Type classes for efficient exact real arithmetic in Coq

Krebbers, Robbert; Spitters, Bas

doi:10.2168/lmcs-9(1:1)2013

Cited by 22 publications

(21 citation statements)

References 51 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Proof developments at Nijmegen led to a different library called C-CoRN [9] and its successor MathClasses [20]. The main idea is to provide Bishop-like constructive mathematics.…”

Section: Another Coq Library: C-corn/mathclassesmentioning

confidence: 99%

See 1 more Smart Citation

Coquelicot: A User-Friendly Library of Real Analysis for Coq

2014

View full text Add to dashboard Cite

Abstract. Real analysis is pervasive to many applications, if only because it is a suitable tool for modeling physical or socio-economical systems. As such, its support is warranted in proof assistants, so that the users have a way to formally verify mathematical theorems and correctness of critical systems. The Coq system comes with an axiomatization of standard real numbers and a library of theorems on real analysis. Unfortunately, this standard library is lacking some widely used results. For instance, power series are not developed further than their definition. Moreover, the definitions of integrals and derivatives are based on dependent types, which make them especially cumbersome to use in practice. To palliate these inadequacies, we have designed a userfriendly library: Coquelicot. An easier way of writing formulas and theorem statements is achieved by relying on total functions in place of dependent types for limits, derivatives, integrals, power series, and so on. To help with the proof process, the library comes with a comprehensive set of theorems that cover not only these notions, but also some extensions such as parametric integrals, two-dimensional differentiability, asymptotic behaviors. It also offers some automations for performing differentiability proofs. Moreover, Coquelicot is a conservative extension of Coq's standard library and we provide correspondence theorems between the two libraries. We have exercised the library on several use cases: in an exam at university entry level, for the definitions and properties of Bessel functions, and for the solution of the one-dimensional wave equation.

show abstract

“…Proof developments at Nijmegen led to a different library called C-CoRN [9] and its successor MathClasses [20]. The main idea is to provide Bishop-like constructive mathematics.…”

Section: Another Coq Library: C-corn/mathclassesmentioning

confidence: 99%

“…The order x ≤ R y is then defined as y − x nonnegative. Abstract interfaces are heavily used to ease statements and proofs [20]. Thanks to type classes, the algebraic and order hierarchies (setoid, group, ring, and so on) easily benefit from inheritance.…”

Section: Another Coq Library: C-corn/mathclassesmentioning

confidence: 99%

Coquelicot: A User-Friendly Library of Real Analysis for Coq

2014

View full text Add to dashboard Cite

show abstract

“…We depend a huge code base, the CoRN library [1] combined with the recent MathClasses library [2,3]. Part of this work 2 is adapting code from the old library to the new coding style.…”

Section: A Computational Library For Analysismentioning

confidence: 99%

The Picard Algorithm for Ordinary Differential Equations in Coq

Makarov

Spitters

2013

Interactive Theorem Proving

Self Cite

View full text Add to dashboard Cite

Abstract. Ordinary Differential Equations (ODEs) are ubiquitous in physical applications of mathematics. The Picard-Lindelöf theorem is the first fundamental theorem in the theory of ODEs. It allows one to solve differential equations numerically. We provide a constructive development of the Picard-Lindelöf theorem which includes a program together with sufficient conditions for its correctness. The proof/program is written in the Coq proof assistant and uses the implementation of efficient real numbers from the CoRN library and the MathClasses library. Our proof makes heavy use of operators and functionals, functions on spaces of functions. This is faithful to the usual mathematical description, but a novel level of abstraction for certified exact real computation.

show abstract

“…It can be seen as a light counterpart of the seminal works about the formalization of exact arithmetic [15,30,19]. Our motivations to do this work come from the difficulties that we faced when showing that the Harthong-Reeb line satisfies the axioms proposed by Bridges [4].…”

Section: Introductionmentioning

confidence: 99%

Formalizing a discrete model of the continuum in Coq from a discrete geometry perspective

Magaud

Chollet

Fuchs

2014

Ann Math Artif Intell

View full text Add to dashboard Cite

Abstract. This work presents a formalization of the discrete model of the continuum introduced by Harthong and Reeb [16], the HarthongReeb line. This model was at the origin of important developments in the Discrete Geometry field [31]. The formalization is based on the work presented in [8,7] where it was shown that the Harthong-Reeb line satisfies the axioms for constructive real numbers introduced by Bridges [4]. Laugwitz-Schmieden numbers [20] are then introduced and their limitations with respect to being a model of the Harthong-Reeb line is investigated [7]. In this paper, we transpose all these definitions and properties into a formal description using the Coq proof assistant. We also show that Laugwitz-Schmieden numbers can be used to actually compute continuous functions. We hope that this work could improve techniques for both implementing numeric computations and reasoning about them in geometric systems.

show abstract

Type classes for efficient exact real arithmetic in Coq

Cited by 22 publications

References 51 publications

Coquelicot: A User-Friendly Library of Real Analysis for Coq

Coquelicot: A User-Friendly Library of Real Analysis for Coq

The Picard Algorithm for Ordinary Differential Equations in Coq

Formalizing a discrete model of the continuum in Coq from a discrete geometry perspective

Contact Info

Product

Resources

About