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% Document

\author{Ewen Le Bihan}
\date{2020-12-29}
\title{Quantification ``inline''}
\begin{document}
\maketitle
\begin{abstract}
Dans de nombreux énoncés de théorèmes et autres, il arrive souvent qu'un grand nombre de variables soit définies
avant l'énoncé du théorème, ce qui peut parfois gravement allonger le temps de lecture et de compréhension.

Cet article propose une nouvelle notation permettant de quantifier des variables dans une expression (d'où le ``inline'')

\end{abstract}

\section{Un exemple}

Considérons le théorème des isomorphismes:

\paragraph{Sans \emph{inlining} }

Soit $E, F$ deux $\K$-espaces vectoriels.
Soit  $p \in \N^{\ast}$.

Notons $\begin{cases}
	E_1 \oplus E_2 \oplus \cdots \oplus E_p := E \\
	\phi_1 \in \cL(E_1, F) \\
						\vdots \\
	\phi_p \in \cL(E_p, F) \\
\end{cases}$

On a alors:

\begin{align*}
	\exists! \phi \in \cL(E, F), \forall i\in \llbracket 1, p \rrbracket, \phi_{|E_i} = \phi_i
\end{align*}

\paragraph{Avec \emph{inlining} }


Soit $E, F$ deux $\K$-espaces vectoriels.
Soit  $p \in \N^{\ast}$.

On a:

\begin{align*}
	\exists !\phi\in \cL(E, F), \phi_{|(E_1 \oplus \cdots \oplus \fbox{E_k} \oplus E_p = E)} = (\fbox{\phi_k} \in \cL(E_k, F))
\end{align*}

\paragraph{Avec  \emph{inlining} excessif}

On peut théoriquement faire rentrer l'énoncé complet en une expression, mais le gain en clarté se transforme en perte:

\begin{align*}
	\exists !\phi\in \cL((\fbox{E} \in \K\text{-evs}), (\fbox{F}\in \K\text{-evs})), \phi_{|(E_1 \oplus \cdots \oplus \fbox{E_k} \oplus E_{(\fbox{p}\in \N^{\ast})} = E)} = (\fbox{\phi_k} \in \cL(E_k, F))
\end{align*}

\end{document}
