Heterogeneous equality

[Wolfgang Jeltsch-3]

Hi!

The base package contains the module Data.Type.Equality, which contains
the type (:~:) for homogeneous equality. I was a bit surprised that
there is no type for heterogeneous equality there. Is there such a type
somewhere else in the standard library?

I tried to define a type for heterogeneous equality myself as follows:

> {-# LANGUAGE GADTs, PolyKinds, TypeOperators #-}
>
> data a :~~: b where
>
>     HRefl :: a :~~: a

To my surprise, the kind of (:~~:) defined this way is k -> k -> *, not
k -> l -> *. Why is this? Apparently, the latter, more general, kind is
possible, since we can define (:~~:) :: k -> l -> * as follows:

> {-# LANGUAGE GADTs, PolyKinds, TypeOperators #-}
>
> data (a :: k) :~~: (b :: l) where
>
>     HRefl :: a :~~: a

All the best,
Wolfgang
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[Richard Eisenberg-4]

> On Jul 5, 2017, at 5:23 PM, Wolfgang Jeltsch <[hidden email]> wrote:
>
> Hi!
>
> The base package contains the module Data.Type.Equality, which contains
> the type (:~:) for homogeneous equality. I was a bit surprised that
> there is no type for heterogeneous equality there. Is there such a type
> somewhere else in the standard library?

I don't believe it is, but (in my opinion) it should be.

>
> I tried to define a type for heterogeneous equality myself as follows:
>
>> {-# LANGUAGE GADTs, PolyKinds, TypeOperators #-}
>>
>> data a :~~: b where
>>
>>     HRefl :: a :~~: a
>
> To my surprise, the kind of (:~~:) defined this way is k -> k -> *, not
> k -> l -> *. Why is this?

Because the definition of heterogeneous equality requires polymorphic recursion, in that the usage of (:~~:) in the type of HRefl has different kind indices than the declaration head. Polymorphic recursion is allowed only when you have a *complete user-supplied kind signature*, as documented here (https://downloads.haskell.org/~ghc/latest/docs/html/users_guide/glasgow_exts.html#complete-user-supplied-kind-signatures-and-polymorphic-recursion).

This surprised me, too, when I first encountered it, but I believe it's the correct behavior.
Richard
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