Heterogeneous equality into base?

[Edward Kmett]

Sent from my iPad

> On Jul 10, 2017, at 1:11 PM, David Feuer <david.feuer at gmail.com> wrote:
> 
> Well, I came up with another notion of heterogeneous equality (much,
> much more limited in practice):
> 
> newtype HEqual (a :: j) (b :: k) = HEqual (forall (f :: forall l. l ->
> Type). f a -> f b)

This is "heterogeneous Leibnizian equality".

> Implementing
> 
> fromHEqual :: forall j k (a :: j) (b :: k). HEqual a b -> a :~~: b
> seems to require something like OtherEquality.
> 
> I'm not sure what you mean about not being able to reorder type
> indices. OtherEquality can also be defined as a newtype around (:~~:):
> 
> newtype OtherEquality :: forall j. j -> forall k. k -> Type where
>  OtherEquality :: a :~~: q -> OtherEquality a q
> 
> But perhaps you mean it's more convenient than :~~:?
> 
>> On Mon, Jul 10, 2017 at 1:03 PM, Richard Eisenberg <rae at cs.brynmawr.edu> wrote:
>> This OtherEquality is actually a tad bit more general, given that GHC can't (currently) reorder type indices. But would anyone take advantage of the generality? I tend to doubt it...
>> 
>> Richard
>> 
>>> On Jul 10, 2017, at 12:20 PM, David Feuer <david.feuer at gmail.com> wrote:
>>> 
>>> There's another version of heterogeneous equality that takes its
>>> arguments in the other order. I'm not sure if this is generally useful
>>> enough to want to include.
>>> 
>>> -- (:~~:) :: forall j k. j -> k -> *
>>> 
>>> data OtherEquality :: forall j. j -> forall k. k -> Type where
>>> OtherRefl :: OtherEquality a a
>>> 
>>>> On Mon, Jul 10, 2017 at 11:24 AM, Ryan Scott <ryan.gl.scott at gmail.com> wrote:
>>>> I also agree that we should keep HRefl a distinct name, and moreover, we
>>>> should keep :~: and :~~: as distinct datatypes.
>>>> 
>>>> I'm also on-board with the idea that we should introduce a separate
>>>> Data.Type.Equality.Hetero module that reexports :~~: and defines
>>>> heterogeneous counterparts for sym, trans, etc. from Data.Type.Equality. I
>>>> don't have a strong opinion on how they should be named (e.g., sym vs.
>>>> hsym).
>>>> 
>>>> Ryan S.
>>>> 
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