
%%%%%%%%%      Pour LL98
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%% tir\'e de a4new
%%%%%%%%%%%%%%%%%%%%%% R\'eglages g\'en\'eraux %%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%% que vous pouvez modifier %%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%% sans faire de degats %%%%%%%%%%%%%%%%%%%%%%
\magnification=1200

\hsize=11.25cm    
\vsize=18cm       
\parskip=0pt     
\parindent=0pt
%yom
\pageno=1

%%%%%%%%%%%%%%%%%%%%%%%%%%%% R\'eglages %%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%% auquels il ne vaut mieux pas toucher %%%%%%%%%%%%%%

\pretolerance=500 \tolerance=1000  \brokenpenalty=5000

%%%%%%%%%%%%%%%%%%%%% D\'ebut des macros priv\'ees %%%%%%%%%%%%%%%%%%%
\catcode`\@=11
%%%%%%%%%%%%%%%%%%%%%% Les fontes %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\font\eightrm=cmr8         \font\eighti=cmmi8
\font\eightsy=cmsy8        \font\eightbf=cmbx8
\font\eighttt=cmtt8        \font\eightit=cmti8
\font\eightsl=cmsl8        \font\sixrm=cmr6
\font\sixi=cmmi6           \font\sixsy=cmsy6
\font\sixbf=cmbx6


%% pour les petits textes (si tu ne joues pas ...)
\def\basehuit{\baselineskip = 8pt\eightit}
%% pour \'ecrire des programmes.
\let\tt = \tentt


% Pour que les accents se placent correctement en mode math en corps 8 et 6

\skewchar\eighti='177 \skewchar\sixi='177
\skewchar\eightsy='60 \skewchar\sixsy='60

%%%%%%%%%%%%%%%%%%%%%%% dactylographie fran\c caise %%%%%%%%%%%%%%%%%
\catcode`\;=\active
\def;{\relax\ifhmode\ifdim\lastskip>\z@\unskip\fi
\kern\fontdimen2 \font\kern -1.2 \fontdimen3 \font\fi\string;}

\catcode`\:=\active
\def:{\relax\ifhmode\ifdim\lastskip>\z@\unskip\fi\penalty\@M\
\fi\string:}

\catcode`\!=\active
\def!{\relax\ifhmode\ifdim\lastskip>\z@
\unskip\fi\kern\fontdimen2 \font \kern -1.1 \fontdimen3
\font\fi\string!}

\catcode`\?=\active
\def?{\relax\ifhmode\ifdim\lastskip>\z@
\unskip\fi\kern\fontdimen2 \font \kern -1.1 \fontdimen3
\font\fi\string?}

\def\^#1{\if#1i{\accent"5E\i}\else{\accent"5E #1}\fi}
\def\"#1{\if#1i{\accent"7F\i}\else{\accent"7F #1}\fi}

\def\og{\leavevmode\raise.3ex\hbox{$\scriptscriptstyle\langle\!\langle\,$}}
\def\fg{\leavevmode\raise.3ex\hbox{$\scriptscriptstyle\,\rangle\!\rangle$}}

\frenchspacing

%%%%%%%%%%%%%%%%% Le format de sortie %%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Haut et bas de page 

\newtoks\auteurcourant      \auteurcourant={\hfil}
\newtoks\titrecourant       \titrecourant={\hfil}

\newtoks\hautpagetitre      \hautpagetitre={\hfil}
\newtoks\baspagetitre       \baspagetitre={\hfil}

\newtoks\hautpagegauche     
\hautpagegauche={{\eightrm\rlap{\eightrm\folio}\hfil\the\auteurcourant\hfil}}
\newtoks\hautpagedroite     
\hautpagedroite={{\eightrm\hfil\the\titrecourant\hfil\llap{\eightrm\folio}}}

\newtoks\baspagegauche      \baspagegauche={\hfil} 
\newtoks\baspagedroite      \baspagedroite={\hfil}

\newif\ifpagetitre          \pagetitretrue  

% \nopagenumbers : c'est un peu violent, mais ?a marche. Alors ...

\def\nopagenumbers{\def\folio{\hfil}}  

\newdimen\decalage\decalage=0pt

\headline={\ifpagetitre\the\hautpagetitre {\global\advance\hoffset by -\decalage\multiply\decalage by 2} \else\ifodd\pageno\the\hautpagedroite {\global\advance\hoffset by -\decalage} \else\the\hautpagegauche {\global\advance\hoffset by \decalage} \fi\fi}

\footline={\ifpagetitre\the\baspagetitre\else \ifodd\pageno\the\baspagedroite\else\the\baspagegauche\fi\fi \global\pagetitrefalse}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% DATE EN FRANCAIS
\newtoks\jour \jour={\the\day /\the\month /\the\year}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Red\'efinition de \raggedbottom pour avoir plus de mou en bas de page
% (necesssaire quand il y a beaucoup de grumeaux, des grosses
% formules centr\'ees et pas beaucoup de texte entre)

\def\raggedbottom{\topskip 10pt plus 36pt\r@ggedbottomtrue}


%%%%%%%%%%%%%%%%%%% Macros de mise en page %%%%%%%%%%%%%%%%%%%%%
% Un point-tiret

\def\pointir{\unskip . --- \ignorespaces}

% Macros \Bigbreak et \Medbreak pour que les blancs verticaux ne s'ajoutent pas

\def\Bigbreak{\vskip-\lastskip\bigbreak}
\def\Medbreak{\vskip-\lastskip\medbreak}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% les diff\'erents retraits, voir aussi \item

\def\decale#1{\smallbreak\hskip 28pt\llap{#1}\kern 5pt}
\def\decaledecale#1{\smallbreak\hskip 34pt\llap{#1}\kern 5pt}
\def\puce{\smallbreak\hskip 6pt{$\scriptstyle\bullet$}\kern 5pt}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ACCENTS DANS LE TEXTE
\input option_keys
% pour avoir des messages raisonnables avec les lettres accentu\'ees

\let\@ldmessage=\message

\def\message#1{{\def\pc{\string\pc\space}%
                \def\'{\string'}\def\`{\string`}%
                \def\^{\string^}\def\"{\string"}%
                \@ldmessage{#1}}}
%%%%%%%%%%%%%%%%%% Divers gadgets %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Pour se rendre la vie facile : \up{er}, \up{i\`eme}, n\up{0}, etc.

\def\up#1{\raise 1ex\hbox{\eightrm#1}}

\def\pv{\ ;\enspace} % point-virgule de ponctuation en maths


% Entoure #2 d'un filet. Le filet est ?cart? tout autour de #1
% Syntaxe \boxit{5pt}{...}. La ligne de base n'est pas perdue.

\def\boxit#1#2{%
\setbox1=\hbox{\kern#1{#2}\kern#1}%
\dimen1=\ht1 \advance\dimen1 by #1 \dimen2=\dp1 \advance\dimen2 by #1 
\setbox1=\hbox{\vrule height\dimen1 depth\dimen2\box1\vrule}%  
\setbox1=\vbox{\hrule\box1\hrule}%
\advance\dimen1 by .4pt \ht1=\dimen1 
\advance\dimen2 by .4pt \dp1=\dimen2  \box1\relax}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% fin des macros priv?es
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\catcode`\@=12

% pour qu'il la ferme
\showboxbreadth=-1  \showboxdepth=-1

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% fin de a4modifi?
%
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LES FIGURES 
% mod?le de d?finition     \def\figXXXX{\scaledpicture 103mm by 91mm (XXXX scaled 600)}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\def\picture #1 by #2 (#3){
  \vbox to #2{
    \hrule width #1 height 0pt depth 0pt
    \vfill
    \special{picture #3} % this is the low-level interface
    }
  }
\def\scaledpicture #1 by #2 (#3 scaled #4){{
  \dimen0=#1 \dimen1=#2
  \divide\dimen0 by 1000 \multiply\dimen0 by #4
  \divide\dimen1 by 1000 \multiply\dimen1 by #4
  \picture \dimen0 by \dimen1 (#3 scaled #4)}
  }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%% PRESENTATIONS DIVERSES %%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\long\def\indenter{\parindent 1cm \par}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% par, page, remm
\def\parn{\par}
\def\pars{\par\smallskip}
\def\parm{\par\medskip}
\def\parb{\par\bigskip}
%
\def\page{\null\vfill\eject}
\long\def\fin{
\ifodd\pageno\page
\hautpagegauche={{\eightrm\rlap{\eightrm\folio}\hfill }}
\baspagegauche={{\eightit Institut Galil?e\kern 1em
\the\annee\hfill}}
\page
\fi
}
%
\def\remm#1\endrem{\medskip\goodbreak{\bf #1}\medskip\nobreak}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% MES EXOS
\newcount\nexo
\def\debexo{\global \nexo = 0}
\def\exo{\global \advance \nexo by 1 \medskip\goodbreak
{\bf Exercice \the\nexo .}\nobreak\smallskip\nobreak}
\def\exot#1{\global \advance \nexo by 1 \medskip\goodbreak
{\bf Exercice \the\nexo . #1}\nobreak\smallskip\nobreak}
\def\rubr#1{\goodbreak\bigskip{\bf #1}\nobreak\smallskip\nobreak}
%%%%%
\def\initexo{\global\nexo = 0}
%
\initexo
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% les questions
\def\Q#1{{\bf  #1)\kern 0.5em}}                 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% propr
% Encadre un texte entre deux traits avec un titre.
\long\def\propr#1#2{{          
\medskip\nobreak
\line{\hrulefill\kern .3em {\bf #1}\kern
.3em\hrulefill}\smallskip\nobreak
#2
\vskip -.6ex\smallskip\nobreak
\line{\hrulefill}\parm
}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ENCADREMENT texte long
% Pour encadrer un texte indent? long 
%
\long\def\boxa#1#2{\setbox45=\hbox{\kern#1\vbox{#2}\kern#1}%
\dimen45=\ht45 \advance\dimen45 by #1 \dimen46=\dp45 \advance\dimen46 by #1
\setbox45=\hbox{\vrule height\dimen45 depth\dimen46\box45\vrule}%
\setbox45=\vbox{\hrule\box45\hrule}%
\advance\dimen45 by .4pt \ht45=\dimen45
\advance\dimen46 by .4pt \dp45=\dimen46 \box45 \relax}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\long\def\boxb#1{\dimen44=\hsize \advance\dimen44 by -.5cm
\setbox45=\hbox{\kern 2pt\vbox{\hsize=\dimen44
\parindent .7cm\parm #1 \parm}\kern .3cm}%
\dimen45=\ht45 \advance\dimen45 by 2pt \dimen46=\dp45
\advance\dimen46 by 2pt
\setbox45=\hbox{\vrule height\dimen45 depth\dimen46\box45\vrule}%
\setbox45=\vbox{\hrule\box45\hrule}%
\advance\dimen45 by .4pt \ht45=\dimen45
\advance\dimen46 by .4pt \dp45=\dimen46 \box45 \relax}
% la commande elle-m?me :
\long\def\boxmm#1{\boxa{1.4pt}{\boxb{#1}}}                   
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% MES BIBLIOS
\long\def\bib#1#2{\par{\hangindent=1cm\hangafter=1{#1}\ 
{\parindent 1cm #2\par}\pars}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PROGRAMMES
\long\def\prog#1\endprog{{
\dimen100=\parindent
\def\dec{\advance \parindent by -2em}
\def\a{\advance \parindent by 2em\par\item{}}
\def\r{\advance \parindent by -2em\par\item{}}
\def\rr{\advance \parindent by -4em\par\item{}}
\def\p{\par\item{}}
\def\init{\parindent = \dimen100}
\goodbreak\medskip\nobreak\tt #1\par\medskip}}
\long\def\comm#1{{\hfill /*\tt #1*/}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%





%%%%%%%%%%%%%%%%%%%%      DES TRAITS %%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% verticaux
\def\tvi{\vrule height 12pt depth 5pt width 0pt}
\def\vertical#1#2{\vrule height #1 depth #2 width 0pt}
\def\htvi{\vrule height 9pt depth 5pt width 0pt}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%pour les tableaux
\def\hfq{\hfil\quad}
\def\cc#1{\hfq#1\hfq}% centr?
\def\tv{\tvi\vrule}
\def\trait{\noalign{\hrule}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%  MODE MATHEMATIQUE DANS LE TEXTE %%%%%%%%%%%%%%%%%%
\def\modemath#1{\ifmmode #1\else$#1$\fi}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%  ENSEMBLES %%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\let\element=\in
\let\vide=\emptyset
\let\inclusdans=\subseteq
\def\parties#1{{\cal P}(#1)}
\let\part=\parties

\font\fivegoth=eufm5
\font\sevengoth=eufm7
\font\tengoth=eufm10
\newfam\gothfam
\scriptscriptfont\gothfam=\fivegoth
\textfont\gothfam=\tengoth
\scriptfont\gothfam=\sevengoth
\def\goth{\fam\gothfam}

%%%%%%%%%%%%%%%%%%%%%%%%% deux fl\`eches
\let\donne=\rightarrow
\let\affect=\leftarrow
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%% LOGIQUE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% s\'equents
\newdimen\hauteur 
\newdimen\profondeur 
%%
\def\crochets#1{\modemath{\lbrack {#1}\rbrack}}
\def\montre{\mathchardef\triangle="312E\mathrel
=\joinrel\mathrel\triangle}
\def\Dem#1#2#3{%
\setbox100=\hbox{${\displaystyle\dem{#1} \montre {#2}\vdash{#3}}$}
\hauteur=\ht100 \profondeur=\dp100%
\advance \hauteur by 2pt \advance \profondeur by 2pt
\modemath{%%
\displaystyle
\vrule height \hauteur depth \profondeur width 0pt
\dem{#1} \montre {#2}\vdash{#3}}%%
}%
%
\def\sequent#1#2{%
\setbox100=\hbox{${\displaystyle {#1}\vdash{#2}}$}
\hauteur=\ht100 \profondeur=\dp100%
\advance \hauteur by .2em \advance \profondeur by .1em
\modemath{%%
\displaystyle
\vrule height \hauteur depth \profondeur width 0pt
{#1}\vdash{#2}}%%
}%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% abr\'viations
\let\r=\regle
\let\u=\underline
\let\s=\sequent
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\sur{{\modemath{\above 0.2pt}}}
\def\dem#1{{\modemath{{\bf #1}}}}
\def\demu#1{{\modemath{\underline{\bf #1}}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% op\'erateurs
\let\existe=\exists
\let\pourtout=\forall
\let\et =\wedge
\let\ou=\vee
\let\non=\neg
\let\implique=\rightarrow
\let\equival=\leftrightarrow
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\def\ssi{\Leftrightarrow}
\def\bar#1{\overline{\! #1}}
%%%%%%%%%%%%%%%%%%%%% LES LANGAGES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\let\eps=\varepsilon
\let\epsilon = \varepsilon
%
\def\lg#1#2{{\mid\mkern -3mu #1\mkern -3mu\mid}_#2}
\def\longueur#1{{\mid\mkern -3mu #1\mkern -3mu\mid}}
\def\concat{\mathop{{\bf .}}}
\def\opdiv{\mathrel{\scriptscriptstyle\diamond}}
\def\comp{\mathrel{\scriptscriptstyle\circ}}
%
\def\somme#1#2{{\displaystyle \sum_{#1}}#2}
%
\def\l#1{{\cal #1}} 
\def\autom#1{{\bf #1}}
\def\lang#1{{\cal L}(#1)}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% transitions, d\'erivations, ...
\def\transx#1{%
\mathrel{\displaystyle\mathop{\hbox to 5mm{\rightarrowfill}}%
\limits^{\kern 1mm #1\kern 1mm}}}
\def\longtransx#1{%
\mathrel{\displaystyle\mathop{\hbox to 15mm{\rightarrowfill}}%
\limits^{#1}}}
\def\trans{\mathop{\hbox to 5mm{\rightarrowfill}}}
\def\transe{\transx{*}}
\def\tr{\trans\cdots\trans}
%
\def\barre_#1^#2{\kern 1.33pt\displaystyle
\mathrel{\mathop{\kern 0pt\vrule 
height .43ex depth -0.37ex width 1.5em}\limits_{#1}^{#2}}}
%
\def\tete{\mathrel{\vrule height 1.2ex depth .4ex width .4pt}}
\def\derive#1#2{\mathrel{\displaystyle\tete\joinrel\barre_{#1}^{#2}}}
\def\der#1{\mathrel{\displaystyle\tete\joinrel\barre_{\kern
1pt}^{#1}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% productions, inf\'erences, ...
\def\infereG#1#2{%
\mathrel{\displaystyle\mathop{\Longrightarrow}%
\limits_{#1}^{\kern .3em #2\kern .3em}}}
\def\infere#1{{\infereG{\ }{#1}}}

\let\prod=\longrightarrow

\def\prodG#1{%
\mathrel{\displaystyle\mathop{\longrightarrow}%
\limits_{#1}^{\ }}}
\def\prodk#1{\buildrel #1 \over\Longrightarrow}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ACTIONS
\def\action{\mathop{\scriptscriptstyle\bullet}}
\def\actionb{\mathop{\scriptscriptstyle\circ}}
\def\baraction{\mathop{\bar{\scriptscriptstyle\bullet}}}
\def\act#1{\displaystyle\action_{\autom #1}}