Formalizing Giles Gardam’s Disproof of Kaplansky’s Unit Conjecture

Abstract

We describe a formalization in Lean 4 of Giles Gardam's disproof of Kaplansky's Unit Conjecture. This makes use of a combination of deductive proving and formally verified computation, using the nature of Lean 4 as a programming language which is also a proof assistant. Our goal in this work, besides formalization of the specific result, is to show what is possible with the current state of the art and illustrate how it can be achieved. Specifically we illustrate real time formalization of an important mathematical result and the seamless integration of proofs and computations in Lean 4.