<!-- =========== Terms, declarations and definitions ============ -->
-<sect1 id="sec_terms">
- <title>Terms, definitions, declarations and proofs</title>
-
- <sect2 id="terms">
- <title>Terms</title>
- <para> &TODO; </para>
+<chapter id="sec_terms">
+ <title>Syntax</title>
+ <para>To describe syntax in this manual we use the following conventions:</para>
+ <orderedlist>
+ <listitem>Non terminal symbols are emphasized and have a link to their definition. E.g.: &term;</listitem>
+ <listitem>Terminal symbols are in bold. E.g.: <emphasis role="bold">theorem</emphasis></listitem>
+ <listitem>Optional sequences of elements are put in square brackets.
+ E.g.: [<emphasis role="bold">in</emphasis> &term;]</listitem>
+ <listitem>Alternatives are put in square brakets and they are separated
+ by vertical bars. E.g.: [<emphasis role="bold"><</emphasis>|<emphasis role="bold">></emphasis>]</listitem>
+ <listitem>Repetition of sequences of elements are given by putting the
+ first sequence in square brackets, that are followed by three dots.
+ E.g.: [<emphasis role="bold">and</emphasis> &term;]…</listitem>
+ </orderedlist>
+ <sect1 id="terms_and_co">
+ <title>Terms & co.</title>
+ <sect2 id="lexical">
+ <title>Lexical conventions</title>
+ <table>
+ <tgroup>
+ <thead />
+ <tbody>
+ <row>
+ <entry id="id">&id;</entry>
+ <entry>::=</entry>
+ <entry><emphasis>〈〈&TODO;〉〉</emphasis></entry>
+ </row>
+ </tbody>
+ </tgroup>
+ </table>
+ <table>
+ <tgroup>
+ <thead />
+ <tbody>
+ <row>
+ <entry id="nat">&nat;</entry>
+ <entry>::=</entry>
+ <entry><emphasis>〈〈&TODO;〉〉</emphasis></entry>
+ </row>
+ </tbody>
+ </tgroup>
+ </table>
+ <table>
+ <tgroup>
+ <thead />
+ <tbody>
+ <row>
+ <entry id="uri">&uri;</entry>
+ <entry>::=</entry>
+ <entry><emphasis>〈〈&TODO;〉〉</emphasis></entry>
+ </row>
+ </tbody>
+ </tgroup>
+ </table>
</sect2>
+ <sect2 id="terms">
+ <title>Terms</title>
+ <table>
+ <tgroup>
+ <thead />
+ <tbody>
+ <row>
+ <entry id="term">&term;</entry>
+ <entry>::=</entry>
+ <entry>&sterm;</entry>
+ <entry>simple or delimited term</entry>
+ </row>
+ <row>
+ <entry/>
+ <entry>|</entry>
+ <entry>&term; &term;</entry>
+ <entry>application</entry>
+ </row>
+ <row>
+ <entry/>
+ <entry>|</entry>
+ <entry><emphasis role="bold">λ</emphasis>&args;<emphasis role="bold">.</emphasis>&term;</entry>
+ <entry>λ-abstraction</entry>
+ </row>
+ <row>
+ <entry/>
+ <entry>|</entry>
+ <entry><emphasis role="bold">Π</emphasis>&args;<emphasis role="bold">.</emphasis>&term;</entry>
+ <entry>dependent product meant to define a datatype</entry>
+ </row>
+ <row>
+ <entry/>
+ <entry>|</entry>
+ <entry><emphasis role="bold">∀</emphasis>&args;<emphasis role="bold">.</emphasis>&term;</entry>
+ <entry>dependent product meant to define a proposition</entry>
+ </row>
+ <row>
+ <entry/>
+ <entry>|</entry>
+ <entry>&term; <emphasis role="bold">→</emphasis> &term;</entry>
+ <entry>non-dependent product (logical implication or function space)</entry>
+ </row>
+ <row>
+ <entry/>
+ <entry>|</entry>
+ <entry><emphasis role="bold">let</emphasis> [&id;|(&id;<emphasis role="bold">:</emphasis> &term;)] <emphasis role="bold">≝</emphasis> &term; <emphasis role="bold">in</emphasis> &term;</entry>
+ <entry>local definition</entry>
+ </row>
+ <row>
+ <entry/>
+ <entry>|</entry>
+ <entry><emphasis role="bold">let</emphasis>
+ [<emphasis role="bold">co</emphasis>]<emphasis role="bold">rec</emphasis>
+ &id; [&id;|<emphasis role="bold">(</emphasis>&id;[<emphasis role="bold">,</emphasis>&term;]… <emphasis role="bold">:</emphasis>&term;<emphasis role="bold">)</emphasis>]… [<emphasis role="bold">on</emphasis> &nat;]
+ [<emphasis role="bold">:</emphasis> &term;]
+ <emphasis role="bold">≝</emphasis> &term;
+ </entry>
+ <entry>(co)recursive definitions</entry>
+ </row>
+ <row>
+ <entry/>
+ <entry/>
+ <entry>
+ [<emphasis role="bold">and</emphasis>
+ [&id;|<emphasis role="bold">(</emphasis>&id;[<emphasis role="bold">,</emphasis>&term;]… <emphasis role="bold">:</emphasis>&term;<emphasis role="bold">)</emphasis>]… [<emphasis role="bold">on</emphasis> &nat;]
+ [<emphasis role="bold">:</emphasis> &term;]
+ <emphasis role="bold">≝</emphasis> &term;]…
+ </entry>
+ <entry/>
+ </row>
+ <row>
+ <entry/>
+ <entry/>
+ <entry>
+ <emphasis role="bold">in</emphasis> &term;
+ </entry>
+ <entry/>
+ </row>
+ <row>
+ <entry/>
+ <entry>|</entry>
+ <entry>…</entry>
+ <entry>user provided notation</entry>
+ </row>
+ </tbody>
+ </tgroup>
+ </table>
- <sect2 id="definitions">
- <title>Definitions</title>
- <para> &TODO; </para>
- </sect2>
+ <table>
+ <tgroup>
+ <thead />
+ <tbody>
+ <row>
+ <entry id="sterm">&sterm;</entry>
+ <entry>::=</entry>
+ <entry><emphasis role="bold">(</emphasis>&term;<emphasis role="bold">)</emphasis></entry>
+ <entry/>
+ </row>
+ <row>
+ <entry/>
+ <entry>|</entry>
+ <entry>&id;[<emphasis role="bold">\subst[</emphasis>
+ &id;<emphasis role="bold">≔</emphasis>&term;
+ [<emphasis role="bold">;</emphasis>&id;<emphasis role="bold">≔</emphasis>&term;]…
+ <emphasis role="bold">]</emphasis>]
+ </entry>
+ <entry>identifier with optional explicit named substitution</entry>
+ </row>
+ <row>
+ <entry/>
+ <entry>|</entry>
+ <entry>&uri;</entry>
+ <entry>a qualified reference</entry>
+ </row>
+ <row>
+ <entry/>
+ <entry>|</entry>
+ <entry><emphasis role="bold">Prop</emphasis></entry>
+ <entry>the impredicative sort of propositions</entry>
+ </row>
+ <row>
+ <entry/>
+ <entry>|</entry>
+ <entry><emphasis role="bold">Set</emphasis></entry>
+ <entry>the impredicate sort of datatypes</entry>
+ </row>
+ <row>
+ <entry/>
+ <entry>|</entry>
+ <entry><emphasis role="bold">Type</emphasis></entry>
+ <entry>one predicative sort of datatypes</entry>
+ </row>
+ <row>
+ <entry/>
+ <entry>|</entry>
+ <entry><emphasis role="bold">?</emphasis></entry>
+ <entry>implicit argument</entry>
+ </row>
+ <row>
+ <entry/>
+ <entry>|</entry>
+ <entry><emphasis role="bold">?n</emphasis>
+ [<emphasis role="bold">[</emphasis>
+ [<emphasis role="bold">_</emphasis>|&term;]…
+ <emphasis role="bold">]</emphasis>]</entry>
+ <entry>metavariable</entry>
+ </row>
+ <row>
+ <entry/>
+ <entry>|</entry>
+ <entry><emphasis role="bold">match</emphasis> &term;
+ [ <emphasis role="bold">in</emphasis> &term; ]
+ [ <emphasis role="bold">return</emphasis> &term; ]
+ <emphasis role="bold">with</emphasis>
+ </entry>
+ <entry>case analysis</entry>
+ </row>
+ <row>
+ <entry/>
+ <entry/>
+ <entry>
+ <emphasis role="bold">[</emphasis>
+ &match_pattern; <emphasis role="bold"> ⇒ </emphasis> &term;
+ [
+ <emphasis role="bold">|</emphasis>
+ &match_pattern; <emphasis role="bold"> ⇒ </emphasis> &term;
+ ]…<emphasis role="bold">]</emphasis> </entry>
+ <entry/>
+ </row>
+ <row>
+ <entry/>
+ <entry>|</entry>
+ <entry><emphasis role="bold">(</emphasis>&term;<emphasis role="bold">:</emphasis>&term;<emphasis role="bold">)</emphasis></entry>
+ <entry>cast</entry>
+ </row>
+ <row>
+ <entry/>
+ <entry>|</entry>
+ <entry>…</entry>
+ <entry>user provided notation at precedence 90</entry>
+ </row>
+ </tbody>
+ </tgroup>
+ </table>
+
+ <table>
+ <tgroup>
+ <thead />
+ <tbody>
+ <row>
+ <entry id="args">&args;</entry>
+ <entry>::=</entry>
+ <entry>
+ <emphasis role="bold">_</emphasis>[<emphasis role="bold">:</emphasis> &term;]
+ </entry>
+ <entry>ignored argument</entry>
+ </row>
+ <row>
+ <entry/>
+ <entry>|</entry>
+ <entry>
+ <emphasis role="bold">(</emphasis><emphasis role="bold">_</emphasis>[<emphasis role="bold">:</emphasis> &term;]<emphasis role="bold">)</emphasis>
+ </entry>
+ <entry>ignored argument</entry>
+ </row>
+ <row>
+ <entry/>
+ <entry>|</entry>
+ <entry>&id;[<emphasis role="bold">,</emphasis>&id;]…[<emphasis role="bold">:</emphasis> &term;]</entry>
+ <entry></entry>
+ </row>
+ <row>
+ <entry/>
+ <entry>|</entry>
+ <entry><emphasis role="bold">(</emphasis>&id;[<emphasis role="bold">,</emphasis>&id;]…[<emphasis role="bold">:</emphasis> &term;]<emphasis role="bold">)</emphasis></entry>
+ <entry/>
+ </row>
+ </tbody>
+ </tgroup>
+ </table>
+
+ <table>
+ <tgroup>
+ <thead />
+ <tbody>
+ <row>
+ <entry id="match_pattern">&match_pattern;</entry>
+ <entry>::=</entry>
+ <entry>&id;</entry>
+ <entry>0-ary constructor</entry>
+ </row>
+ <row>
+ <entry/>
+ <entry>|</entry>
+ <entry><emphasis role="bold">(</emphasis>&id; &id; [&id;]…<emphasis role="bold">)</emphasis></entry>
+ <entry>n-ary constructor (binds the n arguments)</entry>
+ </row>
+ </tbody>
+ </tgroup>
+ </table>
- <sect2 id="declarations">
- <title>Declarations (of inductive types)</title>
- <para> &TODO; </para>
</sect2>
+ </sect1>
- <sect2 id="proofs">
- <title>Proofs</title>
+ <sect1 id="axiom_definition_declaration">
+ <title>Definitions and declarations</title>
+ <sect2 id="axiom">
+ <title>axiom &id;: &term;</title>
+ <titleabbrev>axiom</titleabbrev>
+ <para><userinput>axiom H: P</userinput></para>
+ <para><command>H</command> is declared as an axiom that states <command>P</command></para>
+ </sect2>
+ <sect2 id="definition">
+ <title>definition &id;[: &term;] [≝ &term;]</title>
+ <titleabbrev>definition</titleabbrev>
+ <para><userinput>definition f: T ≝ t</userinput></para>
+ <para><command>f</command> is defined as <command>t</command>;
+ <command>T</command> is its type. An error is raised if the type of
+ <command>t</command> is not convertible to <command>T</command>.</para>
+ <para><command>T</command> is inferred from <command>t</command> if
+ omitted.</para>
+ <para><command>t</command> can be omitted only if <command>T</command> is
+ given. In this case Matita enters in interactive mode and
+ <command>f</command> must be defined by means of tactics.</para>
+ <para>Notice that the command is equivalent to <command>theorem f: T ≝ t</command>.</para>
+ </sect2>
+ <sect2 id="inductive">
+ <title>[co]inductive &id; (of inductive types)</title>
+ <titleabbrev>(co)inductive types declaration</titleabbrev>
<para> &TODO; </para>
</sect2>
+ </sect1>
+
+ <sect1 id="proofs">
+ <title>Proofs</title>
+ <sect2 id="theorem">
+ <title>theorem &id;[: &term;] [≝ &term;]</title>
+ <titleabbrev>theorem</titleabbrev>
+ <para><userinput>theorem f: P ≝ p</userinput></para>
+ <para>Proves a new theorem <command>f</command> whose thesis is
+ <command>P</command>.</para>
+ <para>If <command>p</command> is provided, it must be a proof term for
+ <command>P</command>. Otherwise an interactive proof is started.</para>
+ <para><command>P</command> can be omitted only if the proof is not
+ interactive.</para>
+ <para>Proving a theorem already proved in the library is an error.
+ To provide an alternative name and proof for the same theorem, use
+ <command>variant f: P ≝ p</command>.</para>
+ <para>A warning is raised if the name of the theorem cannot be obtained
+ by mangling the name of the constants in its thesis.</para>
+ <para>Notice that the command is equivalent to <command>definition f: T ≝ t</command>.</para>
+ </sect2>
+ <sect2 id="variant">
+ <title>variant &id;[: &term;] [≝ &term;]</title>
+ <titleabbrev>variant</titleabbrev>
+ <para><userinput>variant f: T ≝ t</userinput></para>
+ <para>Same as <command>theorem f: T ≝ t</command>, but it does not
+ complain if the theorem has already been proved. To be used to give
+ an alternative name or proof to a theorem.</para>
+ </sect2>
+ <sect2 id="lemma">
+ <title>lemma &id;[: &term;] [≝ &term;]</title>
+ <titleabbrev>lemma</titleabbrev>
+ <para><userinput>lemma f: T ≝ t</userinput></para>
+ <para>Same as <command>theorem f: T ≝ t</command></para>
+ </sect2>
+ <sect2 id="fact">
+ <title>fact &id;[: &term;] [≝ &term;]</title>
+ <titleabbrev>fact</titleabbrev>
+ <para><userinput>fact f: T ≝ t</userinput></para>
+ <para>Same as <command>theorem f: T ≝ t</command></para>
+ </sect2>
+ <sect2 id="remark">
+ <title>remark &id;[: &term;] [≝ &term;]</title>
+ <titleabbrev>remark</titleabbrev>
+ <para><userinput>remark f: T ≝ t</userinput></para>
+ <para>Same as <command>theorem f: T ≝ t</command></para>
+ </sect2>
+ </sect1>
-</sect1>
+</chapter>
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