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Proving bounds on realvalued functions with computations
 4th International Joint Conference on Automated Reasoning. Volume 5195 of Lecture Notes in Artificial Intelligence
, 2008
"... Abstract. Intervalbased methods are commonly used for computing numerical bounds on expressions and proving inequalities on real numbers. Yet they are hardly used in proof assistants, as the large amount of numerical computations they require keeps them out of reach from deductive proof processes. ..."
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Abstract. Intervalbased methods are commonly used for computing numerical bounds on expressions and proving inequalities on real numbers. Yet they are hardly used in proof assistants, as the large amount of numerical computations they require keeps them out of reach from deductive proof processes. However, evaluating programs inside proofs is an efficient way for reducing the size of proof terms while performing numerous computations. This work shows how programs combining automatic differentiation with floatingpoint and interval arithmetic can be used as efficient yet certified solvers. They have been implemented in a library for the Coq proof system. This library provides tactics for proving inequalities on realvalued expressions. 1
Formal verification of nonlinear inequalities with taylor interval approximations
 CoRR
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A proof of the dodecahedral conjecture
, 1998
"... This article gives a proof of Fejes Tóth’s Dodecahedral conjecture: the volume of a Voronoi polyhedron in a threedimensional packing of balls of unit radius is at least the volume of a regular dodecahedron of unit inradius. 1 ..."
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This article gives a proof of Fejes Tóth’s Dodecahedral conjecture: the volume of a Voronoi polyhedron in a threedimensional packing of balls of unit radius is at least the volume of a regular dodecahedron of unit inradius. 1
Flyspeck in a Semantic Wiki Collaborating on a Large Scale Formalization of the Kepler Conjecture
"... Abstract. Semantic wikis have been successfully applied to many problems in knowledge management and collaborative authoring. They are particularly appropriate for scientific and mathematical collaboration. In previous work we described an ontology for mathematical knowledge based on the semantic ma ..."
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Abstract. Semantic wikis have been successfully applied to many problems in knowledge management and collaborative authoring. They are particularly appropriate for scientific and mathematical collaboration. In previous work we described an ontology for mathematical knowledge based on the semantic markup language OMDoc and a semantic wiki using both. We are now evaluating these technologies in concrete application scenarios. In this paper we evaluate the applicability of our infrastructure to mathematical knowledge management by focusing on the Flyspeck project, a formalization of Thomas Hales ’ proof of the Kepler Conjecture. After describing the Flyspeck project and its requirements in detail, we evaluate the applicability of two wiki prototypes to Flyspeck, one based on Semantic MediaWiki and another on our mathematicsspecific semantic wiki SWiM. 1 Scientific Communication and the Flyspeck Project Scientific communication consists mainly of exchanging documents, and a great deal of scientific work consists of collaboratively authoring them. Common instances are writing down first hypotheses, commenting on results of experiments or project steps, and structuring, annotating, or reorganizing existing items of knowledge, as depicted in Buchberger’s figure on the right.
A Taylor Function Calculus for Hybrid System Analysis: Validation in Coq (Extended Abstract)
, 2010
"... Abstract. We present a framework for the verification of the numerical algorithms used in Ariadne, a tool for analysis of nonlinear hybrid system. In particular, in Ariadne, smooth functions are approximated by Taylor models based on sparse polynomials. We use the Coq theorem prover for developing T ..."
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Abstract. We present a framework for the verification of the numerical algorithms used in Ariadne, a tool for analysis of nonlinear hybrid system. In particular, in Ariadne, smooth functions are approximated by Taylor models based on sparse polynomials. We use the Coq theorem prover for developing Taylor models as sparse polynomials with floatingpoint coefficients. This development is based on the formalisation of an abstract data type of basic floatingpoint arithmetic. We show how to devise a type of continuous function models and thereby parametrise the framework with respect to the used approximation, which will allow us to plug in alternatives to Taylor models. 1
par Sylvie Boldo Deductive Formal Verification: How To Make Your FloatingPoint Programs Behave
"... First, I would like to thank my reviewers: Yves Bertot, John Harrison, and Philippe Langlois. Thanks for reading this manuscript and writing about it! They found bugs, typos, and English errors. Thanks for making this habilitation better. ..."
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First, I would like to thank my reviewers: Yves Bertot, John Harrison, and Philippe Langlois. Thanks for reading this manuscript and writing about it! They found bugs, typos, and English errors. Thanks for making this habilitation better.
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, 2007
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Verified Reachability Analysis of Continuous Systems
"... Abstract. Ordinary differential equations (ODEs) are often used to model the dynamics of (often safetycritical) continuous systems. This work presents the formal verification of an algorithm for reachability analysis in continuous systems. The algorithm features adaptive RungeKutta methods and rig ..."
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Abstract. Ordinary differential equations (ODEs) are often used to model the dynamics of (often safetycritical) continuous systems. This work presents the formal verification of an algorithm for reachability analysis in continuous systems. The algorithm features adaptive RungeKutta methods and rigorous numerics based on affine arithmetic. It is proved to be sound with respect to the existing formalization of ODEs in Isabelle/HOL. Optimizations like splitting, intersecting and collecting reachable sets are necessary to analyze chaotic systems. Experiments demonstrate the practical usability of our developments.