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I think something along those lines is a reasonable idea; in RedPRL for instance, we considered many different universes with slightly different properties; one of the most useful ones was the "discrete types", which are roughly the types which are both Kan and satisfy equality reflection. There are a lot more types of that kind than you might at first expect!
I'd like to incorporate that, and also strict propositions at some point in the future.
They will be available in Coq and agda, and if I understand things well they are consistent with CTT.
Should we want them in redtt?
coq/ceps#37
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