[Cybiote]

All what you say makes sense, especially as one becomes more comfortable with tactics but I maintain formal proofs are a lot harder to parse than regular math proofs. Having to write for both machine and human distorts a proof's natural flow and reading them often feels like trying to understand partially commented code by executing it in your head.

Well commented proofs are good to communicate intent but if you also want to be able to reproduce a similar proof, it's often useful to understand why tactics were deployed as they were. The existence of the wonderful https://plv.csail.mit.edu/blog/alectryon.html bears out the importance and utility of this.

> For understanding math

My contention, which I believe is shared by the article, is that this understanding is often flimsier than most think, particularly as one leaves well trafficked areas held together by tacit knowledge and knowledge of multiple reinforcing paths shared within communities with relevant expertise. However, even that is insufficient for less used lemmas.

This structuring of knowledge will be useful for mathematics education and research both, especially from the perspective of an individual human trying to safely use results of proofs.

> we really need tools to help us visualize structure of proofs

The output proof is a program, so the problem of visualizing proofs is much the same as visualizing programs and visual programming, sharing the same hurdles. Perhaps friendlier naming and formatting of output code in proof tools will be useful. I believe something more useful would be queryable visualizations of how theorems and lemmas connect and relate to each other.

[fspeech]

The flimsiness of one's understanding tends to manifest as false "lemmas", which can't really be proved because one forgot certain conditions that are typically implicit with human understanding but must be made explicit formally. However if a proof can be completed then it's solid by definition. I would contend when a proof becomes hard to understand instead of documenting tactics one should break up the proof into more lemmas or create more subgoals; preferrably all tactics are "auto". It's nice that the tool you referred to is general and can be used to document any prospect of proof but being based on a markup language it is mostly static. When Gonthier mechanized Feit-Thompson Odd Order Theorem, he had very large graphs showing the relationship of the lemmas. Unfortunately those were also static as I remember and only really comprehensible to the experts. It would be nice to have interactive versions that can aid an average reader on comprehending and exploring the overall relationships. Such a tool is also sorely missing for understanding large code bases. The two are really two sides of the same coin through Curry-Howard. The good thing is a tool that lets users efficiently navigate structures can be done without extra effort of the proof writer (of course documenting intention is always important).