Set Theory or Higher Order Logic to Represent Auction Concepts in Isabelle?

TL;DR

This paper explores how to represent auction concepts in Isabelle/HOL, comparing set-theoretic and higher order logic approaches, and demonstrates formal proofs and verified code for auction theorems.

Contribution

It provides a detailed analysis of representation choices in Isabelle/HOL for auction theory and offers a library of formalized definitions and theorems.

Findings

Set theory aligns closely with economists' representations.

Higher order logic facilitates extraction of computational content.

Trade-offs exist between constructiveness and usability in formalizations.

Abstract

When faced with the question of how to represent properties in a formal proof system any user has to make design decisions. We have proved three of the theorems from Maskin's 2004 survey article on Auction Theory using the Isabelle/HOL system, and we have produced verified code for combinatorial Vickrey auctions. A fundamental question in this was how to represent some basic concepts: since set theory is available inside Isabelle/HOL, when introducing new definitions there is often the issue of balancing the amount of set-theoretical objects and of objects expressed using entities which are more typical of higher order logic such as functions or lists. Likewise, a user has often to answer the question whether to use a constructive or a non-constructive definition. Such decisions have consequences for the proof development and the usability of the formalization. For instance, sets are usually closer to the representation that economists would use and recognize, while the other objects are closer to the extraction of computational content. In this paper we give examples of the advantages and disadvantages for these approaches and their relationships. In addition, we present the corresponding Isabelle library of definitions and theorems, most prominently those dealing with relations and quotients.