[auggierose]

It is more a conceptual thing. In LCF, proofs and terms are different things, and that is how it should be in my opinion. This Curry-Howard confusion is unnecessary, but many people don't realise that, they think it is the only way to do math on a computer. You can still store proofs in LCF if you want, and use them; just as in Lean, you might be able to optimise them away.

[nrds]

You have done no more to show an actual distinction in the approach than TFA and its linked blog post... It sounds like a naming thing to me. On one side we name the term/program as a term and see it as something checked by the kernel, and on the other you name the term/program as a program and see it as something executed by the runtime. What's the difference?

[zozbot234]

There is indeed no difference if your dependent-typing approach is using reflection (where the checked term is actually a program that's logically proven to result in a a correct proof when executed - such as, commonly, by running a decision procedure) but that's not a common approach.

[auggierose]

The difference is that a term is not (necessarily) a program. Also checking is not executing. Its like saying riding a horse is the same as eating a fish. Really just a naming thing, what's the difference?

[nrds]

You're drawing an equivalence between the wrong pair of things. I'm not saying that term=program; I'm saying that the type checker, qua `term -> context -> decision`, bound to a particular term, is a program `context -> decision`, and the other approach is also a program `context -> decision`. I guess it's defunctionalization, not "nothing", but a next-door neighbor of "nothing".