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[helm.git] / matita / help / C / sec_tactics.xml

diff --git a/matita/help/C/sec_tactics.xml b/matita/help/C/sec_tactics.xml
index 4bba23eb6e22c887d99171a63f6776f76dfdf4b5..364401aa5c7b130ded02950e6c72658fc3fb0543 100644 (file)
--- a/matita/help/C/sec_tactics.xml
+++ b/matita/help/C/sec_tactics.xml
@@ -1,14 +1,27 @@
 
 <!-- ============ Tactics ====================== -->
 <chapter id="sec_tactics">
- <title>Tactics</title>
+  <title>Tactics</title>
+
+  <sect1 id="tactics_quickref">
+    <title>Quick reference card</title>
+    <para>
+      &tacticref;
+    </para>
+  </sect1>
 
   <sect1 id="tac_absurd">
-    <title>absurd &lt;term&gt;</title>
+    <title>absurd</title>
     <titleabbrev>absurd</titleabbrev>
     <para><userinput>absurd P</userinput></para>
      <para>
       <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para><emphasis role="bold">absurd</emphasis> &sterm;</para>
+          </listitem>
+        </varlistentry>
         <varlistentry>
           <term>Pre-conditions:</term>
           <listitem>
@@ -33,11 +46,17 @@
      </para>
   </sect1>
   <sect1 id="tac_apply">
-    <title>apply &lt;term&gt;</title>
+    <title>apply</title>
     <titleabbrev>apply</titleabbrev>
     <para><userinput>apply t</userinput></para>
     <para>
       <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para><emphasis role="bold">apply</emphasis> &sterm;</para>
+          </listitem>
+        </varlistentry>
         <varlistentry>
           <term>Pre-conditions:</term>
           <listitem>
@@ -67,12 +86,73 @@
       </variablelist>
     </para>
   </sect1>
+  <sect1 id="tac_applyS">
+    <title>applyS</title>
+    <titleabbrev>applyS</titleabbrev>
+    <para><userinput>applyS t auto_params</userinput></para>
+    <para>
+      <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para><emphasis role="bold">applyS</emphasis> &sterm; &autoparams;</para>
+          </listitem>
+        </varlistentry>
+        <varlistentry>
+          <term>Pre-conditions:</term>
+          <listitem>
+            <para><command>t</command> must have type
+             <command>T<subscript>1</subscript> → ... →
+              T<subscript>n</subscript> → G</command>.</para>
+          </listitem>
+        </varlistentry>
+        <varlistentry>
+          <term>Action:</term>
+          <listitem>
+            <para><command>applyS</command> is useful when
+             <command>apply</command> fails because the current goal
+             and the conclusion of the applied theorems are extensionally
+             equivalent up to instantiation of metavariables, but cannot
+             be unified. E.g. the goal is <command>P(n*O+m)</command> and
+             the theorem to be applied proves <command>∀m.P(m+O)</command>.
+            </para>
+            <para>
+             It tries to automatically rewrite the current goal using
+             <link linkend="tac_auto">auto paramodulation</link>
+             to make it unifiable with <command>G</command>.
+             Then it closes the current sequent by applying
+             <command>t</command> to <command>n</command>
+             implicit arguments (that become new sequents).
+             The <command>auto_params</command> parameters are passed
+             directly to <command>auto paramodulation</command>.
+            </para>
+          </listitem>
+        </varlistentry>
+        <varlistentry>
+          <term>New sequents to prove:</term>
+          <listitem>
+            <para>It opens a new sequent for each premise 
+             <command>T<subscript>i</subscript></command> that is not
+             instantiated by unification. <command>T<subscript>i</subscript></command> is
+             the conclusion of the <command>i</command>-th new sequent to
+             prove.</para>
+          </listitem>
+        </varlistentry>
+      </variablelist>
+    </para>
+  </sect1>
   <sect1 id="tac_assumption">
     <title>assumption</title>
     <titleabbrev>assumption</titleabbrev>
     <para><userinput>assumption </userinput></para>
     <para>
       <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para><emphasis role="bold">assumption</emphasis></para>
+          </listitem>
+        </varlistentry>
         <varlistentry>
           <term>Pre-conditions:</term>
           <listitem>
@@ -96,20 +176,26 @@
     </para>
   </sect1>
   <sect1 id="tac_auto">
-    <title>auto [depth=&lt;int&gt;] [width=&lt;int&gt;] [paramodulation] [full]</title>
+    <title>auto</title>
     <titleabbrev>auto</titleabbrev>
-    <para><userinput>auto depth=d width=w paramodulation full</userinput></para>
+    <para><userinput>auto params</userinput></para>
     <para>
       <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para><emphasis role="bold">auto</emphasis> &autoparams;</para>
+          </listitem>
+        </varlistentry>
         <varlistentry>
           <term>Pre-conditions:</term>
           <listitem>
             <para>None, but the tactic may fail finding a proof if every
              proof is in the search space that is pruned away. Pruning is
-             controlled by <command>d</command> and <command>w</command>.
+             controlled by the optional <command>params</command>.
              Moreover, only lemmas whose type signature is a subset of the
              signature of the current sequent are considered. The signature of
-             a sequent is ...TODO</para>
+             a sequent is ...&TODO;</para>
           </listitem>
         </varlistentry>
         <varlistentry>
@@ -129,24 +215,89 @@
       </variablelist>
     </para>
   </sect1>
+  <sect1 id="tac_cases">
+    <title>cases</title>
+    <titleabbrev>cases</titleabbrev>
+    <para><userinput>
+     cases t hyps
+    </userinput></para>
+    <para>
+      <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para>
+            <emphasis role="bold">cases</emphasis>
+            &term; [<emphasis role="bold">(</emphasis>[&id;]…<emphasis role="bold">)</emphasis>]
+           </para>
+          </listitem>
+        </varlistentry>
+        <varlistentry>
+          <term>Pre-conditions:</term>
+          <listitem>
+            <para>
+            <command>t</command> must inhabit an inductive type
+           </para>
+          </listitem>
+        </varlistentry>
+        <varlistentry>
+          <term>Action:</term>
+          <listitem>
+            <para>
+            It proceed by cases on <command>t</command>. The new generated
+             hypothesis in each branch are named according to
+             <command>hyps</command>.
+           </para>
+          </listitem>
+        </varlistentry>
+        <varlistentry>
+          <term>New sequents to prove:</term>
+          <listitem>
+            <para>One new sequent for each constructor of the type of
+             <command>t</command>. Each sequent has a new hypothesis for
+             each argument of the constructor.</para>
+          </listitem>
+        </varlistentry>
+      </variablelist>
+    </para>
+  </sect1>
   <sect1 id="tac_clear">
-    <title>clear &lt;id&gt;</title>
+    <title>clear</title>
     <titleabbrev>clear</titleabbrev>
-    <para><userinput>clear H</userinput></para>
+    <para><userinput>
+     clear H<subscript>1</subscript> ... H<subscript>m</subscript>
+    </userinput></para>
     <para>
       <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para>
+            <emphasis role="bold">clear</emphasis>
+            &id; [&id;…]
+           </para>
+          </listitem>
+        </varlistentry>
         <varlistentry>
           <term>Pre-conditions:</term>
           <listitem>
-            <para><command>H</command> must be an hypothesis of the
-             current sequent to prove.</para>
+            <para>
+            <command>
+             H<subscript>1</subscript> ... H<subscript>m</subscript>
+            </command> must be hypotheses of the
+             current sequent to prove.
+           </para>
           </listitem>
         </varlistentry>
         <varlistentry>
           <term>Action:</term>
           <listitem>
-            <para>It hides the hypothesis <command>H</command> from the
-             current sequent.</para>
+            <para>
+            It hides the hypotheses 
+             <command>
+             H<subscript>1</subscript> ... H<subscript>m</subscript>
+             </command> from the current sequent.
+           </para>
           </listitem>
         </varlistentry>
         <varlistentry>
@@ -159,11 +310,17 @@
     </para>
   </sect1>
   <sect1 id="tac_clearbody">
-    <title>clearbody &lt;id&gt;</title>
+    <title>clearbody</title>
     <titleabbrev>clearbody</titleabbrev>
     <para><userinput>clearbody H</userinput></para>
     <para>
       <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para><emphasis role="bold">clearbody</emphasis> &id;</para>
+          </listitem>
+        </varlistentry>
         <varlistentry>
           <term>Pre-conditions:</term>
           <listitem>
@@ -187,12 +344,70 @@
       </variablelist>
     </para>
   </sect1>
+  <sect1 id="tac_compose">
+    <title>compose</title>
+    <titleabbrev>compose</titleabbrev>
+    <para><userinput>compose n t1 with t2 hyps</userinput></para>
+    <para>
+      <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para><emphasis role="bold">compose</emphasis> [&nat;] &sterm; [<emphasis role="bold">with</emphasis> &sterm;] [&intros-spec;]</para>
+          </listitem>
+        </varlistentry>
+        <varlistentry>
+          <term>Pre-conditions:</term>
+          <listitem>
+            <para></para>
+          </listitem>
+        </varlistentry>
+        <varlistentry>
+          <term>Action:</term>
+          <listitem>
+            <para>Composes t1 with t2 in every possible way
+              <command>n</command> times introducing generated terms
+              as if <command>intros hyps</command> was issued.</para>
+            <para>If <command>t1:∀x:A.B[x]</command> and
+            <command>t2:∀x,y:A.B[x]→B[y]→C[x,y]</command> it generates:
+             <itemizedlist>
+                <listitem>
+                  <para><command>λx,y:A.t2 x y (t1 x) : ∀x,y:A.B[y]→C[x,y]</command></para>
+                </listitem>
+                <listitem>
+                  <para><command>λx,y:A.λH:B[x].t2 x y H (t1 y) : ∀x,y:A.B[x]→C[x,y]
+                  </command></para>
+                </listitem>
+             </itemizedlist>
+          </para>
+          <para>If <command>t2</command> is omitted it composes 
+            <command>t1</command>
+              with every hypothesis that can be introduced.  
+              <command>n</command> iterates the process.</para>
+          </listitem>
+        </varlistentry>
+        <varlistentry>
+          <term>New sequents to prove:</term>
+          <listitem>
+            <para>The same, but with more hypothesis eventually introduced
+            by the &intros-spec;.</para>
+          </listitem>
+        </varlistentry>
+      </variablelist>
+    </para>
+  </sect1>
   <sect1 id="tac_change">
-    <title>change &lt;pattern&gt; with &lt;term&gt;</title>
+    <title>change</title>
     <titleabbrev>change</titleabbrev>
     <para><userinput>change patt with t</userinput></para>
     <para>
       <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para><emphasis role="bold">change</emphasis> &pattern; <emphasis role="bold">with</emphasis> &sterm;</para>
+          </listitem>
+        </varlistentry>
         <varlistentry>
           <term>Pre-conditions:</term>
           <listitem>
@@ -220,11 +435,17 @@
     </para>
   </sect1>
   <sect1 id="tac_constructor">
-    <title>constructor &lt;int&gt;</title>
+    <title>constructor</title>
     <titleabbrev>constructor</titleabbrev>
     <para><userinput>constructor n</userinput></para>
     <para>
       <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para><emphasis role="bold">constructor</emphasis> &nat;</para>
+          </listitem>
+        </varlistentry>
         <varlistentry>
           <term>Pre-conditions:</term>
           <listitem>
@@ -257,6 +478,12 @@
     <para><userinput>contradiction </userinput></para>
     <para>
       <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para><emphasis role="bold">contradiction</emphasis></para>
+          </listitem>
+        </varlistentry>
         <varlistentry>
           <term>Pre-conditions:</term>
           <listitem>
@@ -281,11 +508,17 @@
     </para>
   </sect1>
   <sect1 id="tac_cut">
-    <title>cut &lt;term&gt; [as &lt;id&gt;]</title>
+    <title>cut</title>
     <titleabbrev>cut</titleabbrev>
     <para><userinput>cut P as H</userinput></para>
     <para>
       <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para><emphasis role="bold">cut</emphasis> &sterm; [<emphasis role="bold">as</emphasis> &id;]</para>
+          </listitem>
+        </varlistentry>
         <varlistentry>
           <term>Pre-conditions:</term>
           <listitem>
@@ -312,50 +545,117 @@
     </para>
   </sect1>
   <sect1 id="tac_decompose">
-    <title>decompose [&lt;ident list&gt;] &lt;ident&gt; [&lt;intros_spec&gt;]</title>
+    <title>decompose</title>
     <titleabbrev>decompose</titleabbrev>
-    <para><userinput>decompose ???</userinput></para>
+    <para><userinput>
+     decompose as H<subscript>1</subscript> ... H<subscript>m</subscript>
+    </userinput></para>
     <para>
       <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para>
+            <emphasis role="bold">decompose</emphasis>
+            [<emphasis role="bold">as</emphasis> &id;…]
+           </para>
+          </listitem>
+        </varlistentry>
         <varlistentry>
           <term>Pre-conditions:</term>
           <listitem>
-            <para>TODO.</para>
+            <para>None.</para>
           </listitem>
         </varlistentry>
         <varlistentry>
           <term>Action:</term>
           <listitem>
-            <para>TODO.</para>
+            <para>
+            For each each premise <command>H</command> of type 
+            <command>T</command> in the current context where
+            <command>T</command> is a non-recursive inductive type without
+            right parameters and of sort Prop or CProp, the tactic runs
+            <command> 
+             elim H as H<subscript>1</subscript> ... H<subscript>m</subscript>
+            </command>, clears <command>H</command>  and runs itself
+            recursively on each new premise introduced by 
+            <command>elim</command> in the opened sequents. 
+           </para>
           </listitem>
         </varlistentry>
         <varlistentry>
           <term>New sequents to prove:</term>
           <listitem>
-            <para>TODO.</para>
+            <para>
+            The ones generated by all the <command>elim</command> tactics run.
+           </para>
           </listitem>
         </varlistentry>
       </variablelist>
     </para>
   </sect1>
-  <sect1 id="tac_discriminate">
-    <title>discriminate &lt;term&gt;</title>
-    <titleabbrev>discriminate</titleabbrev>
-    <para><userinput>discriminate p</userinput></para>
+  <sect1 id="tac_demodulate">
+    <title>demodulate</title>
+    <titleabbrev>demodulate</titleabbrev>
+    <para><userinput>demodulate</userinput></para>
     <para>
       <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para><emphasis role="bold">demodulate</emphasis></para>
+          </listitem>
+        </varlistentry>
         <varlistentry>
           <term>Pre-conditions:</term>
           <listitem>
-            <para><command>p</command> must have type <command>K t<subscript>1</subscript> ... t<subscript>n</subscript> = K' t'<subscript>1</subscript> ... t'<subscript>m</subscript></command> where <command>K</command> and <command>K'</command> must be different constructors of the same inductive type and each argument list can be empty if
-its constructor takes no arguments.</para>
+            <para>None.</para>
           </listitem>
         </varlistentry>
         <varlistentry>
           <term>Action:</term>
           <listitem>
-            <para>It closes the current sequent by proving the absurdity of
-             <command>p</command>.</para>
+            <para>&TODO;</para>
+          </listitem>
+        </varlistentry>
+        <varlistentry>
+          <term>New sequents to prove:</term>
+          <listitem>
+            <para>None.</para>
+          </listitem>
+        </varlistentry>
+      </variablelist>
+    </para>
+  </sect1>
+  <sect1 id="tac_destruct">
+    <title>destruct</title>
+    <titleabbrev>destruct</titleabbrev>
+    <para><userinput>destruct p</userinput></para>
+    <para>
+      <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para><emphasis role="bold">destruct</emphasis> &sterm;</para>
+          </listitem>
+        </varlistentry>
+        <varlistentry>
+          <term>Pre-conditions:</term>
+          <listitem>
+            <para><command>p</command> must have type <command>E<subscript>1</subscript> = E<subscript>2</subscript></command> where the two sides of the equality are possibly applied constructors of an inductive type.</para>
+          </listitem>
+        </varlistentry>
+        <varlistentry>
+          <term>Action:</term>
+          <listitem>
+            <para>The tactic recursively compare the two sides of the equality
+             looking for different constructors in corresponding position.
+             If two of them are found, the tactic closes the current sequent
+             by proving the absurdity of <command>p</command>. Otherwise
+             it adds a new hypothesis for each leaf of the formula that
+             states the equality of the subformulae in the corresponding
+             positions on the two sides of the equality.
+            </para>
           </listitem>
         </varlistentry>
         <varlistentry>
@@ -368,11 +668,17 @@ its constructor takes no arguments.</para>
     </para>
   </sect1>
   <sect1 id="tac_elim">
-    <title>elim &lt;term&gt; [using &lt;term&gt;] [&lt;intros_spec&gt;]</title>
+    <title>elim</title>
     <titleabbrev>elim</titleabbrev>
     <para><userinput>elim t using th hyps</userinput></para>
     <para>
       <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para><emphasis role="bold">elim</emphasis> &sterm; [<emphasis role="bold">using</emphasis> &sterm;] &intros-spec;</para>
+          </listitem>
+        </varlistentry>
         <varlistentry>
           <term>Pre-conditions:</term>
           <listitem>
@@ -405,11 +711,17 @@ its constructor takes no arguments.</para>
     </para>
   </sect1>
   <sect1 id="tac_elimType">
-    <title>elimType &lt;term&gt; [using &lt;term&gt;] [&lt;intros_spec&gt;]</title>
+    <title>elimType</title>
     <titleabbrev>elimType</titleabbrev>
     <para><userinput>elimType T using th hyps</userinput></para>
     <para>
       <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para><emphasis role="bold">elimType</emphasis> &sterm; [<emphasis role="bold">using</emphasis> &sterm;] &intros-spec;</para>
+          </listitem>
+        </varlistentry>
         <varlistentry>
           <term>Pre-conditions:</term>
           <listitem>
@@ -432,11 +744,17 @@ its constructor takes no arguments.</para>
     </para>
   </sect1>
   <sect1 id="tac_exact">
-    <title>exact &lt;term&gt;</title>
+    <title>exact</title>
     <titleabbrev>exact</titleabbrev>
     <para><userinput>exact p</userinput></para>
     <para>
       <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para><emphasis role="bold">exact</emphasis> &sterm;</para>
+          </listitem>
+        </varlistentry>
         <varlistentry>
           <term>Pre-conditions:</term>
           <listitem>
@@ -465,6 +783,12 @@ its constructor takes no arguments.</para>
     <para><userinput>exists </userinput></para>
     <para>
       <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para><emphasis role="bold">exists</emphasis></para>
+          </listitem>
+        </varlistentry>
         <varlistentry>
           <term>Pre-conditions:</term>
           <listitem>
@@ -491,11 +815,17 @@ its constructor takes no arguments.</para>
     </para>
   </sect1>
   <sect1 id="tac_fail">
-    <title>fail </title>
-    <titleabbrev>failt</titleabbrev>
+    <title>fail</title>
+    <titleabbrev>fail</titleabbrev>
     <para><userinput>fail</userinput></para>
     <para>
       <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para><emphasis role="bold">fail</emphasis></para>
+          </listitem>
+        </varlistentry>
         <varlistentry>
           <term>Pre-conditions:</term>
           <listitem>
@@ -518,11 +848,17 @@ its constructor takes no arguments.</para>
     </para>
   </sect1>
   <sect1 id="tac_fold">
-    <title>fold &lt;reduction_kind&gt; &lt;term&gt; &lt;pattern&gt;</title>
+    <title>fold</title>
     <titleabbrev>fold</titleabbrev>
     <para><userinput>fold red t patt</userinput></para>
     <para>
       <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para><emphasis role="bold">fold</emphasis> &reduction-kind; &sterm; &pattern;</para>
+          </listitem>
+        </varlistentry>
         <varlistentry>
           <term>Pre-conditions:</term>
           <listitem>
@@ -555,6 +891,12 @@ its constructor takes no arguments.</para>
     <para><userinput>fourier </userinput></para>
     <para>
       <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para><emphasis role="bold">fourier</emphasis></para>
+          </listitem>
+        </varlistentry>
         <varlistentry>
           <term>Pre-conditions:</term>
           <listitem>
@@ -580,38 +922,67 @@ its constructor takes no arguments.</para>
     </para>
   </sect1>
   <sect1 id="tac_fwd">
-    <title>fwd &lt;ident&gt; [&lt;ident list&gt;]</title>
+    <title>fwd</title>
     <titleabbrev>fwd</titleabbrev>
-    <para><userinput>fwd ...TODO</userinput></para>
+    <para><userinput>fwd H as H<subscript>0</subscript> ... H<subscript>n</subscript></userinput></para>
     <para>
       <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para><emphasis role="bold">fwd</emphasis> &id; [<emphasis role="bold">as</emphasis> &id; [&id;]…]</para>
+          </listitem>
+        </varlistentry>
         <varlistentry>
           <term>Pre-conditions:</term>
           <listitem>
-            <para>TODO.</para>
+            <para>
+              The type of <command>H</command> must be the premise of a
+               forward simplification theorem.
+           </para>
           </listitem>
         </varlistentry>
         <varlistentry>
           <term>Action:</term>
           <listitem>
-            <para>TODO.</para>
+            <para>
+            This tactic is under development.
+             It simplifies the current context by removing
+            <command>H</command> using the following methods:
+            forward application (by <command>lapply</command>) of a suitable
+            simplification theorem, chosen automatically, of which the type
+            of <command>H</command> is a premise, 
+            decomposition (by <command>decompose</command>),
+            rewriting (by <command>rewrite</command>).
+            <command>H<subscript>0</subscript> ... H<subscript>n</subscript></command>
+            are passed to the tactics <command>fwd</command> invokes, as
+             names for the premise they introduce.
+           </para>
           </listitem>
         </varlistentry>
         <varlistentry>
           <term>New sequents to prove:</term>
           <listitem>
-            <para>TODO.</para>
+            <para>
+            The ones opened by the tactics <command>fwd</command> invokes.
+           </para>
           </listitem>
         </varlistentry>
       </variablelist>
     </para>
   </sect1>
   <sect1 id="tac_generalize">
-    <title>generalize &lt;pattern&gt; [as &lt;id&gt;]</title>
+    <title>generalize</title>
     <titleabbrev>generalize</titleabbrev>
     <para><userinput>generalize patt as H</userinput></para>
     <para>
       <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para><emphasis role="bold">generalize</emphasis> &pattern; [<emphasis role="bold">as</emphasis> &id;]</para>
+          </listitem>
+        </varlistentry>
         <varlistentry>
           <term>Pre-conditions:</term>
           <listitem>
@@ -648,64 +1019,45 @@ its constructor takes no arguments.</para>
     <para><userinput>id </userinput></para>
     <para>
       <variablelist>
-        <varlistentry>
-          <term>Pre-conditions:</term>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
           <listitem>
-            <para>None.</para>
-          </listitem>
-        </varlistentry>
-        <varlistentry>
-          <term>Action:</term>
-          <listitem>
-            <para>This identity tactic does nothing without failing.</para>
+            <para><emphasis role="bold">id</emphasis></para>
           </listitem>
         </varlistentry>
-        <varlistentry>
-          <term>New sequents to prove:</term>
-          <listitem>
-            <para>None.</para>
-          </listitem>
-        </varlistentry>
-      </variablelist>
-    </para>
-  </sect1>
-  <sect1 id="tac_injection">
-    <title>injection &lt;term&gt;</title>
-    <titleabbrev>injection</titleabbrev>
-    <para><userinput>injection p</userinput></para>
-    <para>
-      <variablelist>
         <varlistentry>
           <term>Pre-conditions:</term>
           <listitem>
-            <para><command>p</command> must have type <command>K t<subscript>1</subscript> ... t<subscript>n</subscript> = K t'<subscript>1</subscript> ... t'<subscript>n</subscript></command> where both argument lists are empty if
-<command>K</command> takes no arguments.</para>
+            <para>None.</para>
           </listitem>
         </varlistentry>
         <varlistentry>
           <term>Action:</term>
           <listitem>
-            <para>It derives new hypotheses by injectivity of
-             <command>K</command>.</para>
+            <para>This identity tactic does nothing without failing.</para>
           </listitem>
         </varlistentry>
         <varlistentry>
           <term>New sequents to prove:</term>
           <listitem>
-            <para>The new sequent to prove is equal to the current sequent
-             with the additional hypotheses
-             <command>t<subscript>1</subscript>=t'<subscript>1</subscript></command> ... <command>t<subscript>n</subscript>=t'<subscript>n</subscript></command>.</para>
+            <para>None.</para>
           </listitem>
         </varlistentry>
       </variablelist>
     </para>
   </sect1>
   <sect1 id="tac_intro">
-    <title>intro [&lt;ident&gt;]</title>
+    <title>intro</title>
     <titleabbrev>intro</titleabbrev>
     <para><userinput>intro H</userinput></para>
     <para>
       <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para><emphasis role="bold">intro</emphasis> [&id;]</para>
+          </listitem>
+        </varlistentry>
         <varlistentry>
           <term>Pre-conditions:</term>
           <listitem>
@@ -734,11 +1086,17 @@ its constructor takes no arguments.</para>
     </para>
   </sect1>
   <sect1 id="tac_intros">
-    <title>intros &lt;intros_spec&gt;</title>
+    <title>intros</title>
     <titleabbrev>intros</titleabbrev>
     <para><userinput>intros hyps</userinput></para>
     <para>
       <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para><emphasis role="bold">intros</emphasis> &intros-spec;</para>
+          </listitem>
+        </varlistentry>
         <varlistentry>
           <term>Pre-conditions:</term>
           <listitem>
@@ -771,11 +1129,17 @@ its constructor takes no arguments.</para>
     </para>
   </sect1>
   <sect1 id="tac_inversion">
-    <title>inversion &lt;term&gt;</title>
+    <title>inversion</title>
     <titleabbrev>inversion</titleabbrev>
     <para><userinput>inversion t</userinput></para>
     <para>
       <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para><emphasis role="bold">inversion</emphasis> &sterm;</para>
+          </listitem>
+        </varlistentry>
         <varlistentry>
           <term>Pre-conditions:</term>
           <listitem>
@@ -806,27 +1170,70 @@ its constructor takes no arguments.</para>
     </para>
   </sect1>
   <sect1 id="tac_lapply">
-    <title>lapply [depth=&lt;int&gt;] &lt;term&gt; [to &lt;term list] [using &lt;ident&gt;]</title>
+    <title>lapply</title>
     <titleabbrev>lapply</titleabbrev>
-    <para><userinput>lapply ???</userinput></para>
+    <para><userinput>
+     lapply linear depth=d t 
+     to t<subscript>1</subscript>, ..., t<subscript>n</subscript> as H
+    </userinput></para>
     <para>
       <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para>
+            <emphasis role="bold">lapply</emphasis> 
+            [<emphasis role="bold">linear</emphasis>]
+            [<emphasis role="bold">depth=</emphasis>&nat;] 
+            &sterm; 
+            [<emphasis role="bold">to</emphasis>
+             &sterm;
+             [<emphasis role="bold">,</emphasis>&sterm;…]
+            ] 
+            [<emphasis role="bold">as</emphasis> &id;]
+           </para>
+          </listitem>
+        </varlistentry>
         <varlistentry>
           <term>Pre-conditions:</term>
           <listitem>
-            <para>TODO.</para>
+            <para>
+            <command>t</command> must have at least <command>d</command>
+            independent premises and <command>n</command> must not be
+            greater than <command>d</command>.
+           </para>
           </listitem>
         </varlistentry>
         <varlistentry>
           <term>Action:</term>
           <listitem>
-            <para>TODO.</para>
+            <para>
+            Invokes <command>letin H ≝ (t ? ... ?)</command>
+            with enough <command>?</command>'s to reach the 
+            <command>d</command>-th independent premise of
+            <command>t</command>
+            (<command>d</command> is maximum if unspecified).       
+            Then istantiates (by <command>apply</command>) with
+            t<subscript>1</subscript>, ..., t<subscript>n</subscript>
+            the <command>?</command>'s corresponding to the first 
+            <command>n</command> independent premises of
+            <command>t</command>.
+            Usually the other <command>?</command>'s preceding the 
+            <command>n</command>-th independent premise of
+            <command>t</command> are istantiated as a consequence.
+            If the <command>linear</command> flag is specified and if 
+            <command>t, t<subscript>1</subscript>, ..., t<subscript>n</subscript></command>
+            are (applications of) premises in the current context, they are
+             <command>clear</command>ed. 
+           </para>
           </listitem>
         </varlistentry>
         <varlistentry>
           <term>New sequents to prove:</term>
           <listitem>
-            <para>TODO.</para>
+            <para>
+            The ones opened by the tactics <command>lapply</command> invokes.
+           </para>
           </listitem>
         </varlistentry>
       </variablelist>
@@ -838,6 +1245,12 @@ its constructor takes no arguments.</para>
     <para><userinput>left </userinput></para>
     <para>
       <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para><emphasis role="bold">left</emphasis></para>
+          </listitem>
+        </varlistentry>
         <varlistentry>
           <term>Pre-conditions:</term>
           <listitem>
@@ -864,11 +1277,17 @@ its constructor takes no arguments.</para>
     </para>
   </sect1>
   <sect1 id="tac_letin">
-    <title>letin &lt;ident&gt; ≝ &lt;term&gt;</title>
+    <title>letin</title>
     <titleabbrev>letin</titleabbrev>
     <para><userinput>letin x ≝ t</userinput></para>
     <para>
       <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para><emphasis role="bold">letin</emphasis> &id; <emphasis role="bold">≝</emphasis> &sterm;</para>
+          </listitem>
+        </varlistentry>
         <varlistentry>
           <term>Pre-conditions:</term>
           <listitem>
@@ -892,11 +1311,17 @@ its constructor takes no arguments.</para>
     </para>
   </sect1>
   <sect1 id="tac_normalize">
-    <title>normalize &lt;pattern&gt;</title>
+    <title>normalize</title>
     <titleabbrev>normalize</titleabbrev>
     <para><userinput>normalize patt</userinput></para>
     <para>
       <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para><emphasis role="bold">normalize</emphasis> &pattern;</para>
+          </listitem>
+        </varlistentry>
         <varlistentry>
           <term>Pre-conditions:</term>
           <listitem>
@@ -919,39 +1344,18 @@ its constructor takes no arguments.</para>
       </variablelist>
     </para>
   </sect1>
-  <sect1 id="tac_paramodulation">
-    <title>paramodulation &lt;pattern&gt;</title>
-    <titleabbrev>paramodulation</titleabbrev>
-    <para><userinput>paramodulation patt</userinput></para>
-    <para>
-      <variablelist>
-        <varlistentry>
-          <term>Pre-conditions:</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>TODO.</para>
-          </listitem>
-        </varlistentry>
-      </variablelist>
-    </para>
-  </sect1>
   <sect1 id="tac_reduce">
-    <title>reduce &lt;pattern&gt;</title>
+    <title>reduce</title>
     <titleabbrev>reduce</titleabbrev>
     <para><userinput>reduce patt</userinput></para>
     <para>
       <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para><emphasis role="bold">reduce</emphasis> &pattern;</para>
+          </listitem>
+        </varlistentry>
         <varlistentry>
           <term>Pre-conditions:</term>
           <listitem>
@@ -980,6 +1384,12 @@ its constructor takes no arguments.</para>
     <para><userinput>reflexivity </userinput></para>
     <para>
       <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para><emphasis role="bold">reflexivity</emphasis></para>
+          </listitem>
+        </varlistentry>
         <varlistentry>
           <term>Pre-conditions:</term>
           <listitem>
@@ -1004,11 +1414,17 @@ its constructor takes no arguments.</para>
     </para>
   </sect1>
   <sect1 id="tac_replace">
-    <title>replace &lt;pattern&gt; with &lt;term&gt;</title>
+    <title>replace</title>
     <titleabbrev>change</titleabbrev>
     <para><userinput>change patt with t</userinput></para>
     <para>
       <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para><emphasis role="bold">replace</emphasis> &pattern; <emphasis role="bold">with</emphasis> &sterm;</para>
+          </listitem>
+        </varlistentry>
         <varlistentry>
           <term>Pre-conditions:</term>
           <listitem>
@@ -1036,11 +1452,17 @@ its constructor takes no arguments.</para>
     </para>
   </sect1>
   <sect1 id="tac_rewrite">
-    <title>rewrite {&lt;|&gt;} &lt;term&gt; &lt;pattern&gt;</title>
+    <title>rewrite</title>
     <titleabbrev>rewrite</titleabbrev>
     <para><userinput>rewrite dir p patt</userinput></para>
     <para>
       <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para><emphasis role="bold">rewrite</emphasis> [<emphasis role="bold">&lt;</emphasis>|<emphasis role="bold">&gt;</emphasis>] &sterm; &pattern;</para>
+          </listitem>
+        </varlistentry>
         <varlistentry>
           <term>Pre-conditions:</term>
           <listitem>
@@ -1076,6 +1498,12 @@ its constructor takes no arguments.</para>
     <para><userinput>right </userinput></para>
     <para>
       <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para><emphasis role="bold">right</emphasis></para>
+          </listitem>
+        </varlistentry>
         <varlistentry>
           <term>Pre-conditions:</term>
           <listitem>
@@ -1107,6 +1535,12 @@ its constructor takes no arguments.</para>
     <para><userinput>ring </userinput></para>
     <para>
       <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para><emphasis role="bold">ring</emphasis></para>
+          </listitem>
+        </varlistentry>
         <varlistentry>
           <term>Pre-conditions:</term>
           <listitem>
@@ -1133,11 +1567,17 @@ its constructor takes no arguments.</para>
     </para>
   </sect1>
   <sect1 id="tac_simplify">
-    <title>simplify &lt;pattern&gt;</title>
+    <title>simplify</title>
     <titleabbrev>simplify</titleabbrev>
     <para><userinput>simplify patt</userinput></para>
     <para>
       <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para><emphasis role="bold">simplify</emphasis> &pattern;</para>
+          </listitem>
+        </varlistentry>
         <varlistentry>
           <term>Pre-conditions:</term>
           <listitem>
@@ -1166,6 +1606,12 @@ its constructor takes no arguments.</para>
     <para><userinput>split </userinput></para>
     <para>
       <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para><emphasis role="bold">split</emphasis></para>
+          </listitem>
+        </varlistentry>
         <varlistentry>
           <term>Pre-conditions:</term>
           <listitem>
@@ -1191,6 +1637,46 @@ its constructor takes no arguments.</para>
       </variablelist>
     </para>
   </sect1>
+  
+  <sect1 id="tac_subst">
+    <title>subst</title>
+    <titleabbrev>subst</titleabbrev>
+    <para><userinput>subst</userinput></para>
+    <para>
+      <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para><emphasis role="bold">subst</emphasis></para>
+          </listitem>
+        </varlistentry>
+        <varlistentry>
+          <term>Pre-conditions:</term>
+          <listitem><para>
+           None.
+         </para></listitem>
+        </varlistentry>
+        <varlistentry>
+          <term>Action:</term>
+          <listitem><para>
+           For each premise of the form 
+           <command>H: x = t</command> or <command>H: t = x</command>
+           where <command>x</command> is a local variable and 
+           <command>t</command> does not depend on <command>x</command>,
+           the tactic rewrites <command>H</command> wherever 
+           <command>x</command> appears clearing <command>H</command> and
+           <command>x</command> afterwards.
+         </para></listitem>
+        </varlistentry>
+        <varlistentry>
+          <term>New sequents to prove:</term>
+          <listitem><para>
+           The one opened by the applied tactics.
+          </para></listitem>
+        </varlistentry>
+      </variablelist>
+    </para>
+  </sect1>
   <sect1 id="tac_symmetry">
     <title>symmetry</title>
     <titleabbrev>symmetry</titleabbrev>
@@ -1198,6 +1684,12 @@ its constructor takes no arguments.</para>
     <para><userinput>symmetry </userinput></para>
     <para>
       <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para><emphasis role="bold">symmetry</emphasis></para>
+          </listitem>
+        </varlistentry>
         <varlistentry>
           <term>Pre-conditions:</term>
           <listitem>
@@ -1221,11 +1713,17 @@ its constructor takes no arguments.</para>
     </para>
   </sect1>
   <sect1 id="tac_transitivity">
-    <title>transitivity &lt;term&gt;</title>
+    <title>transitivity</title>
     <titleabbrev>transitivity</titleabbrev>
     <para><userinput>transitivity t</userinput></para>
     <para>
       <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para><emphasis role="bold">transitivity</emphasis> &sterm;</para>
+          </listitem>
+        </varlistentry>
         <varlistentry>
           <term>Pre-conditions:</term>
           <listitem>
@@ -1250,11 +1748,17 @@ the current sequent to prove.</para>
     </para>
   </sect1>
   <sect1 id="tac_unfold">
-    <title>unfold [&lt;term&gt;] &lt;pattern&gt;</title>
+    <title>unfold</title>
     <titleabbrev>unfold</titleabbrev>
     <para><userinput>unfold t patt</userinput></para>
     <para>
       <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para><emphasis role="bold">unfold</emphasis> [&sterm;] &pattern;</para>
+          </listitem>
+        </varlistentry>
         <varlistentry>
           <term>Pre-conditions:</term>
           <listitem>
@@ -1282,11 +1786,17 @@ the current sequent to prove.</para>
     </para>
   </sect1>
   <sect1 id="tac_whd">
-    <title>whd &lt;pattern&gt;</title>
+    <title>whd</title>
     <titleabbrev>whd</titleabbrev>
     <para><userinput>whd patt</userinput></para>
     <para>
       <variablelist>
+        <varlistentry role="tactic.synopsis">
+          <term>Synopsis:</term>
+          <listitem>
+            <para><emphasis role="bold">whd</emphasis> &pattern;</para>
+          </listitem>
+        </varlistentry>
         <varlistentry>
           <term>Pre-conditions:</term>
           <listitem>