In this paper we present the Stern{Brocot tree as a basis for performing exact arithmetic on rational and real numbers. We introduce the tree and mention its relation with continued fractions. Based on the tree we present a binary representation of rational numbers and investigate various algorithms to perform exact rational arithmetic using a simpli ed version of the homographic and the quadratic algorithms [19, 12]. We show generalisations of homographic and quadratic algorithms to multilinear forms in n variables and we prove the correctness of the algorithms. Finally we modify the tree to get a redundant representation for real numbers.

It is a well-known fact that algorithms are often hidden inside mathematical proofs. If these proofs are formalized inside a proof assistant, then a mechanism called extraction can generate the corresponding programs automatically. Previous work has focused on the difficulties in obtaining a program from a formalization of the Fundamental Theorem of Algebra inside the Coq proof assistant. In theory, this program allows one to compute approximations of roots of polynomials. However, as we show in this work, there is currently a big gap between theory and practice. We study the complexity of the extracted program and analyze the reasons of its inefficiency, showing that this is a direct consequence of the approach used throughout the formalization.

Abstract. A. Tarski has shown in 1975 that one can perform quantifier elimination in the theory of real closed fields. The introduction of the Cylindrical Algebraic Decomposition (CAD) method has later allowed to design rather feasible algorithms. Our aim is to program a reflectional decision procedure for the Coq system, using the CAD, to decide whether a (possibly multivariate) system of polynomial inequalities with rational coefficients has a solution or not. We have therefore implemented various computer algebra tools like gcd computations, subresultant polynomial or Bernstein polynomials.

We examine a special case of admissible representations of the closed interval, namely those which arise via sequences of a nite number of Mobius transformations. We regard certain sets of Mobius transformations as a generalized notion of digits and introduce sucient conditions that such a \digit set" yields an admissible representation of [0; +1]. Furthermore we establish the productivity and correctness of the homographic algorithm for such \admissible" digit sets. In the second part of the paper we discuss representation of positive real numbers based on the Stern{Brocot tree. We show how we can modify the usual Stern{Brocot representation to yield a ternary admissible digit set.