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better description of declarative tactics
[helm.git] / helm / software / matita / help / C / sec_declarative_tactics.xml
index cc6aeb61de2ae497fff42a6b7ad44909d693baf2..bc0b424e0d3b0bb75ed6613a456fa904315136c8 100644 (file)
    </para>
   </sect1>
 
-  <sect1 id="tac_suppose">
-    <title>suppose</title>
-    <titleabbrev>suppose</titleabbrev>
-    <para><userinput>suppose t(x) </userinput></para>
-  <para>
-     <variablelist>
-       <varlistentry role="tactic.synopsis">
-         <term>Synopsis:</term>
-        <listitem>
-          <para><emphasis role="bold">suppose</emphasis> &term; <emphasis role="bold"> (</emphasis> &id; 
-            <emphasis role="bold">) </emphasis> [ that is equivalent to  &term; ]</para>
-        </listitem>
-       </varlistentry>
-      <varlistentry>
-         <term>Pre-condition:</term>
-        <listitem>
-          <para> &TODO; </para>
-        </listitem>
-      </varlistentry>
-      <varlistentry>
-        <term>Action:</term>
-         <listitem>
-           <para> &TODO; </para>
-         </listitem>
-      </varlistentry>
-      <varlistentry>
-        <term>New sequents to prove:</term>
-        <listitem>
-            <para>None.</para>
-        </listitem>
-      </varlistentry>
-    </variablelist>
-   </para>
-  </sect1>
-
- <sect1 id="tac_bydone">
-   <title>by done</title>
-   <titleabbrev>by done</titleabbrev>
-   <para><userinput>by [ t | _ ] done</userinput></para>
-   <para>
-     <variablelist>
-       <varlistentry role="tactic.synopsis">
-         <term>Synopsis:</term>
-        <listitem>
-          <para><emphasis role="bold">by</emphasis> &term; <emphasis role="bold"> done </emphasis></para>
-        </listitem>
-       </varlistentry>
-       <varlistentry>
-         <term>Pre-condition:</term>
-        <listitem>
-          <para>The term can be omitted with a "_", the running term will come found automatically. </para>
-        </listitem>
-       </varlistentry>
-       <varlistentry>
-         <term>Action:</term>
-          <listitem> 
-            <para>It closes the current sequent by applying <command>t</command>, taht it can be one under proof or the main proof.</para>
-          </listitem>
-       </varlistentry>
-       <varlistentry>
-           <term>New sequents to prove:</term>
-           <listitem>
-               <para>None.</para>
-           </listitem>
-       </varlistentry>
-   </variablelist>
-     </para>
-  </sect1>
-
-  <sect1 id="tac_weneedtoprove">
-    <title> we need to prove</title>
-    <titleabbrev>we need to prove</titleabbrev>
-    <para><userinput>we need to prove t [(id)] [or equivalent t]</userinput></para>
-    <para>
-      <variablelist>
-        <varlistentry role="tactic.synopsis">
-         <term>Synopsis:</term>
-         <listitem>
-            <para><emphasis role="bold">we need to prove</emphasis> &term; </para>
-         </listitem>
-       </varlistentry>
-        <varlistentry>
-          <term>Pre-condition:</term>
-         <listitem>
-           <para> &TODO; </para>
-         </listitem>
-        </varlistentry>
-        <varlistentry>
-          <term>Action:</term>
-           <listitem>
-             <para> &TODO; </para>
-           </listitem>
-        </varlistentry>
-        <varlistentry>
-           <term>New sequents to prove:</term>
-          <listitem>
-            <para>None.</para>
-          </listitem>
-       </varlistentry>     
-     </variablelist>
-    </para>
-  </sect1>
-
-  <sect1 id="tac_weproceedbyinduction">
-    <title> we proceed by induction</title>
-    <titleabbrev>we proceed by induction</titleabbrev>
-    <para><userinput>we proceed by induction on t to prove th</userinput></para>        
-    <para>
-      <variablelist>
-        <varlistentry role="tactic.synopsis">
-         <term>Synopsis:</term>
-         <listitem>
-           <para><emphasis role="bold">we proceed by induction on</emphasis> &term; <emphasis role="bold"> to prove </emphasis> &term; </para>
-         </listitem>
-       </varlistentry>
-        <varlistentry>
-         <term>Pre-condition:</term>
-         <listitem>
-           <para><command>t</command> must inhabit an inductive type and <command>th</command> must be an elimination principle for that inductive type.</para>
-         </listitem>
-       </varlistentry>
-       <varlistentry>
-         <term>Action:</term>
-           <listitem>
-             <para>It proceed on the values of <command>t</command>.</para>
-           </listitem>
-       </varlistentry>
-        <varlistentry>
-            <term>New sequents to prove:</term>
-            <listitem>
-               <para>It opens one new sequent to demonstrate.</para>
-            </listitem>
-        </varlistentry>
-      </variablelist>
-    </para>
-  </sect1>
-
-  <sect1 id="tac_weproceedbycases">
-    <title>we proceed by cases</title>
-      <titleabbrev>we proceed by cases</titleabbrev>
-      <para><userinput>we proceed by cases on t to prove t</userinput></para>
-      <para>
-        <variablelist>
-         <varlistentry role="tactic.synopsis">
-           <term>Synopsis:</term>
-           <listitem>
-             <para><emphasis role="bold">we proceed by cases on </emphasis> &term; <emphasis role="bold">to prove</emphasis> &term; </para>
-            </listitem>
-          </varlistentry>
-         <varlistentry>
-           <term>Pre-condition:</term>
-           <listitem>
-             <para> &TODO; </para>
-           </listitem>
-         </varlistentry>
-         <varlistentry>
-           <term>Action:</term>
-             <listitem>
-               <para> &TODO; </para>
-              </listitem>
-         </varlistentry>
-          <varlistentry>
-            <term>New sequents to prove:</term>
-           <listitem>
-              <para>None.</para>
-           </listitem>
-         </varlistentry>
-         </variablelist>
-       </para>
-    </sect1>
-
     <sect1 id="tac_byinduction">
-      <title>by induction</title>
-      <titleabbrev>by induction</titleabbrev>
+      <title>by induction hypothesis we know</title>
+      <titleabbrev>by induction hypothesis we know</titleabbrev>
       <para><userinput>by induction hypothesis we know t (id)</userinput></para>
       <para>
         <variablelist>
          <varlistentry role="tactic.synopsis">
            <term>Synopsis:</term>
-           <listitem>
-             <para><emphasis role="bold">by induction hypothesis we know</emphasis> &term; <emphasis role="bold"> (</emphasis> &id; <emphasis role="bold">)</emphasis></para>
+           <listitem><para><emphasis role="bold">by induction hypothesis we know</emphasis> &term; <emphasis role="bold"> (</emphasis> &id; <emphasis role="bold">)</emphasis></para>
            </listitem>
          </varlistentry>
          <varlistentry>
            <term>Pre-condition:</term>
            <listitem>
-             <para> &TODO; </para>
+        <para>To be used in a proof by induction to state the inductive
+          hypothesis.</para>
            </listitem>
          </varlistentry>
          <varlistentry>
            <term>Action:</term>
              <listitem>
-               <para> &TODO; </para>
+               <para> Introduces the inductive hypothesis. </para>
              </listitem>
          </varlistentry>
           <varlistentry>
       </para>
     </sect1>  
 
-   <sect1 id="tac_thesisbecomes">
-     <title>thesis becomes</title>
-     <titleabbrev>thesis becomes</titleabbrev>
-     <para><userinput>the thesis becomes t</userinput></para>
-     <para>
-        <variablelist>
-          <varlistentry role="tactic.synopsis">
-            <term>Synopsis:</term>
-            <listitem>
-              <para><emphasis role ="bold">the thesis becomes</emphasis> &sterm; </para>
-            </listitem>
-          </varlistentry>
-          <varlistentry>
-            <term>Pre-condition:</term>
-            <listitem>
-              <para>Each subterm must be convertible with the term <command>t</command> disambiguate in the context of the matched subterm.</para>
-            </listitem>
-          </varlistentry>
-          <varlistentry>
-            <term>Action:</term>
-            <listitem>
-              <para>It replaces the subterm of the current sequent with the new term</para>
-            </listitem>
-          </varlistentry>
-          <varlistentry>
-            <term>New sequence to prove:</term>
-            <listitem>
-              <para>None.</para>
-            </listitem>
-          </varlistentry>
-        </variablelist>
-       </para>
-   </sect1>
-   
    <sect1 id="tac_case">
      <title>case</title>
      <titleabbrev>case</titleabbrev>
-     <para><userinput>case id (id:t)list</userinput></para>
+     <para><userinput>case id (id1:t1) … (idn:tn)</userinput></para>
      <para>
        <variablelist>
          <varlistentry role="tactic.synopsis">
           <term>Synopsis:</term>
           <listitem>
-            <para><emphasis role="bold">case</emphasis> &id; <emphasis role="bold">(</emphasis> &id; <emphasis role="bold">:</emphasis> &term; <emphasis role="bold">)</emphasis></para>
+            <para><emphasis role="bold">case</emphasis> &id; [<emphasis role="bold">(</emphasis> &id; <emphasis role="bold">:</emphasis> &term;  <emphasis role="bold">)</emphasis>] … </para>
           </listitem>
         </varlistentry>
          <varlistentry>
             <term>Pre-condition:</term>
            <listitem>
-               <para> &TODO; </para>
+        <para>To be used in a proof by induction or by cases to start
+          a new case</para>
            </listitem>
         </varlistentry>
          <varlistentry>
             <term>Action:</term>
             <listitem>
-              <para> &TODO; </para>
+              <para>Starts the new case <command>id</command> declaring
+                the local parameters <command>(id1:t1) … (idn:tn)</command></para>
             </listitem>
          </varlistentry>
          <varlistentry>
            <term>New sequents to prove:</term>
            <listitem>
-               <para>&TODO;</para>
+               <para>None</para>
            </listitem>
          </varlistentry>
        </variablelist>
      </para>
   </sect1> 
 
+ <sect1 id="tac_bydone">
+   <title>done</title>
+   <titleabbrev>done</titleabbrev>
+   <para><userinput>justification done</userinput></para>
+   <para>
+     <variablelist>
+       <varlistentry role="tactic.synopsis">
+         <term>Synopsis:</term>
+        <listitem>
+          <para>&justification; <emphasis role="bold">done</emphasis></para>
+        </listitem>
+       </varlistentry>
+       <varlistentry>
+         <term>Pre-condition:</term>
+        <listitem>
+          <para></para>
+        </listitem>
+       </varlistentry>
+       <varlistentry>
+         <term>Action:</term>
+          <listitem> 
+            <para>It closes the current sequent given the justification.</para>
+          </listitem>
+       </varlistentry>
+       <varlistentry>
+           <term>New sequents to prove:</term>
+           <listitem>
+               <para>None.</para>
+           </listitem>
+       </varlistentry>
+   </variablelist>
+     </para>
+  </sect1>
+
+
+  <sect1 id="tac_exitselim">
+    <title>let such that</title>
+    <titleabbrev>let such that</titleabbrev>
+    <para><userinput>justification let x:t such that p (id)</userinput>
+    </para>
+    <para>
+      <variablelist>
+        <varlistentry role="tactic.synopsis">
+         <term>Synopsis:</term>
+         <listitem>
+           <para>&justification; <emphasis role="bold">let</emphasis> &id; 
+                   <emphasis role="bold">:</emphasis> &term; <emphasis role="bold">such that</emphasis> &term; 
+                    <emphasis role="bold">(</emphasis> &id; <emphasis role="bold">)</emphasis></para>
+          </listitem>
+       </varlistentry>
+       <varlistentry>
+         <term>Pre-condition:</term>
+         <listitem>
+      <para>
+           </para>
+         </listitem>  
+       </varlistentry>
+       <varlistentry>
+         <term>Action:</term>
+         <listitem>
+      <para>It derives <command>∃x:t.p</command> using the
+        <command>justification</command> and then it introduces in the context
+        <command>x</command> and the hypothesis  
+        <command>p</command> labelled with
+        <command>id</command>.
+      </para>
+         </listitem>
+       </varlistentry>
+       <varlistentry>
+         <term>New sequent to prove:</term>
+         <listitem>
+      <para>None.</para>
+         </listitem>
+       </varlistentry>
+     </variablelist>
+    </para>
+  </sect1>
+
   <sect1 id="tac_obtain">
-    <title>obtain/conclude</title>
-    <titleabbrev>obtain/conclude</titleabbrev>
-    <para><userinput>obtain H t=t [auto_params | exact t| proof | using t] done</userinput></para>
+    <title>obtain</title>
+    <titleabbrev>obtain</titleabbrev>
+    <para><userinput>obtain H t1 = t2 justification</userinput></para>
     <para>
       <variablelist>
          <varlistentry role="tactic.synopsis">
           <term>Synopsis:</term>
           <listitem>
-            <para>[<emphasis role="bold">obtain</emphasis> &id; | <emphasis role="bold">conclude</emphasis> &term;] <emphasis role="bold">=</emphasis> &term; [&autoparams; | <emphasis role="bold">exact</emphasis> &term; | <emphasis role="bold">using</emphasis> &term; | <emphasis role="bold">proof</emphasis>] [<emphasis role="bold">done</emphasis>]</para>
+            <para>[<emphasis role="bold">obtain</emphasis> &id; | <emphasis role="bold">conclude</emphasis> &term;] <emphasis role="bold">=</emphasis> &term; [&autoparams; | <emphasis role="bold">using</emphasis> &term; | <emphasis role="bold">using once</emphasis> &term; | <emphasis role="bold">proof</emphasis>] [<emphasis role="bold">done</emphasis>]</para>
           </listitem>
         </varlistentry>
         <varlistentry>
           <term>Pre-condition:</term>
             <listitem>
-              <para> &TODO; </para>
-            </listitem>
+         <para><command>conclude</command> can be used only if the current
+           sequent is stating an equality. The left hand side must be omitted
+           in an equality chain.</para> 
+       </listitem>
         </varlistentry>
         <varlistentry>
           <term>Action:</term>
           <listitem>
-             <para> &TODO; </para>
+       <para>Starts or continues an equality chain. If the chain starts 
+         with <command>obtain H</command> a new subproof named 
+         <command>H</command> is started.</para>
            </listitem>
          </varlistentry>
         <varlistentry>
-          <term>New sequence to prove:</term>
+          <term>New sequent to prove:</term>
             <listitem>
-               <para> &TODO; </para>
+               <para>If the chain starts 
+            with <command>obtain H</command> a nre sequent for
+            <command>t2 = ?</command> is opened.
+          </para>
             </listitem>
         </varlistentry>
        </variablelist>
      </para>
    </sect1>
 
- <sect1 id="tac_bytermweproved">
-   <title>by term we proved</title>
-     <titleabbrev>by term we proved</titleabbrev>
-     <para><userinput>by th we proved t [""|(id)]</userinput></para>
+  <sect1 id="tac_suppose">
+    <title>suppose</title>
+    <titleabbrev>suppose</titleabbrev>
+    <para><userinput>suppose t1 (x) that is equivalent to t2</userinput></para>
+  <para>
+     <variablelist>
+       <varlistentry role="tactic.synopsis">
+         <term>Synopsis:</term>
+        <listitem>
+          <para><emphasis role="bold">suppose</emphasis> &term; <emphasis role="bold"> (</emphasis> &id; 
+            <emphasis role="bold">) </emphasis> [ <emphasis role="bold">that is equivalent to</emphasis>  &term; ]</para>
+        </listitem>
+       </varlistentry>
+      <varlistentry>
+         <term>Pre-condition:</term>
+        <listitem>
+          <para>The conclusion of the current proof must be
+          <command>∀x:T.P</command> or
+          <command>T→P</command> where <command>T</command> is
+          a proposition (i.e. <command>T</command> has type
+          <command>Prop</command> or <command>CProp</command>).</para>
+        </listitem>
+      </varlistentry>
+      <varlistentry>
+        <term>Action:</term>
+         <listitem>
+           <para>It adds to the context of the current sequent to prove a new
+          declaration <command>x : T </command>. The new conclusion becomes
+          <command>P</command>.</para>
+         </listitem>
+      </varlistentry>
+      <varlistentry>
+        <term>New sequents to prove:</term>
+        <listitem>
+            <para>None.</para>
+        </listitem>
+      </varlistentry>
+    </variablelist>
+   </para>
+  </sect1>
+
+   <sect1 id="tac_thesisbecomes">
+     <title>the thesis becomes</title>
+     <titleabbrev>the thesis becomes</titleabbrev>
+     <para><userinput>the thesis becomes t</userinput></para>
      <para>
-       <variablelist>
-         <varlistentry role="tactic.synopsis">
-          <term>Synopsis:</term>
-          <listitem>
-            <para><emphasis role="bold">by</emphasis> &term; <emphasis role="bold">we proved</emphasis> &term; 
-                  <emphasis role="bold">(</emphasis> &id; <emphasis role="bold">)</emphasis></para>
-           </listitem>
+        <variablelist>
+          <varlistentry role="tactic.synopsis">
+            <term>Synopsis:</term>
+            <listitem>
+              <para><emphasis role ="bold">the thesis becomes</emphasis> &term; </para>
+            </listitem>
           </varlistentry>
           <varlistentry>
             <term>Pre-condition:</term>
             <listitem>
-              <para><command>th</command>must have type <command>Prop</command>.
-                    <command>id</command>can to be omitted with null</para>
+         <para>The provided term <command>t</command> must be convertible with
+           current sequent.</para>
             </listitem>
           </varlistentry>
           <varlistentry>
             <term>Action:</term>
             <listitem>
-              <para>It closes the current sequent by applying<command>th</command> to <command>t</command>.</para>
+              <para>It changes the current goal to the one provided.</para>
             </listitem>
           </varlistentry>
           <varlistentry>
-            <term>New sequence to prove:</term>
+            <term>New sequent to prove:</term>
             <listitem>
-              <para> &TODO; </para>
+              <para>None.</para>
             </listitem>
           </varlistentry>
         </variablelist>
        </para>
-    </sect1>
-
-  <sect1 id="tac_exitselim">
-    <title>exits elim</title>
-    <titleabbrev>exits elim</titleabbrev>
-    <para><userinput>by t let id:t such that t (id)</userinput></para>
+   </sect1>
+   
+  <sect1 id="tac_weneedtoprove">
+    <title>we need to prove</title>
+    <titleabbrev>we need to prove</titleabbrev>
+    <para><userinput>we need to prove t1 (id) or equivalently t2</userinput></para>
     <para>
       <variablelist>
         <varlistentry role="tactic.synopsis">
          <term>Synopsis:</term>
          <listitem>
-           <para><emphasis role="bold">by</emphasis> &term; <emphasis role="bold">let</emphasis> &id; 
-                   <emphasis role="bold">:</emphasis> &term; <emphasis role="bold">such that</emphasis> &term; 
-                    <emphasis role="bold">(</emphasis> &id; <emphasis role="bold">)</emphasis></para>
-          </listitem>
-       </varlistentry>
-       <varlistentry>
-         <term>Pre-condition:</term>
-         <listitem>
-           <para><command>t</command> must be an inductive type,existential and 
-              <command>t (id)</command> must be an elimanation priciple for that inductive type.
-           </para>
-         </listitem>  
-       </varlistentry>
-       <varlistentry>
-         <term>Action:</term>
-         <listitem>
-           <para>it proceeds on <command>t</command> to the elimination of the existential one </para>
+      <para><emphasis role="bold">we need to prove</emphasis> &term;
+        [<emphasis role="bold">(</emphasis>&id;
+        <emphasis role="bold">)</emphasis>]
+        [ <emphasis role="bold">or equivalently</emphasis> &term;]</para> 
          </listitem>
        </varlistentry>
-       <varlistentry>
-         <term>New sequence to prove:</term>
+        <varlistentry>
+          <term>Pre-condition:</term>
          <listitem>
-           <para>It opens one new sequent. The names of the new hypotheses are picked by<command>t (id)</command> </para>
+           <para></para>
          </listitem>
-       </varlistentry>
+        </varlistentry>
+        <varlistentry>
+          <term>Action:</term>
+           <listitem>
+        <para>If <command>id</command> is provided, starts a subproof that once concluded 
+              will be named <command>id</command>. Otherwise states what needs to be proved.
+              If <command>t2</command> is provided, the new goal is 
+              immediately changed to <command>t2</command> wich must
+              be equivalent to <command>t1</command>.
+            </para>
+           </listitem>
+        </varlistentry>
+        <varlistentry>
+           <term>New sequents to prove:</term>
+          <listitem>
+            <para>The stated one if <command>id</command> is provided</para>
+          </listitem>
+       </varlistentry>     
      </variablelist>
     </para>
   </sect1>
 
+
   <sect1 id="tac_andelim">
-    <title>and elim</title>
-    <titleabbrev>and elim</titleabbrev>
-    <para><userinput>by t we have t (id) and t (id)</userinput></para>
+    <title>we have</title>
+    <titleabbrev>we have</titleabbrev>
+    <para><userinput>justification we have t1 (id1) and t2 (id2)</userinput>
+    </para>
     <para>
       <variablelist>
         <varlistentry role="tactic_synopsis">
          <term>Synopsis:</term>
          <listitem>
-           <para><emphasis role="bold">by</emphasis> &term; <emphasis role="bold">we have</emphasis> &term; 
+           <para>&justification; <emphasis role="bold">we have</emphasis> &term; 
                    <emphasis role="bold">( </emphasis> &id; <emphasis role="bold"> ) and </emphasis> &term; 
                      <emphasis role="bold"> ( </emphasis> &id; <emphasis role="bold">)</emphasis></para>
           </listitem>
        <varlistentry>
          <term>Pre-condition:</term>
          <listitem>
-           <para><command>t</command> must be an logical type,type and  </para>
+           <para></para>
          </listitem>
        </varlistentry>
        <varlistentry>
          <term>Action:</term>
          <listitem>
-           <para>It proceeds on the values of <command>t</command>, according to the elimination principle </para>
+      <para>It derives <command>t1∧t2</command> using the 
+        <command>justification</command> then it introduces in the context
+        <command>t1</command>   labelled with <command>id1</command> and
+        <command>t2</command>   labelled with <command>id2</command>.
+      </para>
          </listitem>
        </varlistentry>
        <varlistentry>
-         <term>New sequence to prove:</term>
+         <term>New sequent to prove:</term>
          <listitem>
-            <para>It opens one new sequent, the names of the new hypotheses are picked by <command>t(id) and t(id)</command> </para>
+            <para>None.</para>
           </listitem>
        </varlistentry>
       </variablelist>
     </para>
   </sect1>
 
+  <sect1 id="tac_weproceedbycases">
+    <title>we proceed by cases on</title>
+      <titleabbrev>we proceed by cases on</titleabbrev>
+      <para><userinput>we proceed by cases on t to prove th</userinput></para>
+      <para>
+        <variablelist>
+         <varlistentry role="tactic.synopsis">
+           <term>Synopsis:</term>
+           <listitem>
+             <para><emphasis role="bold">we proceed by cases on</emphasis> &term; <emphasis role="bold">to prove</emphasis> &term; </para>
+            </listitem>
+          </varlistentry>
+         <varlistentry>
+           <term>Pre-condition:</term>
+           <listitem>
+        <para><command>t</command> must inhabitant of an inductive type and
+          <command>th</command> must be the conclusion to be proved by 
+          cases.</para>
+           </listitem>
+         </varlistentry>
+         <varlistentry>
+           <term>Action:</term>
+             <listitem>
+          <para> It proceeds by cases on <command>t</command> </para>
+              </listitem>
+         </varlistentry>
+          <varlistentry>
+            <term>New sequents to prove:</term>
+           <listitem>
+              <para>It opens one new sequent for each constructor of the
+                type of <command>t</command>.</para>
+           </listitem>
+         </varlistentry>
+         </variablelist>
+       </para>
+    </sect1>
+    
+    <sect1 id="tac_weproceedbyinduction">
+    <title>we proceed by induction on</title>
+    <titleabbrev>we proceed by induction on</titleabbrev>
+    <para><userinput>we proceed by induction on t to prove th</userinput></para>        
+    <para>
+      <variablelist>
+        <varlistentry role="tactic.synopsis">
+         <term>Synopsis:</term>
+         <listitem>
+           <para><emphasis role="bold">we proceed by induction on</emphasis> &term; <emphasis role="bold"> to prove </emphasis> &term; </para>
+         </listitem>
+       </varlistentry>
+        <varlistentry>
+         <term>Pre-condition:</term>
+         <listitem>
+      <para><command>t</command> must inhabitant of an inductive type and 
+        <command>th</command> must be the conclusion to be proved by induction.
+        </para>
+         </listitem>
+       </varlistentry>
+       <varlistentry>
+         <term>Action:</term>
+           <listitem>
+             <para>It proceed by induction on <command>t</command>.</para>
+           </listitem>
+       </varlistentry>
+        <varlistentry>
+            <term>New sequents to prove:</term>
+            <listitem>
+              <para>It opens one new sequent for each constructor of the
+                type of <command>t</command>.</para>
+            </listitem>
+        </varlistentry>
+      </variablelist>
+    </para>
+  </sect1>
+
+
+ <sect1 id="tac_bytermweproved">
+   <title>we proved</title>
+     <titleabbrev>we proved</titleabbrev>
+     <para><userinput>justification we proved t (id)</userinput></para>
+     <para>
+       <variablelist>
+         <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para>&justification; <emphasis role="bold">we proved</emphasis> &term; 
+         <emphasis role="bold">(</emphasis> &id; 
+         <emphasis role="bold">)</emphasis></para>
+           </listitem>
+          </varlistentry>
+          <varlistentry>
+            <term>Pre-condition:</term>
+            <listitem>
+              <para><command>t</command>must have type <command>Prop</command>.
+         </para>
+            </listitem>
+          </varlistentry>
+          <varlistentry>
+            <term>Action:</term>
+            <listitem>
+         <para>It derives <command>t</command>
+           using the justification and labels the conclusion with
+           <command>id</command>.
+         </para>
+            </listitem>
+          </varlistentry>
+          <varlistentry>
+            <term>New sequent to prove:</term>
+            <listitem>
+              <para>None.</para>
+            </listitem>
+          </varlistentry>
+        </variablelist>
+       </para>
+    </sect1>
+
 </chapter>