At the start, the two mathematicians on the team selected small topics to formalise. This allowed them to familiarise themselves with the advantages and limitations of Isabelle/HOL and to plan their next steps. They formalised results on advanced mathematical topics, including quantum computation and irrationality and transcendence criteria for infinite series. We found several errors in mathematics papers.
We worked heavily on consolidating and reorganising our libraries and formalised mathematics, producing a user manual identifying the the main topics in the Analysis library's 150K proof lines and 95 formal theories. Substantial amounts of new material was added to the libraries, covering advanced analysis and algebra: many tens of thousands of lines of formal proofs.
The proposal included the speculative aim of using AI to support users. We developed a tool to perform intelligent search in our mathematical libraries through natural language queries, achieved by indexing all four million lines of Isabelle's Archive of Formal Proofs. We also wanted something like GitHub's Copilot, which could make suggestions based on material found in existing proofs. We needed time to realise these ideas concretely, but towards the end, the project achieved impressive results involving the use of AI to generate proofs, e.g. automatic formalisation (that is, translating normal mathematical text into our formalism).
Verified Computer Algebra is another project focus. Computers have long been able to perform algebraic manipulations, but much of this software is not trustworthy. We developed advanced, formally verified algorithms for root-finding and other operations on polynomials.
As the project progressed, we tackled more and more ambitious pieces of mathematics, and all fairly recent (within the past half century). Separate projects on irrational convergent theories, ordinal partitions and Grothendieck schemes ended up taking over half of a special issue announced by the journal Experimental Mathematics. We tackled advanced topics in extremal graph theory, block designs, additive combinatorics and higher-order category theory, all relevant to today's mathematics.
The theorems we formalised include include Szemerédi's regularity lemma, Roth's theorem on arithmetic progressions, Lucas’s theorem, Fisher's inequality, the Plünnecke-Ruzsa inequality, Kneser's theorem, the Cauchy–Davenport theorem, Khovanskii's theorem, Balog–Szemerédi–Gowers theorem and much more.
• Our main conclusion is that there is no clear limit to what sort of mathematics can be formalised. We have formalised material across the mathematical landscape: combinatorics, analysis, number theory, Ramsey theory. By the end we had formalised the work of some of the greatest mathematicians of our day—Erdős, Gowers, Roth, Szemerédi—and uncovered numerous small errors.
• Another key conclusion: Isabelle/HOL relies on a relatively simple formalism, higher-order logic, thought by some to be unsuitable. This view is now disproved. And the simple formalism has strong advantages: (1) fewer technical quirks to entrap users, (2) better automation, hence better productivity, and (3) legible formal proofs, in which the original mathematical ideas remain visible.
The project developed methodologies that could be applied by mathematicians today, although a major effort of persuasion remains. For that, members of the team have taken on numerous speaking engagements. We launched a seminar series in the Cambridge mathematics department and have delivered lectures in universities and at high-profile conferences.
The project produced 63 outputs, including 13 journal articles, 15 conference papers (including super selective AI conferences), 2 book chapters, and 32 contributions to Isabelle's Archive of Formal Proofs, with more coming.
Dissemination is also taking place over the Internet: the project webpage (
https://www.cl.cam.ac.uk/~lp15/Grants/Alexandria/(s’ouvre dans une nouvelle fenêtre)) includes a comprehensive list of accomplishments, and the PI's blog (
https://lawrencecpaulson.github.io(s’ouvre dans une nouvelle fenêtre)) discusses aspects of the underlying ideas for a more general mathematical audience.
