Terence Tao<p>I recently set myself the exercise of using modern automated tools - in particular, a combination of the <a href="https://mathstodon.xyz/tags/GithubCopilot" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>GithubCopilot</span></a> large language model and the dependent type matching tactic <a href="https://mathstodon.xyz/tags/canonical" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>canonical</span></a> - to try to semi-automatically formalize in <a href="https://mathstodon.xyz/tags/Lean" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>Lean</span></a> a one-page proof provided by a collaborator of the <a href="https://mathstodon.xyz/tags/EquationalTheoriesProject" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>EquationalTheoriesProject</span></a> (Bruno Le Floch). With these tools, I was able to more or less blindly do the formalization in 33 minutes, withou any real high level conception of how the proof proceeded. It was a very different style to how I usually formalize results, but was workable for this type of technical, non-conceptual argument where the main issue is to get the details correct rather than the "big picture".</p><p>I recorded my attempt at <a href="https://www.youtube.com/watch?v=cyyR7j2ChCI" rel="nofollow noopener noreferrer" translate="no" target="_blank"><span class="invisible">https://www.</span><span class="ellipsis">youtube.com/watch?v=cyyR7j2ChC</span><span class="invisible">I</span></a> . See also additional discussion at <a href="https://leanprover.zulipchat.com/#narrow/channel/458659-Equational/topic/Alternative.20proofs.20of.20E1689.E2.8A.A2E2" rel="nofollow noopener noreferrer" translate="no" target="_blank"><span class="invisible">https://</span><span class="ellipsis">leanprover.zulipchat.com/#narr</span><span class="invisible">ow/channel/458659-Equational/topic/Alternative.20proofs.20of.20E1689.E2.8A.A2E2</span></a> . The final proof (which is far from optimized, but got the job done) can be found at <a href="https://github.com/teorth/estimate_tools/blob/master/EstimateTools/test/equational.lean" rel="nofollow noopener noreferrer" translate="no" target="_blank"><span class="invisible">https://</span><span class="ellipsis">github.com/teorth/estimate_too</span><span class="invisible">ls/blob/master/EstimateTools/test/equational.lean</span></a></p>