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Coq is a formal proof management system, also called an interactive theorem prover. It is used to express mathematical assertions, mechanically check proofs of these assertions, find formal proofs, and extract certified programs.

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Are we sure the calculus of inductive constructions and ZFC plus countably many inaccessible...

This answer says, IIRC, the calculus of inductive constructions is equi-interpretable with ZFC plus countably many inaccessibles — see Benjamin Werner's "Sets in types, types in sets". (This is becau …
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