Shallow embedding of type theory into Agda is injective up to definitional equality via a syntactic translation model, with implementation hiding ensuring no illegal propositional equalities arise.
Canonicity and normalisation for Dependent Type Theory
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abstract
We show canonicity and normalization for dependent type theory with a cumulative sequence of universes and a type of Boolean. The argument follows the usual notion of reducibility, going back to Godel's Dialectica interpretation and the work of Tait. A key feature of our approach is the use of a proof relevant notion of reducibility.
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2019 1verdicts
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Shallow Embedding of Type Theory is Morally Correct
Shallow embedding of type theory into Agda is injective up to definitional equality via a syntactic translation model, with implementation hiding ensuring no illegal propositional equalities arise.