How to put constraints on type variable of kind `Constraint`?

Question

I\'m playing around with the ConstraintKinds extension of GHC. I have the following data type, which is just a box for things fulfilling some one parameter constrai

Answer 1

The new QuantifiedConstraints extension allows this.

class (a => b) => Implies a b where
instance (a => b) => Implies a b where
instance (forall a. c a `Implies` Show a) => Show (Some c) where
  show (Some x) = show x

Within the body of the Show instance, it is as if there is a

instance forall a. Implies (c a) (Show a)

in scope. If you then have T :: Type and know c T, then the superclass of c T => Show T of the specialized Implies (c T) (Show T) instance allows you to derive Show T. It is necessary to use Implies instead of a straight forall a. c a => Show a constraint. This incorrect constraint acts like

instance forall a. c a => Show a

which overlaps with every Show instance, causing weird breakage. Forcing an indirection through the superclass of an otherwise useless constraint fixes everything.

Answer 2

You cannot make Some c an instance of Show, except trivially.

You want to show the a inside Some, but that variable is existentially quantified, so we cannot depend on any knowledge of the type of a. In particular, we have no way of knowing that a is an instance of Show.

EDIT: I'll expand on my answer. Even with more machinery, and giving up on a Show instance, I still don't think what you want is possible because of the existential quantification.

First I'll rewrite Some in a more familiar form

data Dict p where
    Dict :: p a => a -> Dict p

The usual way to talk about "constraints implying constraints" is with the concept of constraint entailment.

data p :- q where
    Sub :: p a => Dict q -> p :- q

We can think about a value of type p :- q as a proof that if the constraint forall a. p a holds, then forall a. q a follows.

Now we try to write a sensible show-ish function

showD :: p :- Show -> Dict p -> String
showD (Sub (Dict a)) (Dict b) = show b

At a glance, this might work. We have brought the following constraints into scope (forgive the pseudo-exists syntax)

(0) p :: * -> Constraint
(1) exists a. p a           -- (Dict p)
(2) exists b. p b => Show b -- (p :- Show)

But now things fall apart, GHC rightfully complains:

main.hs:10:33:
    Could not deduce (Show a2) arising from a use of `show' 
    from the context (p a)
      bound by a pattern with constructor
                 Sub :: forall (p :: * -> Constraint) (q :: * -> Constraint) a.
                        (p a) => 
                        Dict q -> p :- q,
               in an equation for `showD' 
      at main.hs:10:8-19                    
    or from (Show a1) 
      bound by a pattern with constructor
                 Dict :: forall (p :: * -> Constraint) a. (p a) => a -> Dict p, 
               in an equation for `showD'
      at main.hs:10:13-18 
    or from (p a2)
      bound by a pattern with constructor
                 Dict :: forall (p :: * -> Constraint) a. (p a) => a -> Dict p,
               in an equation for `showD'
      at main.hs:10:23-28   

because it is impossible to unify the a from (1) with the b from (2).

This is the same essential idea that is used throughout the constraints package mentioned in the comments.

Answer 3

The closest we are able to get is a Class1 class that reifys the relationship between a class and a single superclass constraint as a class. It's based on the Class from constraints.

First, we'll take a short tour of the constraints package. A Dict captures the dictionary for a Constraint

data Dict :: Constraint -> * where
  Dict :: a => Dict a

:- captures that one constraint entails another. If we have a :- b, whenever we have the constraint a we can produce the dictionary for the constraint b.

newtype a :- b = Sub (a => Dict b)

We need a proof similar to :-, we need to know that forall a. h a :- b a, or h a => Dict (b a).

Single Inheritance

Actually implementing this for classes with just single inheritance requires the kitchen sink of language extensions, including OverlappingInstances.

{-# LANGUAGE GADTs #-}
{-# LANGUAGE ConstraintKinds #-}
{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE KindSignatures #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE FunctionalDependencies #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE UndecidableInstances #-}
{-# LANGUAGE OverlappingInstances #-}

import Data.Constraint

We'll define the class of constraints of kind k -> Constraint where the constraint has a single superclass.

class Class1 b h | h -> b where
    cls1 :: h a :- b a

We're now equipped to tackle our example problem. We have a class A that requires a Show instance.

 class Show a => A a
 instance A Int

Show is a superclass of A

instance Class1 Show A where
    cls1 = Sub Dict 

We want to write Show instances for Some

data Some (c :: * -> Constraint) where
    Some :: c a => a -> Some c

We can Show a Some Show.

instance Show (Some Show) where
    showsPrec x (Some a) = showsPrec x a

We can Show a Some h whenever h has a single superclass b and we could show Some b.

instance (Show (Some b), Class1 b h) => Show (Some h) where
    showsPrec x (Some (a :: a)) = 
        case cls1 :: h a :- b a of
            Sub Dict -> showsPrec x ((Some a) :: Some b)

This lets us write

x :: Some A
x = Some (1 :: Int)

main = print x