> no general purpose programming language lets you put an arbitrary expression in the same language in a type as a permanent condition, or lets you state facts that the compiler will prove
Lean can be used as a regular programming language. There's also languages like idris2 and f-star, but they don't seem to have much traction.
Well, they of course not defy computer science. The "trick" is that they are not Turing-complete, they mandate termination of every expression.
Also, most of them are made to prove stuff first and foremost and thus trade off a lot of performance to the point that it makes them practically unusable for many stuff (e.g. numbers may be represented as an object that has n-1 further children recursively), though Lean is an exception as you note.
> The "trick" is that they are not Turing-complete, they mandate termination of every expression.
I don't think this is a big deal for day-to-day programming. You're trying to stay in n, nlogn or maybe n^2 realm most of the time. And the kind of infinite loops you encounter (e.g. event loops) are co-inductive or have some notion of making progress.