A Constructive Proof of the Heine-Borel Covering Theorem for Formal Reals

The continuum is here presented as a formal space by means of a finitary inductive definition. In this setting a constructive proof of the Heine-Borel covering theorem is given. 1 I n t r o d u c t i o n It is well known that the usual classical proofs of the Heine-Borel covering theorem are not acceptable from a constructive point of view (cf. [vS, F]). An intuitionistic alternative proof that relies on the fan theorem was given by Brouwer (cf. [B, H]). In view of the relevance of constructive mathematics for computer science, relying on the connection between constructive proofs and computations, it is natural to look for a completely constructive proof of the theorem in its most general form, namely for intervals with real-valued endpoints. By using formal topology the continuum, as well as the closed intervals of the real line, can be defined by means offinitary inductive definitions. This approach allows a proof of the tIeine-Borel theorem that, besides being constructive, can also be completely formalized and implemented on a computer. Formal topology can be expressed in terms of Martin-Lhf's type theory; a complete formalization of formal topology in the ALF proof editor has been given in [JC]. A development of mathematical results in formal topology will then be a preliminary work for a complete formalization of these results. On the basis of the present work, the first author has implemented the proof of the Heine-Borel theorem for rational intervals. Moreover, here as elsewhere (see for instance [C, C2, N, NV]), the use of a pointfree approach allows to replace non-constructive reasoning by constructive proofs. We point out that a proof similar in spirit to our work was given by MartinLhf in [ML]. The paper is organized as follows: in Section 2 we provide all the preliminary definitions on formM topology to make the exposition self-contained; in Section 3