Contributions à la certification des calculs dans R : théorie, preuves, programmation

Assia Mahboubi 1
1 MARELLE - Mathematical, Reasoning and Software
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : The Coq system is a proof assistant based on the Calculus of Inductive
Constructions.
In this work, we propose to enhance the automation of this system by
providing a reflexive and complete decision procedure for the first
order theory of real numbers.
The Type Theory implemented by the Coq system comprises a
typed functional programming language, which we have used to
implement a Cylindrical Algebraic Decomposition algorithm (CAD). This
algorithm computes a partition of the space into sign-invariant,
semi-algebraic cells for the polynomials of a given family. Hence it
allows to decide all the formulae of this theory.
Then we have to prove formally the correctness of the algorithm and of the
related decision procedure, using the Coq proof assistant.
This work includes a library for certified polynomial arithmetic and
a significant part of the formal proof of correctness of the sub-resultants
algorithm. This last algorithm allows to compute efficiently the
greatest common divisor of polynomials with coefficients in a ring,
and in particular of multivariate polynomials.
We also propose a reflexive tactic for deciding equalities in ring
and semi-ring structures, which enhances the performances of the tool
previously available in the system by taking benefit of the
computational abilities of the system.
In a last part, we study the computational content of a constructive
proof of an elementary lemma of real analysis, called principle of
open induction.
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https://tel.archives-ouvertes.fr/tel-00117409
Contributor : Assia Mahboubi <>
Submitted on : Friday, December 1, 2006 - 3:01:51 PM
Last modification on : Thursday, January 11, 2018 - 4:19:57 PM
Document(s) archivé(s) le : Tuesday, April 6, 2010 - 7:38:01 PM

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Assia Mahboubi. Contributions à la certification des calculs dans R : théorie, preuves, programmation. Génie logiciel [cs.SE]. Université Nice Sophia Antipolis, 2006. Français. ⟨tel-00117409⟩

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