New version of user's manual. Most of files modified.

a4788b26 · Jacques Sakarovitch · Alexandre Duret-Lutz · 0638fc42 · a4788b26 · a4788b26
Commit a4788b26 authored Jul 14, 2012 by Jacques Sakarovitch Committed by Alexandre Duret-Lutz Jul 14, 2012
--- a/doc/manual/TAFK.tex
+++ b/doc/manual/TAFK.tex
@@ -248,9 +248,10 @@
 \newcommand{\tafkit}{\textsc{TAF-Kit}\xspace}
 \newcommand{\vcsn}{\textsc{Vaucanson}\xspace}
 \def\VcsnVersion{1.4.1}
-% \def\VcsnVersion{1.4}
+\def\VcsnVrsnOld{1.4}
 % \def\VcsnVersion{1.3.9}
 \newcommand{\vcsnv}{\vcsn\VcsnVersion\xspace}
+\newcommand{\vcsnvo}{\vcsn\VcsnVrsnOld\xspace}
 \newcommand{\tafkitv}{\tafkit\VcsnVersion\xspace}
 \newcommand{\Vauc}{\vcsn}% for compatibility
 \newcommand{\fsmxml}{\textsc{Fsm\,XML}\xspace}
@@ -265,6 +266,7 @@
 \newcommand{\Comt}{\medskip\noindent\textbf{\textsl{Comments}:}\xspace}
 \newcommand{\Cave}{\medskip\noindent\textbf{\textsl{Caveat}:}\xspace}
 \newcommand{\Exam}{\medskip\noindent\textbf{\textsl{Example}:}\xspace}
+\newcommand{\vrglst}{,\e}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \makeatletter
 \newcommand{\Cxx}{%
@@ -423,24 +425,29 @@
 %%%
 \SetTwClPrm{\TwClOne}%
 \SetVCDirectory{figures/}
-% \newcommand{\VGIFig}{figures/VGI/}
+\newcommand{\VGIFig}{figures/VGI/}
 % \ShowFrame
+%========== hyphenation===========================
+% add words to TeX's hyphenation exception list
+\hyphenation{semi-ring}
+\hyphenation{simu-la-tion}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \includeonly{%
 TFKD-title,%
-% TFKD-intro,%
-% TFKD-ch0,%
-% TFKD-ch1,%
-% TFKD-ch1-1,%
-% TFKD-ch2,%
-% TFKD-bib-ind,% 
-% % TFKD-chZ,%
-% TFKD-chA,%
-% TFKD-chVGI,%
-% % % TFKD-chB,
-% % % TFKD-chC,%
-% TFKD-chD,%
+TFKD-table,%
+TFKD-intro,%
+TFKD-ch0-vm,%
+TFKD-ch1,%
+TFKD-ch1-1,%
+TFKD-ch2,%
+% TFKD-chZ,%
+TFKD-chA,%
+TFKD-chVGI,%
+% % TFKD-chB,
+% % TFKD-chC,%
+TFKD-chD,%
 % % TFKD-chE,%
+TFKD-bib-ind,% 
 }
 \makeindex 
 \markboth{\leftmark}{\rightmark} 
@@ -450,12 +457,12 @@ TFKD-title,%
 %%%%%
 \addtocounter{chapter}{-1}
 \include{TFKD-title}
+\include{TFKD-table}
 \include{TFKD-intro}
-\include{TFKD-ch0}
+\include{TFKD-ch0-vm}
 \include{TFKD-ch1}
 \include{TFKD-ch1-1}
 \include{TFKD-ch2}
-\include{TFKD-bib-ind}
 \appendix
 % \include{TFKD-chZ}
 \include{TFKD-chA}
@@ -464,8 +471,9 @@ TFKD-title,%
 % \include{TFKD-chC}
 \include{TFKD-chD}
 \longonly{\include{TFKD-chE}}
+\include{TFKD-bib-ind}
 %%%%%%%%%%%%%%%%%%%
-\printindex 
+% \printindex 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%
 \end{document}
 %%%%%%%%%%%%%
--- a/doc/manual/TFKD-bib-ind.tex
+++ b/doc/manual/TFKD-bib-ind.tex
@@ -28,10 +28,9 @@ automaton constructions.
 \newblock {\em Theoret. Comput. Sci.\/}~{\bf 155} (1996), 291--319. 

 \bibitem{ClavEtAl05}
-{\sc T. Claveirole, S. Lombardy, S. O'Connor, L.-N. Pouchet, and J.
-  Sakarovitch. }
+{\sc T.~Claveirole, S.~Lombardy, S.~O'Connor, L.-N.~Pouchet, and J.~Sakarovitch. }
 \newblock Inside {V}aucanson.
-\newblock {\em Proc. CIAA 2005} LNCS~3845, Springer (2005) 117--128.
+\newblock {\em Proc. CIAA~2005}, LNCS~3845, Springer (2005) 117--128.

 \bibitem{ChamZiad02}
 {\sc J.-M. Champarnaud, D. Ziadi. }
@@ -85,10 +84,15 @@ multiplicity,
 (J.~Flum, E.~Gr\"adel, Th.~Wilke, eds)
 \newblock Amsterdam Univ. Press, 2007, 457--504. 

+\bibitem{LombSaka12}
+{\sc S.~Lombardy, J.~Sakarovitch. }
+\newblock The Removal of Weighted $\varepsilon$-Transitions,
+\newblock {\em Proc. CIAA~2012} LNCS~7381, Springer (2012) 345--352.
+
 \bibitem{LombSakaXX}
 {\sc S.~Lombardy, J.~Sakarovitch. }
 \newblock On the weighted closure problem,
-\newblock {\em in preparation\/}. 
+\newblock {\em in preparation}.

 \bibitem{Loth83}
 {\sc M.~Lothaire. } 

--- a/doc/manual/TFKD-ch0-vm.tex
+++ b/doc/manual/TFKD-ch0-vm.tex
+\chapter{Getting started}
+
+The \vcsn \textsc{Project}  has a web site:
+\begin{center}
+	\code{http://www.vaucanson-project.org}
+\end{center}
+where all information and documentation on the different components 
+of the project, including the present manual, can easily be found and 
+downloaded.
+
+\section{Getting \vcsnv}
+
+Naturally, the version \VcsnVersion\xspace of the \vcsn platform can be downloaded
+from the \vcsn \textsc{Project} site.
+
+% The version \VcsnVersion\xspace of the \vcsn platform can be downloaded
+% from 
+\medskip 
+Alternatively, it can also be found at\\
+\PushLine
+\code{http://vaucanson.lrde.epita.fr/Vaucanson\VcsnVersion}.
+\PushLine
+
+Other previous versions of the \vcsn platform can be downloaded
+from \\
+\PushLine
+\code{http://vaucanson.lrde.epita.fr/}.
+\PushLine
+% Please note this manual is not meant to be backward compatible with
+% \vcsn versions prior to \VcsnVersion.
+
+\Cave
+This version \VcsnVersion\xspace is not fully backward compatible 
+with earlier versions and 
+this manual is not meant to document the differences with
+\vcsn versions prior to \VcsnVersion\xspace (which had never been really 
+documented).
+In one word, no other version than \vcsnv or \vcsn\VcsnVrsnOld\xspace should be used.
+
+
+
+\section{Licensing}
+\vcsnv is a free software released under the GNU General
+Public Licence version 2. 
+More information on this license is to be found at
+\code{http://www.gnu.org/licenses/gpl-2.0.txt} (a copy of
+this license is included in each copy of \vcsn in the file
+\emph{COPYING}).
+
+Beware that the license for the next versions of \vcsn will
+probably be different (although \vcsn will stay an open and free
+software).
+
+\section{Prerequisites}
+\label{sec:pre-req}%
+
+\begin{description}
+\item[\Cpp compiler] \code{G++ 4.x} or later.
+
+\item[\XML] The \XML I/O system is based on the use of the Apache \code{Xerces}
+  \Cpp library version 2.7 or later (\code{http://apache.org/xerces-c/}). (On
+  Ubuntu/Debian, install packages with names similar to: \code{libxerces28} and \code{libxerces28-dev},
+  or \code{libxerces-c3.1} and \code{libxerces-c-dev}),
+\Indextt{Xerces}%
+
+\item[\code{Boost}] \code{Boost} provides free peer-reviewed
+portable \Cpp source
+  libraries (On Ubuntu/Debian, install the following packages:
+  l\code{ibboost-dev}, \code{libboost-serialization-dev}, \code{libboost-graph},
+  \code{libboost-graph-dev}).
+  \vcsn is compatible with \code{Boost} versions \code{1.34} or later.
+\Indextt{Boost}%
+
+\item[\code{Ncurses}] needed for building \tafkit  (On Ubuntu/Debian, install
+  the following packages: \code{libncurses5}, \code{libncurses-dev}).
+\Indextt{Ncurses}%
+
+\item[\code{Graphviz}] The display of automata is made using AT\&T \code{GraphViz}
+  application (On Ubuntu/Debian, install the following package: \code{graphviz}).
+\Indextt{Graphviz}%
+
+\end{description}
+
+\section{Building \vcsn}
+\label{sec:bui-ld}%
+
+
+Detailed information is provided in both \code{INSTALL} and \code{doc/README.txt}
+files. The following installation commands will install \vcsn in
+'\code{/usr/local}'.
+
+\begin{shell}
+$ cd vaucanson-\VcsnVersion
+$ ./configure
+$ make
+$ sudo make install
+\end{shell}%
+
+
+Depending on the architecture, both \code{Boost} and \code{Xerces} might be located
+in non-standard directories. 
+The location of these
+libraries may known via the following command:
+
+\begin{shell}
+$ whereis boost
+\end{shell}%
+
+This command returns the paths to \code{Boost} headers. 
+These directories are then
+specified  to the \code{configure} file by means of
+two environment variables: \code{CPPFLAGS} for the header files and \code{LDFLAGS}
+for the library files. 
+For instance, if \code{Boost} headers are located
+in:\\
+'\code{/usr/user\_name/home/my\_path\_to\_boost/include}' and its library
+files in:\\
+'\code{/usr/user\_name/home/my\_path\_to\_boost/lib}', the
+following configure line should be invoked:
+
+\begin{shell}
+$ ./configure   CPPFLAGS='-I/usr/user\_name/home/my\_path\_to\_boost/include'
+                LDFLAGS='/usr/user\_name/home/my\_path\_to\_boost/lib'
+\end{shell}%
+
+If \vcsn is not installed but only compiled,
+the \tafkit binaries are to be found in the directory
+'\code{vaucanson-\VcsnVersion/taf-kit/tests/}' (This directory
+contains wrappers
+around the real \tafkit programs from '\code{vaucanson-\VcsnVersion/taf-kit/src/}'
+that enable them to run locally).
+
+\section{Mac\xmd OS\xmd X specifics}
+
+The installation process of \vcsn
+and its dependencies on Mac\xmd OS\xmd X may be less straightforward
+than onto other Linux systems.
+
+% First, the Mac\xmd OS\xmd X system should be up-to-date before
+% going through the rest of the installation process.
+
+First, the \code{macports} software will be used to get all the
+prerequisites and should be installed first on the computer
+(see \code{http://www.macports.org/}).
+A complete guide
+to its installation is available from \code{http://guide.macports.org/}.
+If \code{macports} is already installed, it should be made up-to date
+by synchronising the local port tree with the global \code{macports}
+ports by the following command.
+\begin{shell}
+$ sudo port selfupdate
+\end{shell}%
+
+Three libraries are to be installed in order to
+build \vcsn (see Prerequisite for details):
+\code{Boost},
+\code{Xerces}, and
+\code{Ncurses}.
+\begin{shell}
+$ sudo port install ncurses
+...
+$ sudo port install boost
+...
+$ sudo port install xercesc
+...
+$
+\end{shell}%
+Note that executing each of these commands may take a while
+(especially when installing \code{Boost}).
+%
+By default, \code{macports} will install each of these three
+libraries in the \code{/opt/local} directory, which is not standard
+with respect to the Unix organisation.
+In order to build \vcsn, this directory is therefore to be specified
+to the \code{configure} command by the following options:
+\begin{shell}
+$ ./configure CPPFLAGS='-I/opt/local/include' LDFLAGS='-L/opt/local/lib'
+\end{shell}%
+
+Moreover, if the installed version of \code{Boost}  is greater than
+or equal to~1.44 it is necessary to add another option to the
+\code{configure} command:
+\begin{shell}
+$ ./configure CPPFLAGS='-I/opt/local/include -DBOOST\_SPIRIT\_USE\_OLD\_NAMESPACE'
+              LDFLAGS='-L/opt/local/lib'
+\end{shell}%
+The installation is then to be completed by the usual two commands:
+\begin{shell}
+$ make
+$ sudo make install
+\end{shell}%
+
+\section{Improving the display quality}
+
+As long as a dedicated graphical interface is not fully operational, 
+displaying automata is processed by the \code{Graphviz} application, which 
+is normally launched
+in an \code{X11} window.
+It is to be acknowledged that the call to \code{Graphviz} by \tafkit
+is not well-tuned and that the output is rather poor.
+It is not too difficult however to get a rendering of
+automata of much better quality (\cf \figur{gra-viz}).
+This can be done in three steps and in a slightly different way for 
+Mac and Linux users.
+
+\subsection{\code{Graphviz} for Mac}
+
+First download the \code{Graphviz} application for Mac from
+\code{www.pixelglow.com/graphviz/}.
+Although already old and outdated by the \code{2.xx} versions, the
+\code{1.13 (v16)} version is recommended as the settings is easier to
+handle in that version.
+Complete the installation by putting the \code{Graphviz.app} folder
+in the \code{Applications} folder.
+
+Second, write the following script in a file called \code{dotty}:
+\begin{shell}
+#! /bin/sh
+if [ "x$1" = x- ]; then
+ cat >/tmp/tmpdotty$$.dot
+ open -W -a Graphviz /tmp/tmpdotty$$.dot
+ rm -f /tmp/tmpdotty$$.dot
+else
+ open -W -a Graphviz "$1"
+fi
+\end{shell}%
+
+Finally, make this file executable, store it in a folder, and put the
+full name of this folder in the \code{PATH} variable before
+\code{/usr/local/bin:} and \code{/usr/X11/bin:}.
+The appearance of the automata will be determined by fixing the
+settings in the interface.
+
+
+\subsection{\code{Graphviz} for Linux}
+
+For Linux users, the process is almost same except that the program 
+required is \code{xpdf} and the \code{dotty} script is:
+
+\begin{shell}
+#!/bin/sh
+f=/tmp/pd$$.png
+if test "x$1" = x-; then
+  dot -Tpdf > $f
+else
+  dot -Tpdf "$@" > $f
+fi
+xpdf $f
+rm -f $f
+\end{shell}%
+
+
+\begin{figure}[ht]
+    \centering
+\includegraphics[scale=0.25]{figures/a1-dotty.ps}
+\ee
+\includegraphics[scale=0.5]{figures/a1-gv.ps}
+\caption{Two versions of the \code{Graphviz} application}
+\label{fig:gra-viz}%
+\end{figure}
+
+
+\section{VGI}
+
+The availability of the Graphical User Interface \vgi, developped at 
+NTU, is one of the novelty of \vcsnv with respect to \vcsnvo.
+Every instance of \tafkit has a function \code{gui} that launches 
+\vgi (\cf \secti{taf-io-fun}).
+For this function to work, \tafkit needs to be specified where is \vgi.
+
+Download the \code{vgi.jar} archive from 
+\code{http://vaucanson-project.org/resources/vgi.jar},  store it in a 
+folder denoted by \code{/path/to/vgi/folder}  in the following.
+
+There is two methods to specify to \vcsn where this archive is 
+stored: either by setting the environment variable \code{VGIJAR}
+with the command:
+
+\begin{shell}
+$ export VGIJAR=/path/to/vgi/folder/vgi.jar
+\end{shell}
+
+\noindent
+or by puting the following script named "\code{vgi}" in a directory of the 
+path (for instance \code{/usr/local/bin}): 
+
+\begin{shell}
+#!/bin/sh
+exec java -jar /path/to/vgi/folder/vgi.jar "$@"
+\end{shell}
+
+\noindent
+and making it executable. 
+A call to the \code{gui} function will then open a window such as the 
+one shown at \figur{vgi} (\cf \apndx{vgi}).
+
+
+\begin{figure}[ht]
+    \centering
+\includegraphics[width=3in]{\VGIFig fig14.eps}
+\caption{The \vgi window }
+\label{fig:vgi}%
+\end{figure}
+
+%%%%%%%%%%%
+\endinput
--- a/doc/manual/TFKD-ch0.tex
+++ b/doc/manual/TFKD-ch0.tex
-\chapter{Administrativia}
+\chapter{Getting started}
+
+The \vcsn \textsc{Project}  has a web site:
+\begin{center}
+	\code{http://www.vaucanson-project.org}
+\end{center}
+where all information and documentation on the different components 
+of the project, including the present manual can easily be found and 
+downloaded.

 \section{Getting \vcsnv}
-The version \VcsnVersion\xspace of the \vcsn platform can be downloaded
-from \\
+
+Naturally, the version \VcsnVersion\xspace of the \vcsn platform can be downloaded
+from the \vcsn \textsc{Project} site.
+
+% The version \VcsnVersion\xspace of the \vcsn platform can be downloaded
+% from 
+
+Alternatively, it can also be found at\\
 \code{http://vaucanson.lrde.epita.fr/Vaucanson\VcsnVersion}

 Other previous versions of the \vcsn platform can be downloaded
 from \\
 \code{http://vaucanson.lrde.epita.fr/}

-Please note this manual is not meant to be backward compatible with
-\vcsn versions prior to \VcsnVersion.
+% Please note this manual is not meant to be backward compatible with
+% \vcsn versions prior to \VcsnVersion.
+
+\Cave
+This version \VcsnVersion\xspace is not fully backward compatible 
+with earlier versions and 
+this manual is not meant to document the differences with
+\vcsn versions prior to \VcsnVersion\xspace (which had never been really 
+documented).
+In one word, no other version than \vcsnv or \vcsn\VcsnVrsnOld should be used.
+
+

 \section{Licensing}
 \vcsnv is a free software released under the GNU General

--- a/doc/manual/TFKD-ch1-1.tex
+++ b/doc/manual/TFKD-ch1-1.tex
@@ -249,7 +249,7 @@ written in base~2, and then display the quotient\footnote{%
 	transitions and diplaying it would have been messy.}.

 \begin{shell}
-$ \kbd{vcsn-char-b -a'01\bslash,' exp-to-aut '1(0+1)*+1(0+1)*,(0+1)(0+1)*' > dec-bin.xml}
+$ \kbd{vcsn-char-b -a'01\bslash,' exp-to-aut '1(0+1)*+(1(0+1)*+0),(0+1)(0+1)*' > dec-bin.xml}
 $ \kbd{vcsn-char-b quotient dec-bin.xml \bslash| display -}
 \end{shell}

@@ -484,9 +484,9 @@ $ \kbd{vcsn-char-b -ofsm cat b1.xml}
 1   1   a    0
 1   1   b    0
 1    0
-$ \kbd{vcsn-char-z -ofsm cat b1.xml \bslash| -ifsm eval - 'bab'}
-2
 \end{shell}%
+% $ \kbd{vcsn-char-z -ofsm cat b1.xml \bslash| -ifsm eval - 'bab'}
+% 2

 \Cave 
 The \code{fsm} format is not really implemented in \tafkitv.
@@ -497,16 +497,28 @@ format within \vcsn.
 First, the automata than can be described with the \code{fsm} format 
 must meet several conditions: one initial state only, labels are 
 letters (or integers that refer to a symbol table).
-Second, \vcsn does not code the weights correctly.
+Second, \vcsn does not code the weights correctly for \fst.
 It is thus inadequate to try to use the \code{fsm} format for another 
 automata than `letterized' Boolean automata with a unique initial 
 state.
+The following two examples of commands use the \code{fsm} format for 
+$\Z$-automata.
+The first one gives a correct answer, as the weights are all equal 
+to~``1'';
+the second one demonstrates that \vcsn is not even able to read 
+correctly the \code{fsm} file it has written.
+
+
+\begin{shell}
+$ \kbd{vcsn-char-z -ofsm cat b1.xml \bslash| -ifsm eval - 'bab'}
+2
+\end{shell}%

 \begin{shell}
 $ \kbd{vcsn-char-z -ofsm cat c1.xml}
 0	1	1	 0
 0	0	0+1	 0
-1	1	({2} 0)+({2} 1)	 0
+1	1	(\{2\} 0)+(\{2\} 1)	 0
 1	 0
 $ \kbd{vcsn-char-z -ofsm cat c1.xml | vcsn-char-z -ifsm -ofsm cat -}
 0	1	e	 0
@@ -540,7 +552,7 @@ By default, rational expressions are \emph{read} as strings given on
 the \emph{command line}, 
 and \emph{output} as strings on the \emph{standard output}.
 Both can be diverted in the Unix way, but a \emph{string} written in a file 
-cannot be read by \tafkit in this file.
+cannot be read by \tafkit directly in this file (\cf \sbsct{fct-cat-E}).

 \begin{shell}
 $ \kbd{vcsn-char-b -aab cat-E '(a+b(a(b)*a)*b)*'}
@@ -579,14 +591,6 @@ $ \kbd{vcsn-char-b -ixml cat-E exp.xml}
 (a+b.(a.b*.a)*.b)*
 \end{shell}%$

-\longonly{%
-\begin{ComVd}{110515}
-	Contrarily to what had been written in an earlier version of this 
-	document, ``rational expressions will \emph{not} be read or written as text string if an 
-unsupporting format is requested'' but very logically the reading or 
-writing in that format will be denied..
-\end{ComVd}%
-}%


 \subsubsection{Word formats}
@@ -598,13 +602,17 @@ command line, and written on the standard output.
 \Cave
 Although \emph{words} are, from a formal point of view, a (simple) 
 instance of a rational expression, \tafkitv handles them as objects 
-of different and uninterchangeable types.
+of different and uninterchangeable types (\cf \sbsct{evl-wrd}).
 We come back to the subject in the next section.

 \longonly{%
 \begin{ComVd}{110515}
-	It is not clear whether there exists indeed a command which 
+	The command \Fct{eval} reads a word, and not an expression.
+	It does not seem that there exists indeed a command which 
 	outputs a `word' (and not an automaton or an expression).
+	
+	This will be reworked in the framework where there will be two 
+	kinds of expressions.
 \end{ComVd}%
 }%

@@ -621,6 +629,11 @@ $ \kbd{vcsn-char-z eval c1.xml '101101'}
 The way they are input, as strings as well, as part of a rational 
 expression, is described in the next section.

+In one case (in two similar cases indeed), weights will be input as 
+argument of a function and not as part of a rational expression. They 
+will be written as strings as well, which will raise some problem 
+(\cf \sbsct{aut-sta-ext-mul}).
+

 \subsubsection{Boolean result formats}
 \label{ssc:boo-res-for}
@@ -747,7 +760,7 @@ by other programs.
 \longonly{%
 \begin{ComVd}{110723}
 The \code{-D} option yields a \code{dot} file which is correctly 
-dispalyed on the screen but when exported as a \code{ps} file and 
+displayed on the screen but when exported as a \code{ps} file and 
 incorporated in the \tex output sends the \code{pstopdf} command into 
 error. 
 \end{ComVd}%
@@ -795,10 +808,11 @@ Charge  id:       <name>        total     self   calls   self avg.  total avg.

 \longonly{%
 \begin{ComVd}{110529}
-	I report some of my experiments and findings about \code{cbs} in 
-	a special appendix which will not appear in the distributed 
+	Some experiments and findings about \code{cbs} are reported at
+	\apndx{ben-fun-vcs}, 
+	an appendix which does not appear in the distributed 
 	version of the \tafkit documentation.
-	It should help us to profile the version of \code{cbs} in \vcsn2.
+	It should help us to profile the version of \code{cbs} in \vcsn~2.
 \end{ComVd}%
 }%

@@ -941,46 +955,18 @@ is rewritten as indicated on the right.
 (\{k\}\Ed)\{h\} \Rightarrow \{k\}(\Ed\{h\})
 \tag{$\mathbf{A}_{\K}$}
 \\[.7ex]
-  \und\{k\}  \Rightarrow  \{k\}\und \ee
+%   \und\{k\}  \Rightarrow  \{k\}\und \ee
  \Ed.(\{k\}\und)   \Rightarrow  \Ed\{k\} \ee
  (\{k\}\und).\Ed   \Rightarrow  \{k\}\Ed
 \tag{$\mathbf{U}_{\K}$}\\[.7ex]
  \und \{k\}\Rightarrow \{k\}\und \ee x\{k\}\Rightarrow \{k\}x 
 \tag{$\mathbf{C}_{\mathsf{at}}$}
+\label{equ:c-at}
 \end{gather}
 \caption{The trivial identities}
 \label{tab:tri-ide}%
 \end{table}

-
-\longonly{%
-\begin{ComVd}{110723}
-  La dfinition de l'identit 
-  correspond  ce qui est implment dans
-  Elle est due  un quiproquo sur la signification du mot `atome':
-  \emph{atome} d'une expression rationnelle et \emph{atome} des 
-  \code{labels-are-atoms} pour la dfinition des \emph{kinds} dans 
-  \vcsn~2.xx.
-  
-  Une dfinition plus naturelle serait:
-%   \begin{equation*}
-%   	m\{k\}\Rightarrow \{k\}m