Greg Weber greg at gregweber.info
Sun Apr 1 15:54:32 CEST 2012

```Hi Gershom,

This sounds very interesting even if I have no idea what you are
The first thing you should probably do is explain the programmer's
point of view. That ensures that we are all going through the
requirements phase correctly.
I can assure you that haskell prime would not accept a records change
until it is first implemented in GHC or another Haskell compiler.

Thanks,
Greg Weber

On Sat, Mar 31, 2012 at 11:14 PM, Gershom B <gershomb at gmail.com> wrote:
> The records discussion has been really complicated and confusing. But
> I have a suggestion that should provide a great deal of power to
> records, while being mostly[1] backwards-compatible with Haskell 2010.
> Consider this example:
>
>    data A a = A{a:a, aa::a, aaa :: a -> A (a -> a)}
>    data B a = B{aaa :: a -> A (a -> a), a :: A}
>
> Now what is the type of this?
>
>    aaaa aaaaa a aa = aaaaa{a = a, aaa = aa}
>
> Using standard Haskell typeclasses this is a difficult question to
> answer. The types of aaaa for A and B do not unify in an obvious way.
> However, while they are intensionally quite distinct, they unify
> trivially extensionally. The obvious thing to do is then to extend the
> type system with extensional equality on record functions.
>
> Back when Haskell was invented, extensional equality was thought to be
> hard. But purity was thought to be hard too, and so were Monads. Now,
> we know that function existentionality is easy. In fact, if we add the
> Univalence Axiom to GHC[2], then this is enough to get function
> existensionality. This is a well-known result of Homotopy Type
> Theory[3], which is a well-explored approach that has existed for at
> least a few years and produced more than one paper[4]. Homotopy Type
> Theory is so sound and well understood that it has even been
> formalized in Coq.
>
> Once we extend GHC with homotopies, it turns out that records reduce
> to mere syntactic sugar, and there is a natural proof of their
> soundness (Appendix A). Furthermore, there is a canonical projection
> for any group of fields (Appendix B). Even better, we can make "."
> into the identity path operator, unifying its uses in composition and
> projection. In fact, with extended (parenthesis-free) section rules,
> "." can also be used to terminate expressions, making Haskell friendly
> not only to programmers coming from Java, but also to those coming
> from Prolog!
>
> After some initial feedback, I'm going to create a page for the
> Homotopy Extensional Records Proposal (HERP) on trac. There are really
> only a few remaining questions. 1) Having introduced homotopies, why
> not go all the way and introduce dependent records? In fact, are HERP
> and Dependent Extensional Records Proposal (DERP) already isomorphic?
> My suspicion is that HERP is isomorphic, but DERP is not. However, I
> can only get away with my proof using Scott-free semantics. 2) Which
> trac should I post this too? Given how well understood homotopy type
> theory is, I'm tempted to bypass GHC entirely and propose this for
> haskell-prime. 3) What syntax should we use to represent homotopies?
> See extend discussion in Appendix C.
>
> HTH HAND,
> Gershom
>
> [1] To be precise, 100% of Haskell 2010 programs should, usually, be
> able to be rewritten to work with this proposal with a minimal set of
> changes[1a].
>
> [1a] A minimal set of changes is defined as the smallest set of
> changes necessary to make to a Haskell 2010 program such that it works
> with this proposal. We can arrive at these changes by the following
> procedure: 1) Pick a change[1b]. 2) Is it minimal? If so keep it. 3)
> are we done? If not, make another change.
>
> [1b] To do this constructively, we need an order. I suggest the lo
> mein, since noodles give rise to a free soda.
>
> [2] I haven't looked at the source, but I would suggest putting it in
> the file Axioms.hs.
>
> [3] http://homotopytypetheory.org/
>
> [4] http://arxiv.org/
>
>
> *Appendix A: A Natural Proof of the Soundness of HERP
>
> Take the category of all types in HERP, with functions as morphisms.
> Call it C. Take the category of all sound expressions in HERP, with
> functions as morphisms. Call it D. Define a full functor from C to D.
> Call it F. Define a faithful functor on C and D. Call it G. Draw the
> following diagram.
>
> F(X)----F(Y)
> |          |
> |          |
> |          |
> G(X)----G(Y)
>
> Define the arrows such that everything commutes.
>
>
> *Appendix B: Construction of a Canonical Projection for Any Group of Fields.
>
> 1) Take the fields along the homotopy to an n-ball.
> 2) Pack them loosely with newspaper and gunpowder.
> 3) Project them from a cannon.
>
> In an intuitionistic logic, the following simplification is possible:
>
>
>
> *Appendix C: Homotopy Syntax
>
> Given that we already are using the full unicode set, what syntax
> should we use to distinguish paths and homotopies? At first, I thought
> we could avoid providing any syntax for homotopies at all. Haskell is
> a language with type inference, so we should just be able to infer
> paths and homotopies behind the scenes by adding homotopies to the
> type system. That's a very nice answer in theory. But in the real
> world, when we're writing code that solves actual problems,
> theoretical niceties break down. What if a user wants to use a
> nonstandard homotopy?
>
> Why should we stop them just because we're too lazy to come up with a
> good syntax? I then realized that we keep running out of syntax
> because we've limited ourselves to unicode. Instead, I propose we add
> a potentially infinite universe of identifiers: youtube videos. For
> example, the higher inductive type for a circle can be written as:
>
>    Loop ::: http://www.youtube.com/watch?v=J---aiyznGQ Base Base
>
> Note that the urls do not use SSL. For portability reasons, I propose
> that SSL only be enabled as an optional extension.
>
> _______________________________________________