[flostk]

I'm sure this also depends a lot on the field within mathematics. Areas like mathematical logic or algebra have a pretty formal style anyway, so it's comparatively less effort to translate proofs to lean. I would expect more people from these areas to use proof checkers than say from low-dimensional topology, which has a more "intuitive" style. Of course by "intuitive" I don't mean it's any less rigorous. But a picture proof of some homotopy equivalence is just much harder to translate to something lean can understand than some sequence of inequalities. To add a data point, I'm in geometry/topology and I've never seen or heard of anyone who uses this stuff (so far).

[ocfnash]

If you are curious, I encourage you to look up The Sphere Eversion Project [1].

It was a project in which we formalised a version of Gromov's (open, ample) h-principle for first order differential relations. Part of the reason it was carried out was to demonstrate what is involved formalising something in differential topology.

1. https://leanprover-community.github.io/sphere-eversion/

[enugu]

Pictures can be directly used in formal proofs. See [1] for instance So, the problem is not in principle, but rather, pictures and more direct syntax has not yet been integrated into current proof systems. When that is done, proofs will be much more natural and the formal proof will be closer to the 'proof in the mind'.

[1] https://www.unco.edu/nhs/mathematical-sciences/faculty/mille...