如何对约束类型变量进行约束？约束？

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 constraint c :

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

For example, I could construct a box with some kind of number (arguably not very useful).

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

Now, as long as c includes the constraint Show , I could provide an instance of Show (Some c) .

instance ??? => Show (Some c) where
    show (Some x) = show x    -- Show dictionary for type of x should be in scope here

But how do I express this requirement in the instance context (marked with ??? )?

I cannot use an equality constraint ( c ~ Show ), because the two are not necessarily equal. c could be Num , which implies, but is not equal to, Show .

Edit

I realised that this cannot be possible in general.

If you have two values of type Some Eq , it is not possible to compare them for equality. They could be of different types that each have their own notion of equality.

What applies to Eq applies to any type class in which the type parameter appears on the right hand side of the first function arrow (like the second a in (==) :: a -> a -> Bool ).

Considering that there is no way to create a constraint expressing "this type variable is not used beyond the first arrow", I don't think it is possible to write the instance I want to write.

Answer 1

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. ha :- ba forall a. ha :- ba , or ha => Dict (ba) .

Single Inheritance

Actually implementing this for class es 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

Answer 2

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. ca => Show a forall a. ca => 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 3

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. pa forall a. pa holds, then forall a. qa forall a. qa 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.