Real Vector Spaces and the Cauchy-Schwarz Inequality in ACL2(r)

TL;DR

This paper provides the first formal proof of the Cauchy-Schwarz inequality in ACL2(r), including formalizations of real vector spaces and applications to metric space continuity, advancing formalized mathematics in theorem proving.

Contribution

It introduces the first published formal proof of the Cauchy-Schwarz inequality in ACL2(r) and formalizes the necessary real vector space mathematics.

Findings

Formal proof of Cauchy-Schwarz inequality in ACL2(r)

Formalization of b R^n as an inner product space

Application to continuity in metric spaces

Abstract

We present a mechanical proof of the Cauchy-Schwarz inequality in ACL2(r) and a formalisation of the necessary mathematics to undertake such a proof. This includes the formalisation of $\mathbb{R}^n$ as an inner product space. We also provide an application of Cauchy-Schwarz by formalising $\mathbb R^n$ as a metric space and exhibiting continuity for some simple functions $\mathbb R^n\to\mathbb R$. The Cauchy-Schwarz inequality relates the magnitude of a vector to its projection (or inner product) with another: \[|\langle u,v\rangle| \leq \|u\| \|v\|\] with equality iff the vectors are linearly dependent. It finds frequent use in many branches of mathematics including linear algebra, real analysis, functional analysis, probability, etc. Indeed, the inequality is considered to be among "The Hundred Greatest Theorems" and is listed in the "Formalizing 100 Theorems" project. To the best of our knowledge, our formalisation is the first published proof using ACL2(r) or any other first-order theorem prover.