Formal proofs in real algebraic geometry: from ordered fields to quantifier elimination

Abstract

This paper describes a formalization of discrete real closed fields in the Coq proof assistant. This abstract structure captures for instance the theory of real algebraic numbers, a decidable subset of real numbers with good algorithmic properties. The theory of real algebraic numbers and more generally of semi-algebraic varieties is at the core of a number of effective methods in real analysis, including decision procedures for non linear arithmetic or optimization methods for real valued functions. After defining an abstract structure of discrete real closed field and the elementary theory of real roots of polynomials, we describe the formalization of an algebraic proof of quantifier elimination based on pseudo-remainder sequences following the standard computer algebra literature on the topic. This formalization covers a large part of the theory which underlies the efficient algorithms implemented in practice in computer algebra. The success of this work paves the way for formal certification of these efficient methods.