2 <!-- =========== Terms, declarations and definitions ============ -->
4 <chapter id="sec_terms">
6 <para>To describe syntax in this manual we use the following conventions:</para>
8 <listitem><para>Non terminal symbols are emphasized and have a link to their
9 definition. E.g.: &term;</para></listitem>
10 <listitem><para>Terminal symbols are in bold. E.g.:
11 <emphasis role="bold">theorem</emphasis></para></listitem>
12 <listitem><para>Optional sequences of elements are put in square brackets.
13 E.g.: [<emphasis role="bold">in</emphasis> &term;]</para></listitem>
14 <listitem><para>Alternatives are put in square brakets and they are
15 separated by vertical bars. E.g.: [<emphasis role="bold"><</emphasis>|<emphasis role="bold">></emphasis>]</para></listitem>
16 <listitem><para>Repetitions of a sequence of elements are given by putting the
17 sequence in square brackets, that are followed by three dots. The empty
18 sequence is a valid repetition.
19 E.g.: [<emphasis role="bold">and</emphasis> &term;]…</para></listitem>
20 <listitem><para>Characters belonging to a set of characters are given
21 by listing the set elements in square brackets. Hyphens are used to
22 specify ranges of characters in the set.
23 E.g.: [<emphasis role="bold">a</emphasis>-<emphasis role="bold">zA</emphasis>-<emphasis role="bold">Z0</emphasis>-<emphasis role="bold">9_-</emphasis>]</para></listitem>
25 <sect1 id="terms_and_co">
26 <title>Terms & co.</title>
28 <title>Lexical conventions</title>
29 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
30 <title>qstring</title>
34 <entry id="grammar.qstring">&qstring;</entry>
36 <entry><emphasis role="bold">"</emphasis><emphasis>〈〈any sequence of characters excluded "〉〉</emphasis><emphasis role="bold">"</emphasis></entry>
41 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
46 <entry id="grammar.id">&id;</entry>
48 <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>
53 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
58 <entry id="grammar.nat">&nat;</entry>
60 <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>
65 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
70 <entry id="grammar.char">&char;</entry>
72 <entry>[<emphasis role="bold">a</emphasis>-<emphasis role="bold">zA</emphasis>-<emphasis role="bold">Z0</emphasis>-<emphasis role="bold">9_-</emphasis>]</entry>
77 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
78 <title>uri-step</title>
82 <entry id="grammar.uri-step">&uri-step;</entry>
84 <entry>&char;[&char;]…</entry>
89 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
94 <entry id="grammar.uri">&uri;</entry>
96 <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>
105 <!-- ZACK: Sample EBNF snippet, see:
106 http://www.docbook.org/tdg/en/html/productionset.html -->
110 <production id="grammar.term">
113 <lineannotation></lineannotation>
119 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
124 <entry id="grammar.term">&term;</entry>
126 <entry>&sterm;</entry>
127 <entry>simple or delimited term</entry>
132 <entry>&term; &term;</entry>
133 <entry>application</entry>
138 <entry><emphasis role="bold">λ</emphasis>&args;<emphasis role="bold">.</emphasis>&term;</entry>
139 <entry>λ-abstraction</entry>
144 <entry><emphasis role="bold">Π</emphasis>&args;<emphasis role="bold">.</emphasis>&term;</entry>
145 <entry>dependent product meant to define a datatype</entry>
150 <entry><emphasis role="bold">∀</emphasis>&args;<emphasis role="bold">.</emphasis>&term;</entry>
151 <entry>dependent product meant to define a proposition</entry>
156 <entry>&term; <emphasis role="bold">→</emphasis> &term;</entry>
157 <entry>non-dependent product (logical implication or function space)</entry>
162 <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>
163 <entry>local definition</entry>
169 <emphasis role="bold">let</emphasis>
170 [<emphasis role="bold">co</emphasis>]<emphasis role="bold">rec</emphasis>
173 <entry>(co)recursive definitions</entry>
179 [<emphasis role="bold">and</emphasis> &rec_def;]…
187 <emphasis role="bold">in</emphasis> &term;
195 <entry>user provided notation</entry>
198 <entry id="grammar.rec_def">&rec_def;</entry>
201 &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>]…
209 [<emphasis role="bold">on</emphasis> &id;]
210 [<emphasis role="bold">:</emphasis> &term;]
211 <emphasis role="bold">≝</emphasis> &term;]
219 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
220 <title>Simple terms</title>
224 <entry id="grammar.sterm">&sterm;</entry>
226 <entry><emphasis role="bold">(</emphasis>&term;<emphasis role="bold">)</emphasis></entry>
232 <entry>&id;[<emphasis role="bold">\subst[</emphasis>
233 &id;<emphasis role="bold">≔</emphasis>&term;
234 [<emphasis role="bold">;</emphasis>&id;<emphasis role="bold">≔</emphasis>&term;]…
235 <emphasis role="bold">]</emphasis>]
237 <entry>identifier with optional explicit named substitution</entry>
243 <entry>a qualified reference</entry>
248 <entry><emphasis role="bold">Prop</emphasis></entry>
249 <entry>the impredicative sort of propositions</entry>
254 <entry><emphasis role="bold">Set</emphasis></entry>
255 <entry>the impredicate sort of datatypes</entry>
260 <entry><emphasis role="bold">CProp</emphasis></entry>
261 <entry>one fixed predicative sort of constructive propositions</entry>
266 <entry><emphasis role="bold">Type</emphasis></entry>
267 <entry>one predicative sort of datatypes</entry>
272 <entry><emphasis role="bold">?</emphasis></entry>
273 <entry>implicit argument</entry>
278 <entry><emphasis role="bold">?n</emphasis>
279 [<emphasis role="bold">[</emphasis>
280 [<emphasis role="bold">_</emphasis>|&term;]…
281 <emphasis role="bold">]</emphasis>]</entry>
282 <entry>metavariable</entry>
287 <entry><emphasis role="bold">match</emphasis> &term;
288 [ <emphasis role="bold">in</emphasis> &term; ]
289 [ <emphasis role="bold">return</emphasis> &term; ]
290 <emphasis role="bold">with</emphasis>
292 <entry>case analysis</entry>
298 <emphasis role="bold">[</emphasis>
299 &match_branch;[<emphasis role="bold">|</emphasis>&match_branch;]…
300 <emphasis role="bold">]</emphasis>
307 <entry><emphasis role="bold">(</emphasis>&term;<emphasis role="bold">:</emphasis>&term;<emphasis role="bold">)</emphasis></entry>
314 <entry>user provided notation at precedence 90</entry>
320 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
321 <title>Arguments</title>
325 <entry id="grammar.args">&args;</entry>
328 <emphasis role="bold">_</emphasis>[<emphasis role="bold">:</emphasis> &term;]
330 <entry>ignored argument</entry>
336 <emphasis role="bold">(</emphasis><emphasis role="bold">_</emphasis>[<emphasis role="bold">:</emphasis> &term;]<emphasis role="bold">)</emphasis>
338 <entry>ignored argument</entry>
343 <entry>&id;[<emphasis role="bold">,</emphasis>&id;]…[<emphasis role="bold">:</emphasis> &term;]</entry>
349 <entry><emphasis role="bold">(</emphasis>&id;[<emphasis role="bold">,</emphasis>&id;]…[<emphasis role="bold">:</emphasis> &term;]<emphasis role="bold">)</emphasis></entry>
353 <entry id="grammar.args2">&args2;</entry>
361 <entry><emphasis role="bold">(</emphasis>&id;[<emphasis role="bold">,</emphasis>&id;]…<emphasis role="bold">:</emphasis> &term;<emphasis role="bold">)</emphasis></entry>
368 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
369 <title>Pattern matching</title>
373 <entry id="grammar.match_branch">&match_branch;</entry>
375 <entry>&match_pattern; <emphasis role="bold">⇒</emphasis> &term;</entry>
379 <entry id="grammar.match_pattern">&match_pattern;</entry>
382 <entry>0-ary constructor</entry>
387 <entry><emphasis role="bold">(</emphasis>&id; &id; [&id;]…<emphasis role="bold">)</emphasis></entry>
388 <entry>n-ary constructor (binds the n arguments)</entry>
393 <entry>&id; &id; [&id;]…</entry>
394 <entry>n-ary constructor (binds the n arguments)</entry>
404 <sect1 id="axiom_definition_declaration">
405 <title>Definitions and declarations</title>
407 <title><emphasis role="bold">axiom</emphasis> &id;<emphasis role="bold">:</emphasis> &term;</title>
408 <titleabbrev>axiom</titleabbrev>
409 <para><userinput>axiom H: P</userinput></para>
410 <para><command>H</command> is declared as an axiom that states <command>P</command></para>
412 <sect2 id="definition">
413 <title><emphasis role="bold">definition</emphasis> &id;[<emphasis role="bold">:</emphasis> &term;] [<emphasis role="bold">≝</emphasis> &term;]</title>
414 <titleabbrev>definition</titleabbrev>
415 <para><userinput>definition f: T ≝ t</userinput></para>
416 <para><command>f</command> is defined as <command>t</command>;
417 <command>T</command> is its type. An error is raised if the type of
418 <command>t</command> is not convertible to <command>T</command>.</para>
419 <para><command>T</command> is inferred from <command>t</command> if
421 <para><command>t</command> can be omitted only if <command>T</command> is
422 given. In this case Matita enters in interactive mode and
423 <command>f</command> must be defined by means of tactics.</para>
424 <para>Notice that the command is equivalent to <command>theorem f: T ≝ t</command>.</para>
427 <title><emphasis role="bold">letrec</emphasis> &TODO;</title>
428 <titleabbrev>&TODO;</titleabbrev>
430 <sect2 id="inductive">
431 <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;]…
432 [<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;]…]…
434 <titleabbrev>(co)inductive types declaration</titleabbrev>
435 <para><userinput>inductive i x y z: S ≝ k1:T1 | … | kn:Tn with i' : S' ≝ k1':T1' | … | km':Tm'</userinput></para>
436 <para>Declares a family of two mutually inductive types
437 <command>i</command> and <command>i'</command> whose types are
438 <command>S</command> and <command>S'</command>, which must be convertible
440 <para>The constructors <command>ki</command> of type <command>Ti</command>
441 and <command>ki'</command> of type <command>Ti'</command> are also
442 simultaneously declared. The declared types <command>i</command> and
443 <command>i'</command> may occur in the types of the constructors, but
444 only in strongly positive positions according to the rules of the
446 <para>The whole family is parameterized over the arguments <command>x,y,z</command>.</para>
447 <para>If the keyword <command>coinductive</command> is used, the declared
448 types are considered mutually coinductive.</para>
449 <para>Elimination principles for the record are automatically generated
450 by Matita, if allowed by the typing rules of the calculus according to
451 the sort <command>S</command>. If generated,
452 they are named <command>i_ind</command>, <command>i_rec</command> and
453 <command>i_rect</command> according to the sort of their induction
457 <title><emphasis role="bold">record</emphasis> &id; [&args2;]… <emphasis role="bold">:</emphasis> &term; <emphasis role="bold">≝</emphasis><emphasis role="bold">{</emphasis>[&id; [<emphasis role="bold">:</emphasis>|<emphasis role="bold">:></emphasis>] &term;] [<emphasis role="bold">;</emphasis>&id; [<emphasis role="bold">:</emphasis>|<emphasis role="bold">:></emphasis>] &term;]…<emphasis role="bold">}</emphasis></title>
458 <titleabbrev>record</titleabbrev>
459 <para><userinput>record id x y z: S ≝ { f1: T1; …; fn:Tn }</userinput></para>
460 <para>Declares a new record family <command>id</command> parameterized over
461 <command>x,y,z</command>.</para>
462 <para><command>S</command> is the type of the record
463 and it must be convertible to a sort.</para>
464 <para>Each field <command>fi</command> is declared by giving its type
465 <command>Ti</command>. A record without any field is admitted.</para>
466 <para>Elimination principles for the record are automatically generated
467 by Matita, if allowed by the typing rules of the calculus according to
468 the sort <command>S</command>. If generated,
469 they are named <command>i_ind</command>, <command>i_rec</command> and
470 <command>i_rect</command> according to the sort of their induction
472 <para>For each field <command>fi</command> a record projection
473 <command>fi</command> is also automatically generated if projection
474 is allowed by the typing rules of the calculus according to the
475 sort <command>S</command>, the type <command>T1</command> and
476 the definability of depending record projections.</para>
477 <para>If the type of a field is declared with <command>:></command>,
478 the corresponding record projection becomes an implicit coercion.
479 This is just syntactic sugar and it has the same effect of declaring the
480 record projection as a coercion later on.</para>
485 <title>Proofs</title>
487 <title><emphasis role="bold">theorem</emphasis> &id;[<emphasis role="bold">:</emphasis> &term;] [<emphasis role="bold">≝</emphasis> &term;]</title>
488 <titleabbrev>theorem</titleabbrev>
489 <para><userinput>theorem f: P ≝ p</userinput></para>
490 <para>Proves a new theorem <command>f</command> whose thesis is
491 <command>P</command>.</para>
492 <para>If <command>p</command> is provided, it must be a proof term for
493 <command>P</command>. Otherwise an interactive proof is started.</para>
494 <para><command>P</command> can be omitted only if the proof is not
496 <para>Proving a theorem already proved in the library is an error.
497 To provide an alternative name and proof for the same theorem, use
498 <command>variant f: P ≝ p</command>.</para>
499 <para>A warning is raised if the name of the theorem cannot be obtained
500 by mangling the name of the constants in its thesis.</para>
501 <para>Notice that the command is equivalent to <command>definition f: T ≝ t</command>.</para>
504 <title><emphasis role="bold">variant</emphasis> &id;<emphasis role="bold">:</emphasis> &term; <emphasis role="bold">≝</emphasis> &term;</title>
505 <titleabbrev>variant</titleabbrev>
506 <para><userinput>variant f: T ≝ t</userinput></para>
507 <para>Same as <command>theorem f: T ≝ t</command>, but it does not
508 complain if the theorem has already been proved. To be used to give
509 an alternative name or proof to a theorem.</para>
512 <title><emphasis role="bold">lemma</emphasis> &id;[<emphasis role="bold">:</emphasis> &term;] [<emphasis role="bold">≝</emphasis> &term;]</title>
513 <titleabbrev>lemma</titleabbrev>
514 <para><userinput>lemma f: T ≝ t</userinput></para>
515 <para>Same as <command>theorem f: T ≝ t</command></para>
518 <title><emphasis role="bold">fact</emphasis> &id;[<emphasis role="bold">:</emphasis> &term;] [<emphasis role="bold">≝</emphasis> &term;]</title>
519 <titleabbrev>fact</titleabbrev>
520 <para><userinput>fact f: T ≝ t</userinput></para>
521 <para>Same as <command>theorem f: T ≝ t</command></para>
524 <title><emphasis role="bold">remark</emphasis> &id;[<emphasis role="bold">:</emphasis> &term;] [<emphasis role="bold">≝</emphasis> &term;]</title>
525 <titleabbrev>remark</titleabbrev>
526 <para><userinput>remark f: T ≝ t</userinput></para>
527 <para>Same as <command>theorem f: T ≝ t</command></para>
531 <sect1 id="tacticargs">
532 <title>Tactic arguments</title>
533 <para>This section documents the syntax of some recurring arguments for
536 <sect2 id="introsspec">
537 <title>intros-spec</title>
538 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
539 <title>intros-spec</title>
543 <entry id="grammar.intros-spec">&intros-spec;</entry>
545 <entry>[&nat;] [<emphasis role="bold">(</emphasis>[&id;]…<emphasis role="bold">)</emphasis>]</entry>
550 <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>
554 <title>pattern</title>
555 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
556 <title>pattern</title>
560 <entry id="grammar.pattern">&pattern;</entry>
562 <entry><emphasis role="bold">in</emphasis>
563 [&id;[<emphasis role="bold">:</emphasis> &path;]]…
564 [<emphasis role="bold">⊢</emphasis> &path;]]</entry>
565 <entry>simple pattern</entry>
570 <entry><emphasis role="bold">in match</emphasis> &term;
571 [<emphasis role="bold">in</emphasis>
572 [&id;[<emphasis role="bold">:</emphasis> &path;]]…
573 [<emphasis role="bold">⊢</emphasis> &path;]]</entry>
574 <entry>full pattern</entry>
579 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
584 <entry id="grammar.path">&path;</entry>
586 <entry><emphasis>〈〈any &sterm; whithout 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>
591 <para>A <emphasis>path</emphasis> locates zero or more subterms of a given term by mimicking the term structure up to:</para>
593 <listitem><para>Occurrences of the subterms to locate that are
594 represented by <emphasis role="bold">%</emphasis>.</para></listitem>
595 <listitem><para>Subterms without any occurrence of subterms to locate
596 that can be represented by <emphasis role="bold">?</emphasis>.
599 <para>For instance, the path
600 <userinput>∀_,_:?.(? ? % ?)→(? ? ? %)</userinput>
601 locates at once the subterms
602 <userinput>x+y</userinput> and <userinput>x*y</userinput> in the
603 term <userinput>∀x,y:nat.x+y=1→0=x*y</userinput>
604 (where the notation <userinput>A=B</userinput> hides the term
605 <userinput>(eq T A B)</userinput> for some type <userinput>T</userinput>).
607 <para>A <emphasis>simple pattern</emphasis> extends paths to locate
608 subterms in a whole sequent. In particular, the pattern
609 <userinput>in H: p K: q ⊢ r</userinput> locates at once all the subterms
610 located by the pattern <userinput>r</userinput> in the conclusion of the
611 sequent and by the patterns <userinput>p</userinput> and
612 <userinput>q</userinput> in the hypotheses <userinput>H</userinput>
613 and <userinput>K</userinput> of the sequent.
615 <para>If no list of hypotheses is provided in a simple pattern, no subterm
616 is selected in the hypothesis. If the <userinput>⊢ p</userinput>
617 part of the pattern is not provided, no subterm will be matched in the
618 conclusion if at least one hypothesis is provided; otherwise the whole
619 conclusion is selected.
621 <para>Finally, a <emphasis>full pattern</emphasis> is interpreted in three
622 steps. In the first step the <userinput>match T in</userinput>
623 part is ignored and a set <emphasis>S</emphasis> of subterms is
624 located as for the case of
625 simple patterns. In the second step the term <userinput>T</userinput>
626 is parsed and interpreted in the context of each subterm
627 <emphasis>s ∈ S</emphasis>. In the last term for each
628 <emphasis>s ∈ S</emphasis> the interpreted term <userinput>T</userinput>
629 computed in the previous step is looked for. The final set of subterms
630 located by the full pattern is the set of occurrences of
631 the interpreted <userinput>T</userinput> in the subterms <emphasis>s</emphasis>.
633 <para>A full pattern can always be replaced by a simple pattern,
634 often at the cost of increased verbosity or decreased readability.</para>
635 <para>Example: the pattern
636 <userinput>⊢ in match x+y in ∀_,_:?.(? ? % ?)</userinput>
637 locates only the first occurrence of <userinput>x+y</userinput>
638 in the sequent <userinput>x,y: nat ⊢ ∀z,w:nat. (x+y) * (z+w) =
639 z * (x+y) + w * (x+y)</userinput>. The corresponding simple pattern
640 is <userinput>⊢ ∀_,_:?.(? ? (? % ?) ?)</userinput>.
642 <para>Every tactic that acts on subterms of the selected sequents have
643 a pattern argument for uniformity. To automatically generate a simple
646 <listitem><para>Select in the current goal the subterms to pass to the
647 tactic by using the mouse. In order to perform a multiple selection of
648 subterms, hold the Ctrl key while selecting every subterm after the
649 first one.</para></listitem>
650 <listitem><para>From the contextual menu select "Copy".</para></listitem>
651 <listitem><para>From the "Edit" or the contextual menu select
652 "Paste as pattern"</para></listitem>
656 <sect2 id="reduction-kind">
657 <title>reduction-kind</title>
658 <para>Reduction kinds are normalization functions that transform a term
659 to a convertible but simpler one. Each reduction kind can be used both
660 as a tactic argument and as a stand-alone tactic.</para>
661 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
662 <title>reduction-kind</title>
666 <entry id="grammar.reduction-kind">&reduction-kind;</entry>
668 <entry><emphasis role="bold">normalize</emphasis></entry>
669 <entry>Computes the βδιζ-normal form</entry>
674 <entry><emphasis role="bold">reduce</emphasis></entry>
675 <entry>Computes the βδιζ-normal form</entry>
680 <entry><emphasis role="bold">simplify</emphasis></entry>
681 <entry>Computes a form supposed to be simpler</entry>
686 <entry><emphasis role="bold">unfold</emphasis> [&sterm;]</entry>
687 <entry>δ-reduces the constant or variable if specified, or that
688 in head position</entry>
693 <entry><emphasis role="bold">whd</emphasis></entry>
694 <entry>Computes the βδιζ-weak-head normal form</entry>
701 <sect2 id="auto-params">
702 <title>auto-params</title>
704 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
705 <title>reduction-kind</title>
709 <entry id="grammar.autoparams">&autoparams;</entry>
711 <entry><emphasis role="bold">depth=&nat;</emphasis></entry>
712 <entry>&TODO;</entry>
717 <entry><emphasis role="bold">width=&nat;</emphasis></entry>
718 <entry>&TODO;</entry>
723 <entry><emphasis role="bold">&TODO;</emphasis></entry>
724 <entry>&TODO;</entry>