A HOL Theory of Euclidean Space

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 ${\mathbb R}^{N}$ with the usual notion of distance.) A notable feature is that the HOL type system is used to encode the dimension...

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Bibliographic Details
Published inLecture notes in computer science pp. 114 - 129
Main Author Harrison, John
Format Book Chapter Conference Proceeding
LanguageEnglish
Published Berlin, Heidelberg Springer Berlin Heidelberg 2005
Springer
SeriesLecture Notes in Computer Science
Subjects
Applied sciences
Combinatorial Lemma
Computer science; control theory; systems
Euclidean Space
Exact sciences and technology
Indexing Type
Logical, boolean and switching functions
Real Vector Space
Theoretical computing
Usual Notion
Elementary analysis
Type theory
Euclidean theory
Inversion
High order logic
Systems theory
Topology
Vector space
Theorem proving
Decision theory
Dimensional analysis
Matrix calculus
Euclidean space
Proof theory
Compatibility
Formal verification
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ISBN3540283722
9783540283720
ISSN0302-9743
1611-3349
DOI10.1007/11541868_8

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Summary: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 ${\mathbb R}^{N}$ with the usual notion of distance.) A notable feature is that the HOL type system is used to encode the dimension N 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 N-dimensional Euclidean space, e.g. Brouwer’s fixpoint theorem and the differentiability of inverse functions.
Bibliography:Original 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathbb R}^{N}$\end{document} with the usual notion of distance.) A notable feature is that the HOL type system is used to encode the dimension N 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 N-dimensional Euclidean space, e.g. Brouwer’s fixpoint theorem and the differentiability of inverse functions.
ISBN:3540283722
9783540283720
ISSN:0302-9743
1611-3349
DOI:10.1007/11541868_8