Finite Automata Theory in

We describe here a development in the system Coq of a piece of Finite Automata Theory. The main result is the Kleene's theorem, expressing that regular expressions and nite automata deene the same languages. From a constructive proof of this result, we automatically obtain a functional program that compiles any regular expression into a nite automata, which constitutes the main part of the implementation of grep-like programs. This functional program is obtained by the automatic method of extraction which removes the logical parts of the proof to keep only its informative contents. Starting with an idea of what we would have written in ML, we write the speciication and do the proofs in such a way that we obtain the expected program, which is therefore eecient. R esum e Nous d ecrivons dans cet article un d eveloppement dans le syst eme Coq d'une partie de la Th eorie des Automates Finis. Le principal r esultat est le th eor eme de Kleene, qui etablit que les expressions rationnelles et les automates nis d eenissent la m^ eme classe de langages. A partir d'une preuve constructive de ce r esultat, on obtient automatiquement un programme fonctionnel qui compile une expression rationnelle en un automate ni, ce qui constitue la partie critique de l'impl ementation des programmes du style grep. Ce programme fonctionnel est obtenu par la m ethode automatique d'extraction qui supprime les parties logiques de la preuve pour ne con-server que son contenu informatif. Avec en t^ ete l'id ee du programme ML que l'on aurait ecrit directement, on ecrit la sp eciication et on m ene les preuves de mani ere a obtenir le programme voulu, qui est par cons equent eecace. Abstract We describe here a development in the system Coq of a piece of Finite Automata Theory. The main result is the Kleene's theorem, expressing that regular expressions and nite automata deene the same languages. From a constructive proof of this result, we automatically obtain a functional program that compiles any regular expression into a nite automata, which constitutes the main part of the implementation of grep-like programs. This functional program is obtained by the automatic method of extraction which removes the logical parts of the proof to keep only its informative contents. Starting with an idea of what we would have written in ML, we write the speciication and do the proofs in such a …