A conservativity result for homotopy elementary types in dependent type theory
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We prove a conservativity result for extensional type theories over propositional ones, i.e. dependent type theories with propositional computation rules, using insights from homotopy type theory. The argument exploits a notion of canonical homotopy equivalence between contexts, and uses the notion of a type-category to phrase the semantics of theories of dependent types. Informally, our main result asserts that, for judgements essentially concerning h-sets, reasoning with extensional or propositional type theories is equivalent.
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