Then separately, you might design a language in Iris whose semantics were those of the abstract machine, and try to prove them observationally equivalent (using some sort of contextual refinement, maybe?).
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Tbh this is going a bit over my head. Why would imperative semantics be useful here? Is it because it might lead to a more efficient implementation?
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Well, they're what you're actually *implementing*, so if you want to tie this proof to what you're doing it's necessary. You don't have to do it if you just want to prove things about your theory, but in fact most bugs in Coq are not foundational ones, but implementation errors.
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A huge advantage of having parts of the kernel (especially stuff like the reduction machine) formalized would be to catch those bugs, since the correctness of those modules is relied upon heavily elsewhere and they are used all over the place.
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To avoid really bad bugs, you don't need to prove termination (thank goodness--since nobody's proved it for the full CiC with all the goodies anyway), just absence of UB. That's why I recommend using something like Iris, it is good at such things, e.g. iris-project.org/pdfs/2017-esop.
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What do I do once I have the Iris thingy? Do I extract an implementation from it? Or do I write it separately and try to ensure it matches closely?
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Ideally you'd extract an implementation, but in practice we aren't really there yet, so you just try to make sure it matches closely. It's not a satisfying answer and I hope to improve that! That said, I believe even something like that would catch almost all implementation bugs.
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Understandable! But it would be a whole lot closer than a ton of other systems!
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For dependent type theory I think it's particularly important because the reduction machine is this crazy call by need thing that gets repeatedly *executed* during typechecking rather than being an idealized model used as the source for an optimizing compiler.
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It needs to preserve typing and conversion and properly handle substitutions under bindings and so on, but it *also* usually has a bunch of hacks like caching, configurable reduction strategies, and complex preconditions to make it go fast, making reasoning about it nontrivial.
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Yeah, I can see how this could quickly get out of hand, with a huge combination of different configurations.
And even with how simple Pikelet is right now, I'm still nowhere near confident if I'm evaluation enough or too little, and whether evaluating binders is safe or not.
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