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+<!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>Definitions and declarations</title><link rel="stylesheet" type="text/css" href="docbook.css" /><meta name="generator" content="DocBook XSL Stylesheets Vsnapshot" /><link rel="home" href="index.html" title="Matita V0.99.5 User Manual (rev. 0.99.5 )" /><link rel="up" href="sec_terms.html" title="Chapter 4. Syntax" /><link rel="prev" href="sec_terms.html" title="Chapter 4. Syntax" /><link rel="next" href="proofs.html" title="Proofs" /></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">Definitions and declarations</th></tr><tr><td width="20%" align="left"><a accesskey="p" href="sec_terms.html">Prev</a> </td><th width="60%" align="center">Chapter 4. Syntax</th><td width="20%" align="right"> <a accesskey="n" href="proofs.html">Next</a></td></tr></table><hr /></div><div class="sect1"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a id="axiom_definition_declaration"></a>Definitions and declarations</h2></div></div></div><div class="sect2"><div class="titlepage"><div><div><h3 class="title"><a id="axiom"></a><span class="bold"><strong>axiom</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></h3></div></div></div><p><strong class="userinput"><code>axiom H: P</code></strong></p><p><span class="command"><strong>H</strong></span> is declared as an axiom that states <span class="command"><strong>P</strong></span></p></div><div class="sect2"><div class="titlepage"><div><div><h3 class="title"><a id="definition"></a><span class="bold"><strong>definition</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> <span class="emphasis"><em><a class="link" href="sec_terms.html#grammar.term">term</a></em></span>]</h3></div></div></div><p><strong class="userinput"><code>definition f: T ≝ t</code></strong></p><p><span class="command"><strong>f</strong></span> is defined as <span class="command"><strong>t</strong></span>;
+ <span class="command"><strong>T</strong></span> is its type. An error is raised if the type of
+ <span class="command"><strong>t</strong></span> is not convertible to <span class="command"><strong>T</strong></span>.</p><p><span class="command"><strong>T</strong></span> is inferred from <span class="command"><strong>t</strong></span> if
+ omitted.</p><p><span class="command"><strong>t</strong></span> can be omitted only if <span class="command"><strong>T</strong></span> is
+ given. In this case Matita enters in interactive mode and
+ <span class="command"><strong>f</strong></span> must be defined by means of tactics.</p><p>Notice that the command is equivalent to <span class="command"><strong>theorem f: T ≝ t</strong></span>.</p></div><div class="sect2"><div class="titlepage"><div><div><h3 class="title"><a id="discriminator"></a><span class="bold"><strong>discriminator</strong></span> <span class="emphasis"><em><a class="link" href="sec_terms.html#grammar.id">id</a></em></span></h3></div></div></div><p><strong class="userinput"><code>discriminator i</code></strong></p><p>Defines a new discrimination (injectivity+conflict) principle à la
+ McBride for the inductive type <span class="command"><strong>i</strong></span>.</p><p>The principle will use John
+ Major's equality if such equality is defined, otherwise it will use
+ Leibniz equality; in the former case, it will be called
+ <span class="command"><strong>i_jmdiscr</strong></span>, in the latter, <span class="command"><strong>i_discr</strong></span>.
+ The command will fail if neither equality is available.</p><p>Discrimination principles are used by the destruct tactic and are
+ usually automatically generated by Matita during the definition of the
+ corresponding inductive type. This command is thus especially useful
+ when the correct equality was not loaded at the time of that
+ definition.</p></div><div class="sect2"><div class="titlepage"><div><div><h3 class="title"><a id="inverter"></a><span class="bold"><strong>inverter</strong></span> <span class="emphasis"><em><a class="link" href="sec_terms.html#grammar.id">id</a></em></span> <span class="bold"><strong>for</strong></span> <span class="emphasis"><em><a class="link" href="sec_terms.html#grammar.id">id</a></em></span> (<span class="emphasis"><em><a class="link" href="tacticargs.html#grammar.path">path</a></em></span>) [<span class="emphasis"><em><a class="link" href="sec_terms.html#grammar.term">term</a></em></span>]</h3></div></div></div><p><strong class="userinput"><code>inverter n for i (path) : s</code></strong></p><p>Defines a new induction/inversion principle for the inductive type
+ <span class="command"><strong>i</strong></span>, called <span class="command"><strong>n</strong></span>.</p><p><span class="command"><strong>(path)</strong></span> must be in the form <span class="command"><strong>(# # # ... #)</strong></span>,
+ where each <span class="command"><strong>#</strong></span> can be either <span class="command"><strong>?</strong></span> or
+ <span class="command"><strong>%</strong></span>, and the number of symbols is equal to the number of
+ right parameters (indices) of <span class="command"><strong>i</strong></span>. Parentheses are
+ mandatory. If the j-th symbol is
+ <span class="command"><strong>%</strong></span>, Matita will generate a principle providing
+ equations for reasoning on the j-th index of <span class="command"><strong>i</strong></span>. If the
+ symbol is a <span class="command"><strong>?</strong></span>, no corresponding equation will be
+ provided.</p><p><span class="command"><strong>s</strong></span>, which must be a sort, is the target sort of the
+ induction/inversion principle and defaults to <span class="command"><strong>Prop</strong></span>.</p></div><div class="sect2"><div class="titlepage"><div><div><h3 class="title"><a id="letrec"></a><span class="bold"><strong>letrec</strong></span> <span class="emphasis"><em>TODO</em></span></h3></div></div></div><p><span class="emphasis"><em>TODO</em></span></p></div><div class="sect2"><div class="titlepage"><div><div><h3 class="title"><a id="inductive"></a>[<span class="bold"><strong>inductive</strong></span>|<span class="bold"><strong>coinductive</strong></span>] <span class="emphasis"><em><a class="link" href="sec_terms.html#grammar.id">id</a></em></span> [<span class="emphasis"><em><a class="link" href="sec_terms.html#grammar.args2">args2</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> [<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> <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>with</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> [<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> <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>]…]…
+</h3></div></div></div><p><strong class="userinput"><code>inductive i x y z: S ≝ k1:T1 | … | kn:Tn with i' : S' ≝ k1':T1' | … | km':Tm'</code></strong></p><p>Declares a family of two mutually inductive types
+ <span class="command"><strong>i</strong></span> and <span class="command"><strong>i'</strong></span> whose types are
+ <span class="command"><strong>S</strong></span> and <span class="command"><strong>S'</strong></span>, which must be convertible
+ to sorts.</p><p>The constructors <span class="command"><strong>ki</strong></span> of type <span class="command"><strong>Ti</strong></span>
+ and <span class="command"><strong>ki'</strong></span> of type <span class="command"><strong>Ti'</strong></span> are also
+ simultaneously declared. The declared types <span class="command"><strong>i</strong></span> and
+ <span class="command"><strong>i'</strong></span> may occur in the types of the constructors, but
+ only in strongly positive positions according to the rules of the
+ calculus.</p><p>The whole family is parameterized over the arguments <span class="command"><strong>x,y,z</strong></span>.</p><p>If the keyword <span class="command"><strong>coinductive</strong></span> is used, the declared
+ types are considered mutually coinductive.</p><p>Elimination principles for the record are automatically generated
+ by Matita, if allowed by the typing rules of the calculus according to
+ the sort <span class="command"><strong>S</strong></span>. If generated,
+ they are named <span class="command"><strong>i_ind</strong></span>, <span class="command"><strong>i_rec</strong></span> and
+ <span class="command"><strong>i_rect</strong></span> according to the sort of their induction
+ predicate.</p></div><div class="sect2"><div class="titlepage"><div><div><h3 class="title"><a id="record"></a><span class="bold"><strong>record</strong></span> <span class="emphasis"><em><a class="link" href="sec_terms.html#grammar.id">id</a></em></span> [<span class="emphasis"><em><a class="link" href="sec_terms.html#grammar.args2">args2</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><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="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><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="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></h3></div></div></div><p><strong class="userinput"><code>record id x y z: S ≝ { f1: T1; …; fn:Tn }</code></strong></p><p>Declares a new record family <span class="command"><strong>id</strong></span> parameterized over
+ <span class="command"><strong>x,y,z</strong></span>.</p><p><span class="command"><strong>S</strong></span> is the type of the record
+ and it must be convertible to a sort.</p><p>Each field <span class="command"><strong>fi</strong></span> is declared by giving its type
+ <span class="command"><strong>Ti</strong></span>. A record without any field is admitted.</p><p>Elimination principles for the record are automatically generated
+ by Matita, if allowed by the typing rules of the calculus according to
+ the sort <span class="command"><strong>S</strong></span>. If generated,
+ they are named <span class="command"><strong>i_ind</strong></span>, <span class="command"><strong>i_rec</strong></span> and
+ <span class="command"><strong>i_rect</strong></span> according to the sort of their induction
+ predicate.</p><p>For each field <span class="command"><strong>fi</strong></span> a record projection
+ <span class="command"><strong>fi</strong></span> is also automatically generated if projection
+ is allowed by the typing rules of the calculus according to the
+ sort <span class="command"><strong>S</strong></span>, the type <span class="command"><strong>T1</strong></span> and
+ the definability of depending record projections.</p><p>If the type of a field is declared with <span class="command"><strong>:></strong></span>,
+ the corresponding record projection becomes an implicit coercion.
+ This is just syntactic sugar and it has the same effect of declaring the
+ record projection as a coercion later on.</p></div></div><div class="navfooter"><hr /><table width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p" href="sec_terms.html">Prev</a> </td><td width="20%" align="center"><a accesskey="u" href="sec_terms.html">Up</a></td><td width="40%" align="right"> <a accesskey="n" href="proofs.html">Next</a></td></tr><tr><td width="40%" align="left" valign="top">Chapter 4. Syntax </td><td width="20%" align="center"><a accesskey="h" href="index.html">Home</a></td><td width="40%" align="right" valign="top"> Proofs</td></tr></table></div></body></html>
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