A Coq Formalization of Lebesgue Integration of Nonnegative Functions

Integration, just as much as differentiation, is a fundamental calculus tool that is widely used in many scientific domains. Formalizing the mathematical concept of integration and the associated results in a formal proof assistant helps in providing the highest confidence on the correctness of numerical programs involving the use of integration, directly or indirectly. By its capability to extend the (Riemann) integral to a wide class of irregular functions, and to functions defined on more general spaces than the real line, the Lebesgue integral is perfectly suited for use in mathematical fields such as probability theory, numerical mathematics, and real analysis. In this article, we present the Coq formalization of σ-algebras, measures, simple functions, and integration of nonnegative measurable functions, up to the full formal proofs of the Beppo Levi (monotone convergence) theorem and Fatou’s lemma. More than a plain formalization of the known literature, we present several design choices made to balance the harmony between mathematical readability and usability of Coq theorems. These results are a first milestone toward the formalization of L spaces such as Banach spaces. Key-words: formal proof, Coq, measure theory, Lebesgue integration ∗ Université Paris-Saclay, CNRS, ENS Paris-Saclay, Inria, Laboratoire Méthodes Formelles, 91190, Gif-surYvette, France. sylvie.boldo@inria.fr † a. Inria, 2 rue Simone Iff, 75589 Paris, France. b. CERMICS, École des Ponts, 77455 Marne-la-Vallée, France. francois.clement@inria.fr ‡ Université Paris-Saclay, CNRS, ENS Paris-Saclay, Inria, Laboratoire Méthodes Formelles, 91190, Gif-surYvette, France. F.Faissole@fr.merce.mee.com § LMAC (Laboratory of Applied Mathematics of Compiègne), CS 60319, Université de technologie de Compiègne, 60203 Compiègne Cedex, France. vincent.martin@utc.fr ¶ LIPN, CNRS UMR 7030, Université Paris 13, 93430 Villetaneuse, France. mayero@lipn.univ-paris13.fr Une formalisation en Coq de l’intégrale de Lebesgue des fonctions positives Résumé : Le calcul intégral, tout comme le calcul différentiel, est un outil fondamental utilisé largement dans de nombreux domaines scientifiques. La formalisation de la notion mathématique d’intégrale et de ses propriétés dans un assistant de preuve aide à donner la plus grande confiance sur la correction de programmes numériques utilisant l’intégration, directement ou indirectement. De part sa capacité à étendre l’intégrale (de Riemann) à une large classe de fonctions irrégulières, et à des fonctions définies sur des espaces plus généraux que la droite réelle, l’intégrale de Lebesgue est considérée comme parfaitement adaptée aux domaines mathématiques tels que la théorie des probabilités, les mathématiques numériques et l’analyse réelle. Dans cet article, nous présentons la formalisation en Coq des tribus (ou σ-algèbres), des mesures, des fonctions étagées et de l’intégrale des fonctions mesurables positives, jusqu’aux preuves formelles complètes du théorème de convergence monotone de Beppo Levi et du lemme de Fatou. Plus qu’une simple formalisation de la littérature connue, nous présentons plusieurs choix de design menés pour équilibrer l’harmonie entre la lisibilité mathématique et l’ergonomie des théorèmes Coq. Ces résultats sont un premier jalon vers la formalisation des espaces L comme espaces de Banach. Mots-clés : preuve formelle, Coq, théorie de la mesure, intégrale de Lebesgue A Coq Formalization of Lebesgue Integration of Nonnegative Functions 3

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