Analysis library compatible with Mathematical Components

This repository contains a real analysis library for the Coq proof-assistant. It is based on the Mathematical Components library.

In terms of opam, it comes as the following packages:

• coq-mathcomp-classical: a layer for classical reasoning
• coq-mathcomp-analysis: theories for real analysis

Meta

• Author(s):
  • Reynald Affeldt (initial)
  • Alessandro Bruni
  • Yves Bertot
  • Cyril Cohen (initial)
  • Marie Kerjean
  • Assia Mahboubi (initial)
  • Damien Rouhling (initial)
  • Pierre Roux
  • Kazuhiko Sakaguchi
  • Zachary Stone
  • Pierre-Yves Strub (initial)
  • Laurent Théry
• License: CeCILL-C
• Compatible Coq versions: Coq 8.19 to 8.20 (or dev)
• Additional dependencies:
  • MathComp ssreflect 2.1.0 or later
  • MathComp fingroup 2.1.0 or later
  • MathComp algebra 2.1.0 or later
  • MathComp solvable 2.1.0 or later
  • MathComp field 2.1.0 or later
  • MathComp finmap 2.0.0
  • MathComp bigenough 1.0.0
  • Hierarchy Builder 1.4.0 or later
• Coq namespace: mathcomp.analysis

Building and installation instructions

The easiest way to install the latest released version of MathComp-Analysis library is via the opam package manager:

opam repo add coq-released https://coq.inria.fr/opam/released
opam install coq-mathcomp-analysis

Note that the package coq-mathcomp-classical will be installed as a dependency.

Manual installation

To build and install manually, make sure that the dependencies are met and do:

git clone https://github.com/math-comp/analysis.git
cd analysis
make   # or make -j <number-of-cores-on-your-machine> 
make install

About the stability of this library

Changes are documented systematically in CHANGELOG.md and CHANGELOG_UNRELEASED.md.

We bump the minor part of the version number for breaking changes.

We use deprecation warnings to help transitioning to new versions.

We try to preserve backward compatibility as best as we can.

Documentation

Each file is documented in its header in ASCII.

HTML rendering of the source code (using coq2html).

Overview presentations:

• Classical Analysis with Coq (2018)
• A selection of links to well-known lemmas
• An Introduction to MathComp-Analysis (2024)

Publications about MathComp-Analysis:

• Formalization Techniques for Asymptotic Reasoning in Classical Analysis (2018) doi:10.6092/issn.1972-5787/8124
• Formalisation Tools for Classical Analysis (2019)
• Competing inheritance paths in dependent type theory---a case study in functional analysis (2020) doi:10.1007/978-3-030-51054-1_1
• Measure Construction by Extension in Dependent Type Theory with Application to Integration (2023) doi:10.1007/s10817-023-09671-5
• The Radon-Nikodým Theorem and the Lebesgue-Stieltjes Measure in Coq (2024) doi:10.11309/jssst.41.2_41
• A Comprehensive Overview of the Lebesgue Differentiation Theorem in Coq (2024) doi:10.4230/LIPIcs.ITP.2024.5

Other work using MathComp-Analysis:

• A Formal Classical Proof of Hahn-Banach in Coq (2019)
• Semantics of Probabilistic Programs using s-Finite Kernels in Coq (2023)
• CoqQ: Foundational Verification of Quantum Programs (2023)
• Experimenting with an intrinsically-typed probabilistic programming language in Coq (2023)
• Taming Differentiable Logics with Coq Formalisation (2024)

Mathematical structures

MathComp-Analysis adds mathematical structures on top of MathComp's ones. The following inheritance diagram displays the resulting hierarchy as of version 1.1.0 (excluding most MathComp structures). The structures introduced by MathComp-Analysis are highlighted. (See topology.v, normedtype.v, reals.v, measure.v.)

Hierarchies of functions

Functions	Functions with a finite image	Measures	Kernels
(see functions.v)	(see cardinality.v, lebesgue_integral.v)	(see measure.v, charge.v)	(see kernel.v)

Development information

Detailed requirements and installation procedure

Developping with nix

Contributing

Previous work reused at the time of the first releases

This library was inspired by the Coquelicot library by Sylvie Boldo, Catherine Lelay, and Guillaume Melquiond. In the first releases, topology.v and normedtype.v contained a reimplementation of the file Hierarchy.v from the library Coquelicot.

The instantiation of the mathematical structures of the Mathematical Components library with the real numbers of the standard Coq library used a well-known file (Rstruct.v) from the CoqApprox library (with modifications by various authors).

The proof of Zorn's Lemma in classical_sets.v (NB: new filename) was a reimplementation of the one by Daniel Schepler (https://github.com/coq-community/zorns-lemma) but it has been rewritten for version 1.3.0; we also originally took inspiration from Schepler's work on topology (https://github.com/coq-community/topology) for parts of topology.v.

ORIGINAL_FILES.md gives more details about the files in the first releases.

Acknowledgments

Many thanks to various contributors