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{-# LANGUAGE AllowAmbiguousTypes #-}
{-# LANGUAGE DataKinds #-}
{-# LANGUAGE DeriveTraversable #-}
{-# LANGUAGE GADTs #-}
{-# LANGUAGE PolyKinds #-}
{-# LANGUAGE QuantifiedConstraints #-}
{-# LANGUAGE RankNTypes #-}
{-# LANGUAGE StandaloneDeriving #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE TypeOperators #-}
module Data (module Data, (:~:)(Refl)) where
import Data.Type.Equality
import Unsafe.Coerce (unsafeCoerce)
import Lemmas (Append)
data Dict c where
Dict :: c => Dict c
data SList f l where
SNil :: SList f '[]
SCons :: f a -> SList f l -> SList f (a : l)
deriving instance (forall a. Show (f a)) => Show (SList f l)
infixr `SCons`
slistMap :: (forall t. f t -> g t) -> SList f list -> SList g list
slistMap _ SNil = SNil
slistMap f (SCons x list) = SCons (f x) (slistMap f list)
unSList :: (forall t. f t -> a) -> SList f list -> [a]
unSList _ SNil = []
unSList f (x `SCons` l) = f x : unSList f l
sappend :: SList f l1 -> SList f l2 -> SList f (Append l1 l2)
sappend SNil l = l
sappend (SCons x xs) l = SCons x (sappend xs l)
type family Replicate n x where
Replicate Z x = '[]
Replicate (S n) x = x : Replicate n x
sreplicate :: SNat n -> f t -> SList f (Replicate n t)
sreplicate SZ _ = SNil
sreplicate (SS n) x = x `SCons` sreplicate n x
data Nat = Z | S Nat
deriving (Show, Eq, Ord)
type N0 = Z
type N1 = S N0
type N2 = S N1
type N3 = S N2
data SNat n where
SZ :: SNat Z
SS :: SNat n -> SNat (S n)
deriving instance Show (SNat n)
instance TestEquality SNat where
testEquality SZ SZ = Just Refl
testEquality (SS n) (SS n') | Just Refl <- testEquality n n' = Just Refl
testEquality _ _ = Nothing
fromSNat :: SNat n -> Int
fromSNat SZ = 0
fromSNat (SS n) = succ (fromSNat n)
class KnownNat n where knownNat :: SNat n
instance KnownNat Z where knownNat = SZ
instance KnownNat n => KnownNat (S n) where knownNat = SS knownNat
snatKnown :: SNat n -> Dict (KnownNat n)
snatKnown SZ = Dict
snatKnown (SS n) | Dict <- snatKnown n = Dict
type family n + m where
Z + m = m
S n + m = S (n + m)
snatAdd :: SNat n -> SNat m -> SNat (n + m)
snatAdd SZ m = m
snatAdd (SS n) m = SS (snatAdd n m)
lemPlusSuccRight :: n + S m :~: S (n + m)
lemPlusSuccRight = unsafeCoerceRefl
lemPlusZero :: n + Z :~: n
lemPlusZero = unsafeCoerceRefl
data Vec n t where
VNil :: Vec Z t
(:<) :: t -> Vec n t -> Vec (S n) t
deriving instance Show t => Show (Vec n t)
deriving instance Functor (Vec n)
deriving instance Foldable (Vec n)
deriving instance Traversable (Vec n)
vecLength :: Vec n t -> SNat n
vecLength VNil = SZ
vecLength (_ :< v) = SS (vecLength v)
vecGenerate :: SNat n -> (forall i. SNat i -> t) -> Vec n t
vecGenerate = \n f -> go n f SZ
where
go :: SNat n -> (forall i. SNat i -> t) -> SNat i' -> Vec n t
go SZ _ _ = VNil
go (SS n) f i = f i :< go n f (SS i)
unsafeCoerceRefl :: a :~: b
unsafeCoerceRefl = unsafeCoerce Refl
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