E.g.: [<emphasis role="bold">in</emphasis> &term;]</para></listitem>
<listitem><para>Alternatives are put in square brakets and they are
separated by vertical bars. E.g.: [<emphasis role="bold"><</emphasis>|<emphasis role="bold">></emphasis>]</para></listitem>
- <listitem><para>Repetition of sequences of elements are given by putting the
- first sequence in square brackets, that are followed by three dots.
+ <listitem><para>Repetitions of a sequence of elements are given by putting the
+ sequence in square brackets, that are followed by three dots. The empty
+ sequence is a valid repetition.
E.g.: [<emphasis role="bold">and</emphasis> &term;]…</para></listitem>
+ <listitem><para>Characters belonging to a set of characters are given
+ by listing the set elements in square brackets. Hyphens are used to
+ specify ranges of characters in the set.
+ E.g.: [<emphasis role="bold">a</emphasis>-<emphasis role="bold">zA</emphasis>-<emphasis role="bold">Z0</emphasis>-<emphasis role="bold">9_-</emphasis>]</para></listitem>
</orderedlist>
<sect1 id="terms_and_co">
<title>Terms & co.</title>
<sect2 id="lexical">
<title>Lexical conventions</title>
- <para>
- <table frame="all" rowsep="0" colsep="0">
+ <table frame="topbot" rowsep="0" colsep="0" role="grammar">
+ <title>qstring</title>
+ <tgroup cols="4">
+ <tbody>
+ <row>
+ <entry id="grammar.qstring">&qstring;</entry>
+ <entry>::=</entry>
+ <entry><emphasis role="bold">"</emphasis><emphasis>〈〈any sequence of characters excluded "〉〉</emphasis><emphasis role="bold">"</emphasis></entry>
+ </row>
+ </tbody>
+ </tgroup>
+ </table>
+ <table frame="topbot" rowsep="0" colsep="0" role="grammar">
<title>id</title>
<tgroup cols="4">
<tbody>
<row>
- <entry id="id">&id;</entry>
+ <entry id="grammar.id">&id;</entry>
<entry>::=</entry>
- <entry><emphasis>〈〈&TODO;〉〉</emphasis></entry>
+ <entry><emphasis>〈〈any sequence of letters, underscores or valid <ulink type="http" url="http://www.w3.org/TR/2004/REC-xml-20040204/#NT-Digit">XML digits</ulink> prefixed by a latin letter ([a-zA-Z]) and post-fixed by a possible empty sequence of decorators ([?'`])〉〉</emphasis></entry>
</row>
</tbody>
</tgroup>
</table>
- <table frame="all" rowsep="0" colsep="0">
+ <table frame="topbot" rowsep="0" colsep="0" role="grammar">
<title>nat</title>
<tgroup cols="4">
<tbody>
<row>
- <entry id="nat">&nat;</entry>
+ <entry id="grammar.nat">&nat;</entry>
<entry>::=</entry>
- <entry><emphasis>〈〈&TODO;〉〉</emphasis></entry>
+ <entry><emphasis>〈〈any sequence of valid <ulink type="http" url="http://www.w3.org/TR/2004/REC-xml-20040204/#NT-Digit">XML digits</ulink>〉〉</emphasis></entry>
</row>
</tbody>
</tgroup>
</table>
- <table frame="all" rowsep="0" colsep="0">
+ <table frame="topbot" rowsep="0" colsep="0" role="grammar">
+ <title>char</title>
+ <tgroup cols="4">
+ <tbody>
+ <row>
+ <entry id="grammar.char">&char;</entry>
+ <entry>::=</entry>
+ <entry>[<emphasis role="bold">a</emphasis>-<emphasis role="bold">zA</emphasis>-<emphasis role="bold">Z0</emphasis>-<emphasis role="bold">9_-</emphasis>]</entry>
+ </row>
+ </tbody>
+ </tgroup>
+ </table>
+ <table frame="topbot" rowsep="0" colsep="0" role="grammar">
+ <title>uri-step</title>
+ <tgroup cols="4">
+ <tbody>
+ <row>
+ <entry id="grammar.uri-step">&uri-step;</entry>
+ <entry>::=</entry>
+ <entry>&char;[&char;]…</entry>
+ </row>
+ </tbody>
+ </tgroup>
+ </table>
+ <table frame="topbot" rowsep="0" colsep="0" role="grammar">
<title>uri</title>
<tgroup cols="4">
<tbody>
<row>
- <entry id="uri">&uri;</entry>
+ <entry id="grammar.uri">&uri;</entry>
<entry>::=</entry>
- <entry><emphasis>〈〈&TODO;〉〉</emphasis></entry>
+ <entry>[<emphasis role="bold">cic:/</emphasis>|<emphasis role="bold">theory:/</emphasis>]&uri-step;[<emphasis role="bold">/</emphasis>&uri-step;]…<emphasis role="bold">.</emphasis>&id;[<emphasis role="bold">.</emphasis>&id;]…[<emphasis role="bold">#xpointer(</emphasis>&nat;<emphasis role="bold">/</emphasis>&nat;[<emphasis role="bold">/</emphasis>&nat;]…<emphasis role="bold">)</emphasis>]</entry>
</row>
</tbody>
</tgroup>
</table>
- </para>
</sect2>
<sect2 id="terms">
<title>Terms</title>
+
+ <!-- ZACK: Sample EBNF snippet, see:
+ http://www.docbook.org/tdg/en/html/productionset.html -->
+ <!--
+ <productionset>
+ <title>Terms</title>
+ <production id="grammar.term">
+ <lhs>&term;</lhs>
+ <rhs>&sterm;</rhs>
+ <lineannotation></lineannotation>
+ </production>
+ </productionset>
+ -->
+
<para>
- <table frame="all" rowsep="0" colsep="0">
+ <table frame="topbot" rowsep="0" colsep="0" role="grammar">
<title>Terms</title>
<tgroup cols="4">
<tbody>
<row>
- <entry id="term">&term;</entry>
+ <entry id="grammar.term">&term;</entry>
<entry>::=</entry>
<entry>&sterm;</entry>
<entry>simple or delimited term</entry>
<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>
+ <emphasis role="bold">let</emphasis>
+ [<emphasis role="bold">co</emphasis>]<emphasis role="bold">rec</emphasis>
+ &rec_def;
</entry>
<entry>(co)recursive definitions</entry>
</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;]…
+ [<emphasis role="bold">and</emphasis> &rec_def;]…
</entry>
<entry/>
</row>
<entry>…</entry>
<entry>user provided notation</entry>
</row>
+ <row>
+ <entry id="grammar.rec_def">&rec_def;</entry>
+ <entry>::=</entry>
+ <entry>
+ &id; [&id;|<emphasis role="bold">_</emphasis>|<emphasis role="bold">(</emphasis>&id;[<emphasis role="bold">,</emphasis>&id;]… <emphasis role="bold">:</emphasis>&term;<emphasis role="bold">)</emphasis>]…
+ </entry>
+ <entry />
+ </row>
+ <row>
+ <entry />
+ <entry />
+ <entry>
+ [<emphasis role="bold">on</emphasis> &id;]
+ [<emphasis role="bold">:</emphasis> &term;]
+ <emphasis role="bold">≝</emphasis> &term;]
+ </entry>
+ <entry />
+ </row>
</tbody>
</tgroup>
</table>
- <table frame="all" rowsep="0" colsep="0">
+ <table frame="topbot" rowsep="0" colsep="0" role="grammar">
<title>Simple terms</title>
<tgroup cols="4">
<tbody>
<row>
- <entry id="sterm">&sterm;</entry>
+ <entry id="grammar.sterm">&sterm;</entry>
<entry>::=</entry>
<entry><emphasis role="bold">(</emphasis>&term;<emphasis role="bold">)</emphasis></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">CProp</emphasis></entry>
+ <entry>one fixed predicative sort of constructive propositions</entry>
+ </row>
<row>
<entry/>
<entry>|</entry>
<entry/>
<entry>|</entry>
<entry><emphasis role="bold">match</emphasis> &term;
- [ <emphasis role="bold">in</emphasis> &term; ]
+ [ <emphasis role="bold">in</emphasis> &id; ]
[ <emphasis role="bold">return</emphasis> &term; ]
<emphasis role="bold">with</emphasis>
</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>
+ &match_branch;[<emphasis role="bold">|</emphasis>&match_branch;]…
+ <emphasis role="bold">]</emphasis>
+ </entry>
<entry/>
</row>
<row>
</tgroup>
</table>
- <table frame="all" rowsep="0" colsep="0">
+ <table frame="topbot" rowsep="0" colsep="0" role="grammar">
<title>Arguments</title>
<tgroup cols="4">
<tbody>
<row>
- <entry id="args">&args;</entry>
+ <entry id="grammar.args">&args;</entry>
<entry>::=</entry>
<entry>
<emphasis role="bold">_</emphasis>[<emphasis role="bold">:</emphasis> &term;]
<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 frame="all" rowsep="0" colsep="0">
- <title>Miscellaneous arguments</title>
- <tgroup cols="4">
- <tbody>
<row>
- <entry id="args2">&args2;</entry>
+ <entry id="grammar.args2">&args2;</entry>
<entry>::=</entry>
<entry>&id;</entry>
<entry/>
</tgroup>
</table>
- <table frame="all" rowsep="0" colsep="0">
+ <table frame="topbot" rowsep="0" colsep="0" role="grammar">
<title>Pattern matching</title>
<tgroup cols="4">
<tbody>
+ <row>
+ <entry id="grammar.match_branch">&match_branch;</entry>
+ <entry>::=</entry>
+ <entry>&match_pattern; <emphasis role="bold">⇒</emphasis> &term;</entry>
+ <entry />
+ </row>
<row>
- <entry id="match_pattern">&match_pattern;</entry>
+ <entry id="grammar.match_pattern">&match_pattern;</entry>
<entry>::=</entry>
<entry>&id;</entry>
<entry>0-ary constructor</entry>
<entry><emphasis role="bold">(</emphasis>&id; &id; [&id;]…<emphasis role="bold">)</emphasis></entry>
<entry>n-ary constructor (binds the n arguments)</entry>
</row>
+ <row>
+ <entry/>
+ <entry>|</entry>
+ <entry>&id; &id; [&id;]…</entry>
+ <entry>n-ary constructor (binds the n arguments)</entry>
+ </row>
+ <row>
+ <entry/>
+ <entry>|</entry>
+ <entry><emphasis role="bold">_</emphasis></entry>
+ <entry>any remaining constructor (ignoring its arguments)</entry>
+ </row>
</tbody>
</tgroup>
</table>
<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="letrec">
+ <title><emphasis role="bold">letrec</emphasis> &TODO;</title>
+ <titleabbrev>&TODO;</titleabbrev>
+ <para>&TODO;</para>
+ </sect2>
<sect2 id="inductive">
<title>[<emphasis role="bold">inductive</emphasis>|<emphasis role="bold">coinductive</emphasis>] &id; [&args2;]… <emphasis role="bold">:</emphasis> &term; <emphasis role="bold">≝</emphasis> [<emphasis role="bold">|</emphasis>] [&id;<emphasis role="bold">:</emphasis>&term;] [<emphasis role="bold">|</emphasis> &id;<emphasis role="bold">:</emphasis>&term;]…
[<emphasis role="bold">with</emphasis> &id; <emphasis role="bold">:</emphasis> &term; <emphasis role="bold">≝</emphasis> [<emphasis role="bold">|</emphasis>] [&id;<emphasis role="bold">:</emphasis>&term;] [<emphasis role="bold">|</emphasis> &id;<emphasis role="bold">:</emphasis>&term;]…]…
<para>Notice that the command is equivalent to <command>definition f: T ≝ t</command>.</para>
</sect2>
<sect2 id="variant">
- <title><emphasis role="bold">variant</emphasis> &id;[<emphasis role="bold">:</emphasis> &term;] [<emphasis role="bold">≝</emphasis> &term;]</title>
+ <title><emphasis role="bold">variant</emphasis> &id;<emphasis role="bold">:</emphasis> &term; <emphasis role="bold">≝</emphasis> &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
</sect2>
</sect1>
+ <sect1 id="tacticargs">
+ <title>Tactic arguments</title>
+ <para>This section documents the syntax of some recurring arguments for
+ tactics.</para>
+
+ <sect2 id="introsspec">
+ <title>intros-spec</title>
+ <table frame="topbot" rowsep="0" colsep="0" role="grammar">
+ <title>intros-spec</title>
+ <tgroup cols="4">
+ <tbody>
+ <row>
+ <entry id="grammar.intros-spec">&intros-spec;</entry>
+ <entry>::=</entry>
+ <entry>[&nat;] [<emphasis role="bold">(</emphasis>[&id;]…<emphasis role="bold">)</emphasis>]</entry>
+ </row>
+ </tbody>
+ </tgroup>
+ </table>
+ <para>The natural number is the number of new hypotheses to be introduced. The list of identifiers gives the name for the first hypotheses.</para>
+ </sect2>
+
+ <sect2 id="pattern">
+ <title>pattern</title>
+ <table frame="topbot" rowsep="0" colsep="0" role="grammar">
+ <title>pattern</title>
+ <tgroup cols="4">
+ <tbody>
+ <row>
+ <entry id="grammar.pattern">&pattern;</entry>
+ <entry>::=</entry>
+ <entry><emphasis role="bold">in</emphasis>
+ [&id;[<emphasis role="bold">:</emphasis> &path;]]…
+ [<emphasis role="bold">⊢</emphasis> &path;]]</entry>
+ <entry>simple pattern</entry>
+ </row>
+ <row>
+ <entry/>
+ <entry>|</entry>
+ <entry><emphasis role="bold">in match</emphasis> &path;
+ [<emphasis role="bold">in</emphasis>
+ [&id;[<emphasis role="bold">:</emphasis> &path;]]…
+ [<emphasis role="bold">⊢</emphasis> &path;]]</entry>
+ <entry>full pattern</entry>
+ </row>
+ </tbody>
+ </tgroup>
+ </table>
+ <table frame="topbot" rowsep="0" colsep="0" role="grammar">
+ <title>path</title>
+ <tgroup cols="4">
+ <tbody>
+ <row>
+ <entry id="grammar.path">&path;</entry>
+ <entry>::=</entry>
+ <entry><emphasis>〈〈any &sterm; without occurrences of <emphasis role="bold">Set</emphasis>, <emphasis role="bold">Prop</emphasis>, <emphasis role="bold">CProp</emphasis>, <emphasis role="bold">Type</emphasis>, &id;, &uri; and user provided notation; however, <emphasis role="bold">%</emphasis> is now an additional production for &sterm;〉〉</emphasis></entry>
+ </row>
+ </tbody>
+ </tgroup>
+ </table>
+ <para>A <emphasis>path</emphasis> locates zero or more subterms of a given term by mimicking the term structure up to:</para>
+ <orderedlist>
+ <listitem><para>Occurrences of the subterms to locate that are
+ represented by <emphasis role="bold">%</emphasis>.</para></listitem>
+ <listitem><para>Subterms without any occurrence of subterms to locate
+ that can be represented by <emphasis role="bold">?</emphasis>.
+ </para></listitem>
+ </orderedlist>
+ <para>Warning: the format for a path for a <emphasis role="bold">match</emphasis> … <emphasis role="bold">with</emphasis>
+ expression is restricted to: <emphasis role="bold">match</emphasis> &path;
+ <emphasis role="bold">with</emphasis>
+ <emphasis role="bold">[</emphasis>
+ <emphasis role="bold">_</emphasis>
+ <emphasis role="bold">⇒</emphasis>
+ &path;
+ <emphasis role="bold">|</emphasis> …
+ <emphasis role="bold">|</emphasis>
+ <emphasis role="bold">_</emphasis>
+ <emphasis role="bold">⇒</emphasis>
+ &path;
+ <emphasis role="bold">]</emphasis>
+ Its semantics is the following: the n-th
+ "<emphasis role="bold">_</emphasis>
+ <emphasis role="bold">⇒</emphasis>
+ &path;" branch is matched against the n-th constructor of the
+ inductive data type. The head λ-abstractions of &path; are matched
+ against the corresponding constructor arguments.
+ </para>
+ <para>For instance, the path
+ <userinput>∀_,_:?.(? ? % ?)→(? ? ? %)</userinput>
+ locates at once the subterms
+ <userinput>x+y</userinput> and <userinput>x*y</userinput> in the
+ term <userinput>∀x,y:nat.x+y=1→0=x*y</userinput>
+ (where the notation <userinput>A=B</userinput> hides the term
+ <userinput>(eq T A B)</userinput> for some type <userinput>T</userinput>).
+ </para>
+ <para>A <emphasis>simple pattern</emphasis> extends paths to locate
+ subterms in a whole sequent. In particular, the pattern
+ <userinput>in H: p K: q ⊢ r</userinput> locates at once all the subterms
+ located by the pattern <userinput>r</userinput> in the conclusion of the
+ sequent and by the patterns <userinput>p</userinput> and
+ <userinput>q</userinput> in the hypotheses <userinput>H</userinput>
+ and <userinput>K</userinput> of the sequent.
+ </para>
+ <para>If no list of hypotheses is provided in a simple pattern, no subterm
+ is selected in the hypothesis. If the <userinput>⊢ p</userinput>
+ part of the pattern is not provided, no subterm will be matched in the
+ conclusion if at least one hypothesis is provided; otherwise the whole
+ conclusion is selected.
+ </para>
+ <para>Finally, a <emphasis>full pattern</emphasis> is interpreted in three
+ steps. In the first step the <userinput>match T in</userinput>
+ part is ignored and a set <emphasis>S</emphasis> of subterms is
+ located as for the case of
+ simple patterns. In the second step the term <userinput>T</userinput>
+ is parsed and interpreted in the context of each subterm
+ <emphasis>s ∈ S</emphasis>. In the last term for each
+ <emphasis>s ∈ S</emphasis> the interpreted term <userinput>T</userinput>
+ computed in the previous step is looked for. The final set of subterms
+ located by the full pattern is the set of occurrences of
+ the interpreted <userinput>T</userinput> in the subterms <emphasis>s</emphasis>.
+ </para>
+ <para>A full pattern can always be replaced by a simple pattern,
+ often at the cost of increased verbosity or decreased readability.</para>
+ <para>Example: the pattern
+ <userinput>⊢ in match x+y in ∀_,_:?.(? ? % ?)</userinput>
+ locates only the first occurrence of <userinput>x+y</userinput>
+ in the sequent <userinput>x,y: nat ⊢ ∀z,w:nat. (x+y) * (z+w) =
+ z * (x+y) + w * (x+y)</userinput>. The corresponding simple pattern
+ is <userinput>⊢ ∀_,_:?.(? ? (? % ?) ?)</userinput>.
+ </para>
+ <para>Every tactic that acts on subterms of the selected sequents have
+ a pattern argument for uniformity. To automatically generate a simple
+ pattern:</para>
+ <orderedlist>
+ <listitem><para>Select in the current goal the subterms to pass to the
+ tactic by using the mouse. In order to perform a multiple selection of
+ subterms, hold the Ctrl key while selecting every subterm after the
+ first one.</para></listitem>
+ <listitem><para>From the contextual menu select "Copy".</para></listitem>
+ <listitem><para>From the "Edit" or the contextual menu select
+ "Paste as pattern"</para></listitem>
+ </orderedlist>
+ </sect2>
+
+ <sect2 id="reduction-kind">
+ <title>reduction-kind</title>
+ <para>Reduction kinds are normalization functions that transform a term
+ to a convertible but simpler one. Each reduction kind can be used both
+ as a tactic argument and as a stand-alone tactic.</para>
+ <table frame="topbot" rowsep="0" colsep="0" role="grammar">
+ <title>reduction-kind</title>
+ <tgroup cols="4">
+ <tbody>
+ <row>
+ <entry id="grammar.reduction-kind">&reduction-kind;</entry>
+ <entry>::=</entry>
+ <entry><emphasis role="bold">normalize</emphasis></entry>
+ <entry>Computes the βδιζ-normal form</entry>
+ </row>
+ <row>
+ <entry/>
+ <entry>|</entry>
+ <entry><emphasis role="bold">reduce</emphasis></entry>
+ <entry>Computes the βδιζ-normal form</entry>
+ </row>
+ <row>
+ <entry/>
+ <entry>|</entry>
+ <entry><emphasis role="bold">simplify</emphasis></entry>
+ <entry>Computes a form supposed to be simpler</entry>
+ </row>
+ <row>
+ <entry/>
+ <entry>|</entry>
+ <entry><emphasis role="bold">unfold</emphasis> [&sterm;]</entry>
+ <entry>δ-reduces the constant or variable if specified, or that
+ in head position</entry>
+ </row>
+ <row>
+ <entry/>
+ <entry>|</entry>
+ <entry><emphasis role="bold">whd</emphasis></entry>
+ <entry>Computes the βδιζ-weak-head normal form</entry>
+ </row>
+ </tbody>
+ </tgroup>
+ </table>
+ </sect2>
+
+ <sect2 id="auto-params">
+ <title>auto-params</title>
+ <para>&TODO;</para>
+ <table frame="topbot" rowsep="0" colsep="0" role="grammar">
+ <title>reduction-kind</title>
+ <tgroup cols="4">
+ <tbody>
+ <row>
+ <entry id="grammar.autoparams">&autoparams;</entry>
+ <entry>::=</entry>
+ <entry><emphasis role="bold">depth=&nat;</emphasis></entry>
+ <entry>&TODO;</entry>
+ </row>
+ <row>
+ <entry/>
+ <entry>|</entry>
+ <entry><emphasis role="bold">width=&nat;</emphasis></entry>
+ <entry>&TODO;</entry>
+ </row>
+ <row>
+ <entry/>
+ <entry>|</entry>
+ <entry><emphasis role="bold">&TODO;</emphasis></entry>
+ <entry>&TODO;</entry>
+ </row>
+ </tbody>
+ </tgroup>
+ </table>
+ </sect2>
+ </sect1>
+
</chapter>
projects / helm.git / blobdiff