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fin du rapport, sujet à modification

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@ -61,9 +61,10 @@ Nous indiquons dans la partie~\ref{s:opt} comment compiler,
lancer et utiliser notre programme. lancer et utiliser notre programme.
Nous exposons ensuite à la partie~\ref{s:orga} comment notre Nous exposons ensuite à la partie~\ref{s:orga} comment notre
programme est structuré. programme est structuré.
Nous présentons enfin dans~\ref{s:stlc} notre implémentation Nous présentons enfin dans la partie~\ref{s:stlc} notre
du $\lambda$-calcul simplement typé puis \ref{s:proof} implémentation du $\lambda$-calcul simplement typé puis dans
notre implémentation des preuves. les parties~\ref{s:proof} et~\ref{s:tactic} notre
implémentation des preuves et des tactiques.
\section{Compiler et exécuter} \section{Compiler et exécuter}
\label{s:opt} \label{s:opt}
@ -129,6 +130,9 @@ pour les types et pour les termes:
\alt \lstinline{fun (x : A) => M} \alt \lstinline{fun (x : A) => M}
\alt \lstinline{M N} \alt \lstinline{M N}
\alt \lstinline{exf (M : A)} \alt \lstinline{exf (M : A)}
\alt \lstinline{l(M : A \/ B)}
\alt \lstinline{r(M : A \/ B)}
\alt \lstinline{(M, N)}
\end{grammar} \end{grammar}
Les termes ainsi définis se réduisent à l'aide des règles de Les termes ainsi définis se réduisent à l'aide des règles de
$\beta$-réduction rappelées dans les notes de $\beta$-réduction rappelées dans les notes de
@ -139,12 +143,12 @@ $\alpha$-convertis avec \lstinline{alpha_convert} afin d'éviter
les problèmes de substitution. les problèmes de substitution.
On remarque que l'on impose d'annoter le type de l'argument On remarque que l'on impose d'annoter le type de l'argument
d'une $\lambda$-abstraction. Tous les termes sont alors d'une $\lambda$-abstraction, des constructeurs de termes somme
typables avec les règles bidirectionnelles suivantes: et produit.
On peut alors vérifier le type de tous les termes de manière
déterministe avec les règles de typage suivantes:
\TODO{règles de typage}
\[ \[
\dfrac \dfrac
{\lstinline{x : A} \in \Gamma} {\lstinline{x : A} \in \Gamma}
@ -153,13 +157,13 @@ typables avec les règles bidirectionnelles suivantes:
\[ \[
\dfrac \dfrac
{\Gamma \vdash \lstinline{M : B}} {\Gamma, \lstinline{x : A} \vdash \lstinline{M : B}}
{\Gamma \vdash \lstinline{fun (x : A) => M : A -> B}} {\Gamma \vdash \lstinline{fun (x : A) => M : A -> B}}
\] \]
\[ \[
\dfrac{\Gamma \vdash M : \lstinline{A -> B} \dfrac{\Gamma \vdash \lstinline{M : A -> B}
\quad \Gamma \vdash N : \lstinline{A}} \quad \Gamma \vdash \lstinline{N : A}}
{\Gamma \vdash \lstinline{M N} : \lstinline{B}} {\Gamma \vdash \lstinline{M N} : \lstinline{B}}
\] \]
@ -169,17 +173,116 @@ typables avec les règles bidirectionnelles suivantes:
{\Gamma \vdash \lstinline{exf (M : A) : A}} {\Gamma \vdash \lstinline{exf (M : A) : A}}
\] \]
\[
\dfrac
{\Gamma \vdash \lstinline{M : A}
\quad \Gamma \vdash \lstinline{N : B}}
{\Gamma \vdash \lstinline{(M, N) : A /\\ B}}
\]
\[
\dfrac
{\Gamma \vdash \lstinline{M : A}}
{\Gamma \vdash \lstinline{l(M : A \\/ B) : A \\/ B}}
\quad \quad
\dfrac
{\Gamma \vdash \lstinline{M : B}}
{\Gamma \vdash \lstinline{r(M : A \\/ B) : A \\/ B}}
\]
\section{Preuves} \section{Preuves}
\label{s:proof} \label{s:proof}
\TODO{Augustin} Pour construire la preuve d'une proposition,
on se base sur la correspondance de
Curry-Howard :
ainsi, construire une preuve correspond à créer
un~$\lambda$-terme dont le type correspond à la proposition.
\section{Vérification des preuves} Les~$\lambda$-termes de preuves sont des~$\lambda$-termes
\TODO{parler de la vérification de preuve, particuliers au sens où ils sont \textbf{creux}.
quelles difficultés ? quelles certitudes ?} Intuitivement, les~$\lambda$-termes creux, ou~\lstinline{hlam},
sont des termes avec des trous.
Compléter la preuve correspond
à remplir ces trous à l'aide de tactique.
Lorsqu'on déclare un but, on lui assigne donc :
\begin{itemize}
\item un lambda terme creux (\verb|Ref (ref Hole)|) que l'on construit petit à petit à l'aide des tactiques.
\item le type associé à ce que l'on souhaite prouver
\item son contexte, c'est à dire un liste d'hypothèses (nom de l'hypothèse, type, $\lambda$-terme possiblement encore creux qui sera mis à jour facilement grâce aux \verb|ref|s)
\end{itemize}
Au moment du \verb|Qed|, ce $\lambda$-terme de preuve qui a
été gardé de côté est $\beta$-réduit puis typé contre le type
du but définit au moment de \verb|Goal|.
\section{Tactiques}
\label{s:tactic}
Dans le mode générique, le programme démarre un assistant de preuve interactif semblable à Coq.
Voici les commandes et tactiques implémentées:
\begin{itemize}
\item \verb|Goal| pour émettre un nouveau but
\item \verb|Undo| pour annuler la dernière tactique
\item \verb|Qed| pour clore la preuve d'un but
\item \verb|exact| si une hypothèse correspond parfaitement à notre but courant
\item \verb|assumption| recherche automatique d'une hypothèse correspondant au but courant
\item \verb|intro| pour défaire une implication
\item \verb|intros| pour répéter \verb|intro| autant que possible
\item \verb|cut| permet de faire une élimination
de la coupure.
\item \verb|apply H| où \verb|H| est une implication
avec le but courant en conclusion permet de prouver
directement les prémisses de l'implication.
\item \verb|split| permet de prouver un côté puis l'autre
d'une conjonction
\item \verb|left| et \verb|right| permettent de choisir
quel côté de la conjonction prouver
\item \verb|try| pour essayer une tactique. Si elle échoue,
rien ne se passe.
\end{itemize}
\section{Conlusion}
Nous avons présenté notre assistant de preuve \textbf{Pieuvre}.
Il permet de prouver des prédicats logiques simples.
Plusieurs extensions (et améliorations) sont envisageables.
Tout d'abord, on pourrait évidemment enrichir la grammaire
et les tactiques pour avoir un langage plus expressif.
On pourrait également laisser la possibilité à l'utilisateur
de prouver des \verb|lemmes| et des \verb|théorèmes| pour les
sauvegarder et les réutiliser plus tard.
Il faudrait alors également proposer les types dépendants.
Enfin, il faut se poser la question de la fiabilité de notre
assistant de preuve. Il serait difficile de prouver formellement
toutes les composantes du programme (la~$\beta$-réduction,
l'~$\alpha$-conversion, le \verb|typecheck|, etc.).
Ainsi, on propose de sauvegarder les termes de preuves puis de
les envoyer à Coq pour valider la preuve.
\bibliographystyle{plain} \bibliographystyle{plain}
\bibliography{ex-biblio} \bibliography{ex-biblio}
\section{Répartition du travail}
\begin{itemize}
\item parser et lexer pour le~$\lambda$-calcul et les tactiques (Marwan)
\item exfalso et typage bidirectionnel (Marwan)
\item $\beta$-réduction (Marwan)
\item système de preuve (définition des \verb|hlam|,
des contextes, des buts, etc.) (Marwan)
\item les tactiques exact, assumption, intro, intros,
cut, apply et try (Marwan)
\item $\alpha$-équivalence / renommage (Augustin)
\item fonctionnement du \verb|main| (options, arguments, etc.)
(Augustin)
\item boucle interactive et interface utilisateur
(Augustin)
\item les tactiques Goal, Qed, left, right et split (Augustin)
\item Undo (Augustin)
\end{itemize}
\end{document} \end{document}