I continue the discussion of Mitchell's paper Type Inference with Simple Subtypes.  Coming soon: a discussion of semantics of subtyping.

In this episode, I wax rhapsodic for the potential of subtyping to improve the practice of pure functional programming, in particular by allowing functional programmers to drop various irritating function calls that are needed just to make types work out.  Examples are lifting functions with monad transformers, or even just the pure/return functions for applicative functors/monads.

In this episode, I begin discussing a paper titled "Type Inference with Simple Subtypes," by John C. Mitchell.  The paper presents algorithms for computing a type and set of subtype constraints for any term of the pure lambda calculus.  I mostly focus here on how subtype constraints allow typing any term (which seems surprising).

You can join the telegram group for discussion related to the podcast.

In this episode, I discuss a few of the basics for what we expect from a subtyping relation on types: reflexivity, transitivity, and the variances for arrow types.

We begin a discussion of subtyping in functional programming.  In this episode, I talk about how subtyping is a neglected feature in implemented functional programming languages (for example, not found in Haskell), and how it could be very useful for writing lighter, more elegant code.  I also talk about how subtyping could help realize a new vision for practical strong functional programming, where the language has a pure, terminating core language, then a monad for pure but possibly diverging computation, and finally a monad for impurity and divergence.

In this episode, I conclude my discussion of some (but hardly all!) points from Pujet and Tabareau's POPL 2022 paper, "Observational Equality -- Now for Good!".  I talk a bit about the structure of the normalization proof in the paper, which uses induction recursion.  See this paper by Peter Dybjer for more about that feature.  Also, feel free to join the new Telegram group for the podcast if you want to discuss episodes.

I continue discussing the Puject and Tabareau paper, "Observational Equality -- Now for Good", in particular discussing more about how the equality type simplifies based on its index (which is the type of the terms being equated by the equality type), and how proofs of equalities can be used to cast terms from one type to another.

Also, in exciting news, I created a Telegram group that you can join if you want to discuss topics related to the podcast or particularly podcast episodes.  I will be monitoring the group.  I believe you have to request to join, and then I approve (it might take me until later in the day to do that, just fyi).  The invitation link is here.  

In this episode, I introduce an important paper by Pujet and Tabareau, titled "Observational Equality: Now for Good", that develops earlier work of McBride, Swierstra, and Altenkirch (which I will cover in a later episode) on a new approach to making a type theory extensional.  The idea is to have equality types reduce, within the theory, to statements of extensional equality for the type of the values being equated.

I pause the chapter on extensionality in type theory to talk about something very exciting that I just learned about (though the project was completed Summer 2022): the so-called Liquid Tensor Experiment, to formalize a recent very difficult proof by a mathematician named Peter Scholze, in Lean.  This is the first time in history, that I know of, when a theorem was formalized in a theorem prover, in order to resolve doubts of the mathematician who proved it.  An amazing achievement.  This episode tells the story, as I have understood it on line.  The result apparently sparked this recent workshop.

In this episode, I discuss the basic distinguishing rule of Extensional Martin-Loef Type Theory, namely equality reflection.  This rule says that propositional equality implies definitional equality.  Algorithmically, it would imply that the type checker should do arbitrary proof search during type checking, to see if two expressions are definitionally equal.  This immediately gives us undecidability of type checking for the theory, at least as usually realized.