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

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.Comment: In Proceedings ACL2 2018, arXiv:1810.0376

Convex Functions in ACL2(r)

This paper builds upon our prior formalisation of R^n in ACL2(r) by presenting a set of theorems for reasoning about convex functions. This is a demonstration of the higher-dimensional analytical reasoning possible in our metric space formalisation of R^n. Among the introduced theorems is a set of equivalent conditions for convex functions with Lipschitz continuous gradients from Yurii Nesterov's classic text on convex optimisation. To the best of our knowledge a full proof of the theorem has yet to be published in a single piece of literature. We also explore "proof engineering" issues, such as how to state Nesterov's theorem in a manner that is both clear and useful.Comment: In Proceedings ACL2 2018, arXiv:1810.0376