A HOL theory of Euclidean space

Abstract

We describe a formalization of the elementary algebra, topology and analysis of finite-dimensional Euclidean space in the HOL Light theorem prover. (Euclidean space is RN with the usual notion of distance.) A notable feature is that the HOL type system is used to encode the dimension TV in a simple and useful way, even though HOL does not permit dependent types. In the resulting theory the HOL type system, far from getting in the way, naturally imposes the correct dimensional constraints, e.g. checking compatibility in matrix multiplication. Among the interesting later developments of the theory are a partial decision procedure for the theory of vector spaces (based on a more general algorithm due to Solovay) and a formal proof of various classic theorems of topology and analysis for arbitrary TV-dimensional Euclidean space, e.g. Brouwer's fixpoint theorem and the differentiability of inverse functions. © Springer-Verlag Berlin Heidelberg 2005.