A Formalization of Metric Spaces in HOL Light

Abstract

We present a computer formalization of metric spaces in the HOL Light theorem prover. Basic results of the theory of complete metric spaces are provided, including the Banach Fixed-Point Theorem, the Baire Category Theorem and the completeness of the space of continuous bounded functions. A decision procedure for a fragment of the elementary theory of metric spaces is also implemented. As an application, the Picard–Lindelof theorem on the existence of the solutions of ordinary differential equations is proved by using the well-known argument which appeals to the Banach theorem.