Native Type Theory
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We present a method to construct "native" type systems for a broad class of languages, in which types are built from term constructors by predicate logic and dependent types. Any language with products and functions can be modelled as a higher-order algebraic theory with rewrites, and the internal language of its presheaf topos provides total specification of the structure and behavior of terms. This construction is functorial, so translations of languages give translations of type systems. The construction therefore provides a shared framework of higher-order reasoning for most existing programming languages.
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