[alex-g]

Yes, the interface idea is basically the right thing, but there are technical difficulties which aren't immediately obvious. For example, we'd like to be able to prove that two different implementations of the natural numbers are equivalent (one can do this mathematically, so if our system is going to handle general mathematics then it has to be capable of doing this). So we have to think quite hard about what this "equivalence" actually means. It's not enough to say that they satisfy the same axioms, because in general there can be all kinds of models for some set of axioms. It's closer to say that all number-theoretic statements about numbers-v1 are true of numbers-v2, and vice versa: but you can see that this is starting to get a bit hairy in terms of computable proofs.

A related problem, which speaks to the leaky abstraction issue, is "proof irrelevance". Typically, if I've proved something, it shouldn't matter exactly how I did it. But it turns out to be tricky to make sure that the proof objects in the system don't accidentally carry too much information about where they came from. Sure, you can define a way to erase the details, but you still have to prove that erasing doesn't mess up the deductive system.

None of this is insurmountable, but it's a glimpse into the reasons why encoding mathematics computationally is not trivial.