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A Coq Formalization of Lebesgue Integration of Nonnegative Functions

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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.

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Notes

  1. The inductive set from https://isabelle.in.tum.de/dist/library/HOL/HOL-Analysis/Sigma_Algebra.html.

  2. The inductive type from https://leanprover-community.github.io/mathlib_docs/measure_theory/measurable_space_def.html#measurable_space.generate_measurable.

  3. The operator from https://leanprover-community.github.io/mathlib_docs/topology/algebra/infinite_sum.html#tsum.

  4. https://www.cs.ru.nl/~freek/100/.

  5. https://isabelle.in.tum.de/dist/library/HOL/HOL-Analysis/Nonnegative_Lebesgue_Integration.html.

  6. https://shemesh.larc.nasa.gov/fm/ftp/larc/PVS-library/library/measure_integration.html, https://shemesh.larc.nasa.gov/fm/ftp/larc/PVS-library/library/lebesgue.html.

  7. https://leanprover-community.github.io/mathlib_docs/measure_theory/constructions/borel_space.html#borel_space.

  8. https://leanprover-community.github.io/mathlib_docs/measure_theory/integral/lebesgue.html#measure_theory.simple_func.

  9. https://github.com/math-comp/analysis.

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Acknowledgements

We are deeply grateful to Stéphane Aubry for some proofs and tactics on properties, and to Assia Mahboubi, Cyril Cohen, Guillaume Melquiond, and Vincent Semeria for fruitful discussions about and in . This work was partly supported by the Paris Île-de-France Region (DIM RFSI MILC). This work was partly supported by Labex DigiCosme (project ANR-11-LABEX-0045-DIGICOSME) operated by ANR as part of the program “Investissement d’Avenir” Idex Paris-Saclay (ANR-11-IDEX-0003-02).

This work was partly supported by the European Research Council (ERC) under the European Union’s Horizon 2020 Research and Innovation Programme—Grant Agreement n\(^\circ \)810367.

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Authors and Affiliations

  1. Université Paris-Saclay, CNRS, ENS Paris-Saclay, Inria, Laboratoire Méthodes Formelles, 91190, Gif-sur-Yvette, France

    Sylvie Boldo & Florian Faissole

  2. Inria, 2 rue Simone Iff, 75589, Paris, France

    François Clément

  3. CERMICS, École des Ponts, 77455, Marne-la-Vallée, France

    François Clément

  4. LMAC (Laboratory of Applied Mathematics of Compiègne), CS 60319, Université de technologie de Compiègne, 60203, Compiègne Cedex, France

    Vincent Martin

  5. LIPN, CNRS UMR 7030, Université Paris 13, 93430, Villetaneuse, France

    Micaela Mayero

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  1. Sylvie Boldo
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Boldo, S., Clément, F., Faissole, F. et al. A Coq Formalization of Lebesgue Integration of Nonnegative Functions. J Autom Reasoning 66, 175–213 (2022). https://doi.org/10.1007/s10817-021-09612-0

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