A Coq Formalization of Lebesgue Integration of Nonnegative Functions - Archive ouverte HAL Access content directly
Journal Articles Journal of Automated Reasoning Year : 2022

A Coq Formalization of Lebesgue Integration of Nonnegative Functions

(1) , (2, 3) , (1) , (4) , (5)
1
2
3
4
5

Abstract

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 $\sigma$-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 $L^p$~spaces such as Banach spaces.
Fichier principal
Vignette du fichier
article.pdf (516.14 Ko) Télécharger le fichier
Vignette du fichier
depend.pdf (23.71 Ko) Télécharger le fichier
Origin : Files produced by the author(s)
Origin : Files produced by the author(s)

Dates and versions

hal-03471095 , version 1 (09-12-2021)

Identifiers

Cite

Sylvie Boldo, François Clément, Florian Faissole, Vincent Martin, Micaela Mayero. A Coq Formalization of Lebesgue Integration of Nonnegative Functions. Journal of Automated Reasoning, 2022, 66, pp.175-213. ⟨10.1007/s10817-021-09612-0⟩. ⟨hal-03471095⟩

Relations

159 View
84 Download

Altmetric

Share

Gmail Facebook Twitter LinkedIn More