Laurence Rideau

Packaging Mathematical Structures

This paper proposes generic design patterns to define and combine algebraic structures, using dependent records, coercions and type inference, inside the Coq system. This alternative to telescopes in particular supports multiple inheritance, maximal sharing of notations and theories, and automated structure inference. Our methodology is robust enough to handle a hierarchy comprising a broad variety of algebraic structures, from types with a choice operator to algebraically closed fields. Interfaces for the structures enjoy the convenience of a classical setting, without requiring any axiom. Finally, we present two applications of our proof techniques: a key lemma for characterising the discrete logarithm, and a matrix decomposition problem.

A Generic Library for Floating-Point Numbers and Its Application to Exact Computing

In this paper we present a general library to reason about floating-point numbers within the Coq system. Most of the results of the library are proved for an arbitrary floating-point format and an arbitrary base. A special emphasis has been put on proving properties for exact computing, i.e. computing without rounding errors.

Tilting at Windmills with Coq: Formal Verification of a Compilation Algorithm for Parallel Moves

This article describes the formal verification of a compilation algorithm that transforms parallel moves (parallel assignments between variables) into a semantically-equivalent sequence of elementary moves. Two different specifications of the algorithm are given: an inductive specification and a functional one, each with its correctness proofs. A functional program can then be extracted and integrated in the Compcert verified compiler.

Rigorous polynomial approximation using taylor models in Coq

One of the most common and practical ways of representing a real function on machines is by using a polynomial approximation. It is then important to properly handle the error introduced by such an approximation. The purpose of this work is to offer guaranteed error bounds for a specific kind of rigorous polynomial approximation called Taylor model. We carry out this work in the Coq proof assistant, with a special focus on genericity and efficiency for our implementation. We give an abstract interface for rigorous polynomial approximations, parameterized by the type of coefficients and the implementation of polynomials, and we instantiate this interface to the case of Taylor models with interval coefficients, while providing all the machinery for computing them. We compare the performances of our implementation in Coq with those of the Sollya tool, which contains an implementation of Taylor models written in C. This is a milestone in our long-term goal of providing fully formally proved and efficient Taylor models.

Incidence simplicial matrices formalized in Coq/SSReflect

Simplicial complexes are at the heart of Computational Algebraic Topology, since they give a concrete, combinatorial description of otherwise rather abstract objects which makes many important topological computations possible. The whole theory has many applications such as coding theory, robotics or digital image analysis. In this paper we present a formalization in the Coq theorem prover of simplicial complexes and their incidence matrices as well as the main theorem that gives meaning to the definition of homology groups and is a first step towards their computation.

Formal proofs of transcendence for e and pi as an application of multivariate and symmetric polynomials

We describe the formalisation in Coq of a proof that the numbers `e` and `pi` are transcendental. This proof lies at the interface of two domains of mathematics that are often considered separately: calculus (real and elementary complex analysis) and algebra. For the work on calculus, we rely on the Coquelicot library and for the work on algebra, we rely on the Mathematical Components library. Moreover, some of the elements of our formalized proof originate in the more ancient library for real numbers included in the Coq distribution. The case of `pi` relies extensively on properties of multivariate polynomials and this experiment was also an occasion to put to test a newly developed library for these multivariate polynomials.

Certified, Efficient and Sharp Univariate Taylor Models in COQ

We present a library for univariate Taylor models that has been developed with the COQ proof assistant. Each algorithm of this library is executable and has been formally proved correct. Using this library, one can then effectively compute rigorous and sharp approximations of univariate functions composed of usual functions such as reciprocal, square root, exponential, or sine among others. In this paper, we present the key parts of the formalisation as well as of the proofs of correctness, and we evaluate the quality of our certified library on a set of examples.