Constructing the real numbers in HOL

This paper describes a construction of the real numbers in the HOL theorem-prover by strictly deenitional means using a version of Dedekind's method. It also outlines the theory of mathematical analysis that has been built on top of it and discusses current and potential applications in veriication and computer algebra. 1 The real numbers For some mathematical tasks, the natural numbers N = f0; 1; 2; : : :g are suucient. However for many purposes it is convenient to use a more extensive system, such as the integers (Z) or the rational (Q), real (R) or complex (C) numbers. In particular the real numbers are normally used for the measurement of physical quantities which (at least in abstract models) are continuously variable, and are therefore ubiquitous in scientiic applications. 1.1 Properties of the real numbers We can characterize the reals as the uniquècomplete ordered eld'. More precisely, the reals are a set R together with two distinguished constants 0 2 R and 1 2 R and the operations having all the properties in the list given below. In what follows we use the more conventional notation xy for x:y and x ?1 for inv(x). The use of such symbolism, including 0 and 1, is not intended to carry any connotations about what the symbols actually denote.