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

Carl Kwan
(University of British Columbia)
Mark R. Greenstreet
(University of British Columbia)

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 R^n as an inner product space. We also provide an application of Cauchy-Schwarz by formalising R^n as a metric space and exhibiting continuity for some simple functions R^n –> R.

The Cauchy-Schwarz inequality relates the magnitude of a vector to its projection (or inner product) with another:

|<u, v>| <= ||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.

In Shilpi Goel and Matt Kaufmann: Proceedings of the 15th International Workshop on the ACL2 Theorem Prover and Its Applications (ACL2 2018), Austin, Texas, USA, November 5-6, 2018, Electronic Proceedings in Theoretical Computer Science 280, pp. 111–127.
Published: 29th October 2018.

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