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   3       </p><div class="variablelist"><dl class="variablelist"><dt><span class="term">Synopsis:</span></dt><dd><p><span class="bold"><strong>we proceed by induction on</strong></span> <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> </p></dd><dt><span class="term">Pre-condition:</span></dt><dd><p>The type of <span class="command"><strong>t</strong></span> must be an inductive type and <span class="command"><strong>P</strong></span> must be identical to the current conclusion.
   4         </p></dd><dt><span class="term">Action:</span></dt><dd><p>It applies the induction principle on <span class="command"><strong>t</strong></span> to prove <span class="command"><strong>P</strong></span>.</p></dd><dt><span class="term">New sequents to prove:</span></dt><dd><p>It opens a new sequent for each constructor of the type of <span class="command"><strong>t</strong></span>, each with the conclusion <span class="command"><strong>P</strong></span> instantiated for the constructor. For the inductive constructors (i.e. if the inductive type is <span class="command"><strong>T</strong></span>, constructors with an argument of type <span class="command"><strong>T</strong></span>), the conclusion is a logical implication, where the antecedent is the inductive hypothesis for the constructor, and the consequent is <span class="command"><strong>P</strong></span>.</p></dd></dl></div><p>
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