The Picard Algorithm for Ordinary Differential Equations in Coq
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.
KeywordsCoq Exact real computation Ordinary Differential Equations Constructive mathematics Type classes
Unable to display preview. Download preview PDF.
- 2.Spitters, B., van der Weegen, E.: Type classes for mathematics in type theory. MSCS, Special Issue on “Interactive Theorem Proving and the Formalization of Mathematics” 21, 1–31 (2011)Google Scholar
- 3.Krebbers, R., Spitters, B.: Type classes for efficient exact real arithmetic in Coq. LMCS 9(1:1) (2013), doi:10.2168/LMCS-9(1:01)2013Google Scholar
- 5.Gonthier, G., Ziliani, B., Nanevski, A., Dreyer, D.: How to make ad hoc proof automation less ad hoc. In: ICFP, pp. 163–175 (2011)Google Scholar
- 7.Bridger, M.: Real analysis, a constructive approach. Pure and Applied Mathematics (New York). Wiley (2007)Google Scholar
- 13.Boldo, S., Clément, F., Filliâtre, J., Mayero, M., Melquiond, G., Weis, P.: Wave equation numerical resolution: a comprehensive mechanized proof of a C program. Journal of Automated Reasoning, 1–34 (2011)Google Scholar