使用实例约束中的量化类型等式约束

Question

为了设置场景，这里有一堆我们将使用的语言扩展，以及来自 CLaSH 的一些简化定义：

{-# LANGUAGE GADTs, StandaloneDeriving #-}
{-# LANGUAGE TypeOperators, DataKinds, PolyKinds #-}
{-# LANGUAGE TypeFamilyDependencies, FlexibleContexts, FlexibleInstances #-}
{-# LANGUAGE PatternSynonyms #-}
{-# LANGUAGE QuantifiedConstraints #-}

data Signal dom a
instance Functor (Signal dom) where
instance Applicative (Signal dom) where

class Bundle a where
    type Unbundled dom a = res | res -> dom a

    bundle :: Unbundled dom a -> Signal dom a
    unbundle :: Signal dom a -> Unbundled dom a

我想为 n 元产品类型创建 Bundle 实例。类型本身定义如下：

import Control.Monad.Identity

data ProductF (f :: * -> *) (ts :: [*]) where
    NilP :: ProductF f '[]
    ConsP :: f a -> ProductF f ts -> ProductF f (a : ts)
deriving instance (Show (f t), Show (ProductF f ts)) => Show (ProductF f (t : ts))

headPF :: ProductF f (t : ts) -> f t
headPF (ConsP x xs) = x

tailP :: ProductF f (t : ts) -> ProductF f ts
tailP (ConsP x xs) = xs

-- Utilities for the simple case    
type Product = ProductF Identity

infixr 5 ::>    
pattern (::>) :: t -> Product ts -> Product (t : ts)
pattern x ::> xs = ConsP (Identity x) xs

headP :: Product (t : ts) -> t
headP (x ::> xs) = x

我想写的是一个Bundle 实例，它简单地将Identity 替换为Signal dom。不幸的是，我们不能一口气做到这一点：

instance Bundle (Product ts) where
    type Unbundled dom (Product ts) = ProductF (Signal dom) ts

    bundle NilP = pure NilP
    bundle (ConsP x xs) = (::>) <$> x <*> bundle xs

    unbundle = _ -- Can't implement this, since it would require splitting on ts

在这里，unbundle 需要为ts ~ [] 和ts ~ t : ts' 做一些不同的事情。好的，让我们尝试在两种情况下编写它：

instance Bundle (Product '[]) where
    type Unbundled dom (Product '[]) = ProductF (Signal dom) '[]

    bundle NilP = pure NilP
    unbundle _ = NilP

instance (Bundle (Product ts), forall dom. Unbundled dom (Product ts) ~ ProductF (Signal dom) ts) => Bundle (Product (t : ts)) where
    type Unbundled dom (Product (t : ts)) = ProductF (Signal dom) (t : ts)

    bundle (ConsP x xs) = (::>) <$> x <*> bundle xs
    unbundle xs = ConsP (headP <$> xs) (unbundle $ tailP <$> xs)

所以问题就出现在第二个例子中。即使我们在实例约束中有一个（量化的）类型相等 forall dom. Unbundled dom (Product ts) ~ ProductF (Signal dom) ts，GHC 8.6.3 在类型检查期间也不会使用它：

对于bundle：

• Couldn't match type ‘Unbundled dom (Product ts)’
                 with ‘ProductF (Signal dom) ts’
  Expected type: Unbundled dom (Product ts)
    Actual type: ProductF (Signal dom) ts1
• In the first argument of ‘bundle’, namely ‘xs’
  In the second argument of ‘(<*>)’, namely ‘bundle xs’
  In the expression: (::>) <$> x <*> bundle xs

对于unbundle：

• Couldn't match expected type ‘ProductF (Signal dom) ts’
              with actual type ‘Unbundled dom (ProductF Identity ts)’
• In the second argument of ‘ConsP’, namely
    ‘(unbundle $ tailP <$> xs)’
  In the expression: ConsP (headP <$> xs) (unbundle $ tailP <$> xs)
  In an equation for ‘unbundle’:
      unbundle xs = ConsP (headP <$> xs) (unbundle $ tailP <$> xs)

一种可能的解决方法

当然，我们可以只走很长的路：专门为Product 创建我们自己的课程，并将所有实际工作委托给它。我在这里介绍该解决方案，但我对比这更简洁和临时的东西特别感兴趣。

class IsProduct (ts :: [*]) where
    type UnbundledProd dom ts = (ts' :: [*]) | ts' -> dom ts

    bundleProd :: Product (UnbundledProd dom ts) -> Signal dom (Product ts)
    unbundleProd :: Signal dom (Product ts) -> Product (UnbundledProd dom ts)

instance (IsProduct ts) => Bundle (Product ts) where
    type Unbundled dom (Product ts) = Product (UnbundledProd dom ts)

    bundle = bundleProd
    unbundle = unbundleProd

然后IsProduct的好处是可以实际实现：

type (:::) (name :: k) (a :: k1) = (a :: k1)

instance IsProduct '[] where
    type UnbundledProd dom '[] = dom ::: '[]

    bundleProd NilP = pure NilP
    unbundleProd _ = NilP

instance (IsProduct ts) => IsProduct (t : ts) where
    type UnbundledProd dom (t : ts) = Signal dom t : UnbundledProd dom ts

    bundleProd (x ::> xs) = (::>) <$> x <*> bundleProd xs
    unbundleProd xs = (headP <$> xs) ::> (unbundleProd $ tailP <$> xs)

Answer 1

嗯，原则上的解决方案是单例：

-- | Reifies the length of a list
data SLength :: [a] -> Type where
   SLenNil :: SLength '[]
   SLenCons :: SLength xs -> SLength (x : xs)

-- | Implicitly provides @kLength@: the length of the list @xs@
class KLength xs where kLength :: SLength xs
instance KLength '[] where kLength = SLenNil
instance KLength xs => KLength (x : xs) where kLength = SLenCons kLength

单例背后的核心思想（至少是其中之一）是隐式单例类KLength 可以排除对像您这样的临时类的需求。 “classiness”进入KLength，可以重复使用； “caseiness”进入文字case，SLength 是将它们粘合在一起的数据类型。

instance KLength ts => Bundle (Product ts) where
    type Unbundled dom (Product ts) = ProductF (Signal dom) ts

    bundle = impl
        -- the KLength xs constraint is unnecessary for bundle
        -- but the recursive call would still need it, and we wouldn't have it
        -- there's a rather unholy unsafeCoerce trick you can pull
        -- but it's not necessary yet
        where impl :: forall dom us. ProductF (Signal dom) us -> Signal dom (Product us)
              impl NilP = pure NilP
              impl (ConsP x xs) = (::>) <$> x <*> impl xs

    unbundle = impl kLength
        -- impl explicitly manages the length of the list
        -- unbundle just fetches the length of ts from the givens and passes it on
        where impl :: forall dom us. SLength us -> Signal dom (Product us) -> ProductF (Signal dom) us
              impl SLenNil _ = NilP
              impl (SLenCons n) xs = ConsP (headP <$> xs) (impl n $ tailP <$> xs)

Answer 2

等式表示在'[] 和t ': ts 两种情况下，Unbundled 系列被定义为ProductF。一种更简单的方法是在生成 ProductF 之前不对列表进行模式匹配。这涉及拆分类的Unbundled 家族：

type family Unbundled dom a = res | res -> dom a
class Bundle a where
    bundle :: Unbundled dom a -> Signal dom a
    unbundle :: Signal dom a -> Unbundled dom a

所以你可以为两个类实例使用一个类型实例：

type instance Unbundled dom (Product ts) = ProductF (Signal dom) ts
instance Bundle (Product '[]) where
    bundle NilP = pure NilP
    unbundle _ = NilP

instance (Bundle (Product ts), forall dom. Unbundled dom (Product ts) ~ ProductF (Signal dom) ts) => Bundle (Product (t : ts)) where
    bundle (ConsP x xs) = (::>) <$> x <*> bundle xs
    unbundle xs = ConsP (headP <$> xs) (unbundle $ tailP <$> xs)