A Type Theory for Defining Logics and Proofs

TL;DR

This paper introduces Cocon, a dependent type theory that combines intensional and extensional functions, enabling higher-order syntax, computation, and metaprogramming, with proven normalization and consistency.

Contribution

It develops a Kripke-style model for Cocon, establishing normalization, subject reduction, and consistency, bridging logical frameworks with systems like Agda.

Findings

Proven normalization of Cocon

Established subject reduction and consistency

Supports higher-order syntax and metaprogramming

Abstract

We describe a Martin-L\"of-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.