这个问题实际上是由于尝试将几个数学组实现为类型而引起的。
循环组没有问题(Data.Group
的实例在其他地方定义)
newtype Cyclic (n :: Nat) = Cyclic {cIndex :: Integer} deriving (Eq, Ord)
cyclic :: forall n. KnownNat n => Integer -> Cyclic n
cyclic x = Cyclic $ x `mod` toInteger (natVal (Proxy :: Proxy n))
但是对称组在定义某些实例时存在一些问题(通过阶乘数系统实现):
infixr 6 :.
data Symmetric (n :: Nat) where
S1 :: Symmetric 1
(:.) :: (KnownNat n, 2 <= n) => Cyclic n -> Symmetric (n-1) -> Symmetric n
instance {-# OVERLAPPING #-} Enum (Symmetric 1) where
toEnum _ = S1
fromEnum S1 = 0
instance (KnownNat n, 2 <= n) => Enum (Symmetric n) where
toEnum n = let
(q,r) = divMod n (1 + fromEnum (maxBound :: Symmetric (n-1)))
in toEnum q :. toEnum r
fromEnum (x :. y) = fromInteger (cIndex x) * (1 + fromEnum (maxBound `asTypeOf` y)) + fromEnum y
instance {-# OVERLAPPING #-} Bounded (Symmetric 1) where
minBound = S1
maxBound = S1
instance (KnownNat n, 2 <= n) => Bounded (Symmetric n) where
minBound = minBound :. minBound
maxBound = maxBound :. maxBound
ghci的错误消息(仅简短地):
Overlapping instances for Enum (Symmetric (n - 1))
Overlapping instances for Bounded (Symmetric (n - 1))
那么,GHC如何知道n-1
是否等于1?我也想知道是否可以不用FlexibleInstances
来编写解决方案。
答案 0 :(得分:3)
添加Bounded (Symmetric (n-1))
和Enum (Symmetric (n-1))
作为约束,因为要完全解决这些约束将需要知道n的确切值。
instance (KnownNat n, 2 <= n, Bounded (Symmetric (n-1)), Enum (Symmetric (n-1))) =>
Enum (Symmetric n) where
...
instance (KnownNat n, 2 <= n, Bounded (Symmetric (n-1))) =>
Bounded (Symmetric n) where
...
为避免使用FlexibleInstances
(这不值得IMO使用,FlexibleInstances
是良性扩展),请使用Peano数字data Nat = Z | S Nat
代替GHC的原始表示形式。首先用Bounded (Symmetric n)
替换实例头Bounded (Symmetric (S (S n')))
(这起约束2 <= n
的作用),然后用辅助类(可能更多)分解实例以满足标准要求在实例头上。可能看起来像这样:
instance Bounded_Symmetric n => Bounded (Symmetric n) where ...
instance Bounded_Symmetric O where ...
instance Bounded_Symmetric n => Bounded_Symmetric (S n) where ...