使用Haskell的类型系统指定一个类服从额外的属性（即类型类的类型类）

Question

当我们创建一个类型类时，通常会假设它的函数必须服从某些属性。因此，我们得到了它们各自类型类的么半群和么单律。但是，如果有某种规律，比如结合性，我想要指定多个类可以服从，也可以不服从这个规律，那该怎么办？在Haskell的类型系统中有办法做到这一点吗？这种类型类对类型类的想法在实践中是否可行？

下面是代数中一个很有启发性的例子：

class Addition x where
    add :: x -> x -> x

class Multiplication x where
    mult :: x -> x -> x

instance Addition Int where
    add = (+)

instance Multiplication Int where
    add = (*)

class (Addition x) => AssociativeAddition x where
class (Addition x) => CommutativeAddition x where

instance AssociativeAddition Int where
instance CommutativeAddition Int where

class Associative x where

instance (Associative Addition, Commutative Addition) => Addition Int where
    add = (+)

instance (Commutative Multiplication) => Multiplication Int where
    mult = (*)

Answer 1

您实际上可以通过最近的一些GHC扩展来实现这一点：

{-# LANGUAGE ConstraintKinds, KindSignatures, MultiParamTypeClasses #-}
import GHC.Exts (Constraint)

class Addition (a :: *) where
    plus :: a -> a -> a

instance Addition Integer where
    plus = (+)

class (c a) => Commutative (a :: *) (c :: * -> Constraint) where
    op :: a -> a -> a

instance Commutative Integer Addition where
    op = plus