Initial

5 files changed, 454 insertions, 0 deletions

diff --git a/.gitignore b/.gitignore
new file mode 100644
index 0000000..c33954f
--- /dev/null
+++ b/.gitignore
@@ -0,0 +1 @@
+dist-newstyle/
diff --git a/cabal.project b/cabal.project
new file mode 100644
index 0000000..a13761a
--- /dev/null
+++ b/cabal.project
@@ -0,0 +1,2 @@
+packages: .
+with-compiler: ghc-9.6.4
diff --git a/ox-arrays.cabal b/ox-arrays.cabal
new file mode 100644
index 0000000..aea0a94
--- /dev/null
+++ b/ox-arrays.cabal
@@ -0,0 +1,19 @@
+cabal-version:   3.0
+name:            ox-arrays
+version:         0.1.0.0
+author:          Tom Smeding
+license:         BSD-3-Clause
+build-type:      Simple
+
+library
+  exposed-modules:
+    Array
+    Fancy
+  build-depends:
+    base >=4.18,
+    ghc-typelits-knownnat,
+    orthotope,
+    vector
+  hs-source-dirs: src
+  default-language: Haskell2010
+  ghc-options: -Wall
diff --git a/src/Array.hs b/src/Array.hs
new file mode 100644
index 0000000..25db19e
--- /dev/null
+++ b/src/Array.hs
@@ -0,0 +1,195 @@
+{-# LANGUAGE DataKinds #-}
+{-# LANGUAGE FlexibleInstances #-}
+{-# LANGUAGE GADTs #-}
+{-# LANGUAGE PolyKinds #-}
+{-# LANGUAGE ScopedTypeVariables #-}
+{-# LANGUAGE StandaloneDeriving #-}
+{-# LANGUAGE StandaloneKindSignatures #-}
+{-# LANGUAGE TypeApplications #-}
+{-# LANGUAGE TypeFamilies #-}
+{-# LANGUAGE TypeOperators #-}
+{-# LANGUAGE UndecidableInstances #-}
+{-# OPTIONS_GHC -fplugin GHC.TypeLits.KnownNat.Solver #-}
+module Array where
+
+import qualified Data.Array.RankedU as U
+import Data.Kind
+import Data.Proxy
+import Data.Type.Equality
+import qualified Data.Vector.Unboxed as VU
+import GHC.TypeLits
+import Unsafe.Coerce (unsafeCoerce)
+
+
+data Dict c a where
+  Dict :: c a => Dict c a
+
+type family l1 ++ l2 where
+  '[] ++ l2 = l2
+  (x : xs) ++ l2 = x : xs ++ l2
+
+lemAppNil :: l ++ '[] :~: l
+lemAppNil = unsafeCoerce Refl
+
+lemAppAssoc :: Proxy a -> Proxy b -> Proxy c -> (a ++ b) ++ c :~: a ++ (b ++ c)
+lemAppAssoc _ _ _ = unsafeCoerce Refl
+
+
+type IxX :: [Maybe Nat] -> Type
+data IxX sh where
+  IZX :: IxX '[]
+  (::@) :: Int -> IxX sh -> IxX (Just n : sh)
+  (::?) :: Int -> IxX sh -> IxX (Nothing : sh)
+deriving instance Show (IxX sh)
+
+type StaticShapeX :: [Maybe Nat] -> Type
+data StaticShapeX sh where
+  SZX :: StaticShapeX '[]
+  (:$@) :: SNat n -> StaticShapeX sh -> StaticShapeX (Just n : sh)
+  (:$?) :: () -> StaticShapeX sh -> StaticShapeX (Nothing : sh)
+
+type KnownShapeX :: [Maybe Nat] -> Constraint
+class KnownShapeX sh where
+  knownShapeX :: StaticShapeX sh
+instance KnownShapeX '[] where
+  knownShapeX = SZX
+instance (KnownNat n, KnownShapeX sh) => KnownShapeX (Just n : sh) where
+  knownShapeX = natSing @n :$@ knownShapeX
+instance KnownShapeX sh => KnownShapeX (Nothing : sh) where
+  knownShapeX = () :$? knownShapeX
+
+type family Rank sh where
+  Rank '[] = 0
+  Rank (_ : sh) = 1 + Rank sh
+
+type XArray :: [Maybe Nat] -> Type -> Type
+data XArray sh a = XArray (U.Array (Rank sh) a)
+
+zeroIdx :: StaticShapeX sh -> IxX sh
+zeroIdx SZX = IZX
+zeroIdx (_ :$@ ssh) = 0 ::@ zeroIdx ssh
+zeroIdx (_ :$? ssh) = 0 ::? zeroIdx ssh
+
+zeroIdx' :: IxX sh -> IxX sh
+zeroIdx' IZX = IZX
+zeroIdx' (_ ::@ sh) = 0 ::@ zeroIdx' sh
+zeroIdx' (_ ::? sh) = 0 ::? zeroIdx' sh
+
+ixAppend :: IxX sh -> IxX sh' -> IxX (sh ++ sh')
+ixAppend IZX idx' = idx'
+ixAppend (i ::@ idx) idx' = i ::@ ixAppend idx idx'
+ixAppend (i ::? idx) idx' = i ::? ixAppend idx idx'
+
+ixDrop :: IxX (sh ++ sh') -> IxX sh -> IxX sh'
+ixDrop sh IZX = sh
+ixDrop (_ ::@ sh) (_ ::@ idx) = ixDrop sh idx
+ixDrop (_ ::? sh) (_ ::? idx) = ixDrop sh idx
+
+ssxAppend :: StaticShapeX sh -> StaticShapeX sh' -> StaticShapeX (sh ++ sh')
+ssxAppend SZX idx' = idx'
+ssxAppend (n :$@ idx) idx' = n :$@ ssxAppend idx idx'
+ssxAppend (() :$? idx) idx' = () :$? ssxAppend idx idx'
+
+shapeSize :: IxX sh -> Int
+shapeSize IZX = 1
+shapeSize (n ::@ sh) = n * shapeSize sh
+shapeSize (n ::? sh) = n * shapeSize sh
+
+fromLinearIdx :: IxX sh -> Int -> IxX sh
+fromLinearIdx = \sh i -> case go sh i of
+  (idx, 0) -> idx
+  _ -> error $ "fromLinearIdx: out of range (" ++ show i ++
+               " in array of shape " ++ show sh ++ ")"
+  where
+    -- returns (index in subarray, remaining index in enclosing array)
+    go :: IxX sh -> Int -> (IxX sh, Int)
+    go IZX i = (IZX, i)
+    go (n ::@ sh) i =
+      let (idx, i') = go sh i
+          (upi, locali) = i' `quotRem` n
+      in (locali ::@ idx, upi)
+    go (n ::? sh) i =
+      let (idx, i') = go sh i
+          (upi, locali) = i' `quotRem` n
+      in (locali ::? idx, upi)
+
+toLinearIdx :: IxX sh -> IxX sh -> Int
+toLinearIdx = \sh i -> fst (go sh i)
+  where
+    -- returns (index in subarray, size of subarray)
+    go :: IxX sh -> IxX sh -> (Int, Int)
+    go IZX IZX = (0, 1)
+    go (n ::@ sh) (i ::@ ix) =
+      let (lidx, sz) = go sh ix
+      in (sz * i + lidx, n * sz)
+    go (n ::? sh) (i ::? ix) =
+      let (lidx, sz) = go sh ix
+      in (sz * i + lidx, n * sz)
+
+enumShape :: IxX sh -> [IxX sh]
+enumShape = \sh -> go 0 sh id []
+  where
+    go :: Int -> IxX sh -> (IxX sh -> a) -> [a] -> [a]
+    go _ IZX _ = id
+    go i (n ::@ sh) f
+      | i < n = go (i + 1) (n ::@ sh) f . go 0 sh (f . (i ::@))
+      | otherwise = id
+    go i (n ::? sh) f
+      | i < n = go (i + 1) (n ::? sh) f . go 0 sh (f . (i ::?))
+      | otherwise = id
+
+shapeLshape :: IxX sh -> U.ShapeL
+shapeLshape IZX = []
+shapeLshape (n ::@ sh) = n : shapeLshape sh
+shapeLshape (n ::? sh) = n : shapeLshape sh
+
+lemKnownNatRank :: IxX sh -> Dict KnownNat (Rank sh)
+lemKnownNatRank IZX = Dict
+lemKnownNatRank (_ ::@ sh) | Dict <- lemKnownNatRank sh = Dict
+lemKnownNatRank (_ ::? sh) | Dict <- lemKnownNatRank sh = Dict
+
+lemKnownShapeX :: StaticShapeX sh -> Dict KnownShapeX sh
+lemKnownShapeX SZX = Dict
+lemKnownShapeX (SNat :$@ ssh) | Dict <- lemKnownShapeX ssh = Dict
+lemKnownShapeX (() :$? ssh) | Dict <- lemKnownShapeX ssh = Dict
+lemKnownShapeX (_ :$@ _) = error "SNat does not have a COMPLETE pragma"
+
+lemAppKnownShapeX :: StaticShapeX sh1 -> StaticShapeX sh2 -> Dict KnownShapeX (sh1 ++ sh2)
+lemAppKnownShapeX SZX ssh' = lemKnownShapeX ssh'
+lemAppKnownShapeX (SNat :$@ ssh) ssh' | Dict <- lemAppKnownShapeX ssh ssh' = Dict
+lemAppKnownShapeX (() :$? ssh) ssh' | Dict <- lemAppKnownShapeX ssh ssh' = Dict
+lemAppKnownShapeX (_ :$@ _) _ = error "SNat does not have a COMPLETE pragma"
+
+shape :: forall sh a. KnownShapeX sh => XArray sh a -> IxX sh
+shape (XArray arr) = go (knownShapeX @sh) (U.shapeL arr)
+  where
+    go :: StaticShapeX sh' -> [Int] -> IxX sh'
+    go SZX [] = IZX
+    go (n :$@ ssh) (_ : l) = fromIntegral (fromSNat n) ::@ go ssh l
+    go (() :$? ssh) (n : l) = n ::? go ssh l
+    go _ _ = error "Invalid shapeL"
+
+fromVector :: U.Unbox a => IxX sh -> VU.Vector a -> XArray sh a 
+fromVector sh v | Dict <- lemKnownNatRank sh = XArray (U.fromVector (shapeLshape sh) v)
+
+toVector :: U.Unbox a => XArray sh a -> VU.Vector a
+toVector (XArray arr) = U.toVector arr
+
+generate :: U.Unbox a => IxX sh -> (IxX sh -> a) -> XArray sh a
+generate sh f = fromVector sh $ VU.generate (shapeSize sh) (f . fromLinearIdx sh)
+
+-- generateM :: (Monad m, U.Unbox a) => IxX sh -> (IxX sh -> m a) -> m (XArray sh a)
+-- generateM sh f | Dict <- lemKnownNatRank sh =
+--   XArray . U.fromVector (shapeLshape sh)
+--     <$> VU.generateM (shapeSize sh) (f . fromLinearIdx sh)
+
+indexPartial :: U.Unbox a => XArray (sh ++ sh') a -> IxX sh -> XArray sh' a
+indexPartial (XArray arr) IZX = XArray arr
+indexPartial (XArray arr) (i ::@ idx) = indexPartial (XArray (U.index arr i)) idx
+indexPartial (XArray arr) (i ::? idx) = indexPartial (XArray (U.index arr i)) idx
+
+index :: forall sh a. U.Unbox a => XArray sh a -> IxX sh -> a
+index xarr i
+  | Refl <- lemAppNil @sh
+  = let XArray arr' = indexPartial xarr i :: XArray '[] a
+    in U.unScalar arr'
diff --git a/src/Fancy.hs b/src/Fancy.hs
new file mode 100644
index 0000000..821073e
--- /dev/null
+++ b/src/Fancy.hs
@@ -0,0 +1,237 @@
+{-# LANGUAGE DataKinds #-}
+{-# LANGUAGE GADTs #-}
+{-# LANGUAGE InstanceSigs #-}
+{-# LANGUAGE PolyKinds #-}
+{-# LANGUAGE ScopedTypeVariables #-}
+{-# LANGUAGE StandaloneKindSignatures #-}
+{-# LANGUAGE TypeFamilies #-}
+{-# LANGUAGE TypeOperators #-}
+{-# LANGUAGE UndecidableInstances #-}
+{-# LANGUAGE TypeApplications #-}
+{-# LANGUAGE FlexibleInstances #-}
+module Fancy where
+
+import Control.Monad (forM_)
+import Control.Monad.ST
+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 GHC.TypeLits
+
+import Array (XArray, IxX(..), KnownShapeX(..), StaticShapeX(..), type (++))
+import qualified Array as X
+
+
+type family Replicate n a where
+  Replicate 0 a = '[]
+  Replicate n a = a : Replicate (n - 1) a
+
+type family MapJust l where
+  MapJust '[] = '[]
+  MapJust (x : xs) = Just x : MapJust xs
+
+
+type Mixed :: [Maybe Nat] -> Type -> Type
+data family Mixed sh a
+
+newtype instance Mixed sh Int = M_Int (XArray sh Int)
+newtype instance Mixed sh Double = M_Double (XArray sh Double)
+-- etc.
+
+newtype instance Mixed sh () = M_Nil (IxX sh)  -- store the shape
+data instance Mixed sh (a, b) = M_Tup2 (Mixed sh a) (Mixed sh b)
+data instance Mixed sh (a, b, c) = M_Tup3 (Mixed sh a) (Mixed sh b) (Mixed sh c)
+data instance Mixed sh (a, b, c, d) = M_Tup4 (Mixed sh a) (Mixed sh b) (Mixed sh c) (Mixed sh d)
+
+newtype instance Mixed sh1 (Mixed sh2 a) = M_Nest (Mixed (sh1 ++ sh2) a)
+
+
+type MixedVecs :: Type -> [Maybe Nat] -> Type -> Type
+data family MixedVecs s sh 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)
+-- etc.
+
+data instance MixedVecs s sh () = MV_Nil
+data instance MixedVecs s sh (a, b) = MV_Tup2 (MixedVecs s sh a) (MixedVecs s sh b)
+data instance MixedVecs s sh (a, b, c) = MV_Tup3 (MixedVecs s sh a) (MixedVecs s sh b) (MixedVecs s sh c)
+data instance MixedVecs s sh (a, b, c, d) = MV_Tup4 (MixedVecs s sh a) (MixedVecs s sh b) (MixedVecs s sh c) (MixedVecs s sh d)
+
+data instance MixedVecs s sh1 (Mixed sh2 a) = MV_Nest (IxX sh2) (MixedVecs s (sh1 ++ sh2) a)
+
+
+class GMixed a where
+  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
+
+  -- | 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)
+
+-- TODO: this use of toVector is suboptimal
+mvecsWritePartialPrimitive
+  :: forall sh' sh a s. (KnownShapeX sh', VU.Unbox a)
+  => IxX (sh ++ sh') -> IxX sh -> XArray sh' a -> VU.MVector s a -> ST s ()
+mvecsWritePartialPrimitive sh i arr 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)
+
+instance GMixed Int where
+  mshape (M_Int a) = X.shape a
+  mindex (M_Int a) i = X.index a i
+  mindexPartial (M_Int a) i = M_Int (X.indexPartial a i)
+  memptyArray sh = M_Int (X.generate sh (error "memptyArray Int: shape was not empty"))
+
+  mvecsNumElts _ = 1
+  mvecsUnsafeNew sh _ = MV_Int <$> VUM.unsafeNew (X.shapeSize sh)
+  mvecsWrite sh i x (MV_Int v) = VUM.write v (X.toLinearIdx sh i) x
+  mvecsWritePartial sh i (M_Int @sh' arr) (MV_Int v) = mvecsWritePartialPrimitive @sh' sh i arr v
+  mvecsFreeze sh (MV_Int v) = M_Int . X.fromVector sh <$> VU.freeze v
+
+instance GMixed Double where
+  mshape (M_Double a) = X.shape a
+  mindex (M_Double a) i = X.index a i
+  mindexPartial (M_Double a) i = M_Double (X.indexPartial a i)
+  memptyArray sh = M_Double (X.generate sh (error "memptyArray Double: shape was not empty"))
+
+  mvecsNumElts _ = 1
+  mvecsUnsafeNew sh _ = MV_Double <$> VUM.unsafeNew (X.shapeSize sh)
+  mvecsWrite sh i x (MV_Double v) = VUM.write v (X.toLinearIdx sh i) x
+  mvecsWritePartial sh i (M_Double @sh' arr) (MV_Double v) = mvecsWritePartialPrimitive @sh' sh i arr v
+  mvecsFreeze sh (MV_Double v) = M_Double . X.fromVector sh <$> VU.freeze v
+
+instance GMixed () where
+  mshape (M_Nil sh) = sh
+  mindex _ _ = ()
+  mindexPartial = \(M_Nil sh) i -> M_Nil (X.ixDrop sh i)
+  memptyArray sh = M_Nil sh
+
+  mvecsNumElts _ = 1
+  mvecsUnsafeNew _ _ = return MV_Nil
+  mvecsWrite _ _ _ _ = return ()
+  mvecsWritePartial _ _ _ _ = return ()
+  mvecsFreeze sh _ = return (M_Nil sh)
+
+instance (GMixed a, GMixed b) => GMixed (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)
+  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
+
+instance (GMixed a, KnownShapeX sh') => GMixed (Mixed sh' a) where
+  -- TODO: this is quadratic in the nesting level
+  mshape :: forall sh. KnownShapeX sh => Mixed sh (Mixed sh' a) -> IxX sh
+  mshape (M_Nest arr)
+    | X.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)
+
+  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' <$> mvecsUnsafeNew (X.ixAppend sh (mshape example))
+                                                 (mindex example (X.zeroIdx (knownShapeX @sh')))
+    where
+      sh' = mshape example
+
+  mvecsWrite sh idx val (MV_Nest sh' vecs) = mvecsWritePartial (X.ixAppend sh sh') idx val vecs
+
+  mvecsWritePartial :: forall sh1 sh2 s. KnownShapeX sh2
+                    => IxX (sh1 ++ sh2) -> IxX sh1 -> Mixed sh2 (Mixed sh' a)
+                    -> MixedVecs s (sh1 ++ sh2) (Mixed sh' a)
+                    -> ST s ()
+  mvecsWritePartial sh12 idx (M_Nest arr) (MV_Nest sh' vecs)
+    | X.Dict <- X.lemKnownShapeX (X.ssxAppend (knownShapeX @sh2) (knownShapeX @sh'))
+    , Refl <- X.lemAppAssoc (Proxy @sh1) (Proxy @sh2) (Proxy @sh')
+    = mvecsWritePartial @a @(sh2 ++ sh') @sh1 (X.ixAppend sh12 sh') idx arr vecs
+
+  mvecsFreeze sh (MV_Nest sh' vecs) = M_Nest <$> mvecsFreeze (X.ixAppend sh sh') vecs
+
+mgenerate :: GMixed a => IxX sh -> (IxX sh -> a) -> Mixed sh a
+mgenerate sh f
+  -- 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
+                  forM_ (tail (X.enumShape sh)) $ \idx ->
+                    mvecsWrite sh idx (f idx) vecs
+                  mvecsFreeze sh vecs
+
+
+type Ranked :: Nat -> Type -> Type
+newtype Ranked n a = Ranked (Mixed (Replicate n Nothing) a)
+
+type Shaped :: [Nat] -> Type -> Type
+newtype Shaped sh a = Shaped (Mixed (MapJust sh) a)
+
+
+type IxR :: Nat -> Type
+data IxR n where
+  IZR :: IxR 0
+  (:::) :: Int -> IxR n -> IxR (n + 1)
+
+type IxS :: [Nat] -> Type
+data IxS sh where
+  IZS :: IxS '[]
+  (::$) :: Int -> IxS sh -> IxS (n : sh)
+
+