🤖 AI Summary
This paper addresses the conservativity of extensional dependent type theory over propositional dependent type theory with respect to the h-set fragment. We introduce **canonical homotopy equivalence**—a novel semantic tool—and employ categories with attributes (CwAs) to model the computational rules of both theories at the level of contexts. We establish, via a rigorous proof, that all judgmental consequences of extensional type theory restricted to h-sets are fully captured by propositional type theory. This constitutes the first application of homotopy equivalence in proving conservativity for dependent type theories, unifying the semantic treatment of extensional and propositional computation rules and revealing their essential equivalence on h-sets. The result provides a new paradigm for type theory design and formal verification, and bridges homotopical semantics with logical reasoning.
📝 Abstract
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 category with attributes 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.