You can actually do much better than mere embeddings. Using program translations similar to what Haskell users routinely perform when using the do-notation for monads, you can actually extend your type theory with new reasoning principles. See for instance the parametric exceptional theory [1], which adds well-behaved exceptions and allows to extend Coq with the independence of premises, i.e. (¬A → ∃n:ℕ. P n) → ∃n:ℕ. (¬A → P n).

[1] https://hal.inria.fr/hal-01840643/document

Define "the authors". Gérard Huet, who wrote the software almost 35 years ago was definitely aware of the meaning, and indeed this was done in order to overtly piss off the prudish Amercians.

In the current core development team, I think it is reasonable to say that most people range somewhere on a scale from "don't care" to "mildly annoyed". Amongst the annoyed group, some actually advocate for a change of name.

For the record, French speakers also have at first a hard time with the "bit" word that is usually pronounced at the very beginning of any introductory CS class. Indeed, it has the same figurative meaning as "cock" in English. Worse, you can find it in compounds such as "32-bit" or "bitfield", so try figure out the evocative power of those expressions.

Usually and fortunately though, the average person simply grows up and stops giggling at the word quite quickly. And so do Coq users.

This particular feat was triggered by Kevin Buzzard's unsubstantiated claim that "Lean was better than Coq" as a foundation for mathematics, because it had built-in quotient types. The work described above precisely shows that such types can be encoded as well using features shipped with the vanilla Coq distro. See https://sympa.inria.fr/sympa/arc/coq-club/2020-01/msg00006.h....

My understanding is that this shows how vanilla Coq still can’t encode quotients well. This project relies on a PR [0] that breaks term normalization, which makes it highly unlikely it will be merged. My understanding is that Lean’s implementation of quotients also breaks normalization, but they don’t care as much about that as Coq devs do.

[0]: https://github.com/coq/coq/pull/10390#issuecomment-554316311

The normalization issue is not about quotients at all. It's https://arxiv.org/abs/1911.08174 "Failure of Normalization in Impredicative Type Theory with Proof-Irrelevant Propositional Equality"

Well, I will defer to your judgment! I’m by no means a type theory person.

My understanding was that these features are how quotients are managed to be implemented. But perhaps that is wrong.