如何解构SNat(单身人士)

Question

我正在试验Haskell中的从属类型,并在"单身人士"包的文章中遇到以下内容:

replicate2 :: forall n a. SingI n => a -> Vec a n
replicate2 a = case (sing :: Sing n) of
  SZero -> VNil
  SSucc _ -> VCons a (replicate2 a)

所以我试着自己实现这个,只是想知道它是如何工作的:

{-# LANGUAGE DataKinds           #-}
{-# LANGUAGE GADTs               #-}
{-# LANGUAGE KindSignatures      #-}
{-# LANGUAGE TypeOperators       #-}
{-# LANGUAGE RankNTypes          #-}
{-# LANGUAGE ScopedTypeVariables #-}

import           Data.Singletons
import           Data.Singletons.Prelude
import           Data.Singletons.TypeLits

data V :: Nat -> * -> * where
  Nil  :: V 0 a
  (:>) :: a -> V n a -> V (n :+ 1) a

infixr 5 :>

replicateV :: SingI n => a -> V n a
replicateV = replicateV' sing
  where replicateV' :: Sing n -> a -> V n a
        replicateV' sn a = case sn of
            SNat -> undefined -- what can I do with this?

现在问题是Sing实例Nat没有SZero或SSucc.只有一个构造函数被调用SNat.

> :info Sing
data instance Sing n where
  SNat :: KnownNat n => Sing n

这与允许匹配的其他单例不同,例如STrue和SFalse以下(无用)示例:

data Foo :: Bool -> * -> * where
  T :: a -> Foo True a
  F :: a -> Foo False a

foo :: forall a b. SingI b => a -> Foo b a
foo a = case (sing :: Sing b) of
  STrue -> T a
  SFalse -> F a

您可以使用fromSing获取基本类型,但这当然允许GHC检查输出向量的类型:

-- does not typecheck
replicateV2 :: SingI n => a -> V n a
replicateV2 = replicateV' sing
  where replicateV' :: Sing n -> a -> V n a
        replicateV' sn a = case fromSing sn of
              0 -> Nil
              n -> a :> replicateV2 a

所以我的问题:如何实施replicateV？

编辑

erisco给出的答案解释了为什么我的解构方法SNat不起作用.不过,即使有type-natural图书馆,我无法实现replicateV的V数据类型使用GHC的内置的Nat类型.

例如,以下代码编译:

replicateV :: SingI n => a -> V n a
replicateV = replicateV' sing
  where replicateV' :: Sing n -> a -> V n a
        replicateV' sn a = case TN.sToPeano sn of
            TN.SZ       -> undefined
            (TN.SS sn') -> undefined

但是,这似乎并没有提供足够的信息给编译器来推断是否n是0或不是.例如,以下给出了编译器错误:

replicateV :: SingI n => a -> V n a
replicateV = replicateV' sing
  where replicateV' :: Sing n -> a -> V n a
        replicateV' sn a = case TN.sToPeano sn of
            TN.SZ       -> Nil
            (TN.SS sn') -> undefined

这会出现以下错误:

src/Vec.hs:25:28: error:
    • Could not deduce: n1 ~ 0
      from the context: TN.ToPeano n1 ~ 'TN.Z
        bound by a pattern with constructor:
                   TN.SZ :: forall (z0 :: TN.Nat). z0 ~ 'TN.Z => Sing z0,
                 in a case alternative
        at src/Vec.hs:25:13-17
      ‘n1’ is a rigid type variable bound by
        the type signature for:
          replicateV' :: forall (n1 :: Nat) a1. Sing n1 -> a1 -> V n1 a1
        at src/Vec.hs:23:24
      Expected type: V n1 a1
        Actual type: V 0 a1
    • In the expression: Nil
      In a case alternative: TN.SZ -> Nil
      In the expression:
        case TN.sToPeano sn of {
          TN.SZ -> Nil
          (TN.SS sn') -> undefined }
    • Relevant bindings include
        sn :: Sing n1 (bound at src/Vec.hs:24:21)
        replicateV' :: Sing n1 -> a1 -> V n1 a1 (bound at src/Vec.hs:24:9)

所以,我原来的问题仍然存在,我仍然无法做任何有用的事情SNat.

Answer 1

从评论中，我担心我一定错过了一些非常明显的东西，但这是我的看法。整个要点是：

replicate2 :: forall n a. SingI n => a -> Vec a n
replicate2 a = case (sing :: Sing n) of
  SZero -> VNil
  SSucc _ -> VCons a (replicate2 a)

是，为了在VNil :: Vec a 0函数具有通用返回类型时返回Vec a n，您需要专门化nto 0，并且 GADT 上的模式匹配提供了一种方法来执行此操作，只要您有一个构造函数，例如SZero，这意味着n ~ 0。

现在SNatsingleton包中的s没有这样的构造函数。据我所知，获得它的唯一方法是为自然对象构建一个全新的单例类型并实现必要的类型系列。也许你可以用一种包裹 s 的方式来做到这一点Nat，这样你就比皮亚诺结构更接近SZero :: Sing (SN 0),SNonZero :: Sing (SN n)但我不知道。

Vec a n当然，还有另一种方法可以专门化返回 return的函数Vec a 0，即类型类。

如果您愿意放弃一些显式单例机制并切换到类型类（并且还允许重叠和不可判定的实例），那么以下似乎可行。我不得不稍微修改 的定义来V使用n :- 1而不是，但我认为n :+ 1这不会造成问题。

{-# LANGUAGE DataKinds             #-}
{-# LANGUAGE GADTs                 #-}
{-# LANGUAGE KindSignatures        #-}
{-# LANGUAGE TypeOperators         #-}
{-# LANGUAGE RankNTypes            #-}
{-# LANGUAGE ScopedTypeVariables   #-}

{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE FlexibleInstances     #-}
{-# LANGUAGE FlexibleContexts      #-}
{-# LANGUAGE OverlappingInstances  #-}
{-# LANGUAGE UndecidableInstances  #-}

import           Data.Singletons
import           Data.Singletons.Prelude
import           Data.Singletons.TypeLits

data V :: Nat -> * -> * where
  Nil  :: V 0 a
  (:>) :: a -> V (n :- 1) a -> V n a

infixr 5 :>

class VC n a where
  replicateV :: a -> V n a

instance VC 0 a where
  replicateV _ = Nil

instance VC (n :- 1) a => VC n a where
  replicateV x = x :> replicateV x

instance (Show a) => Show (V n a) where
  show Nil = "Nil"
  show (x :> v) = show x ++ " :> " ++ show v

headV :: V (n :+ 1) a -> a
headV (x :> _) = x

tailV :: ((n :+ 1) :- 1) ~ n => V (n :+ 1) a -> V n a
tailV (_ :> v) = v

main = do print (replicateV False   :: V 0 Bool)
          print (replicateV 1       :: V 1 Int)
          print (replicateV "Three" :: V 3 String)