Category Archives: Programming
The HoTT Game
What is the HoTT Game? The Homotopy Type Theory (HoTT) Game is a project written by mathematicians for mathematicians interested in HoTT and no experience in proof verification, with the aim of introducing cubical agda as a tool for trying … Continue reading
Combinatorial Species and Finite Sets in HoTT
(Post by Brent Yorgey) My dissertation was on the topic of combinatorial species, and specifically on the idea of using species as a foundation for thinking about generalized notions of algebraic data types. (Species are sort of dual to containers; … Continue reading
The Lean Theorem Prover
Lean is a new player in the field of proof assistants for Homotopy Type Theory. It is being developed by Leonardo de Moura working at Microsoft Research, and it is still under active development for the foreseeable future. The code … Continue reading
Modules for Modalities
As defined in chapter 7 of the book, a modality is an operation on types that behaves somewhat like the n-truncation. Specifically, it consists of a collection of types, called the modal ones, together with a way to turn any … Continue reading
A Formalized Interpreter
I’d like to announce a result that might be interesting: an interpreter for a very small dependent type system in Coq, assuming uniqueness of identity proofs (UIP). Because it assumes UIP, it’s not immediately compatible with HoTT, but it seems … Continue reading
Abstract Types with Isomorphic Types
Here’s a cute little example of programming in HoTT that I worked out for a recent talk. Abstract Types One of the main ideas used to structure large programs is abstract types. You give a specification for a component, and … Continue reading
Canonicity for 2-Dimensional Type Theory
A consequence of the univalence axiom is that isomorphic types are equivalent (propositionally equal), and therefore interchangable in any context (by the identity eliminaiton rule J). Type isomorphisms arise frequently in dependently typed programming, where types are often refined to … Continue reading
Homotopy Type Theory 