A Typed Operational Semantics for Type Theory
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Untyped reduction provides a natural operational semantics for type theory. Normalization results say that such a semantics is sound. However, this reduction does not take type information into account and gives no information about canonical forms for terms. We introduce a new operational semantics, which we call typed operational semantics, which defines a reduction to normal form for terms which are well-typed in the type theory. The central result of the thesis is soundness of the typed operational semantics for the original system. Completeness of the semantics is straightforward. We demonstrate that this equivalence between the declarative and operational presentations of type theory has important metatheoretic consequences: results such as strengthening, subject reduction and strong normalization follow by straightforward induction on derivations in the new system. We introduce these ideas in the setting of the simply typed lambda calculus. We then extend the techniques to Luo's system UTT, which is Martin-Löf's Logical Framework extended by a general mechanism for inductive types, a predicative universe and an impredicative universe of propositions. We also give a proof-irrelevant set-theoretic semantics for UTT.