Chain Bounding, the leanest proof of Zorn's lemma, and an illustration of computerized proof formalization

We present an exposition of the Chain Bounding Lemma, which is a common generalization of both Zorn’s Lemma and the Bourbaki-Witt fixed point theorem. The proofs of these results through the use of Chain Bounding are amongst the simplest ones that we are aware of. As a by-product, we show that for every poset $P$ and function $f$ from the powerset of $P$ into $P$, there exists a maximal well-ordered chain whose family of initial segments is appropriately closed under $f$. We also provide an introduction to the process of “computer formalization” of mathematical proofs by using proofs assistants. As an illustration, we verify our main results with the Lean proof assistant.