Foundational, Compositional (Co)datatypes for Higher-Order Logic

Category Theory Applied to Theorem Proving

Dmitriy Traytel, Andrei Popescu, Jasmin Christian Blanchette


Higher-order logic (HOL) forms the basis of several popular interactive theorem provers. These follow the definitional approach, reducing high-level specifications to logical primitives. This also applies to the support for datatype definitions. However, the internal datatype construction used in HOL4, HOL Light, and Isabelle/HOL is fundamentally noncompositional, limiting its efficiency and flexibility, and it does not cater for codatatypes.

We present a fully modular framework for constructing (co)datatypes in HOL, with support for mixed mutual and nested (co)recursion. Mixed (co)recursion enables type definitions involving both datatypes and codatatypes, such as the type of finitely branching trees of possibly infinite depth. Our framework draws heavily from category theory. The key notion is that of a rich type constructor—a functor satisfying specific properties preserved by interesting categorical operations. Our ideas are formalized in Isabelle and implemented as a new definitional package, addressing long-standing user requests.

Keywords: Category theory, higher-order logic, interactive theorem proving, (co)datatypes, cardinals
Paper draft LICS 2012 slides (Co)datatype package development A HOL theory of cardinals