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   2 <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"><html xmlns="http://www.w3.org/1999/xhtml"><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /><title>Chapter 8. Declarative Tactics</title><link rel="stylesheet" type="text/css" href="docbook.css" /><meta name="generator" content="DocBook XSL Stylesheets V1.78.1" /><link rel="home" href="index.html" title="Matita V0.5.9 User Manual (rev. 0.5.9 )" /><link rel="up" href="index.html" title="Matita V0.5.9 User Manual (rev. 0.5.9 )" /><link rel="prev" href="tac_whd.html" title="whd" /><link rel="next" href="tac_assume.html" title="assume" /></head><body><a xmlns="" href="../../../"><div class="matita_logo"><img src="figures/matita.png" alt="Tiny Matita logo" /><span>Matita Home</span></div></a><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3" align="center">Chapter 8. Declarative Tactics</th></tr><tr><td width="20%" align="left"><a accesskey="p" href="tac_whd.html">Prev</a> </td><th width="60%" align="center"> </th><td width="20%" align="right"> <a accesskey="n" href="tac_assume.html">Next</a></td></tr></table><hr /></div><div class="chapter"><div class="titlepage"><div><div><h1 class="title"><a id="sec_declarative_tactics"></a>Chapter 8. Declarative Tactics</h1></div></div></div><div class="toc"><p><strong>Table of Contents</strong></p><dl class="toc"><dt><span class="sect1"><a href="sec_declarative_tactics.html#declarative_tactics_quickref">Quick reference card</a></span></dt><dt><span class="sect1"><a href="tac_assume.html">assume</a></span></dt><dt><span class="sect1"><a href="tac_byinduction.html">by induction hypothesis we know</a></span></dt><dt><span class="sect1"><a href="tac_case.html">case</a></span></dt><dt><span class="sect1"><a href="tac_bydone.html">done</a></span></dt><dt><span class="sect1"><a href="tac_exitselim.html">let such that</a></span></dt><dt><span class="sect1"><a href="tac_obtain.html">obtain</a></span></dt><dt><span class="sect1"><a href="tac_suppose.html">suppose</a></span></dt><dt><span class="sect1"><a href="tac_thesisbecomes.html">the thesis becomes</a></span></dt><dt><span class="sect1"><a href="tac_weneedtoprove.html">we need to prove</a></span></dt><dt><span class="sect1"><a href="tac_andelim.html">we have</a></span></dt><dt><span class="sect1"><a href="tac_weproceedbycases.html">we proceed by cases on</a></span></dt><dt><span class="sect1"><a href="tac_weproceedbyinduction.html">we proceed by induction on</a></span></dt><dt><span class="sect1"><a href="tac_bytermweproved.html">we proved</a></span></dt></dl></div><div class="sect1"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a id="declarative_tactics_quickref"></a>Quick reference card</h2></div></div></div><p>
   3       </p><div class="table"><a id="idp73048816"></a><p class="title"><strong>Table 8.1. tactics</strong></p><div class="table-contents"><table summary="tactics" style="border-collapse: collapse;border-top: 0.5pt solid ; border-bottom: 0.5pt solid ; "><colgroup><col /><col /><col /></colgroup><tbody><tr><td style=""><a id="grammar.declarative_tactic"></a><span class="emphasis"><em><a class="link" href="sec_tactics.html#grammar.tactic">tactic</a></em></span></td><td style="">::=</td><td style=""><a class="link" href="tac_assume.html" title="assume"><span class="bold"><strong>assume</strong></span></a> <span class="emphasis"><em><a class="link" href="sec_terms.html#grammar.id">id</a></em></span> <span class="bold"><strong> : </strong></span> <span class="emphasis"><em><a class="link" href="sec_terms.html#grammar.sterm">sterm</a></em></span></td></tr><tr><td style=""> </td><td style="">|</td><td style=""><a class="link" href=""><span class="bold"><strong>by induction hypothesis we know</strong></span></a> <span class="emphasis"><em><a class="link" href="sec_terms.html#grammar.term">term</a></em></span> <span class="bold"><strong> (</strong></span> <span class="emphasis"><em><a class="link" href="sec_terms.html#grammar.id">id</a></em></span> <span class="bold"><strong>)</strong></span></td></tr><tr><td style=""> </td><td style="">|</td><td style=""><a class="link" href="tac_case.html" title="case"><span class="bold"><strong>case</strong></span></a> <span class="emphasis"><em><a class="link" href="sec_terms.html#grammar.id">id</a></em></span> [<span class="bold"><strong>(</strong></span> <span class="emphasis"><em><a class="link" href="sec_terms.html#grammar.id">id</a></em></span> <span class="bold"><strong>:</strong></span> <span class="emphasis"><em><a class="link" href="sec_terms.html#grammar.term">term</a></em></span>  <span class="bold"><strong>)</strong></span>] … </td></tr><tr><td style=""> </td><td style="">|</td><td style=""><span class="emphasis"><em><a class="link" href="tacticargs.html#grammar.justification">justification</a></em></span> <a class="link" href=""><span class="bold"><strong>done</strong></span></a></td></tr><tr><td style=""> </td><td style="">|</td><td style=""><span class="emphasis"><em><a class="link" href="tacticargs.html#grammar.justification">justification</a></em></span> <span class="bold"><strong>let</strong></span> <span class="emphasis"><em><a class="link" href="sec_terms.html#grammar.id">id</a></em></span>
   4                     <span class="bold"><strong>:</strong></span> <span class="emphasis"><em><a class="link" href="sec_terms.html#grammar.term">term</a></em></span> <span class="bold"><strong>such that</strong></span> <span class="emphasis"><em><a class="link" href="sec_terms.html#grammar.term">term</a></em></span>
   5                      <span class="bold"><strong>(</strong></span> <span class="emphasis"><em><a class="link" href="sec_terms.html#grammar.id">id</a></em></span> <span class="bold"><strong>)</strong></span></td></tr><tr><td style=""> </td><td style="">|</td><td style="">[<a class="link" href="tac_obtain.html" title="obtain"><span class="bold"><strong>obtain</strong></span></a> <span class="emphasis"><em><a class="link" href="sec_terms.html#grammar.id">id</a></em></span> | <span class="bold"><strong>conclude</strong></span> <span class="emphasis"><em><a class="link" href="sec_terms.html#grammar.term">term</a></em></span>] <span class="bold"><strong>=</strong></span> <span class="emphasis"><em><a class="link" href="sec_terms.html#grammar.term">term</a></em></span> [<span class="emphasis"><em><a class="link" href="tacticargs.html#grammar.autoparams">auto_params</a></em></span> | <span class="bold"><strong>using</strong></span> <span class="emphasis"><em><a class="link" href="sec_terms.html#grammar.term">term</a></em></span> | <span class="bold"><strong>using once</strong></span> <span class="emphasis"><em><a class="link" href="sec_terms.html#grammar.term">term</a></em></span> | <span class="bold"><strong>proof</strong></span>] [<span class="bold"><strong>done</strong></span>]</td></tr><tr><td style=""> </td><td style="">|</td><td style=""><a class="link" href="tac_suppose.html" title="suppose"><span class="bold"><strong>suppose</strong></span></a> <span class="emphasis"><em><a class="link" href="sec_terms.html#grammar.term">term</a></em></span> <span class="bold"><strong> (</strong></span> <span class="emphasis"><em><a class="link" href="sec_terms.html#grammar.id">id</a></em></span>
   6              <span class="bold"><strong>) </strong></span> [ <span class="bold"><strong>that is equivalent to</strong></span>  <span class="emphasis"><em><a class="link" href="sec_terms.html#grammar.term">term</a></em></span> ]</td></tr><tr><td style=""> </td><td style="">|</td><td style=""><a class="link" href=""><span class="bold"><strong>the thesis becomes</strong></span></a> <span class="emphasis"><em><a class="link" href="sec_terms.html#grammar.term">term</a></em></span> </td></tr><tr><td style=""> </td><td style="">|</td><td style=""><a class="link" href=""><span class="bold"><strong>we need to prove</strong></span></a> <span class="emphasis"><em><a class="link" href="sec_terms.html#grammar.term">term</a></em></span>
   7         [<span class="bold"><strong>(</strong></span><span class="emphasis"><em><a class="link" href="sec_terms.html#grammar.id">id</a></em></span>
   8         <span class="bold"><strong>)</strong></span>]
   9         [ <span class="bold"><strong>or equivalently</strong></span> <span class="emphasis"><em><a class="link" href="sec_terms.html#grammar.term">term</a></em></span>]</td></tr><tr><td style=""> </td><td style="">|</td><td style=""><a class="link" href=""><span class="bold"><strong>we proceed by cases on</strong></span></a> <span class="emphasis"><em><a class="link" href="sec_terms.html#grammar.term">term</a></em></span> <span class="bold"><strong>to prove</strong></span> <span class="emphasis"><em><a class="link" href="sec_terms.html#grammar.term">term</a></em></span> </td></tr><tr><td style=""> </td><td style="">|</td><td style=""><a class="link" href=""><span class="bold"><strong>we proceed by induction on</strong></span></a> <span class="emphasis"><em><a class="link" href="sec_terms.html#grammar.term">term</a></em></span> <span class="bold"><strong> to prove </strong></span> <span class="emphasis"><em><a class="link" href="sec_terms.html#grammar.term">term</a></em></span> </td></tr><tr><td style=""> </td><td style="">|</td><td style=""><span class="emphasis"><em><a class="link" href="tacticargs.html#grammar.justification">justification</a></em></span> <a class="link" href=""><span class="bold"><strong>we proved</strong></span></a> <span class="emphasis"><em><a class="link" href="sec_terms.html#grammar.term">term</a></em></span>
  10          <span class="bold"><strong>(</strong></span> <span class="emphasis"><em><a class="link" href="sec_terms.html#grammar.id">id</a></em></span>
  11          <span class="bold"><strong>)</strong></span></td></tr></tbody></table></div></div><p><br class="table-break" />
  12
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