Yeah, I just meant proposition-as-types. Sloppy terminology by me. I meant to complain about the general conflation of the "proving system" and the "typechecking system".

Generally in dependent type systems you are limited in some way in what sort of values a type can refer to. The fancy type checking happens at compile time, and then at run time some or most of the type information is erased. The implementation is tricky but the more difficult problem IMO is the developer experience; it involves a lot of extra work to express dependent types and help the compiler verify them.