## 5. New Results.............................................................................. 3

### BibTeX

@MISC{Méditerranée_5.new,

author = {Sophia Antipolis Méditerranée},

title = {5. New Results.............................................................................. 3},

year = {}

}

### OpenURL

### Abstract

c t i v it y e p o r t 2007 Table of contents

### Citations

763 | publications by the team in recent years - Major |

508 | Interactive Theorem Proving and Program Development. Coq’Art: The Calculus of Inductive Constructions - Bertot, Castéran - 2004 |

18 | A modular formalisation of finite group theory
- Gonthier, Mahboubi, et al.
- 2007
(Show Context)
Citation Context ...ave completed a formal description of all the concepts and theorems found in an introductory course on group theory (group, subgroup, coset, Sylow group, permutation group). This work is described in =-=[4]-=-.4 Activity Report INRIA 2007 5.1.4. Formalizing matrix algebra Participants: Yves Bertot, Georges Gonthier [Microsoft Research], Sidi Ould Biha. In the context of the “Mathematical Components” proje... |

14 |
Canonical big operators
- Bertot, Gonthier, et al.
- 2008
(Show Context)
Citation Context ...amlining a framework for iterated operators (big operators), like repeated sums, or products, where the index ranges over a finite domain. The library of operators we obtain is described in the paper =-=[6]-=-. 5.1.5. Formalising Geometric Algebras Participants: Sylvain Charneau [Université de Poitiers], Laurent Fuchs [Université de Poitiers], Laurent Théry. Geometric algebras are effective tools to reason... |

11 | A Machine-Checked Implementation of Buchberger’s Algorithm - Théry - 2001 |

8 |
Fixed point semantics and partial recursion in Coq
- Bertot, Komendantsky
- 2008
(Show Context)
Citation Context ...we found an ad hoc method of proving monotonicity and continuity of functionals, for which we provided a few illustrative examples. A description of this work has been submitted as a conference paper =-=[10]-=-. Also we studied the theory of fixed points in a generic category-theoretic setting, revisiting results due to P. Mulry and P. Freyd. We have found some ideas quite helpful for the purposes of our Co... |

6 | G.: Primality proving with elliptic curves
- Théry, Hanrot
- 2007
(Show Context)
Citation Context ...ertificates inside COQ. Thanks to a relatively efficient computation mechanism, it is now possible to prove prime in COQ any prime number with less than 200 decimal digits. This works is described in =-=[9]-=-. 5.2.4. Certified compilation Participants: Xavier Leroy [project-team GALLIUM], Laurence Rideau, Bernard Serpette. In previous years, we worked on the formal description of an algorithm for the para... |

6 | Inductiveand coinductive components of corecursive functions in coq - Bertot, Komendantskaya |

5 |
Formalisation des mathématiques : une preuve du théorème de Cayley-Hamilton
- Biha
- 2008
(Show Context)
Citation Context ...eterminants: multi-linearity, alternatedness, and the Cramer formula. As the next step, we addressed the more significant theorem about characteristic polynomials known as the Cayley-Hamilton theorem =-=[6]-=-. We were able to formalize an elegant proof of this theorem by proving that the natural bijection between matrix with polynomial coefficients and polynomial with matrix coefficients is an isomorphism... |

5 |
Tilting at windmills with Coq: Formal verification of a compilation algorithm for parallel moves
- Rideau, Serpette, et al.
(Show Context)
Citation Context ... previous years, we worked on the formal description of an algorithm for the parallel move from collections of registers to collections of registers. This work appeared this year in a journal article =-=[4]-=-. 5.2.4. Formally verified structural abstract interpretation Participant: Yves Bertot. We implemented a structurally recursive abstract interpreter for a simple language of while constructs, sequence... |

4 | Certified exact real arithmetic using co-induction in arbitrary integer base
- Julien
- 2008
(Show Context)
Citation Context ...proach to computations based on Newton’s algorithm, for instance for the square-root function, and to start the implementation of trigonometric functions. Part of this work was published in the paper =-=[10]-=-. 5.2.2. The Kantorovitch theorem Participants: Yves Bertot, Ioana Pasca. Newton’s method is frequently used to find the roots of continuous, twice differentiable multivariate functions. In previous w... |

3 | Connecting gröbner bases programs with coq to do proofs in algebra, geometry and arithmetics
- Pottier
(Show Context)
Citation Context ...bcoq. The first two are C programs developed by J.C. Faugere. The third was partially extracted from the proof of Buchberger’s algorithm by L.Théry. This lead to a Coq tactic, called gb, described in =-=[11]-=-. After that, we remarked that the computation of the whole Gröbner basis was not necessary. So we adapted gbcoq and, with the help of L.Théry and B.Grégoire, obtained a much more efficient tactic, wh... |

1 |
Proofs for freshmen with Coqweb
- BLANC, GIACOMETTI, et al.
(Show Context)
Citation Context ...urrently porting the wiki encapsulation to Mediawiki, the same wiki framework as Wikipedia. http://pcmath170.unice.fr/wikimath This work was described in a paper presented at PATE’07 in Paris in June =-=[3]-=- and at the Types workshop in Edinburgh in October. We expect to participate to a European project proposal on this topic.6 Activity Report INRIA 2007 5.3.2. Connecting to computer algebra systems Pa... |

1 |
Arithmétique réelle exacte certifiée, co-induction et base arbitraire, in "Journées francophones pour les langages applicatifs, JFLA’07
- JULIEN
- 2007
(Show Context)
Citation Context ... functions like exponentials, sine and cosine. The main work of this year was to define the inverse function and optimize the existing algorithms. The first stages of this work were also published in =-=[5]-=-. 5.2.2. The Kantorovitch theorem Participants: Yves Bertot, Ioana Pasca. Newton’s method is frequently used to find the roots of continuous, twice differentiable multivariate functions. However, this... |

1 |
A Formal Verification for Kantorovitch’s Theorem, in "Journées Francophones des Langages Applicatifs", to appear
- ¸SCA
- 2008
(Show Context)
Citation Context ...e long run we expect that a formal description of the convergence theorems makes it possible to propose new tools for the verification of controling software in this domain. This work is published as =-=[7]-=-. 5.2.3. Primality Proving with Elliptic Curves Participants: Laurent Théry, Guillaume Hanrot [project-team CACAO]. We applied our formalisation of elliptic curves inside COQ to primality proving. Ell... |

1 |
Types, Logique et Coercions Implicites, in "Journées Francophones des Langages Applicatifs", to appear
- ROUX
- 2008
(Show Context)
Citation Context ... coercions. This problem is open even for simple types. We propose a novative rule system that makes it possible to infer types in many cases. An article will be published in the next JFLA conference =-=[8]-=-. 5.1.3. Group theory Participants: Georges Gonthier [Microsoft Research], Assia Mahboubi [INRIA MSR common laboratory], Laurence Rideau, Laurent Théry, Sidi Ould Biha. We participate in the collabora... |

1 | A Short Presentation of Coq, in "Theorem Proving in Higher Order Logics - BERTOT |

1 | Dependent Types, Theorem Proving, and Applications for a Verifying Compiler - BERTOT, THÉRY |

1 | Revisiting the Mini-Rubik in Coq - Pearl - 2008 |