Wave Equation Numerical Resolution: A Comprehensive Mechanized Proof of a C Program

Boldo, Sylvie; Clément, François; Filliâtre, Jean-Christophe; Mayero, Micaela; Melquiond, Guillaume; Weis, Pierre

doi:10.1007/s10817-012-9255-4

Cited by 64 publications

(63 citation statements)

References 32 publications

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“…Hence, all our computations are actually verified. Finally, we would like to mention the work on verifying a C-program for the wave equation [12,13].…”

Section: Resultsmentioning

confidence: 99%

The Picard Algorithm for Ordinary Differential Equations in Coq

Makarov

Spitters

2013

Interactive Theorem Proving

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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.

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“…Hence, all our computations are actually verified. Finally, we would like to mention the work on verifying a C-program for the wave equation [12,13].…”

Section: Resultsmentioning

confidence: 99%

The Picard Algorithm for Ordinary Differential Equations in Coq

Makarov

Spitters

2013

Interactive Theorem Proving

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“…The full C program was verified for a finite rope [5], including both the rounding error and the method error. The proofs of that development were relying on two axioms: the existence of a solution to the partial differential equation and its regularity.…”

Section: D'alembert Formulamentioning

confidence: 99%

“…about partial differential equations [5]. This calls for a user-friendly library that provides limits of sequences and functions, derivatives, integrals, power series, and numerous theorems that relate these notions.…”

Section: Introductionmentioning

confidence: 99%

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

2014

Self Cite

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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.

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“…These goals are generated for Why3 (a framework for expressing multi-sorted first-order theories) that can discharge them using a wide set of automated theorem provers, or generate Coq goals as a last resort. The whole stack has been used for example to verify C functions for numerical analysis [10]. No experiment about communication protocols seems to have been carried out yet.…”

Section: Related Workmentioning

confidence: 99%

Towards formal verification of TLS network packet processing written in C

Affeldt

Marti

2013

Proceedings of the 7th Workshop on Programming Languages Meets Program Verification

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TLS is such a widespread security protocol that errors in its implementation can have disastrous consequences. This responsibility is mostly borne by programmers, caught between specifications with the ambiguities of natural language and error-prone low-level parsing of network packets. We provide new Coq libraries for the formal verification of TLS packet processing written in C. The originality of our encoding of the core subset of C is its use of dependent types to guarantee statically well-formedness of datatypes and correct typing. We further equip this encoding with a Separation logic that enables byte-level reasoning and also provide a logical view of data structures. We also formalize a significant part of the RFC for TLS, again using dependent types to capture succinctly constraints that are left implicit in the prose document. Finally, we apply the above framework to an existing implementation of TLS of which we specify and verify a parsing function for network packets.Though not yet completed, this experiment already led us to spot correctness issues with the RFC and the C source code.

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Wave Equation Numerical Resolution: A Comprehensive Mechanized Proof of a C Program

Cited by 64 publications

References 32 publications

The Picard Algorithm for Ordinary Differential Equations in Coq

The Picard Algorithm for Ordinary Differential Equations in Coq

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

Towards formal verification of TLS network packet processing written in C

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