A Type Theory for Defining Logics and Proofs

A Type Theory for Defining Logics and Proofs

We describe a Martin-Lof-style dependent type theory, called COCON, that allows us to mix the intensional function space that is used to represent higher-order abstract syntax (HOAS) trees with the extensional function space that describes (recursive) computations. We mediate between HOAS representations and computations using contextual modal types. Our type theory also supports an infinite hierarchy of universes and hence supports type-level computation thereby providing metaprogramming and (small-scale) reflection. Our main contribution is the development of a Kripke-style model for COCON that allows us to prove normalization. From the normalization proof, we derive subject reduction and consistency. Our work lays the foundation to incorporate the methodology of logical frameworks into systems such as Agda and bridges the longstanding gap between these two worlds.

我们描述了一种称为Cocon的Martin-LOF风格的依赖性类型理论，它使我们能够混合用于代表高阶抽象语法（HOAS）树的强化函数空间与描述（递归）计算的扩展函数空间。我们使用上下文模态类型在HOA表示和计算之间进行调解。我们的类型理论还支持宇宙的无限层次结构，因此支持类型级计算，从而提供元编程和（小规模）反射。我们的主要贡献是开发用于Cocon的Kripke风格模型，该模型使我们能够证明标准化。根据归一化的证明，我们得出了受试者的降低和一致性。我们的工作奠定了将逻辑框架的方法纳入AGDA等系统的基础，并弥合了这两个世界之间的长期差距。