[Tainnor]

I'm familiar with norm_cast and push_cast, but they don't always do what I expect them to do or solve all the problems. Then there's also the issue (in my experience at least) that in order to e.g. define the real exp or cos function you need to compose the complex function with the real projection and then manually use theorems such as "for all reals r, exp(r_inj_c(r)) = r_inj_c(re_exp(r))", which are easy enough to prove but it's still more work than in "regular" mathematics where you just say "re_exp = exp restricted to the reals" (and then implicitly use the fact that exp maps reals to reals).

I can usually find a way around such things but it does make proofs more tedious. As said, I'm sure I'm not using Lean optimally, but I wouldn't know what to ask for specifically, it's a case of "unknown unknowns".

> One way you can focus a question like this if you don't know what techniques to use but just have a feeling that formalization is too hard, is to isolate an example where you really had to work hard to get a proof and your proof is unsatisfying to you, and challenge people to golf it down.

Fair, that's something I could try. For example, my proof that cos 2 ≤ -1/3 (which is used for showing that cos has a zero in (0,2) with which I can define pi) is unreasonably complicated, while the proof on paper is just a couple of lines. There are a bunch of techniques used when estimating series, such as (re)grouping terms, that I found difficult to do rigorously.