{-# LANGUAGE DataKinds #-}
{-# LANGUAGE DerivingVia #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE GADTs #-}
{-# LANGUAGE InstanceSigs #-}
{-# LANGUAGE PolyKinds #-}
{-# LANGUAGE QuantifiedConstraints #-}
{-# LANGUAGE RankNTypes #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE StandaloneDeriving #-}
{-# LANGUAGE StandaloneKindSignatures #-}
{-# LANGUAGE TypeApplications #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE UndecidableInstances #-}

{-|
TODO:
* This module needs better structure with an Internal module and less public
  exports etc.

* We should be more consistent in whether functions take a 'StaticShapeX'
  argument or a 'KnownShapeX' constraint.

-}

module Data.Array.Nested.Internal where

import Control.Monad (forM_)
import Control.Monad.ST
import Data.Coerce (coerce, Coercible)
import Data.Kind
import Data.Proxy
import Data.Type.Equality
import qualified Data.Vector.Unboxed as VU
import qualified Data.Vector.Unboxed.Mutable as VUM

import Data.Array.Mixed (XArray, IxX(..), KnownShapeX(..), StaticShapeX(..), type (++))
import qualified Data.Array.Mixed as X
import Data.Nat


type family Replicate n a where
  Replicate Z a = '[]
  Replicate (S n) a = a : Replicate n a

type family MapJust l where
  MapJust '[] = '[]
  MapJust (x : xs) = Just x : MapJust xs

lemKnownReplicate :: forall n. KnownNat n => Proxy n -> Dict KnownShapeX (Replicate n Nothing)
lemKnownReplicate _ = X.lemKnownShapeX (go (knownNat @n))
  where
    go :: SNat m -> StaticShapeX (Replicate m Nothing)
    go SZ = SZX
    go (SS n) = () :$? go n

lemRankReplicate :: forall n. KnownNat n => Proxy n -> X.Rank (Replicate n (Nothing @Nat)) :~: n
lemRankReplicate _ = go (knownNat @n)
  where
    go :: SNat m -> X.Rank (Replicate m (Nothing @Nat)) :~: m
    go SZ = Refl
    go (SS n) | Refl <- go n = Refl

lemReplicatePlusApp :: forall n m a. KnownNat n => Proxy n -> Proxy m -> Proxy a
                    -> Replicate (n + m) a :~: Replicate n a ++ Replicate m a
lemReplicatePlusApp _ _ _ = go (knownNat @n)
  where
    go :: SNat n' -> Replicate (n' + m) a :~: Replicate n' a ++ Replicate m a
    go SZ = Refl
    go (SS n) | Refl <- go n = Refl


-- | Wrapper type used as a tag to attach instances on. The instances on arrays
-- of @'Primitive' a@ are more polymorphic than the direct instances for arrays
-- of scalars; this means that if @orthotope@ supports an element type @T@ that
-- this library does not (directly), it may just work if you use an array of
-- @'Primitive' T@ instead.
newtype Primitive a = Primitive a


-- | Mixed arrays: some dimensions are size-typed, some are not. Distributes
-- over product-typed elements using a data family so that the full array is
-- always in struct-of-arrays format.
--
-- Built on top of 'XArray' which is built on top of @orthotope@, meaning that
-- dimension permutations (e.g. 'transpose') are typically free.
--
-- Many of the methods for working on 'Mixed' arrays come from the 'Elt' type
-- class.
type Mixed :: [Maybe Nat] -> Type -> Type
data family Mixed sh a

newtype instance Mixed sh (Primitive a) = M_Primitive (XArray sh a)
  deriving (Show)

newtype instance Mixed sh Int = M_Int (XArray sh Int)
  deriving (Show)
newtype instance Mixed sh Double = M_Double (XArray sh Double)
  deriving (Show)
newtype instance Mixed sh () = M_Nil (XArray sh ())  -- no content, orthotope optimises this (via Vector)
  deriving (Show)
-- etc.

data instance Mixed sh (a, b) = M_Tup2 (Mixed sh a) (Mixed sh b)
deriving instance (Show (Mixed sh a), Show (Mixed sh b)) => Show (Mixed sh (a, b))
-- etc.

newtype instance Mixed sh1 (Mixed sh2 a) = M_Nest (Mixed (sh1 ++ sh2) a)
deriving instance Show (Mixed (sh1 ++ sh2) a) => Show (Mixed sh1 (Mixed sh2 a))


-- | Internal helper data family mirrorring 'Mixed' that consists of mutable
-- vectors instead of 'XArray's.
type MixedVecs :: Type -> [Maybe Nat] -> Type -> Type
data family MixedVecs s sh a

newtype instance MixedVecs s sh (Primitive a) = MV_Primitive (VU.MVector s a)

newtype instance MixedVecs s sh Int = MV_Int (VU.MVector s Int)
newtype instance MixedVecs s sh Double = MV_Double (VU.MVector s Double)
newtype instance MixedVecs s sh () = MV_Nil (VU.MVector s ())  -- no content, MVector optimises this
-- etc.

data instance MixedVecs s sh (a, b) = MV_Tup2 (MixedVecs s sh a) (MixedVecs s sh b)
-- etc.

data instance MixedVecs s sh1 (Mixed sh2 a) = MV_Nest (IxX sh2) (MixedVecs s (sh1 ++ sh2) a)


-- | Allowable scalar types in a mixed array, and by extension in a 'Ranked' or
-- 'Shaped' array. Note the polymorphic instance for 'Elt' of @'Primitive'
-- a@; see the documentation for 'Primitive' for more details.
class Elt a where
  -- ====== PUBLIC METHODS ====== --

  mshape :: KnownShapeX sh => Mixed sh a -> IxX sh
  mindex :: Mixed sh a -> IxX sh -> a
  mindexPartial :: forall sh sh'. Mixed (sh ++ sh') a -> IxX sh -> Mixed sh' a

  mlift :: forall sh1 sh2. KnownShapeX sh2
        => (forall sh' b. (KnownShapeX sh', VU.Unbox b) => Proxy sh' -> XArray (sh1 ++ sh') b -> XArray (sh2 ++ sh') b)
        -> Mixed sh1 a -> Mixed sh2 a

  -- ====== PRIVATE METHODS ====== --
  -- Remember I said that this module needed better management of exports?

  -- | Create an empty array. The given shape must have size zero; this may or may not be checked.
  memptyArray :: IxX sh -> Mixed sh a

  -- | Return the size of the individual (SoA) arrays in this value. If @a@
  -- does not contain tuples, this coincides with the total number of scalars
  -- in the given value; if @a@ contains tuples, then it is some multiple of
  -- this number of scalars.
  mvecsNumElts :: a -> Int

  -- | Create uninitialised vectors for this array type, given the shape of
  -- this vector and an example for the contents. The shape must not have size
  -- zero; an error may be thrown otherwise.
  mvecsUnsafeNew :: IxX sh -> a -> ST s (MixedVecs s sh a)

  -- | Given the shape of this array, an index and a value, write the value at
  -- that index in the vectors.
  mvecsWrite :: IxX sh -> IxX sh -> a -> MixedVecs s sh a -> ST s ()

  -- | Given the shape of this array, an index and a value, write the value at
  -- that index in the vectors.
  mvecsWritePartial :: KnownShapeX sh' => IxX (sh ++ sh') -> IxX sh -> Mixed sh' a -> MixedVecs s (sh ++ sh') a -> ST s ()

  -- | Given the shape of this array, finalise the vectors into 'XArray's.
  mvecsFreeze :: IxX sh -> MixedVecs s sh a -> ST s (Mixed sh a)


-- Arrays of scalars are basically just arrays of scalars.
instance VU.Unbox a => Elt (Primitive a) where
  mshape (M_Primitive a) = X.shape a
  mindex (M_Primitive a) i = Primitive (X.index a i)
  mindexPartial (M_Primitive a) i = M_Primitive (X.indexPartial a i)

  mlift :: forall sh1 sh2.
           (Proxy '[] -> XArray (sh1 ++ '[]) a -> XArray (sh2 ++ '[]) a)
        -> Mixed sh1 (Primitive a) -> Mixed sh2 (Primitive a)
  mlift f (M_Primitive a)
    | Refl <- X.lemAppNil @sh1
    , Refl <- X.lemAppNil @sh2
    = M_Primitive (f Proxy a)

  memptyArray sh = M_Primitive (X.generate sh (error "memptyArray Int: shape was not empty"))
  mvecsNumElts _ = 1
  mvecsUnsafeNew sh _ = MV_Primitive <$> VUM.unsafeNew (X.shapeSize sh)
  mvecsWrite sh i (Primitive x) (MV_Primitive v) = VUM.write v (X.toLinearIdx sh i) x

  -- TODO: this use of toVector is suboptimal
  mvecsWritePartial
    :: forall sh' sh s. (KnownShapeX sh', VU.Unbox a)
    => IxX (sh ++ sh') -> IxX sh -> Mixed sh' (Primitive a) -> MixedVecs s (sh ++ sh') (Primitive a) -> ST s ()
  mvecsWritePartial sh i (M_Primitive arr) (MV_Primitive v) = do
    let offset = X.toLinearIdx sh (X.ixAppend i (X.zeroIdx' (X.shape arr)))
    VU.copy (VUM.slice offset (X.shapeSize (X.shape arr)) v) (X.toVector arr)

  mvecsFreeze sh (MV_Primitive v) = M_Primitive . X.fromVector sh <$> VU.freeze v

deriving via Primitive Int instance Elt Int
deriving via Primitive Double instance Elt Double
deriving via Primitive () instance Elt ()

-- Arrays of pairs are pairs of arrays.
instance (Elt a, Elt b) => Elt (a, b) where
  mshape (M_Tup2 a _) = mshape a
  mindex (M_Tup2 a b) i = (mindex a i, mindex b i)
  mindexPartial (M_Tup2 a b) i = M_Tup2 (mindexPartial a i) (mindexPartial b i)
  mlift f (M_Tup2 a b) = M_Tup2 (mlift f a) (mlift f b)

  memptyArray sh = M_Tup2 (memptyArray sh) (memptyArray sh)
  mvecsNumElts (x, y) = mvecsNumElts x * mvecsNumElts y
  mvecsUnsafeNew sh (x, y) = MV_Tup2 <$> mvecsUnsafeNew sh x <*> mvecsUnsafeNew sh y
  mvecsWrite sh i (x, y) (MV_Tup2 a b) = do
    mvecsWrite sh i x a
    mvecsWrite sh i y b
  mvecsWritePartial sh i (M_Tup2 x y) (MV_Tup2 a b) = do
    mvecsWritePartial sh i x a
    mvecsWritePartial sh i y b
  mvecsFreeze sh (MV_Tup2 a b) = M_Tup2 <$> mvecsFreeze sh a <*> mvecsFreeze sh b

-- Arrays of arrays are just arrays, but with more dimensions.
instance (Elt a, KnownShapeX sh') => Elt (Mixed sh' a) where
  mshape :: forall sh. KnownShapeX sh => Mixed sh (Mixed sh' a) -> IxX sh
  mshape (M_Nest arr)
    | Dict <- X.lemAppKnownShapeX (knownShapeX @sh) (knownShapeX @sh')
    = ixAppPrefix (knownShapeX @sh) (mshape arr)
    where
      ixAppPrefix :: StaticShapeX sh1 -> IxX (sh1 ++ sh') -> IxX sh1
      ixAppPrefix SZX _ = IZX
      ixAppPrefix (_ :$@ ssh) (i ::@ idx) = i ::@ ixAppPrefix ssh idx
      ixAppPrefix (_ :$? ssh) (i ::? idx) = i ::? ixAppPrefix ssh idx

  mindex (M_Nest arr) i = mindexPartial arr i

  mindexPartial :: forall sh1 sh2.
                   Mixed (sh1 ++ sh2) (Mixed sh' a) -> IxX sh1 -> Mixed sh2 (Mixed sh' a)
  mindexPartial (M_Nest arr) i
    | Refl <- X.lemAppAssoc (Proxy @sh1) (Proxy @sh2) (Proxy @sh')
    = M_Nest (mindexPartial @a @sh1 @(sh2 ++ sh') arr i)

  mlift :: forall sh1 sh2. KnownShapeX sh2
        => (forall sh3 b. (KnownShapeX sh3, VU.Unbox b) => Proxy sh3 -> XArray (sh1 ++ sh3) b -> XArray (sh2 ++ sh3) b)
        -> Mixed sh1 (Mixed sh' a) -> Mixed sh2 (Mixed sh' a)
  mlift f (M_Nest arr)
    | Dict <- X.lemKnownShapeX (X.ssxAppend (knownShapeX @sh2) (knownShapeX @sh'))
    = M_Nest (mlift f' arr)
    where
      f' :: forall sh3 b. (KnownShapeX sh3, VU.Unbox b) => Proxy sh3 -> XArray ((sh1 ++ sh') ++ sh3) b -> XArray ((sh2 ++ sh') ++ sh3) b
      f' _
        | Refl <- X.lemAppAssoc (Proxy @sh1) (Proxy @sh') (Proxy @sh3)
        , Refl <- X.lemAppAssoc (Proxy @sh2) (Proxy @sh') (Proxy @sh3)
        , Dict <- X.lemKnownShapeX (X.ssxAppend (knownShapeX @sh') (knownShapeX @sh3))
        = f (Proxy @(sh' ++ sh3))

  memptyArray sh = M_Nest (memptyArray (X.ixAppend sh (X.zeroIdx (knownShapeX @sh'))))

  mvecsNumElts arr =
    let n = X.shapeSize (mshape arr)
    in if n == 0 then 0 else n * mvecsNumElts (mindex arr (X.zeroIdx (knownShapeX @sh')))

  mvecsUnsafeNew sh example
    | X.shapeSize sh' == 0 = error "mvecsUnsafeNew: empty example"
    | otherwise = MV_Nest sh' <$> mvecsUnsafeNew (X.ixAppend sh (mshape example))
                                                 (mindex example (X.zeroIdx (knownShapeX @sh')))
    where
      sh' = mshape example

  mvecsWrite sh idx val (MV_Nest sh' vecs) = mvecsWritePartial (X.ixAppend sh sh') idx val vecs

  mvecsWritePartial :: forall sh1 sh2 s. KnownShapeX sh2
                    => IxX (sh1 ++ sh2) -> IxX sh1 -> Mixed sh2 (Mixed sh' a)
                    -> MixedVecs s (sh1 ++ sh2) (Mixed sh' a)
                    -> ST s ()
  mvecsWritePartial sh12 idx (M_Nest arr) (MV_Nest sh' vecs)
    | Dict <- X.lemKnownShapeX (X.ssxAppend (knownShapeX @sh2) (knownShapeX @sh'))
    , Refl <- X.lemAppAssoc (Proxy @sh1) (Proxy @sh2) (Proxy @sh')
    = mvecsWritePartial @a @(sh2 ++ sh') @sh1 (X.ixAppend sh12 sh') idx arr vecs

  mvecsFreeze sh (MV_Nest sh' vecs) = M_Nest <$> mvecsFreeze (X.ixAppend sh sh') vecs


-- Public method. Turns out this doesn't have to be in the type class!
-- | Create an array given a size and a function that computes the element at a
-- given index.
mgenerate :: forall sh a. (KnownShapeX sh, Elt a) => IxX sh -> (IxX sh -> a) -> Mixed sh a
mgenerate sh f
  -- TODO: Do we need this checkBounds check elsewhere as well?
  | not (checkBounds sh (knownShapeX @sh)) =
      error $ "mgenerate: Shape " ++ show sh ++ " not valid for shape type " ++ show (knownShapeX @sh)
  -- We need to be very careful here to ensure that neither 'sh' nor
  -- 'firstelem' that we pass to 'mvecsUnsafeNew' are empty.
  | X.shapeSize sh == 0 = memptyArray sh
  | otherwise =
      let firstidx = X.zeroIdx' sh
          firstelem = f (X.zeroIdx' sh)
      in if mvecsNumElts firstelem == 0
           then memptyArray sh
           else runST $ do
                  vecs <- mvecsUnsafeNew sh firstelem
                  mvecsWrite sh firstidx firstelem vecs
                  -- TODO: This is likely fine if @a@ is big, but if @a@ is a
                  -- scalar this feels inefficient. Should improve this.
                  forM_ (tail (X.enumShape sh)) $ \idx ->
                    mvecsWrite sh idx (f idx) vecs
                  mvecsFreeze sh vecs
  where
    checkBounds :: IxX sh' -> StaticShapeX sh' -> Bool
    checkBounds IZX SZX = True
    checkBounds (n ::@ sh') (n' :$@ ssh') = n == fromIntegral (unSNat n') && checkBounds sh' ssh'
    checkBounds (_ ::? sh') (() :$? ssh') = checkBounds sh' ssh'

mtranspose :: forall sh a. (KnownShapeX sh, Elt a) => [Int] -> Mixed sh a -> Mixed sh a
mtranspose perm =
  mlift (\(Proxy @sh') -> X.rerankTop (knownShapeX @sh) (knownShapeX @sh) (knownShapeX @sh')
                            (X.transpose perm))


-- | A rank-typed array: the number of dimensions of the array (its /rank/) is
-- represented on the type level as a 'Nat'.
--
-- Valid elements of a ranked arrays are described by the 'Elt' type class.
-- Because 'Ranked' itself is also an instance of 'Elt', nested arrays are
-- supported (and are represented as a single, flattened, struct-of-arrays
-- array internally).
--
-- Note that this 'Nat' is not a "GHC.TypeLits" natural, because we want a
-- type-level natural that supports induction.
--
-- 'Ranked' is a newtype around a 'Mixed' of 'Nothing's.
type Ranked :: Nat -> Type -> Type
newtype Ranked n a = Ranked (Mixed (Replicate n Nothing) a)
deriving instance Show (Mixed (Replicate n Nothing) a) => Show (Ranked n a)

-- | A shape-typed array: the full shape of the array (the sizes of its
-- dimensions) is represented on the type level as a list of 'Nat's.
--
-- Like for 'Ranked', the valid elements are described by the 'Elt' type class,
-- and 'Shaped' itself is again an instance of 'Elt' as well.
--
-- 'Shaped' is a newtype around a 'Mixed' of 'Just's.
type Shaped :: [Nat] -> Type -> Type
newtype Shaped sh a = Shaped (Mixed (MapJust sh) a)
deriving instance Show (Mixed (MapJust sh) a) => Show (Shaped sh a)

-- just unwrap the newtype and defer to the general instance for nested arrays
newtype instance Mixed sh (Ranked n   a) = M_Ranked (Mixed sh (Mixed (Replicate n Nothing) a))
deriving instance Show (Mixed sh (Mixed (Replicate n Nothing) a)) => Show (Mixed sh (Ranked n a))
newtype instance Mixed sh (Shaped sh' a) = M_Shaped (Mixed sh (Mixed (MapJust sh'        ) a))
deriving instance Show (Mixed sh (Mixed (MapJust sh'        ) a)) => Show (Mixed sh (Shaped sh' a))

newtype instance MixedVecs s sh (Ranked n   a) = MV_Ranked (MixedVecs s sh (Mixed (Replicate n Nothing) a))
newtype instance MixedVecs s sh (Shaped sh' a) = MV_Shaped (MixedVecs s sh (Mixed (MapJust sh'        ) a))


-- 'Ranked' and 'Shaped' can already be used at the top level of an array nest;
-- these instances allow them to also be used as elements of arrays, thus
-- making them first-class in the API.
instance (KnownNat n, Elt a) => Elt (Ranked n a) where
  mshape (M_Ranked arr) | Dict <- lemKnownReplicate (Proxy @n) = mshape arr
  mindex (M_Ranked arr) i | Dict <- lemKnownReplicate (Proxy @n) = Ranked (mindex arr i)

  mindexPartial :: forall sh sh'. Mixed (sh ++ sh') (Ranked n a) -> IxX sh -> Mixed sh' (Ranked n a)
  mindexPartial (M_Ranked arr) i
    | Dict <- lemKnownReplicate (Proxy @n)
    = coerce @(Mixed sh' (Mixed (Replicate n Nothing) a)) @(Mixed sh' (Ranked n a)) $
        mindexPartial arr i

  mlift :: forall sh1 sh2. KnownShapeX sh2
        => (forall sh' b. (KnownShapeX sh', VU.Unbox b) => Proxy sh' -> XArray (sh1 ++ sh') b -> XArray (sh2 ++ sh') b)
        -> Mixed sh1 (Ranked n a) -> Mixed sh2 (Ranked n a)
  mlift f (M_Ranked arr)
    | Dict <- lemKnownReplicate (Proxy @n)
    = coerce @(Mixed sh2 (Mixed (Replicate n Nothing) a)) @(Mixed sh2 (Ranked n a)) $
        mlift f arr

  memptyArray :: forall sh. IxX sh -> Mixed sh (Ranked n a)
  memptyArray i
    | Dict <- lemKnownReplicate (Proxy @n)
    = coerce @(Mixed sh (Mixed (Replicate n Nothing) a)) @(Mixed sh (Ranked n a)) $
        memptyArray i

  mvecsNumElts (Ranked arr)
    | Dict <- lemKnownReplicate (Proxy @n)
    = mvecsNumElts arr

  mvecsUnsafeNew idx (Ranked arr)
    | Dict <- lemKnownReplicate (Proxy @n)
    = MV_Ranked <$> mvecsUnsafeNew idx arr

  mvecsWrite :: forall sh s. IxX sh -> IxX sh -> Ranked n a -> MixedVecs s sh (Ranked n a) -> ST s ()
  mvecsWrite sh idx (Ranked arr) vecs
    | Dict <- lemKnownReplicate (Proxy @n)
    = mvecsWrite sh idx arr
        (coerce @(MixedVecs s sh (Ranked n a)) @(MixedVecs s sh (Mixed (Replicate n Nothing) a))
           vecs)

  mvecsWritePartial :: forall sh sh' s. KnownShapeX sh'
                    => IxX (sh ++ sh') -> IxX sh -> Mixed sh' (Ranked n a)
                    -> MixedVecs s (sh ++ sh') (Ranked n a)
                    -> ST s ()
  mvecsWritePartial sh idx arr vecs
    | Dict <- lemKnownReplicate (Proxy @n)
    = mvecsWritePartial sh idx
        (coerce @(Mixed sh' (Ranked n a))
                @(Mixed sh' (Mixed (Replicate n Nothing) a))
           arr)
        (coerce @(MixedVecs s (sh ++ sh') (Ranked n a))
                @(MixedVecs s (sh ++ sh') (Mixed (Replicate n Nothing) a))
           vecs)

  mvecsFreeze :: forall sh s. IxX sh -> MixedVecs s sh (Ranked n a) -> ST s (Mixed sh (Ranked n a))
  mvecsFreeze sh vecs
    | Dict <- lemKnownReplicate (Proxy @n)
    = coerce @(Mixed sh (Mixed (Replicate n Nothing) a))
             @(Mixed sh (Ranked n a))
        <$> mvecsFreeze sh
              (coerce @(MixedVecs s sh (Ranked n a))
                      @(MixedVecs s sh (Mixed (Replicate n Nothing) a))
                      vecs)


-- | The shape of a shape-typed array given as a list of 'SNat' values.
data SShape sh where
  ShNil :: SShape '[]
  ShCons :: SNat n -> SShape sh -> SShape (n : sh)
deriving instance Show (SShape sh)
infixr 5 `ShCons`

-- | A statically-known shape of a shape-typed array.
class KnownShape sh where knownShape :: SShape sh
instance KnownShape '[] where knownShape = ShNil
instance (KnownNat n, KnownShape sh) => KnownShape (n : sh) where knownShape = ShCons knownNat knownShape

sshapeKnown :: SShape sh -> Dict KnownShape sh
sshapeKnown ShNil = Dict
sshapeKnown (ShCons n sh) | Dict <- snatKnown n, Dict <- sshapeKnown sh = Dict

lemKnownMapJust :: forall sh. KnownShape sh => Proxy sh -> Dict KnownShapeX (MapJust sh)
lemKnownMapJust _ = X.lemKnownShapeX (go (knownShape @sh))
  where
    go :: SShape sh' -> StaticShapeX (MapJust sh')
    go ShNil = SZX
    go (ShCons n sh) = n :$@ go sh

lemMapJustPlusApp :: forall sh1 sh2. KnownShape sh1 => Proxy sh1 -> Proxy sh2
                  -> MapJust (sh1 ++ sh2) :~: MapJust sh1 ++ MapJust sh2
lemMapJustPlusApp _ _ = go (knownShape @sh1)
  where
    go :: SShape sh1' -> MapJust (sh1' ++ sh2) :~: MapJust sh1' ++ MapJust sh2
    go ShNil = Refl
    go (ShCons _ sh) | Refl <- go sh = Refl

instance (KnownShape sh, Elt a) => Elt (Shaped sh a) where
  mshape (M_Shaped arr) | Dict <- lemKnownMapJust (Proxy @sh) = mshape arr
  mindex (M_Shaped arr) i | Dict <- lemKnownMapJust (Proxy @sh) = Shaped (mindex arr i)

  mindexPartial :: forall sh1 sh2. Mixed (sh1 ++ sh2) (Shaped sh a) -> IxX sh1 -> Mixed sh2 (Shaped sh a)
  mindexPartial (M_Shaped arr) i
    | Dict <- lemKnownMapJust (Proxy @sh)
    = coerce @(Mixed sh2 (Mixed (MapJust sh) a)) @(Mixed sh2 (Shaped sh a)) $
        mindexPartial arr i

  mlift :: forall sh1 sh2. KnownShapeX sh2
        => (forall sh' b. (KnownShapeX sh', VU.Unbox b) => Proxy sh' -> XArray (sh1 ++ sh') b -> XArray (sh2 ++ sh') b)
        -> Mixed sh1 (Shaped sh a) -> Mixed sh2 (Shaped sh a)
  mlift f (M_Shaped arr)
    | Dict <- lemKnownMapJust (Proxy @sh)
    = coerce @(Mixed sh2 (Mixed (MapJust sh) a)) @(Mixed sh2 (Shaped sh a)) $
        mlift f arr

  memptyArray :: forall sh'. IxX sh' -> Mixed sh' (Shaped sh a)
  memptyArray i
    | Dict <- lemKnownMapJust (Proxy @sh)
    = coerce @(Mixed sh' (Mixed (MapJust sh) a)) @(Mixed sh' (Shaped sh a)) $
        memptyArray i

  mvecsNumElts (Shaped arr)
    | Dict <- lemKnownMapJust (Proxy @sh)
    = mvecsNumElts arr

  mvecsUnsafeNew idx (Shaped arr)
    | Dict <- lemKnownMapJust (Proxy @sh)
    = MV_Shaped <$> mvecsUnsafeNew idx arr

  mvecsWrite :: forall sh' s. IxX sh' -> IxX sh' -> Shaped sh a -> MixedVecs s sh' (Shaped sh a) -> ST s ()
  mvecsWrite sh idx (Shaped arr) vecs
    | Dict <- lemKnownMapJust (Proxy @sh)
    = mvecsWrite sh idx arr
        (coerce @(MixedVecs s sh' (Shaped sh a)) @(MixedVecs s sh' (Mixed (MapJust sh) a))
           vecs)

  mvecsWritePartial :: forall sh1 sh2 s. KnownShapeX sh2
                    => IxX (sh1 ++ sh2) -> IxX sh1 -> Mixed sh2 (Shaped sh a)
                    -> MixedVecs s (sh1 ++ sh2) (Shaped sh a)
                    -> ST s ()
  mvecsWritePartial sh idx arr vecs
    | Dict <- lemKnownMapJust (Proxy @sh)
    = mvecsWritePartial sh idx
        (coerce @(Mixed sh2 (Shaped sh a))
                @(Mixed sh2 (Mixed (MapJust sh) a))
           arr)
        (coerce @(MixedVecs s (sh1 ++ sh2) (Shaped sh a))
                @(MixedVecs s (sh1 ++ sh2) (Mixed (MapJust sh) a))
           vecs)

  mvecsFreeze :: forall sh' s. IxX sh' -> MixedVecs s sh' (Shaped sh a) -> ST s (Mixed sh' (Shaped sh a))
  mvecsFreeze sh vecs
    | Dict <- lemKnownMapJust (Proxy @sh)
    = coerce @(Mixed sh' (Mixed (MapJust sh) a))
             @(Mixed sh' (Shaped sh a))
        <$> mvecsFreeze sh
              (coerce @(MixedVecs s sh' (Shaped sh a))
                      @(MixedVecs s sh' (Mixed (MapJust sh) a))
                      vecs)


-- Utility function to satisfy the type checker sometimes
rewriteMixed :: sh1 :~: sh2 -> Mixed sh1 a -> Mixed sh2 a
rewriteMixed Refl x = x


-- ====== API OF RANKED ARRAYS ====== --

-- | An index into a rank-typed array.
type IxR :: Nat -> Type
data IxR n where
  IZR :: IxR Z
  (:::) :: Int -> IxR n -> IxR (S n)
infixr 5 :::

ixCvtXR :: IxX sh -> IxR (X.Rank sh)
ixCvtXR IZX = IZR
ixCvtXR (n ::@ idx) = n ::: ixCvtXR idx
ixCvtXR (n ::? idx) = n ::: ixCvtXR idx

ixCvtRX :: IxR n -> IxX (Replicate n Nothing)
ixCvtRX IZR = IZX
ixCvtRX (n ::: idx) = n ::? ixCvtRX idx


rshape :: forall n a. (KnownNat n, Elt a) => Ranked n a -> IxR n
rshape (Ranked arr)
  | Dict <- lemKnownReplicate (Proxy @n)
  , Refl <- lemRankReplicate (Proxy @n)
  = ixCvtXR (mshape arr)

rindex :: Elt a => Ranked n a -> IxR n -> a
rindex (Ranked arr) idx = mindex arr (ixCvtRX idx)

rindexPartial :: forall n m a. (KnownNat n, Elt a) => Ranked (n + m) a -> IxR n -> Ranked m a
rindexPartial (Ranked arr) idx =
  Ranked (mindexPartial @a @(Replicate n Nothing) @(Replicate m Nothing)
            (rewriteMixed (lemReplicatePlusApp (Proxy @n) (Proxy @m) (Proxy @Nothing)) arr)
            (ixCvtRX idx))

rgenerate :: forall n a. (KnownNat n, Elt a) => IxR n -> (IxR n -> a) -> Ranked n a
rgenerate sh f
  | Dict <- lemKnownReplicate (Proxy @n)
  , Refl <- lemRankReplicate (Proxy @n)
  = Ranked (mgenerate (ixCvtRX sh) (f . ixCvtXR))

rlift :: forall n1 n2 a. (KnownNat n2, Elt a)
      => (forall sh' b. (KnownShapeX sh', VU.Unbox b) => Proxy sh' -> XArray (Replicate n1 Nothing ++ sh') b -> XArray (Replicate n2 Nothing ++ sh') b)
      -> Ranked n1 a -> Ranked n2 a
rlift f (Ranked arr)
  | Dict <- lemKnownReplicate (Proxy @n2)
  = Ranked (mlift f arr)

rsumOuter1 :: forall n a.
              (VU.Unbox a, Num a, KnownNat n, forall sh. Coercible (Mixed sh a) (XArray sh a))
           => Ranked (S n) a -> Ranked n a
rsumOuter1 (Ranked arr)
  | Dict <- lemKnownReplicate (Proxy @n)
  = Ranked
    . coerce @(XArray (Replicate n Nothing) a) @(Mixed (Replicate n Nothing) a)
    . X.sumOuter (() :$? SZX) (knownShapeX @(Replicate n Nothing))
    . coerce @(Mixed (Replicate (S n) Nothing) a) @(XArray (Replicate (S n) Nothing) a)
    $ arr

rtranspose :: forall n a. (KnownNat n, Elt a) => [Int] -> Ranked n a -> Ranked n a
rtranspose perm (Ranked arr)
  | Dict <- lemKnownReplicate (Proxy @n)
  = Ranked (mtranspose perm arr)


-- ====== API OF SHAPED ARRAYS ====== --

-- | An index into a shape-typed array.
--
-- For convenience, this contains regular 'Int's instead of bounded integers
-- (traditionally called \"@Fin@\"). Note that because the shape of a
-- shape-typed array is known statically, you can also retrieve the array shape
-- from a 'KnownShape' dictionary.
type IxS :: [Nat] -> Type
data IxS sh where
  IZS :: IxS '[]
  (::$) :: Int -> IxS sh -> IxS (n : sh)
infixr 5 ::$

cvtSShapeIxS :: SShape sh -> IxS sh
cvtSShapeIxS ShNil = IZS
cvtSShapeIxS (ShCons n sh) = fromIntegral (unSNat n) ::$ cvtSShapeIxS sh

ixCvtXS :: SShape sh -> IxX (MapJust sh) -> IxS sh
ixCvtXS ShNil IZX = IZS
ixCvtXS (ShCons _ sh) (n ::@ idx) = n ::$ ixCvtXS sh idx

ixCvtSX :: IxS sh -> IxX (MapJust sh)
ixCvtSX IZS = IZX
ixCvtSX (n ::$ sh) = n ::@ ixCvtSX sh


sshape :: forall sh a. (KnownShape sh, Elt a) => Shaped sh a -> IxS sh
sshape _ = cvtSShapeIxS (knownShape @sh)

sindex :: Elt a => Shaped sh a -> IxS sh -> a
sindex (Shaped arr) idx = mindex arr (ixCvtSX idx)

sindexPartial :: forall sh1 sh2 a. (KnownShape sh1, Elt a) => Shaped (sh1 ++ sh2) a -> IxS sh1 -> Shaped sh2 a
sindexPartial (Shaped arr) idx =
  Shaped (mindexPartial @a @(MapJust sh1) @(MapJust sh2)
            (rewriteMixed (lemMapJustPlusApp (Proxy @sh1) (Proxy @sh2)) arr)
            (ixCvtSX idx))

sgenerate :: forall sh a. (KnownShape sh, Elt a) => IxS sh -> (IxS sh -> a) -> Shaped sh a
sgenerate sh f
  | Dict <- lemKnownMapJust (Proxy @sh)
  = Shaped (mgenerate (ixCvtSX sh) (f . ixCvtXS (knownShape @sh)))

slift :: forall sh1 sh2 a. (KnownShape sh2, Elt a)
      => (forall sh' b. (KnownShapeX sh', VU.Unbox b) => Proxy sh' -> XArray (MapJust sh1 ++ sh') b -> XArray (MapJust sh2 ++ sh') b)
      -> Shaped sh1 a -> Shaped sh2 a
slift f (Shaped arr)
  | Dict <- lemKnownMapJust (Proxy @sh2)
  = Shaped (mlift f arr)

ssumOuter1 :: forall sh n a.
              (VU.Unbox a, Num a, KnownNat n, KnownShape sh, forall sh'. Coercible (Mixed sh' a) (XArray sh' a))
           => Shaped (n : sh) a -> Shaped sh a
ssumOuter1 (Shaped arr)
  | Dict <- lemKnownMapJust (Proxy @sh)
  = Shaped
    . coerce @(XArray (MapJust sh) a) @(Mixed (MapJust sh) a)
    . X.sumOuter (knownNat @n :$@ SZX) (knownShapeX @(MapJust sh))
    . coerce @(Mixed (Just n : MapJust sh) a) @(XArray (Just n : MapJust sh) a)
    $ arr

stranspose :: forall sh a. (KnownShape sh, Elt a) => [Int] -> Shaped sh a -> Shaped sh a
stranspose perm (Shaped arr)
  | Dict <- lemKnownMapJust (Proxy @sh)
  = Shaped (mtranspose perm arr)