+                           ∃∃V2,T2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[c, h] ⓑ{p,I}V2.T2 &
                                     T = -ⓑ{I}V2.T2.
 #c #h #I #G #L #V1 #T1 #T #H #p elim (cpg_inv_bind1 … H) -H *
 [ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct /3 width=4 by cpg_bind, ex2_2_intro/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg_drops.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg_drops.ma
index c8f0b394f..779ea690b 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg_drops.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg_drops.ma
@@ -21,8 +21,8 @@ include "basic_2/rt_transition/cpg.ma".
 
 (* Advanced properties ******************************************************)
 
-lemma cpg_delta_drops: ∀c,h,G,K,V,V2,i,L,T2. ⬇*[i] L ≡ K.ⓓV → ⦃G, K⦄ ⊢ V ➡[c, h] V2 →
-                       ⬆*[⫯i] V2 ≡ T2 →  ⦃G, L⦄ ⊢ #i ➡[c, h] T2.
+lemma cpg_delta_drops: ∀c,h,G,K,V,V2,i,L,T2. ⬇*[i] L ≡ K.ⓓV → ⦃G, K⦄ ⊢ V ⬈[c, h] V2 →
+                       ⬆*[⫯i] V2 ≡ T2 →  ⦃G, L⦄ ⊢ #i ⬈[c, h] T2.
 #c #h #G #K #V #V2 #i elim i -i
 [ #L #T2 #HLK lapply (drops_fwd_isid … HLK ?) // #H destruct /3 width=3 by cpg_delta/
 | #i #IH #L0 #T0 #H0 #HV2 #HVT2
@@ -31,8 +31,8 @@ lemma cpg_delta_drops: ∀c,h,G,K,V,V2,i,L,T2. ⬇*[i] L ≡ K.ⓓV → ⦃G, K
 ]
 qed.
 
-lemma cpg_ell_drops: ∀c,h,G,K,V,V2,i,L,T2. ⬇*[i] L ≡ K.ⓛV → ⦃G, K⦄ ⊢ V ➡[c, h] V2 →
-                     ⬆*[⫯i] V2 ≡ T2 →  ⦃G, L⦄ ⊢ #i ➡[(↓c)+𝟘𝟙, h] T2.
+lemma cpg_ell_drops: ∀c,h,G,K,V,V2,i,L,T2. ⬇*[i] L ≡ K.ⓛV → ⦃G, K⦄ ⊢ V ⬈[c, h] V2 →
+                     ⬆*[⫯i] V2 ≡ T2 →  ⦃G, L⦄ ⊢ #i ⬈[(↓c)+𝟘𝟙, h] T2.
 #c #h #G #K #V #V2 #i elim i -i
 [ #L #T2 #HLK lapply (drops_fwd_isid … HLK ?) // #H destruct /3 width=3 by cpg_ell/
 | #i #IH #L0 #T0 #H0 #HV2 #HVT2
@@ -43,11 +43,11 @@ qed.
 
 (* Advanced inversion lemmas ************************************************)
 
-lemma cpg_inv_lref1_drops: ∀c,h,G,i,L,T2. ⦃G, L⦄ ⊢ #i ➡[c, h] T2 →
+lemma cpg_inv_lref1_drops: ∀c,h,G,i,L,T2. ⦃G, L⦄ ⊢ #i ⬈[c, h] T2 →
                            ∨∨ T2 = #i ∧ c = 𝟘𝟘
-                            | ∃∃cV,K,V,V2. ⬇*[i] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V ➡[cV, h] V2 &
+                            | ∃∃cV,K,V,V2. ⬇*[i] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V ⬈[cV, h] V2 &
                                            ⬆*[⫯i] V2 ≡ T2 & c = cV
-                            | ∃∃cV,K,V,V2. ⬇*[i] L ≡ K.ⓛV & ⦃G, K⦄ ⊢ V ➡[cV, h] V2 &
+                            | ∃∃cV,K,V,V2. ⬇*[i] L ≡ K.ⓛV & ⦃G, K⦄ ⊢ V ⬈[cV, h] V2 &
                                            ⬆*[⫯i] V2 ≡ T2 & c = (↓cV) + 𝟘𝟙.
 #c #h #G #i elim i -i
 [ #L #T2 #H elim (cpg_inv_zero1 … H) -H * /3 width=1 by or3_intro0, conj/
@@ -61,12 +61,12 @@ lemma cpg_inv_lref1_drops: ∀c,h,G,i,L,T2. ⦃G, L⦄ ⊢ #i ➡[c, h] T2 →
 ]
 qed-.
 
-lemma cpg_inv_atom1_drops: ∀c,h,I,G,L,T2. ⦃G, L⦄ ⊢ ⓪{I} ➡[c, h] T2 →
+lemma cpg_inv_atom1_drops: ∀c,h,I,G,L,T2. ⦃G, L⦄ ⊢ ⓪{I} ⬈[c, h] T2 →
                            ∨∨ T2 = ⓪{I} ∧ c = 𝟘𝟘
                             | ∃∃s. T2 = ⋆(next h s) & I = Sort s & c = 𝟘𝟙
-                            | ∃∃cV,i,K,V,V2. ⬇*[i] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V ➡[cV, h] V2 &
+                            | ∃∃cV,i,K,V,V2. ⬇*[i] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V ⬈[cV, h] V2 &
                                              ⬆*[⫯i] V2 ≡ T2 & I = LRef i & c = cV
-                            | ∃∃cV,i,K,V,V2. ⬇*[i] L ≡ K.ⓛV & ⦃G, K⦄ ⊢ V ➡[cV, h] V2 &
+                            | ∃∃cV,i,K,V,V2. ⬇*[i] L ≡ K.ⓛV & ⦃G, K⦄ ⊢ V ⬈[cV, h] V2 &
                                              ⬆*[⫯i] V2 ≡ T2 & I = LRef i & c = (↓cV) + 𝟘𝟙.
 #c #h * #n #G #L #T2 #H
 [ elim (cpg_inv_sort1 … H) -H *
diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg_simple.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg_simple.ma
index 5dcf8ece0..a53be69b0 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg_simple.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg_simple.ma
@@ -20,8 +20,8 @@ include "basic_2/rt_transition/cpg.ma".
 (* Properties with simple terms *********************************************)
 
 (* Note: the main property of simple terms *)
-lemma cpg_inv_appl1_simple: ∀c,h,G,L,V1,T1,U. ⦃G, L⦄ ⊢ ⓐV1.T1 ➡[c, h] U → 𝐒⦃T1⦄ →
-                            ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ➡[cV, h] V2 & ⦃G, L⦄ ⊢ T1 ➡[cT, h] T2 &
+lemma cpg_inv_appl1_simple: ∀c,h,G,L,V1,T1,U. ⦃G, L⦄ ⊢ ⓐV1.T1 ⬈[c, h] U → 𝐒⦃T1⦄ →
+                            ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[cV, h] V2 & ⦃G, L⦄ ⊢ T1 ⬈[cT, h] T2 &
                                            U = ⓐV2.T2 & c = (↓cV)+cT.
 #c #h #G #L #V1 #T1 #U #H #HT1 elim (cpg_inv_appl1 … H) -H *
 [ /2 width=8 by ex4_4_intro/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx.ma
index c40733142..ac0248923 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx.ma
@@ -12,81 +12,81 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/notation/relations/pred_5.ma".
+include "basic_2/notation/relations/predty_5.ma".
 include "basic_2/rt_transition/cpg.ma".
 
-(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL REDUCTION FOR TERMS *****************)
+(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS *************)
 
 definition cpx (h): relation4 genv lenv term term ≝
-                    λG,L,T1,T2. ∃c. ⦃G, L⦄ ⊢ T1 ➡[c, h] T2.
+                    λG,L,T1,T2. ∃c. ⦃G, L⦄ ⊢ T1 ⬈[c, h] T2.
 
 interpretation
    "uncounted context-sensitive parallel reduction (term)"
-   'PRed h G L T1 T2 = (cpx h G L T1 T2).
+   'PRedTy h G L T1 T2 = (cpx h G L T1 T2).
 
 (* Basic properties *********************************************************)
 
-lemma cpx_atom: ∀h,I,G,L. ⦃G, L⦄ ⊢ ⓪{I} ➡[h] ⓪{I}.
+lemma cpx_atom: ∀h,I,G,L. ⦃G, L⦄ ⊢ ⓪{I} ⬈[h] ⓪{I}.
 /2 width=2 by cpg_atom, ex_intro/ qed.
 
 (* Basic_2A1: was: cpx_st *)
-lemma cpx_ess: ∀h,G,L,s. ⦃G, L⦄ ⊢ ⋆s ➡[h] ⋆(next h s).
+lemma cpx_ess: ∀h,G,L,s. ⦃G, L⦄ ⊢ ⋆s ⬈[h] ⋆(next h s).
 /2 width=2 by cpg_ess, ex_intro/ qed.
 
-lemma cpx_delta: ∀h,I,G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ➡[h] V2 →
-                 ⬆*[1] V2 ≡ W2 → ⦃G, K.ⓑ{I}V1⦄ ⊢ #0 ➡[h] W2.
+lemma cpx_delta: ∀h,I,G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ⬈[h] V2 →
+                 ⬆*[1] V2 ≡ W2 → ⦃G, K.ⓑ{I}V1⦄ ⊢ #0 ⬈[h] W2.
 #h * #G #K #V1 #V2 #W2 *
 /3 width=4 by cpg_delta, cpg_ell, ex_intro/
 qed.
 
-lemma cpx_lref: ∀h,I,G,K,V,T,U,i. ⦃G, K⦄ ⊢ #i ➡[h] T →
-                ⬆*[1] T ≡ U → ⦃G, K.ⓑ{I}V⦄ ⊢ #⫯i ➡[h] U.
+lemma cpx_lref: ∀h,I,G,K,V,T,U,i. ⦃G, K⦄ ⊢ #i ⬈[h] T →
+                ⬆*[1] T ≡ U → ⦃G, K.ⓑ{I}V⦄ ⊢ #⫯i ⬈[h] U.
 #h #I #G #K #V #T #U #i *
 /3 width=4 by cpg_lref, ex_intro/
 qed.
 
 lemma cpx_bind: ∀h,p,I,G,L,V1,V2,T1,T2.
-                 ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡[h] T2 →
-                 ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[h] ⓑ{p,I}V2.T2.
+                 ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬈[h] T2 →
+                 ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] ⓑ{p,I}V2.T2.
 #h #p #I #G #L #V1 #V2 #T1 #T2 * #cV #HV12 *
 /3 width=2 by cpg_bind, ex_intro/
 qed.
 
 lemma cpx_flat: ∀h,I,G,L,V1,V2,T1,T2.
-                 ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ T1 ➡[h] T2 →
-                 ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡[h] ⓕ{I}V2.T2.
+                 ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L⦄ ⊢ T1 ⬈[h] T2 →
+                 ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬈[h] ⓕ{I}V2.T2.
 #h #I #G #L #V1 #V2 #T1 #T2 * #cV #HV12 *
 /3 width=2 by cpg_flat, ex_intro/
 qed.
 
-lemma cpx_zeta: ∀h,G,L,V,T1,T,T2. ⦃G, L.ⓓV⦄ ⊢ T1 ➡[h] T →
-                ⬆*[1] T2 ≡ T → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡[h] T2.
+lemma cpx_zeta: ∀h,G,L,V,T1,T,T2. ⦃G, L.ⓓV⦄ ⊢ T1 ⬈[h] T →
+                ⬆*[1] T2 ≡ T → ⦃G, L⦄ ⊢ +ⓓV.T1 ⬈[h] T2.
 #h #G #L #V #T1 #T #T2 *
 /3 width=4 by cpg_zeta, ex_intro/
 qed.
 
-lemma cpx_eps: ∀h,G,L,V,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h] T2 → ⦃G, L⦄ ⊢ ⓝV.T1 ➡[h] T2.
+lemma cpx_eps: ∀h,G,L,V,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → ⦃G, L⦄ ⊢ ⓝV.T1 ⬈[h] T2.
 #h #G #L #V #T1 #T2 *
 /3 width=2 by cpg_eps, ex_intro/
 qed.
 
 (* Basic_2A1: was: cpx_ct *)
-lemma cpx_ee: ∀h,G,L,V1,V2,T. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ ⓝV1.T ➡[h] V2.
+lemma cpx_ee: ∀h,G,L,V1,V2,T. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L⦄ ⊢ ⓝV1.T ⬈[h] V2.
 #h #G #L #V1 #V2 #T *
 /3 width=2 by cpg_ee, ex_intro/
 qed.
 
 lemma cpx_beta: ∀h,p,G,L,V1,V2,W1,W2,T1,T2.
-                ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ W1 ➡[h] W2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[h] T2 →
-                ⦃G, L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ➡[h] ⓓ{p}ⓝW2.V2.T2.
+                ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L⦄ ⊢ W1 ⬈[h] W2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈[h] T2 →
+                ⦃G, L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ⬈[h] ⓓ{p}ⓝW2.V2.T2.
 #h #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 * #cV #HV12 * #cW #HW12 * 
 /3 width=2 by cpg_beta, ex_intro/
 qed.
 
 lemma cpx_theta: ∀h,p,G,L,V1,V,V2,W1,W2,T1,T2.
-                 ⦃G, L⦄ ⊢ V1 ➡[h] V → ⬆*[1] V ≡ V2 → ⦃G, L⦄ ⊢ W1 ➡[h] W2 →
-                 ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[h] T2 →
-                 ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ➡[h] ⓓ{p}W2.ⓐV2.T2.
+                 ⦃G, L⦄ ⊢ V1 ⬈[h] V → ⬆*[1] V ≡ V2 → ⦃G, L⦄ ⊢ W1 ⬈[h] W2 →
+                 ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈[h] T2 →
+                 ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ⬈[h] ⓓ{p}W2.ⓐV2.T2.
 #h #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 * #cV #HV1 #HV2 * #cW #HW12 * 
 /3 width=4 by cpg_theta, ex_intro/
 qed.
@@ -94,119 +94,119 @@ qed.
 lemma cpx_refl: ∀h,G,L. reflexive … (cpx h G L).
 /2 width=2 by ex_intro/ qed.
 
-lemma cpx_pair_sn: ∀h,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 →
-                   ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ➡[h] ②{I}V2.T.
+lemma cpx_pair_sn: ∀h,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 →
+                   ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ⬈[h] ②{I}V2.T.
 #h #I #G #L #V1 #V2 *
 /3 width=2 by cpg_pair_sn, ex_intro/
 qed.
 
 (* Basic inversion lemmas ***************************************************)
 
-lemma cpx_inv_atom1: ∀h,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ➡[h] T2 →
+lemma cpx_inv_atom1: ∀h,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ⬈[h] T2 →
                      ∨∨ T2 = ⓪{J}
                       | ∃∃s. T2 = ⋆(next h s) & J = Sort s
-                      | ∃∃I,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[h] V2 & ⬆*[1] V2 ≡ T2 &
+                      | ∃∃I,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[h] V2 & ⬆*[1] V2 ≡ T2 &
                                      L = K.ⓑ{I}V1 & J = LRef 0
-                      | ∃∃I,K,V,T,i. ⦃G, K⦄ ⊢ #i ➡[h] T & ⬆*[1] T ≡ T2 &
+                      | ∃∃I,K,V,T,i. ⦃G, K⦄ ⊢ #i ⬈[h] T & ⬆*[1] T ≡ T2 &
                                      L = K.ⓑ{I}V & J = LRef (⫯i).
 #h #J #G #L #T2 * #c #H elim (cpg_inv_atom1 … H) -H *
 /4 width=9 by or4_intro0, or4_intro1, or4_intro2, or4_intro3, ex4_5_intro, ex4_4_intro, ex2_intro, ex_intro/
 qed-.
 
-lemma cpx_inv_sort1: ∀h,G,L,T2,s. ⦃G, L⦄ ⊢ ⋆s ➡[h] T2 →
+lemma cpx_inv_sort1: ∀h,G,L,T2,s. ⦃G, L⦄ ⊢ ⋆s ⬈[h] T2 →
                      T2 = ⋆s ∨ T2 = ⋆(next h s).
 #h #G #L #T2 #s * #c #H elim (cpg_inv_sort1 … H) -H *
 /2 width=1 by or_introl, or_intror/
 qed-.
 
-lemma cpx_inv_zero1: ∀h,G,L,T2. ⦃G, L⦄ ⊢ #0 ➡[h] T2 →
+lemma cpx_inv_zero1: ∀h,G,L,T2. ⦃G, L⦄ ⊢ #0 ⬈[h] T2 →
                      T2 = #0 ∨
-                     ∃∃I,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[h] V2 & ⬆*[1] V2 ≡ T2 &
+                     ∃∃I,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[h] V2 & ⬆*[1] V2 ≡ T2 &
                                   L = K.ⓑ{I}V1.
 #h #G #L #T2 * #c #H elim (cpg_inv_zero1 … H) -H *
 /4 width=7 by ex3_4_intro, ex_intro, or_introl, or_intror/
 qed-.
 
-lemma cpx_inv_lref1: ∀h,G,L,T2,i. ⦃G, L⦄ ⊢ #⫯i ➡[h] T2 →
+lemma cpx_inv_lref1: ∀h,G,L,T2,i. ⦃G, L⦄ ⊢ #⫯i ⬈[h] T2 →
                      T2 = #(⫯i) ∨
-                     ∃∃I,K,V,T. ⦃G, K⦄ ⊢ #i ➡[h] T & ⬆*[1] T ≡ T2 & L = K.ⓑ{I}V.
+                     ∃∃I,K,V,T. ⦃G, K⦄ ⊢ #i ⬈[h] T & ⬆*[1] T ≡ T2 & L = K.ⓑ{I}V.
 #h #G #L #T2 #i * #c #H elim (cpg_inv_lref1 … H) -H *
 /4 width=7 by ex3_4_intro, ex_intro, or_introl, or_intror/
 qed-.
 
-lemma cpx_inv_gref1: ∀h,G,L,T2,l. ⦃G, L⦄ ⊢ §l ➡[h] T2 → T2 = §l.
+lemma cpx_inv_gref1: ∀h,G,L,T2,l. ⦃G, L⦄ ⊢ §l ⬈[h] T2 → T2 = §l.
 #h #G #L #T2 #l * #c #H elim (cpg_inv_gref1 … H) -H //
 qed-.
 
-lemma cpx_inv_bind1: ∀h,p,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[h] U2 → (
-                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡[h] T2 &
+lemma cpx_inv_bind1: ∀h,p,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] U2 → (
+                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬈[h] T2 &
                               U2 = ⓑ{p,I}V2.T2
                      ) ∨
-                     ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[h] T & ⬆*[1] U2 ≡ T &
+                     ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ⬈[h] T & ⬆*[1] U2 ≡ T &
                           p = true & I = Abbr.
 #h #p #I #G #L #V1 #T1 #U2 * #c #H elim (cpg_inv_bind1 … H) -H *
 /4 width=5 by ex4_intro, ex3_2_intro, ex_intro, or_introl, or_intror/
 qed-.
 
-lemma cpx_inv_abbr1: ∀h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{p}V1.T1 ➡[h] U2 → (
-                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[h] T2 &
+lemma cpx_inv_abbr1: ∀h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{p}V1.T1 ⬈[h] U2 → (
+                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ⬈[h] T2 &
                               U2 = ⓓ{p}V2.T2
                      ) ∨
-                     ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[h] T & ⬆*[1] U2 ≡ T & p = true.
+                     ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ⬈[h] T & ⬆*[1] U2 ≡ T & p = true.
 #h #p #G #L #V1 #T1 #U2 * #c #H elim (cpg_inv_abbr1 … H) -H *
 /4 width=5 by ex3_2_intro, ex3_intro, ex_intro, or_introl, or_intror/
 qed-.
 
-lemma cpx_inv_abst1: ∀h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{p}V1.T1 ➡[h] U2 →
-                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ➡[h] T2 &
+lemma cpx_inv_abst1: ∀h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{p}V1.T1 ⬈[h] U2 →
+                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ⬈[h] T2 &
                               U2 = ⓛ{p}V2.T2.
 #h #p #G #L #V1 #T1 #U2 * #c #H elim (cpg_inv_abst1 … H) -H
 /3 width=5 by ex3_2_intro, ex_intro/
 qed-.
 
-lemma cpx_inv_flat1: ∀h,I,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ➡[h] U2 →
-                     ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L⦄ ⊢ U1 ➡[h] T2 &
+lemma cpx_inv_flat1: ∀h,I,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ⬈[h] U2 →
+                     ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[h] T2 &
                                  U2 = ⓕ{I}V2.T2
-                      | (⦃G, L⦄ ⊢ U1 ➡[h] U2 ∧ I = Cast)
-                      | (⦃G, L⦄ ⊢ V1 ➡[h] U2 ∧ I = Cast)
-                      | ∃∃p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L⦄ ⊢ W1 ➡[h] W2 &
-                                            ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[h] T2 &
+                      | (⦃G, L⦄ ⊢ U1 ⬈[h] U2 ∧ I = Cast)
+                      | (⦃G, L⦄ ⊢ V1 ⬈[h] U2 ∧ I = Cast)
+                      | ∃∃p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L⦄ ⊢ W1 ⬈[h] W2 &
+                                            ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈[h] T2 &
                                             U1 = ⓛ{p}W1.T1 &
                                             U2 = ⓓ{p}ⓝW2.V2.T2 & I = Appl
-                      | ∃∃p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V & ⬆*[1] V ≡ V2 &
-                                              ⦃G, L⦄ ⊢ W1 ➡[h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[h] T2 &
+                      | ∃∃p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V & ⬆*[1] V ≡ V2 &
+                                              ⦃G, L⦄ ⊢ W1 ⬈[h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈[h] T2 &
                                               U1 = ⓓ{p}W1.T1 &
                                               U2 = ⓓ{p}W2.ⓐV2.T2 & I = Appl.
 #h #I #G #L #V1 #U1 #U2 * #c #H elim (cpg_inv_flat1 … H) -H *
 /4 width=14 by or5_intro0, or5_intro1, or5_intro2, or5_intro3, or5_intro4, ex7_7_intro, ex6_6_intro, ex3_2_intro, ex_intro, conj/
 qed-.
 
-lemma cpx_inv_appl1: ∀h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐ V1.U1 ➡[h] U2 →
-                     ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L⦄ ⊢ U1 ➡[h] T2 &
+lemma cpx_inv_appl1: ∀h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐ V1.U1 ⬈[h] U2 →
+                     ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[h] T2 &
                                  U2 = ⓐV2.T2
-                      | ∃∃p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L⦄ ⊢ W1 ➡[h] W2 &
-                                            ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[h] T2 &
+                      | ∃∃p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L⦄ ⊢ W1 ⬈[h] W2 &
+                                            ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈[h] T2 &
                                             U1 = ⓛ{p}W1.T1 & U2 = ⓓ{p}ⓝW2.V2.T2
-                      | ∃∃p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V & ⬆*[1] V ≡ V2 &
-                                              ⦃G, L⦄ ⊢ W1 ➡[h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[h] T2 &
+                      | ∃∃p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V & ⬆*[1] V ≡ V2 &
+                                              ⦃G, L⦄ ⊢ W1 ⬈[h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈[h] T2 &
                                               U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2.
 #h #G #L #V1 #U1 #U2 * #c #H elim (cpg_inv_appl1 … H) -H *
 /4 width=13 by or3_intro0, or3_intro1, or3_intro2, ex6_7_intro, ex5_6_intro, ex3_2_intro, ex_intro/
 qed-.
 
-lemma cpx_inv_cast1: ∀h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝV1.U1 ➡[h] U2 →
-                     ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L⦄ ⊢ U1 ➡[h] T2 &
+lemma cpx_inv_cast1: ∀h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝV1.U1 ⬈[h] U2 →
+                     ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[h] T2 &
                                  U2 = ⓝV2.T2
-                      | ⦃G, L⦄ ⊢ U1 ➡[h] U2
-                      | ⦃G, L⦄ ⊢ V1 ➡[h] U2.
+                      | ⦃G, L⦄ ⊢ U1 ⬈[h] U2
+                      | ⦃G, L⦄ ⊢ V1 ⬈[h] U2.
 #h #G #L #V1 #U1 #U2 * #c #H elim (cpg_inv_cast1 … H) -H *
 /4 width=5 by or3_intro0, or3_intro1, or3_intro2, ex3_2_intro, ex_intro/
 qed-.
 
 (* Basic forward lemmas *****************************************************)
 
-lemma cpx_fwd_bind1_minus: ∀h,I,G,L,V1,T1,T. ⦃G, L⦄ ⊢ -ⓑ{I}V1.T1 ➡[h] T → ∀p.
-                           ∃∃V2,T2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[h] ⓑ{p,I}V2.T2 &
+lemma cpx_fwd_bind1_minus: ∀h,I,G,L,V1,T1,T. ⦃G, L⦄ ⊢ -ⓑ{I}V1.T1 ⬈[h] T → ∀p.
+                           ∃∃V2,T2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] ⓑ{p,I}V2.T2 &
                                     T = -ⓑ{I}V2.T2.
 #h #I #G #L #V1 #T1 #T * #c #H #p elim (cpg_fwd_bind1_minus … H p) -H
 /3 width=4 by ex2_2_intro, ex_intro/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_drops.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_drops.ma
index 6caaa1d76..98c2b491f 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_drops.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_drops.ma
@@ -15,14 +15,14 @@
 include "basic_2/rt_transition/cpg_drops.ma".
 include "basic_2/rt_transition/cpx.ma".
 
-(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL REDUCTION FOR TERMS *****************)
+(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS *************)
 
 (* Advanced properties ******************************************************)
 
 (* Basic_2A1: was: cpx_delta *)
 lemma cpx_delta_drops: ∀h,I,G,L,K,V,V2,W2,i.
-                       ⬇*[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ➡[h] V2 →
-                       ⬆*[⫯i] V2 ≡ W2 → ⦃G, L⦄ ⊢ #i ➡[h] W2.
+                       ⬇*[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ⬈[h] V2 →
+                       ⬆*[⫯i] V2 ≡ W2 → ⦃G, L⦄ ⊢ #i ⬈[h] W2.
 #h * #G #L #K #V #V2 #W2 #i #HLK *
 /3 width=7 by cpg_ell_drops, cpg_delta_drops, ex_intro/
 qed.
@@ -30,19 +30,19 @@ qed.
 (* Advanced inversion lemmas ************************************************)
 
 (* Basic_2A1: was: cpx_inv_atom1 *)
-lemma cpx_inv_atom1_drops: ∀h,I,G,L,T2. ⦃G, L⦄ ⊢ ⓪{I} ➡[h] T2 →
+lemma cpx_inv_atom1_drops: ∀h,I,G,L,T2. ⦃G, L⦄ ⊢ ⓪{I} ⬈[h] T2 →
                            ∨∨ T2 = ⓪{I}
                             | ∃∃s. T2 = ⋆(next h s) & I = Sort s
-                            | ∃∃J,K,V,V2,i. ⬇*[i] L ≡ K.ⓑ{J}V & ⦃G, K⦄ ⊢ V ➡[h] V2 &
+                            | ∃∃J,K,V,V2,i. ⬇*[i] L ≡ K.ⓑ{J}V & ⦃G, K⦄ ⊢ V ⬈[h] V2 &
                                             ⬆*[⫯i] V2 ≡ T2 & I = LRef i.
 #h #I #G #L #T2 * #c #H elim (cpg_inv_atom1_drops … H) -H *
 /4 width=9 by or3_intro0, or3_intro1, or3_intro2, ex4_5_intro, ex2_intro, ex_intro/
 qed-.
 
 (* Basic_2A1: was: cpx_inv_lref1 *)
-lemma cpx_inv_lref1_drops: ∀h,G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡[h] T2 →
+lemma cpx_inv_lref1_drops: ∀h,G,L,T2,i. ⦃G, L⦄ ⊢ #i ⬈[h] T2 →
                            T2 = #i ∨
-                           ∃∃J,K,V,V2. ⬇*[i] L ≡ K. ⓑ{J}V & ⦃G, K⦄ ⊢ V ➡[h] V2 &
+                           ∃∃J,K,V,V2. ⬇*[i] L ≡ K. ⓑ{J}V & ⦃G, K⦄ ⊢ V ⬈[h] V2 &
                                        ⬆*[⫯i] V2 ≡ T2.
 #h #G #L #T1 #i * #c #H elim (cpg_inv_lref1_drops … H) -H *
 /4 width=7 by ex3_4_intro, ex_intro, or_introl, or_intror/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_fqus.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_fqus.ma
index 645f59987..41763dfa2 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_fqus.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_fqus.ma
@@ -12,11 +12,13 @@
 (*                                                                        *)
 (**************************************************************************)
 
+(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS *************)
+
 (* Properties on supclosure *************************************************)
 
 lemma fqu_cpx_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
-                     ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, o] U2 →
-                     ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, o] U1 & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
+                     ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h, o] U2 →
+                     ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h, o] U1 & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
 #h #o #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 
 /3 width=3 by fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, cpx_pair_sn, cpx_bind, cpx_flat, ex2_intro/
 [ #I #G #L #V2 #U2 #HVU2
@@ -29,8 +31,8 @@ lemma fqu_cpx_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T
 qed-.
 
 lemma fquq_cpx_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
-                      ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, o] U2 →
-                      ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, o] U1 & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
+                      ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h, o] U2 →
+                      ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h, o] U1 & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
 #h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fquq_inv_gen … H) -H
 [ #HT12 elim (fqu_cpx_trans … HT12 … HTU2) /3 width=3 by fqu_fquq, ex2_intro/
 | * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
@@ -38,8 +40,8 @@ lemma fquq_cpx_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L
 qed-.
 
 lemma fqup_cpx_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
-                      ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, o] U2 →
-                      ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, o] U1 & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
+                      ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h, o] U2 →
+                      ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h, o] U1 & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
 #h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
 [ #G2 #L2 #T2 #H12 #U2 #HTU2 elim (fqu_cpx_trans … H12 … HTU2) -T2
   /3 width=3 by fqu_fqup, ex2_intro/
@@ -50,8 +52,8 @@ lemma fqup_cpx_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2,
 qed-.
 
 lemma fqus_cpx_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
-                      ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, o] U2 →
-                      ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, o] U1 & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
+                      ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h, o] U2 →
+                      ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h, o] U1 & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
 #h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fqus_inv_gen … H) -H
 [ #HT12 elim (fqup_cpx_trans … HT12 … HTU2) /3 width=3 by fqup_fqus, ex2_intro/
 | * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
@@ -59,8 +61,8 @@ lemma fqus_cpx_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2,
 qed-.
 
 lemma fqu_cpx_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
-                         ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, o] U2 → (T2 = U2 → ⊥) →
-                         ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
+                         ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h, o] U2 → (T2 = U2 → ⊥) →
+                         ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
 #h #o #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
 [ #I #G #L #V1 #V2 #HV12 #_ elim (lift_total V2 0 1)
   #U2 #HVU2 @(ex3_intro … U2)
@@ -88,8 +90,8 @@ lemma fqu_cpx_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L
 qed-.
 
 lemma fquq_cpx_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
-                          ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, o] U2 → (T2 = U2 → ⊥) →
-                          ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
+                          ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h, o] U2 → (T2 = U2 → ⊥) →
+                          ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
 #h #o #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fquq_inv_gen … H12) -H12
 [ #H12 elim (fqu_cpx_trans_neq … H12 … HTU2 H) -T2
   /3 width=4 by fqu_fquq, ex3_intro/
@@ -98,8 +100,8 @@ lemma fquq_cpx_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G
 qed-.
 
 lemma fqup_cpx_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
-                          ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, o] U2 → (T2 = U2 → ⊥) →
-                          ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
+                          ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h, o] U2 → (T2 = U2 → ⊥) →
+                          ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
 #h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1
 [ #G1 #L1 #T1 #H12 #U2 #HTU2 #H elim (fqu_cpx_trans_neq … H12 … HTU2 H) -T2
   /3 width=4 by fqu_fqup, ex3_intro/
@@ -110,8 +112,8 @@ lemma fqup_cpx_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2,
 qed-.
 
 lemma fqus_cpx_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
-                          ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, o] U2 → (T2 = U2 → ⊥) →
-                          ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
+                          ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h, o] U2 → (T2 = U2 → ⊥) →
+                          ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
 #h #o #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fqus_inv_gen … H12) -H12
 [ #H12 elim (fqup_cpx_trans_neq … H12 … HTU2 H) -T2
   /3 width=4 by fqup_fqus, ex3_intro/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_lleq.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_lleq.ma
index 2f608db01..946975957 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_lleq.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_lleq.ma
@@ -15,12 +15,12 @@
 include "basic_2/multiple/lleq_drop.ma".
 include "basic_2/reduction/cpx_llpx_sn.ma".
 
-(* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
+(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS *************)
 
 (* Properties on lazy equivalence for local environments ********************)
 
-lemma lleq_cpx_trans: ∀h,o,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[h, o] T2 →
-                      ∀L1. L1 ≡[T1, 0] L2 → ⦃G, L1⦄ ⊢ T1 ➡[h, o] T2.
+lemma lleq_cpx_trans: ∀h,o,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ⬈[h, o] T2 →
+                      ∀L1. L1 ≡[T1, 0] L2 → ⦃G, L1⦄ ⊢ T1 ⬈[h, o] T2.
 #h #o #G #L2 #T1 #T2 #H elim H -G -L2 -T1 -T2 /2 width=2 by cpx_st/
 [ #I #G #L2 #K2 #V1 #V2 #W2 #i #HLK2 #_ #HVW2 #IHV12 #L1 #H elim (lleq_fwd_lref_dx … H … HLK2) -L2
   [ #H elim (ylt_yle_false … H) //
@@ -43,13 +43,13 @@ lemma lleq_cpx_trans: ∀h,o,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[h, o] T2 →
 ]
 qed-.
 
-lemma cpx_lleq_conf: ∀h,o,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[h, o] T2 →
-                     ∀L1. L2 ≡[T1, 0] L1 → ⦃G, L1⦄ ⊢ T1 ➡[h, o] T2.
+lemma cpx_lleq_conf: ∀h,o,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ⬈[h, o] T2 →
+                     ∀L1. L2 ≡[T1, 0] L1 → ⦃G, L1⦄ ⊢ T1 ⬈[h, o] T2.
 /3 width=3 by lleq_cpx_trans, lleq_sym/ qed-.
 
 lemma cpx_lleq_conf_sn: ∀h,o,G. b_c_confluent1 … (cpx h o G) (lleq 0).
 /3 width=6 by cpx_llpx_sn_conf, lift_mono, ex2_intro/ qed-.
 
-lemma cpx_lleq_conf_dx: ∀h,o,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[h, o] T2 →
+lemma cpx_lleq_conf_dx: ∀h,o,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ⬈[h, o] T2 →
                         ∀L1. L1 ≡[T1, 0] L2 → L1 ≡[T2, 0] L2.
 /4 width=6 by cpx_lleq_conf_sn, lleq_sym/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_llpx_sn.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_llpx_sn.ma
index 6be896ddc..ec6255dfc 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_llpx_sn.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_llpx_sn.ma
@@ -15,7 +15,7 @@
 include "basic_2/multiple/llpx_sn_drop.ma".
 include "basic_2/reduction/cpx.ma".
 
-(* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
+(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS *************)
 
 (* Properties on lazy sn pointwise extensions *******************************)
 
diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_lreq.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_lreq.ma
index 04d85b66f..9c41e252a 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_lreq.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_lreq.ma
@@ -15,7 +15,7 @@
 include "basic_2/substitution/drop_lreq.ma".
 include "basic_2/reduction/cpx.ma".
 
-(* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
+(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS *************)
 
 (* Properties on equivalence for local environments *************************)
 
diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_lsubr.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_lsubr.ma
index cd50906ec..c2e3d8ad7 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_lsubr.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_lsubr.ma
@@ -15,7 +15,7 @@
 include "basic_2/rt_transition/cpg_lsubr.ma".
 include "basic_2/rt_transition/cpx.ma".
 
-(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL REDUCTION FOR TERMS *****************)
+(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS *************)
 
 lemma lsubr_cpx_trans: ∀h,G. lsub_trans … (cpx h G) lsubr.
 #h #G #L1 #T1 #T2 * /3 width=4 by lsubr_cpg_trans, ex_intro/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_simple.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_simple.ma
index 70e2d3dd3..fbc9eb644 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_simple.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_simple.ma
@@ -15,10 +15,10 @@
 include "basic_2/rt_transition/cpg_simple.ma".
 include "basic_2/rt_transition/cpx.ma".
 
-(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL REDUCTION FOR TERMS *****************)
+(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS *************)
 
-lemma cpx_inv_appl1_simple: ∀h,G,L,V1,T1,U. ⦃G, L⦄ ⊢ ⓐV1.T1 ➡[h] U → 𝐒⦃T1⦄ →
-                            ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L⦄ ⊢ T1 ➡[h] T2 &
+lemma cpx_inv_appl1_simple: ∀h,G,L,V1,T1,U. ⦃G, L⦄ ⊢ ⓐV1.T1 ⬈[h] U → 𝐒⦃T1⦄ →
+                            ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L⦄ ⊢ T1 ⬈[h] T2 &
                                      U = ⓐV2.T2.
 #h #G #L #V1 #T1 #U * #c #H #HT1 elim (cpg_inv_appl1_simple … H) -H
 /3 width=5 by ex3_2_intro, ex_intro/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/partial.txt b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/partial.txt
new file mode 100644
index 000000000..e69bf6b0a
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/partial.txt
@@ -0,0 +1,2 @@
+cpg.ma cpg_simple.ma cpg_drops.ma cpg_lsubr.ma
+cpx.ma cpx_simple.ma cpx_drops.ma cpx_lsubr.ma
diff --git a/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl b/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl
index a5aeb85db..6e479eaa3 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl
+++ b/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl
@@ -149,11 +149,11 @@ table {
         ]
 *)
         [ { "uncounted context-sensitive rt-transition" * } {
-             [ "cpx ( ⦃?,?⦄ ⊢ ? ➡[?] ? )" "cpx_simple" + "cpx_drops" + "cpx_lsubr" * ]
+             [ "cpx ( ⦃?,?⦄ ⊢ ? ⬈[?] ? )" "cpx_simple" + "cpx_drops" + "cpx_lsubr" * ]
           }
         ]
         [ { "counted context-sensitive rt-transition" * } {
-             [ "cpg ( ⦃?,?⦄ ⊢ ? ➡[?,?] ? )" "cpg_simple" + "cpg_drops" + "cpg_lsubr" * ]
+             [ "cpg ( ⦃?,?⦄ ⊢ ? ⬈[?,?] ? )" "cpg_simple" + "cpg_drops" + "cpg_lsubr" * ]
           }
         ]
      }
@@ -215,8 +215,8 @@ table {
    class "orange"
    [ { "relocation" * } {
         [ { "generic slicing for local environments" * } {
-             [ "drops_vector ( ⬇*[?,?] ? ≡ ? )" * ]
-             [ "drops ( ⬇*[?,?] ? ≡ ? )" "drops_lstar" + "drops_weight" + "drops_length" + "drops_ceq" + "drops_lexs" + "drops_lreq" + "drops_drops" * ]
+             [ "drops_vector ( ⬇*[?,?] ? ≡ ? ) ( ⬇*[?] ? ≡ ? )" * ]
+             [ "drops ( ⬇*[?,?] ? ≡ ? ) ( ⬇*[?] ? ≡ ? )" "drops_lstar" + "drops_weight" + "drops_length" + "drops_ceq" + "drops_lexs" + "drops_lreq" + "drops_drops" * ]
           }
         ]
         [ { "generic relocation for terms" * } {
diff --git a/matita/matita/contribs/lambdadelta/replace.sh b/matita/matita/contribs/lambdadelta/replace.sh
index 5e281b251..739dd9871 100644
--- a/matita/matita/contribs/lambdadelta/replace.sh
+++ b/matita/matita/contribs/lambdadelta/replace.sh
@@ -1,5 +1,6 @@
 #!/bin/sh
-for MA in `find -name "*.ma"`; do
+#for MA in `find -name "*.ma"`; do
+for MA in `find -name "cpg*.ma" -or -name "cpx*.ma"`; do
    echo ${MA}; sed "s!$1!$2!g" ${MA} > ${MA}.new
    if diff ${MA} ${MA}.new > /dev/null; 
       then rm -f ${MA}.new; 
diff --git a/matita/matita/predefined_virtuals.ml b/matita/matita/predefined_virtuals.ml
index 768850779..6b259a253 100644
--- a/matita/matita/predefined_virtuals.ml
+++ b/matita/matita/predefined_virtuals.ml
@@ -1514,7 +1514,7 @@ let predefined_classes = [
  ["-"; "÷"; "⊢"; "⊩"; "⊟"; ];
  ["="; "≝"; "≡"; "⩬"; "≂"; "≃"; "≈"; "≅"; "≗"; "≐"; "≑"; "≚"; "≙"; "⌆"; "⊜"; ];  
  ["→"; "↦"; "⇝"; "⤞"; "⇾"; "⤍"; "⤏"; "⤳"; ] ;
- ["⇒"; "⤇"; "➾"; "⇨"; "➡"; "➤"; "➸"; "⇉"; "⥰"; ] ;
+ ["⇒"; "⤇"; "➾"; "⇨"; "➡"; "⬈"; "➤"; "➸"; "⇉"; "⥰"; ] ;
  ["^"; "↑"; ] ;
  ["⇑"; "⇧"; "⬆"; ] ; 
  ["⇓"; "⇩"; "⬇"; "⬊"; "➷"; ] ;