From a65306ba5d80891b20ac86fa3a3242f9497751e6 Mon Sep 17 00:00:00 2001
From: Tom Smeding <tom@tomsmeding.com>
Date: Thu, 30 May 2024 11:58:40 +0200
Subject: Refactor Mixed (modules, regular function names)

---
 src/Data/Array/Mixed.hs                          | 757 ++---------------------
 src/Data/Array/Mixed/Internal/Arith.hs           | 435 +++++++++++++
 src/Data/Array/Mixed/Internal/Arith/Foreign.hs   |  55 ++
 src/Data/Array/Mixed/Internal/Arith/Lists.hs     |  78 +++
 src/Data/Array/Mixed/Internal/Arith/Lists/TH.hs  |  82 +++
 src/Data/Array/Mixed/Lemmas.hs                   |  47 ++
 src/Data/Array/Mixed/Permutation.hs              | 252 ++++++++
 src/Data/Array/Mixed/Shape.hs                    | 455 ++++++++++++++
 src/Data/Array/Mixed/Types.hs                    | 110 ++++
 src/Data/Array/Nested.hs                         |  10 +-
 src/Data/Array/Nested/Internal.hs                | 326 +++++-----
 src/Data/Array/Nested/Internal/Arith.hs          | 435 -------------
 src/Data/Array/Nested/Internal/Arith/Foreign.hs  |  55 --
 src/Data/Array/Nested/Internal/Arith/Lists.hs    |  78 ---
 src/Data/Array/Nested/Internal/Arith/Lists/TH.hs |  82 ---
 15 files changed, 1729 insertions(+), 1528 deletions(-)
 create mode 100644 src/Data/Array/Mixed/Internal/Arith.hs
 create mode 100644 src/Data/Array/Mixed/Internal/Arith/Foreign.hs
 create mode 100644 src/Data/Array/Mixed/Internal/Arith/Lists.hs
 create mode 100644 src/Data/Array/Mixed/Internal/Arith/Lists/TH.hs
 create mode 100644 src/Data/Array/Mixed/Lemmas.hs
 create mode 100644 src/Data/Array/Mixed/Permutation.hs
 create mode 100644 src/Data/Array/Mixed/Shape.hs
 create mode 100644 src/Data/Array/Mixed/Types.hs
 delete mode 100644 src/Data/Array/Nested/Internal/Arith.hs
 delete mode 100644 src/Data/Array/Nested/Internal/Arith/Foreign.hs
 delete mode 100644 src/Data/Array/Nested/Internal/Arith/Lists.hs
 delete mode 100644 src/Data/Array/Nested/Internal/Arith/Lists/TH.hs

(limited to 'src')

diff --git a/src/Data/Array/Mixed.hs b/src/Data/Array/Mixed.hs
index 4ae89a1..0100ec8 100644
--- a/src/Data/Array/Mixed.hs
+++ b/src/Data/Array/Mixed.hs
@@ -30,300 +30,23 @@ module Data.Array.Mixed where
 import Control.DeepSeq (NFData(..))
 import qualified Data.Array.RankedS as S
 import qualified Data.Array.Ranked as ORB
-import Data.Bifunctor (first)
 import Data.Coerce
-import qualified Data.Foldable as Foldable
-import Data.Functor.Const
 import Data.Kind
-import Data.List (sort)
-import Data.Monoid (Sum(..))
 import Data.Proxy
-import Data.Type.Bool
 import Data.Type.Equality
 import Data.Type.Ord
 import qualified Data.Vector.Storable as VS
 import Foreign.Storable (Storable)
 import GHC.Generics (Generic)
-import GHC.IsList (IsList)
-import qualified GHC.IsList as IsList
-import GHC.TypeError
 import GHC.TypeLits
-import qualified GHC.TypeNats as TypeNats
-import Unsafe.Coerce (unsafeCoerce)
 
-import Data.Array.Nested.Internal.Arith
+import Data.Array.Mixed.Internal.Arith
+import Data.Array.Mixed.Lemmas
+import Data.Array.Mixed.Permutation
+import Data.Array.Mixed.Shape
+import Data.Array.Mixed.Types
 
 
--- | Evidence for the constraint @c a@.
-data Dict c a where
-  Dict :: c a => Dict c a
-
-fromSNat' :: SNat n -> Int
-fromSNat' = fromIntegral . fromSNat
-
-pattern SZ :: () => (n ~ 0) => SNat n
-pattern SZ <- ((\sn -> testEquality sn (SNat @0)) -> Just Refl)
-  where SZ = SNat
-
-pattern SS :: forall np1. () => forall n. (n + 1 ~ np1) => SNat n -> SNat np1
-pattern SS sn <- (snatPred -> Just (SNatPredResult sn Refl))
-  where SS = snatSucc
-
-{-# COMPLETE SZ, SS #-}
-
-snatSucc :: SNat n -> SNat (n + 1)
-snatSucc SNat = SNat
-
-data SNatPredResult np1 = forall n. SNatPredResult (SNat n) (n + 1 :~: np1)
-snatPred :: forall np1. SNat np1 -> Maybe (SNatPredResult np1)
-snatPred snp1 =
-  withKnownNat snp1 $
-    case cmpNat (Proxy @1) (Proxy @np1) of
-      LTI -> Just (SNatPredResult (SNat @(np1 - 1)) Refl)
-      EQI -> Just (SNatPredResult (SNat @(np1 - 1)) Refl)
-      GTI -> Nothing
-
-
--- | Type-level list append.
-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 family Replicate n a where
-  Replicate 0 a = '[]
-  Replicate n a = a : Replicate (n - 1) a
-
-
-type role ListX nominal representational
-type ListX :: [Maybe Nat] -> (Maybe Nat -> Type) -> Type
-data ListX sh f where
-  ZX :: ListX '[] f
-  (::%) :: f n -> ListX sh f -> ListX (n : sh) f
-deriving instance (forall n. Eq (f n)) => Eq (ListX sh f)
-deriving instance (forall n. Ord (f n)) => Ord (ListX sh f)
-infixr 3 ::%
-
-instance (forall n. Show (f n)) => Show (ListX sh f) where
-  showsPrec _ = showListX shows
-
-instance (forall n. NFData (f n)) => NFData (ListX sh f) where
-  rnf ZX = ()
-  rnf (x ::% l) = rnf x `seq` rnf l
-
-data UnconsListXRes f sh1 =
-  forall n sh. (n : sh ~ sh1) => UnconsListXRes (ListX sh f) (f n)
-unconsListX :: ListX sh1 f -> Maybe (UnconsListXRes f sh1)
-unconsListX (i ::% shl') = Just (UnconsListXRes shl' i)
-unconsListX ZX = Nothing
-
-fmapListX :: (forall n. f n -> g n) -> ListX sh f -> ListX sh g
-fmapListX _ ZX = ZX
-fmapListX f (x ::% xs) = f x ::% fmapListX f xs
-
-foldListX :: Monoid m => (forall n. f n -> m) -> ListX sh f -> m
-foldListX _ ZX = mempty
-foldListX f (x ::% xs) = f x <> foldListX f xs
-
-lengthListX :: ListX sh f -> Int
-lengthListX = getSum . foldListX (\_ -> Sum 1)
-
-snatLengthListX :: ListX sh f -> SNat (Rank sh)
-snatLengthListX ZX = SNat
-snatLengthListX (_ ::% l) | SNat <- snatLengthListX l = SNat
-
-showListX :: forall sh f. (forall n. f n -> ShowS) -> ListX sh f -> ShowS
-showListX f l = showString "[" . go "" l . showString "]"
-  where
-    go :: String -> ListX sh' f -> ShowS
-    go _ ZX = id
-    go prefix (x ::% xs) = showString prefix . f x . go "," xs
-
-listXToList :: ListX sh' (Const i) -> [i]
-listXToList ZX = []
-listXToList (Const i ::% is) = i : listXToList is
-
-
-type role IxX nominal representational
-type IxX :: [Maybe Nat] -> Type -> Type
-newtype IxX sh i = IxX (ListX sh (Const i))
-  deriving (Eq, Ord, Generic)
-
-pattern ZIX :: forall sh i. () => sh ~ '[] => IxX sh i
-pattern ZIX = IxX ZX
-
-pattern (:.%)
-  :: forall {sh1} {i}.
-     forall n sh. (n : sh ~ sh1)
-  => i -> IxX sh i -> IxX sh1 i
-pattern i :.% shl <- IxX (unconsListX -> Just (UnconsListXRes (IxX -> shl) (getConst -> i)))
-  where i :.% IxX shl = IxX (Const i ::% shl)
-infixr 3 :.%
-
-{-# COMPLETE ZIX, (:.%) #-}
-
-type IIxX sh = IxX sh Int
-
-instance Show i => Show (IxX sh i) where
-  showsPrec _ (IxX l) = showListX (\(Const i) -> shows i) l
-
-instance Functor (IxX sh) where
-  fmap f (IxX l) = IxX (fmapListX (Const . f . getConst) l)
-
-instance Foldable (IxX sh) where
-  foldMap f (IxX l) = foldListX (f . getConst) l
-
-instance NFData i => NFData (IxX sh i)
-
-
-data SMayNat i f n where
-  SUnknown :: i -> SMayNat i f Nothing
-  SKnown :: f n -> SMayNat i f (Just n)
-deriving instance (Show i, forall m. Show (f m)) => Show (SMayNat i f n)
-deriving instance (Eq i, forall m. Eq (f m)) => Eq (SMayNat i f n)
-deriving instance (Ord i, forall m. Ord (f m)) => Ord (SMayNat i f n)
-
-instance (NFData i, forall m. NFData (f m)) => NFData (SMayNat i f n) where
-  rnf (SUnknown i) = rnf i
-  rnf (SKnown x) = rnf x
-
-fromSMayNat :: (n ~ Nothing => i -> r) -> (forall m. n ~ Just m => f m -> r) -> SMayNat i f n -> r
-fromSMayNat f _ (SUnknown i) = f i
-fromSMayNat _ g (SKnown s) = g s
-
-fromSMayNat' :: SMayNat Int SNat n -> Int
-fromSMayNat' = fromSMayNat id fromSNat'
-
-type role ShX nominal representational
-type ShX :: [Maybe Nat] -> Type -> Type
-newtype ShX sh i = ShX (ListX sh (SMayNat i SNat))
-  deriving (Eq, Ord, Generic)
-
-pattern ZSX :: forall sh i. () => sh ~ '[] => ShX sh i
-pattern ZSX = ShX ZX
-
-pattern (:$%)
-  :: forall {sh1} {i}.
-     forall n sh. (n : sh ~ sh1)
-  => SMayNat i SNat n -> ShX sh i -> ShX sh1 i
-pattern i :$% shl <- ShX (unconsListX -> Just (UnconsListXRes (ShX -> shl) i))
-  where i :$% ShX shl = ShX (i ::% shl)
-infixr 3 :$%
-
-{-# COMPLETE ZSX, (:$%) #-}
-
-type IShX sh = ShX sh Int
-
-instance Show i => Show (ShX sh i) where
-  showsPrec _ (ShX l) = showListX (fromSMayNat shows (shows . fromSNat)) l
-
-instance Functor (ShX sh) where
-  fmap f (ShX l) = ShX (fmapListX (fromSMayNat (SUnknown . f) SKnown) l)
-
-instance NFData i => NFData (ShX sh i) where
-  rnf (ShX ZX) = ()
-  rnf (ShX (SUnknown i ::% l)) = rnf i `seq` rnf (ShX l)
-  rnf (ShX (SKnown SNat ::% l)) = rnf (ShX l)
-
-lengthShX :: ShX sh i -> Int
-lengthShX (ShX l) = lengthListX l
-
-shXToList :: IShX sh -> [Int]
-shXToList ZSX = []
-shXToList (smn :$% sh) = fromSMayNat' smn : shXToList sh
-
-
--- | The part of a shape that is statically known.
-type StaticShX :: [Maybe Nat] -> Type
-newtype StaticShX sh = StaticShX (ListX sh (SMayNat () SNat))
-  deriving (Eq, Ord)
-
-pattern ZKX :: forall sh. () => sh ~ '[] => StaticShX sh
-pattern ZKX = StaticShX ZX
-
-pattern (:!%)
-  :: forall {sh1}.
-     forall n sh. (n : sh ~ sh1)
-  => SMayNat () SNat n -> StaticShX sh -> StaticShX sh1
-pattern i :!% shl <- StaticShX (unconsListX -> Just (UnconsListXRes (StaticShX -> shl) i))
-  where i :!% StaticShX shl = StaticShX (i ::% shl)
-infixr 3 :!%
-
-{-# COMPLETE ZKX, (:!%) #-}
-
-instance Show (StaticShX sh) where
-  showsPrec _ (StaticShX l) = showListX (fromSMayNat shows (shows . fromSNat)) l
-
-lengthStaticShX :: StaticShX sh -> Int
-lengthStaticShX (StaticShX l) = lengthListX l
-
-geqStaticShX :: StaticShX sh -> StaticShX sh' -> Maybe (sh :~: sh')
-geqStaticShX ZKX ZKX = Just Refl
-geqStaticShX (SKnown n@SNat :!% sh) (SKnown m@SNat :!% sh')
-  | Just Refl <- sameNat n m
-  , Just Refl <- geqStaticShX sh sh'
-  = Just Refl
-geqStaticShX (SUnknown () :!% sh) (SUnknown () :!% sh')
-  | Just Refl <- geqStaticShX sh sh'
-  = Just Refl
-geqStaticShX _ _ = Nothing
-
-
--- | Evidence for the static part of a shape. This pops up only when you are
--- polymorphic in the element type of an array.
-type KnownShX :: [Maybe Nat] -> Constraint
-class KnownShX sh where knownShX :: StaticShX sh
-instance KnownShX '[] where knownShX = ZKX
-instance (KnownNat n, KnownShX sh) => KnownShX (Just n : sh) where knownShX = SKnown natSing :!% knownShX
-instance KnownShX sh => KnownShX (Nothing : sh) where knownShX = SUnknown () :!% knownShX
-
-
--- | Very untyped: only length is checked (at runtime).
-instance KnownShX sh => IsList (ListX sh (Const i)) where
-  type Item (ListX sh (Const i)) = i
-  fromList topl = go (knownShX @sh) topl
-    where
-      go :: StaticShX sh' -> [i] -> ListX sh' (Const i)
-      go ZKX [] = ZX
-      go (_ :!% sh) (i : is) = Const i ::% go sh is
-      go _ _ = error $ "IsList(ListX): Mismatched list length (type says "
-                         ++ show (lengthStaticShX (knownShX @sh)) ++ ", list has length "
-                         ++ show (length topl) ++ ")"
-  toList = listXToList
-
--- | Very untyped: only length is checked (at runtime), index bounds are __not checked__.
-instance KnownShX sh => IsList (IxX sh i) where
-  type Item (IxX sh i) = i
-  fromList = IxX . IsList.fromList
-  toList = Foldable.toList
-
--- | Untyped: length and known dimensions are checked (at runtime).
-instance KnownShX sh => IsList (ShX sh Int) where
-  type Item (ShX sh Int) = Int
-  fromList topl = ShX (go (knownShX @sh) topl)
-    where
-      go :: StaticShX sh' -> [Int] -> ListX sh' (SMayNat Int SNat)
-      go ZKX [] = ZX
-      go (SKnown sn :!% sh) (i : is)
-        | i == fromSNat' sn = SKnown sn ::% go sh is
-        | otherwise = error $ "IsList(ShX): Value does not match typing (type says "
-                                ++ show (fromSNat' sn) ++ ", list contains " ++ show i ++ ")"
-      go (SUnknown () :!% sh) (i : is) = SUnknown i ::% go sh is
-      go _ _ = error $ "IsList(ShX): Mismatched list length (type says "
-                         ++ show (lengthStaticShX (knownShX @sh)) ++ ", list has length "
-                         ++ show (length topl) ++ ")"
-  toList = shXToList
-
-
-type family Rank sh where
-  Rank '[] = 0
-  Rank (_ : sh) = Rank sh + 1
-
 type XArray :: [Maybe Nat] -> Type -> Type
 newtype XArray sh a = XArray (S.Array (Rank sh) a)
   deriving (Show, Eq, Generic)
@@ -333,180 +56,6 @@ deriving instance (Ord a, Storable a) => Ord (XArray '[] a)
 
 instance NFData a => NFData (XArray sh a)
 
-zeroIxX :: StaticShX sh -> IIxX sh
-zeroIxX ZKX = ZIX
-zeroIxX (_ :!% ssh) = 0 :.% zeroIxX ssh
-
-zeroIxX' :: IShX sh -> IIxX sh
-zeroIxX' ZSX = ZIX
-zeroIxX' (_ :$% sh) = 0 :.% zeroIxX' sh
-
--- This is a weird operation, so it has a long name
-completeShXzeros :: StaticShX sh -> IShX sh
-completeShXzeros ZKX = ZSX
-completeShXzeros (SUnknown () :!% ssh) = SUnknown 0 :$% completeShXzeros ssh
-completeShXzeros (SKnown n :!% ssh) = SKnown n :$% completeShXzeros ssh
-
-listxAppend :: ListX sh f -> ListX sh' f -> ListX (sh ++ sh') f
-listxAppend ZX idx' = idx'
-listxAppend (i ::% idx) idx' = i ::% listxAppend idx idx'
-
-ixAppend :: forall sh sh' i. IxX sh i -> IxX sh' i -> IxX (sh ++ sh') i
-ixAppend = coerce (listxAppend @_ @(Const i))
-
-shAppend :: forall sh sh' i. ShX sh i -> ShX sh' i -> ShX (sh ++ sh') i
-shAppend = coerce (listxAppend @_ @(SMayNat i SNat))
-
-listxDrop :: forall f g sh sh'. ListX (sh ++ sh') f -> ListX sh g -> ListX sh' f
-listxDrop long ZX = long
-listxDrop long (_ ::% short) = case long of _ ::% long' -> listxDrop long' short
-
-ixDrop :: forall sh sh' i. IxX (sh ++ sh') i -> IxX sh i -> IxX sh' i
-ixDrop = coerce (listxDrop @(Const i) @(Const i))
-
-shDropSSX :: forall sh sh' i. ShX (sh ++ sh') i -> StaticShX sh -> ShX sh' i
-shDropSSX = coerce (listxDrop @(SMayNat i SNat) @(SMayNat () SNat))
-
-shDropIx :: forall sh sh' i j. ShX (sh ++ sh') i -> IxX sh j -> ShX sh' i
-shDropIx = coerce (listxDrop @(SMayNat i SNat) @(Const j))
-
-shDropSh :: forall sh sh' i. ShX (sh ++ sh') i -> ShX sh i -> ShX sh' i
-shDropSh = coerce (listxDrop @(SMayNat i SNat) @(SMayNat i SNat))
-
-shTakeSSX :: forall sh sh' i. Proxy sh' -> ShX (sh ++ sh') i -> StaticShX sh -> ShX sh i
-shTakeSSX _ = flip go
-  where
-    go :: StaticShX sh1 -> ShX (sh1 ++ sh') i -> ShX sh1 i
-    go ZKX _ = ZSX
-    go (_ :!% ssh1) (n :$% sh) = n :$% go ssh1 sh
-
-ssxDropIx :: forall sh sh' i. StaticShX (sh ++ sh') -> IxX sh i -> StaticShX sh'
-ssxDropIx = coerce (listxDrop @(SMayNat () SNat) @(Const i))
-
--- TODO: generalise all these things to arbitrary @i@
-shTail :: IShX (n : sh) -> IShX sh
-shTail (_ :$% sh) = sh
-
-ssxTail :: StaticShX (n : sh) -> StaticShX sh
-ssxTail (_ :!% ssh) = ssh
-
-shAppSplit :: Proxy sh' -> StaticShX sh -> IShX (sh ++ sh') -> (IShX sh, IShX sh')
-shAppSplit _ ZKX idx = (ZSX, idx)
-shAppSplit p (_ :!% ssh) (i :$% idx) = first (i :$%) (shAppSplit p ssh idx)
-
-ssxAppend :: StaticShX sh -> StaticShX sh' -> StaticShX (sh ++ sh')
-ssxAppend ZKX sh' = sh'
-ssxAppend (n :!% sh) sh' = n :!% ssxAppend sh sh'
-
-shapeSize :: IShX sh -> Int
-shapeSize ZSX = 1
-shapeSize (n :$% sh) = fromSMayNat' n * shapeSize sh
-
--- | This may fail if @sh@ has @Nothing@s in it.
-ssxToShape' :: StaticShX sh -> Maybe (IShX sh)
-ssxToShape' ZKX = Just ZSX
-ssxToShape' (SKnown n :!% sh) = (SKnown n :$%) <$> ssxToShape' sh
-ssxToShape' (SUnknown _ :!% _) = Nothing
-
-lemReplicateSucc :: (a : Replicate n a) :~: Replicate (n + 1) a
-lemReplicateSucc = unsafeCoerce Refl
-
-ssxReplicate :: SNat n -> StaticShX (Replicate n Nothing)
-ssxReplicate SZ = ZKX
-ssxReplicate (SS (n :: SNat n'))
-  | Refl <- lemReplicateSucc @(Nothing @Nat) @n'
-  = SUnknown () :!% ssxReplicate n
-
-fromLinearIdx :: IShX sh -> Int -> IIxX 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 :: IShX sh -> Int -> (IIxX sh, Int)
-    go ZSX i = (ZIX, i)
-    go (n :$% sh) i =
-      let (idx, i') = go sh i
-          (upi, locali) = i' `quotRem` fromSMayNat' n
-      in (locali :.% idx, upi)
-
-toLinearIdx :: IShX sh -> IIxX sh -> Int
-toLinearIdx = \sh i -> fst (go sh i)
-  where
-    -- returns (index in subarray, size of subarray)
-    go :: IShX sh -> IIxX sh -> (Int, Int)
-    go ZSX ZIX = (0, 1)
-    go (n :$% sh) (i :.% ix) =
-      let (lidx, sz) = go sh ix
-      in (sz * i + lidx, fromSMayNat' n * sz)
-
-enumShape :: IShX sh -> [IIxX sh]
-enumShape = \sh -> go sh id []
-  where
-    go :: IShX sh -> (IIxX sh -> a) -> [a] -> [a]
-    go ZSX f = (f ZIX :)
-    go (n :$% sh) f = foldr (.) id [go sh (f . (i :.%)) | i <- [0 .. fromSMayNat' n - 1]]
-
-shapeLshape :: IShX sh -> S.ShapeL
-shapeLshape ZSX = []
-shapeLshape (n :$% sh) = fromSMayNat' n : shapeLshape sh
-
-ssxLength :: StaticShX sh -> Int
-ssxLength ZKX = 0
-ssxLength (_ :!% ssh) = 1 + ssxLength ssh
-
-ssxIotaFrom :: Int -> StaticShX sh -> [Int]
-ssxIotaFrom _ ZKX = []
-ssxIotaFrom i (_ :!% ssh) = i : ssxIotaFrom (i+1) ssh
-
-type Flatten sh = Flatten' 1 sh
-
-type family Flatten' acc sh where
-  Flatten' acc '[] = Just acc
-  Flatten' acc (Nothing : sh) = Nothing
-  Flatten' acc (Just n : sh) = Flatten' (acc * n) sh
-
-flattenSSX :: StaticShX sh -> SMayNat () SNat (Flatten sh)
-flattenSSX = go (SNat @1)
-  where
-    go :: SNat acc -> StaticShX sh -> SMayNat () SNat (Flatten' acc sh)
-    go acc ZKX = SKnown acc
-    go _ (SUnknown () :!% _) = SUnknown ()
-    go acc (SKnown sn :!% sh) = go (mulSNat acc sn) sh
-
-flattenSh :: IShX sh -> SMayNat Int SNat (Flatten sh)
-flattenSh = go (SNat @1)
-  where
-    go :: SNat acc -> IShX sh -> SMayNat Int SNat (Flatten' acc sh)
-    go acc ZSX = SKnown acc
-    go acc (SUnknown n :$% sh) = SUnknown (goUnknown (fromSNat' acc * n) sh)
-    go acc (SKnown sn :$% sh) = go (mulSNat acc sn) sh
-
-    goUnknown :: Int -> IShX sh -> Int
-    goUnknown acc ZSX = acc
-    goUnknown acc (SUnknown n :$% sh) = goUnknown (acc * n) sh
-    goUnknown acc (SKnown sn :$% sh) = goUnknown (acc * fromSNat' sn) sh
-
-staticShapeFrom :: IShX sh -> StaticShX sh
-staticShapeFrom ZSX = ZKX
-staticShapeFrom (n :$% sh) = fromSMayNat (\_ -> SUnknown ()) SKnown n :!% staticShapeFrom sh
-
-lemRankApp :: StaticShX sh1 -> StaticShX sh2
-           -> Rank (sh1 ++ sh2) :~: Rank sh1 + Rank sh2
-lemRankApp _ _ = unsafeCoerce Refl  -- TODO improve this
-
-lemRankAppComm :: StaticShX sh1 -> StaticShX sh2
-               -> Rank (sh1 ++ sh2) :~: Rank (sh2 ++ sh1)
-lemRankAppComm _ _ = unsafeCoerce Refl  -- TODO improve this
-
-lemKnownNatRank :: IShX sh -> Dict KnownNat (Rank sh)
-lemKnownNatRank ZSX = Dict
-lemKnownNatRank (_ :$% sh) | Dict <- lemKnownNatRank sh = Dict
-
-lemKnownNatRankSSX :: StaticShX sh -> Dict KnownNat (Rank sh)
-lemKnownNatRankSSX ZKX = Dict
-lemKnownNatRankSSX (_ :!% ssh) | Dict <- lemKnownNatRankSSX ssh = Dict
 
 shape :: forall sh a. StaticShX sh -> XArray sh a -> IShX sh
 shape = \ssh (XArray arr) -> go ssh (S.shapeL arr)
@@ -519,7 +68,7 @@ shape = \ssh (XArray arr) -> go ssh (S.shapeL arr)
 fromVector :: forall sh a. Storable a => IShX sh -> VS.Vector a -> XArray sh a
 fromVector sh v
   | Dict <- lemKnownNatRank sh
-  = XArray (S.fromVector (shapeLshape sh) v)
+  = XArray (S.fromVector (shxToList sh) v)
 
 toVector :: Storable a => XArray sh a -> VS.Vector a
 toVector (XArray arr) = S.toVector arr
@@ -527,23 +76,18 @@ toVector (XArray arr) = S.toVector arr
 scalar :: Storable a => a -> XArray '[] a
 scalar = XArray . S.scalar
 
-eqShX :: IShX sh1 -> IShX sh2 -> Bool
-eqShX ZSX ZSX = True
-eqShX (n :$% sh1) (m :$% sh2) = fromSMayNat' n == fromSMayNat' m && eqShX sh1 sh2
-eqShX _ _ = False
-
 -- | Will throw if the array does not have the casted-to shape.
 cast :: forall sh1 sh2 sh' a. Rank sh1 ~ Rank sh2
      => StaticShX sh1 -> IShX sh2 -> StaticShX sh'
      -> XArray (sh1 ++ sh') a -> XArray (sh2 ++ sh') a
 cast ssh1 sh2 ssh' (XArray arr)
   | Refl <- lemRankApp ssh1 ssh'
-  , Refl <- lemRankApp (staticShapeFrom sh2) ssh'
+  , Refl <- lemRankApp (ssxFromShape sh2) ssh'
   = let arrsh :: IShX sh1
-        (arrsh, _) = shAppSplit (Proxy @sh') ssh1 (shape (ssxAppend ssh1 ssh') (XArray arr))
-    in if eqShX arrsh sh2
-         then XArray arr
-         else error $ "Data.Array.Mixed.cast: Cannot cast (" ++ show arrsh ++ ") to (" ++ show sh2 ++ ")"
+        (arrsh, _) = shxSplitApp (Proxy @sh') ssh1 (shape (ssxAppend ssh1 ssh') (XArray arr))
+    in case shxEqual arrsh sh2 of
+         Just _ -> XArray arr
+         Nothing -> error $ "Data.Array.Mixed.cast: Cannot cast (" ++ show arrsh ++ ") to (" ++ show sh2 ++ ")"
 
 unScalar :: Storable a => XArray '[] a -> a
 unScalar (XArray a) = S.unScalar a
@@ -551,24 +95,24 @@ unScalar (XArray a) = S.unScalar a
 replicate :: forall sh sh' a. Storable a => IShX sh -> StaticShX sh' -> XArray sh' a -> XArray (sh ++ sh') a
 replicate sh ssh' (XArray arr)
   | Dict <- lemKnownNatRankSSX ssh'
-  , Dict <- lemKnownNatRankSSX (ssxAppend (staticShapeFrom sh) ssh')
-  , Refl <- lemRankApp (staticShapeFrom sh) ssh'
-  = XArray (S.stretch (shapeLshape sh ++ S.shapeL arr) $
-            S.reshape (map (const 1) (shapeLshape sh) ++ S.shapeL arr) $
+  , Dict <- lemKnownNatRankSSX (ssxAppend (ssxFromShape sh) ssh')
+  , Refl <- lemRankApp (ssxFromShape sh) ssh'
+  = XArray (S.stretch (shxToList sh ++ S.shapeL arr) $
+            S.reshape (map (const 1) (shxToList sh) ++ S.shapeL arr) $
               arr)
 
 replicateScal :: forall sh a. Storable a => IShX sh -> a -> XArray sh a
 replicateScal sh x
   | Dict <- lemKnownNatRank sh
-  = XArray (S.constant (shapeLshape sh) x)
+  = XArray (S.constant (shxToList sh) x)
 
 generate :: Storable a => IShX sh -> (IIxX sh -> a) -> XArray sh a
-generate sh f = fromVector sh $ VS.generate (shapeSize sh) (f . fromLinearIdx sh)
+generate sh f = fromVector sh $ VS.generate (shxSize sh) (f . ixxFromLinear sh)
 
 -- generateM :: (Monad m, Storable a) => IShX sh -> (IIxX sh -> m a) -> m (XArray sh a)
 -- generateM sh f | Dict <- lemKnownNatRank sh =
---   XArray . S.fromVector (shapeLshape sh)
---     <$> VS.generateM (shapeSize sh) (f . fromLinearIdx sh)
+--   XArray . S.fromVector (shxShapeL sh)
+--     <$> VS.generateM (shxSize sh) (f . ixxFromLinear sh)
 
 indexPartial :: Storable a => XArray (sh ++ sh') a -> IIxX sh -> XArray sh' a
 indexPartial (XArray arr) ZIX = XArray arr
@@ -580,24 +124,6 @@ index xarr i
   = let XArray arr' = indexPartial xarr i :: XArray '[] a
     in S.unScalar arr'
 
-type family AddMaybe n m where
-  AddMaybe Nothing _ = Nothing
-  AddMaybe (Just _) Nothing = Nothing
-  AddMaybe (Just n) (Just m) = Just (n + m)
-
--- This should be a function in base
-plusSNat :: SNat n -> SNat m -> SNat (n + m)
-plusSNat n m = TypeNats.withSomeSNat (TypeNats.fromSNat n + TypeNats.fromSNat m) unsafeCoerce
-
--- This should be a function in base
-mulSNat :: SNat n -> SNat m -> SNat (n * m)
-mulSNat n m = TypeNats.withSomeSNat (TypeNats.fromSNat n * TypeNats.fromSNat m) unsafeCoerce
-
-smnAddMaybe :: SMayNat Int SNat n -> SMayNat Int SNat m -> SMayNat Int SNat (AddMaybe n m)
-smnAddMaybe (SUnknown n) m = SUnknown (n + fromSMayNat' m)
-smnAddMaybe (SKnown n) (SUnknown m) = SUnknown (fromSNat' n + m)
-smnAddMaybe (SKnown n) (SKnown m) = SKnown (plusSNat n m)
-
 append :: forall n m sh a. Storable a
        => StaticShX sh -> XArray (n : sh) a -> XArray (m : sh) a -> XArray (AddMaybe n m : sh) a
 append ssh (XArray a) (XArray b)
@@ -639,9 +165,9 @@ rerank :: forall sh sh1 sh2 a b.
        -> XArray (sh ++ sh1) a -> XArray (sh ++ sh2) b
 rerank ssh ssh1 ssh2 f xarr@(XArray arr)
   | Dict <- lemKnownNatRankSSX (ssxAppend ssh ssh2)
-  = let (sh, _) = shAppSplit (Proxy @sh1) ssh (shape (ssxAppend ssh ssh1) xarr)
-    in if any (== 0) (shapeLshape sh)
-         then XArray (S.fromList (shapeLshape (shAppend sh (completeShXzeros ssh2))) [])
+  = let (sh, _) = shxSplitApp (Proxy @sh1) ssh (shape (ssxAppend ssh ssh1) xarr)
+    in if any (== 0) (shxToList sh)
+         then XArray (S.fromList (shxToList (shxAppend sh (shxCompleteZeros ssh2))) [])
          else case () of
            () | Dict <- lemKnownNatRankSSX ssh
               , Dict <- lemKnownNatRankSSX ssh2
@@ -666,9 +192,9 @@ rerank2 :: forall sh sh1 sh2 a b c.
         -> XArray (sh ++ sh1) a -> XArray (sh ++ sh1) b -> XArray (sh ++ sh2) c
 rerank2 ssh ssh1 ssh2 f xarr1@(XArray arr1) (XArray arr2)
   | Dict <- lemKnownNatRankSSX (ssxAppend ssh ssh2)
-  = let (sh, _) = shAppSplit (Proxy @sh1) ssh (shape (ssxAppend ssh ssh1) xarr1)
-    in if any (== 0) (shapeLshape sh)
-         then XArray (S.fromList (shapeLshape (shAppend sh (completeShXzeros ssh2))) [])
+  = let (sh, _) = shxSplitApp (Proxy @sh1) ssh (shape (ssxAppend ssh ssh1) xarr1)
+    in if any (== 0) (shxToList sh)
+         then XArray (S.fromList (shxToList (shxAppend sh (shxCompleteZeros ssh2))) [])
          else case () of
            () | Dict <- lemKnownNatRankSSX ssh
               , Dict <- lemKnownNatRankSSX ssh2
@@ -678,211 +204,14 @@ rerank2 ssh ssh1 ssh2 f xarr1@(XArray arr1) (XArray arr2)
                           (\a b -> let XArray r = f (XArray a) (XArray b) in r)
                           arr1 arr2)
 
-type family Elem x l where
-  Elem x '[] = 'False
-  Elem x (x : _) = 'True
-  Elem x (_ : ys) = Elem x ys
-
-type family AllElem' as bs where
-  AllElem' '[] bs = 'True
-  AllElem' (a : as) bs = Elem a bs && AllElem' as bs
-
-type AllElem as bs = Assert (AllElem' as bs)
-  (TypeError (Text "The elements of " :<>: ShowType as :<>: Text " are not all in " :<>: ShowType bs))
-
-type family Count i n where
-  Count n n = '[]
-  Count i n = i : Count (i + 1) n
-
-type Permutation as = (AllElem as (Count 0 (Rank as)), AllElem (Count 0 (Rank as)) as)
-
-type family Index i sh where
-  Index 0 (n : sh) = n
-  Index i (_ : sh) = Index (i - 1) sh
-
-type family Permute is sh where
-  Permute '[] sh = '[]
-  Permute (i : is) sh = Index i sh : Permute is sh
-
-type PermutePrefix is sh = Permute is (TakeLen is sh) ++ DropLen is sh
-
-data HList f list where
-  HNil :: HList f '[]
-  HCons :: f a -> HList f l -> HList f (a : l)
-infixr 5 `HCons`
-deriving instance (forall a. Show (f a)) => Show (HList f list)
-deriving instance (forall a. Eq (f a)) => Eq (HList f list)
-
-foldHList :: Monoid m => (forall a. f a -> m) -> HList f list -> m
-foldHList _ HNil = mempty
-foldHList f (x `HCons` l) = f x <> foldHList f l
-
-snatLengthHList :: HList f list -> SNat (Rank list)
-snatLengthHList HNil = SNat
-snatLengthHList (_ `HCons` l) | SNat <- snatLengthHList l = SNat
-
-permFromList :: [Int] -> (forall list. HList SNat list -> r) -> r
-permFromList [] k = k HNil
-permFromList (x : xs) k = withSomeSNat (fromIntegral x) $ \case
-  Just sn -> permFromList xs $ \list -> k (sn `HCons` list)
-  Nothing -> error $ "Data.Array.Mixed.permFromList: negative number in list: " ++ show x
-
-permToList :: HList SNat list -> [Int]
-permToList = foldHList (pure . fromSNat')
-
-type family TakeLen ref l where
-  TakeLen '[] l = '[]
-  TakeLen (_ : ref) (x : xs) = x : TakeLen ref xs
-
-type family DropLen ref l where
-  DropLen '[] l = l
-  DropLen (_ : ref) (_ : xs) = DropLen ref xs
-
-lemRankPermute :: Proxy sh -> HList SNat is -> Rank (Permute is sh) :~: Rank is
-lemRankPermute _ HNil = Refl
-lemRankPermute p (_ `HCons` is) | Refl <- lemRankPermute p is = Refl
-
-lemRankDropLen :: forall is sh. (Rank is <= Rank sh)
-               => StaticShX sh -> HList SNat is -> Rank (DropLen is sh) :~: Rank sh - Rank is
-lemRankDropLen ZKX HNil = Refl
-lemRankDropLen (_ :!% sh) (_ `HCons` is) | Refl <- lemRankDropLen sh is = Refl
-lemRankDropLen (_ :!% _) HNil = Refl
-lemRankDropLen ZKX (_ `HCons` _) = error "1 <= 0"
-
-lemIndexSucc :: Proxy i -> Proxy a -> Proxy l -> Index (i + 1) (a : l) :~: Index i l
-lemIndexSucc _ _ _ = unsafeCoerce Refl
-
-listxTakeLen :: forall f is sh. HList SNat is -> ListX sh f -> ListX (TakeLen is sh) f
-listxTakeLen HNil _ = ZX
-listxTakeLen (_ `HCons` is) (n ::% sh) = n ::% listxTakeLen is sh
-listxTakeLen (_ `HCons` _) ZX = error "Permutation longer than shape"
-
-listxDropLen :: forall f is sh. HList SNat is -> ListX sh f -> ListX (DropLen is sh) f
-listxDropLen HNil sh = sh
-listxDropLen (_ `HCons` is) (_ ::% sh) = listxDropLen is sh
-listxDropLen (_ `HCons` _) ZX = error "Permutation longer than shape"
-
-listxPermute :: forall f is sh. HList SNat is -> ListX sh f -> ListX (Permute is sh) f
-listxPermute HNil _ = ZX
-listxPermute (i `HCons` (is :: HList SNat is')) (sh :: ListX sh f) = listxIndex (Proxy @is') (Proxy @sh) i sh (listxPermute is sh)
-
-listxIndex :: forall f is shT i sh. Proxy is -> Proxy shT -> SNat i -> ListX sh f -> ListX (Permute is shT) f -> ListX (Index i sh : Permute is shT) f
-listxIndex _ _ SZ (n ::% _) rest = n ::% rest
-listxIndex p pT (SS (i :: SNat i')) ((_ :: f n) ::% (sh :: ListX sh' f)) rest
-  | Refl <- lemIndexSucc (Proxy @i') (Proxy @n) (Proxy @sh')
-  = listxIndex p pT i sh rest
-listxIndex _ _ _ ZX _ = error "Index into empty shape"
-
-listxPermutePrefix :: forall f is sh. HList SNat is -> ListX sh f -> ListX (PermutePrefix is sh) f
-listxPermutePrefix perm sh = listxAppend (listxPermute perm (listxTakeLen perm sh)) (listxDropLen perm sh)
-
-ssxTakeLen :: forall is sh. HList SNat is -> StaticShX sh -> StaticShX (TakeLen is sh)
-ssxTakeLen = coerce (listxTakeLen @(SMayNat () SNat))
-
-ssxDropLen :: HList SNat is -> StaticShX sh -> StaticShX (DropLen is sh)
-ssxDropLen = coerce (listxDropLen @(SMayNat () SNat))
-
-ssxPermute :: HList SNat is -> StaticShX sh -> StaticShX (Permute is sh)
-ssxPermute = coerce (listxPermute @(SMayNat () SNat))
-
-ssxIndex :: Proxy is -> Proxy shT -> SNat i -> StaticShX sh -> StaticShX (Permute is shT) -> StaticShX (Index i sh : Permute is shT)
-ssxIndex p1 p2 = coerce (listxIndex @(SMayNat () SNat) p1 p2)
-
-ssxPermutePrefix :: HList SNat is -> StaticShX sh -> StaticShX (PermutePrefix is sh)
-ssxPermutePrefix = coerce (listxPermutePrefix @(SMayNat () SNat))
-
-shPermutePrefix :: HList SNat is -> IShX sh -> IShX (PermutePrefix is sh)
-shPermutePrefix = coerce (listxPermutePrefix @(SMayNat Int SNat))
-
--- TODO: test this thing more properly
-invertPermutation :: HList SNat is
-                  -> (forall is'.
-                           Permutation is'
-                        => HList SNat is'
-                        -> (forall sh. Rank sh ~ Rank is => StaticShX sh -> Permute is' (Permute is sh) :~: sh)
-                        -> r)
-                  -> r
-invertPermutation = \perm k ->
-  genPerm perm $ \(invperm :: HList SNat is') ->
-    let sn = snatLengthHList invperm
-    in case (provePerm1 (Proxy @is') sn invperm, provePerm2 (SNat @0) sn invperm) of
-         (Just Refl, Just Refl) ->
-           k invperm
-             (\ssh -> case provePermInverse perm invperm ssh of
-                        Just eq -> eq
-                        Nothing -> error $ "invertPermutation: did not generate inverse? perm = " ++ show perm
-                                           ++ " ; invperm = " ++ show invperm)
-         _ -> error $ "invertPermutation: did not generate permutation? perm = " ++ show perm
-                      ++ " ; invperm = " ++ show invperm
-  where
-    genPerm :: HList SNat is -> (forall is'. HList SNat is' -> r) -> r
-    genPerm perm =
-      let permList = foldHList (pure . fromSNat) perm
-      in toHList $ map snd (sort (zip permList [0..]))
-      where
-        toHList :: [Natural] -> (forall is'. HList SNat is' -> r) -> r
-        toHList [] k = k HNil
-        toHList (n : ns) k = toHList ns $ \l -> TypeNats.withSomeSNat n $ \sn -> k (HCons sn l)
-
-    lemElemCount :: (0 <= n, Compare n m ~ LT) => proxy n -> proxy m -> Elem n (Count 0 m) :~: True
-    lemElemCount _ _ = unsafeCoerce Refl
-
-    lemCount :: (OrdCond (Compare i n) True False True ~ True) => proxy i -> proxy n -> Count i n :~: i : Count (i + 1) n
-    lemCount _ _ = unsafeCoerce Refl
-
-    lemElem :: Elem x ys ~ True => proxy x -> proxy' (y : ys) -> Elem x (y : ys) :~: True
-    lemElem _ _ = unsafeCoerce Refl
-
-    provePerm1 :: Proxy isTop -> SNat (Rank isTop) -> HList SNat is'
-               -> Maybe (AllElem' is' (Count 0 (Rank isTop)) :~: True)
-    provePerm1 _ _ HNil = Just (Refl)
-    provePerm1 p rtop@SNat (HCons sn@SNat perm)
-      | Just Refl <- provePerm1 p rtop perm
-      = case (cmpNat (SNat @0) sn, cmpNat sn rtop) of
-          (LTI, LTI) | Refl <- lemElemCount sn rtop -> Just Refl
-          (EQI, LTI) | Refl <- lemElemCount sn rtop -> Just Refl
-          _ -> Nothing
-      | otherwise
-      = Nothing
-
-    provePerm2 :: SNat i -> SNat n -> HList SNat is' -> Maybe (AllElem' (Count i n) is' :~: True)
-    provePerm2 = \i@(SNat :: SNat i) n@SNat perm ->
-      case cmpNat i n of
-        EQI -> Just Refl
-        LTI | Refl <- lemCount i n
-            , Just Refl <- provePerm2 (SNat @(i + 1)) n perm
-            -> checkElem i perm
-            | otherwise -> Nothing
-        GTI -> error "unreachable"
-      where
-        checkElem :: SNat i -> HList SNat is' -> Maybe (Elem i is' :~: True)
-        checkElem _ HNil = Nothing
-        checkElem i@SNat (HCons k@SNat perm :: HList SNat is') =
-          case sameNat i k of
-            Just Refl -> Just Refl
-            Nothing | Just Refl <- checkElem i perm, Refl <- lemElem i (Proxy @is') -> Just Refl
-                    | otherwise -> Nothing
-
-    provePermInverse :: HList SNat is -> HList SNat is' -> StaticShX sh -> Maybe (Permute is' (Permute is sh) :~: sh)
-    provePermInverse perm perminv ssh = geqStaticShX (ssxPermute perminv (ssxPermute perm ssh)) ssh
-
-applyPermX :: forall f is sh. HList SNat is -> ListX sh f -> ListX (PermutePrefix is sh) f
-applyPermX perm sh = listxAppend (listxPermute perm (listxTakeLen perm sh)) (listxDropLen perm sh)
-
-applyPermIxX :: forall i is sh. HList SNat is -> IxX sh i -> IxX (PermutePrefix is sh) i
-applyPermIxX = coerce (applyPermX @(Const i))
-
-applyPermShX :: forall i is sh. HList SNat is -> ShX sh i -> ShX (PermutePrefix is sh) i
-applyPermShX = coerce (applyPermX @(SMayNat i SNat))
-
-class KnownNatList l where makeNatList :: HList SNat l
-instance KnownNatList '[] where makeNatList = HNil
-instance (KnownNat n, KnownNatList l) => KnownNatList (n : l) where makeNatList = natSing `HCons` makeNatList
+class KnownNatList l where makeNatList :: Perm l
+instance KnownNatList '[] where makeNatList = PNil
+instance (KnownNat n, KnownNatList l) => KnownNatList (n : l) where makeNatList = natSing `PCons` makeNatList
 
 -- | The list argument gives indices into the original dimension list.
-transpose :: forall is sh a. (Permutation is, Rank is <= Rank sh)
+transpose :: forall is sh a. (IsPermutation is, Rank is <= Rank sh)
           => StaticShX sh
-          -> HList SNat is
+          -> Perm is
           -> XArray sh a
           -> XArray (PermutePrefix is sh) a
 transpose ssh perm (XArray arr)
@@ -890,7 +219,7 @@ transpose ssh perm (XArray arr)
   , Refl <- lemRankApp (ssxPermute perm (ssxTakeLen perm ssh)) (ssxDropLen perm ssh)
   , Refl <- lemRankPermute (Proxy @(TakeLen is sh)) perm
   , Refl <- lemRankDropLen ssh perm
-  = XArray (S.transpose (permToList perm) arr)
+  = XArray (S.transpose (permToList' perm) arr)
 
 -- | The list argument gives indices into the original dimension list.
 --
@@ -929,14 +258,14 @@ sumInner :: forall sh sh' a. (Storable a, NumElt a)
          => StaticShX sh -> StaticShX sh' -> XArray (sh ++ sh') a -> XArray sh a
 sumInner ssh ssh' arr
   | Refl <- lemAppNil @sh
-  = let (_, sh') = shAppSplit (Proxy @sh') ssh (shape (ssxAppend ssh ssh') arr)
-        sh'F = flattenSh sh' :$% ZSX
-        ssh'F = staticShapeFrom sh'F
+  = let (_, sh') = shxSplitApp (Proxy @sh') ssh (shape (ssxAppend ssh ssh') arr)
+        sh'F = shxFlatten sh' :$% ZSX
+        ssh'F = ssxFromShape sh'F
 
         go :: XArray (sh ++ '[Flatten sh']) a -> XArray sh a
         go (XArray arr')
           | Refl <- lemRankApp ssh ssh'F
-          , let sn = snatLengthListX (let StaticShX l = ssh in l)
+          , let sn = listxLengthSNat (let StaticShX l = ssh in l)
           = XArray (numEltSum1Inner sn arr')
 
     in go $
@@ -949,10 +278,10 @@ sumOuter :: forall sh sh' a. (Storable a, NumElt a)
          => StaticShX sh -> StaticShX sh' -> XArray (sh ++ sh') a -> XArray sh' a
 sumOuter ssh ssh' arr
   | Refl <- lemAppNil @sh
-  = let (sh, _) = shAppSplit (Proxy @sh') ssh (shape (ssxAppend ssh ssh') arr)
-        shF = flattenSh sh :$% ZSX
-    in sumInner ssh' (staticShapeFrom shF) $
-       transpose2 (staticShapeFrom shF) ssh' $
+  = let (sh, _) = shxSplitApp (Proxy @sh') ssh (shape (ssxAppend ssh ssh') arr)
+        shF = shxFlatten sh :$% ZSX
+    in sumInner ssh' (ssxFromShape shF) $
+       transpose2 (ssxFromShape shF) ssh' $
        reshapePartial ssh ssh' shF $
          arr
 
@@ -988,7 +317,7 @@ toList1 (XArray arr) = S.toList arr
 empty :: forall sh a. Storable a => IShX sh -> XArray sh a
 empty sh
   | Dict <- lemKnownNatRank sh
-  = XArray (S.constant (shapeLshape sh)
+  = XArray (S.constant (shxToList sh)
                        (error "Data.Array.Mixed.empty: shape was not empty"))
 
 slice :: SNat i -> SNat n -> XArray (Just (i + n + k) : sh) a -> XArray (Just n : sh) a
@@ -1005,14 +334,14 @@ reshape :: forall sh1 sh2 a. Storable a => StaticShX sh1 -> IShX sh2 -> XArray s
 reshape ssh1 sh2 (XArray arr)
   | Dict <- lemKnownNatRankSSX ssh1
   , Dict <- lemKnownNatRank sh2
-  = XArray (S.reshape (shapeLshape sh2) arr)
+  = XArray (S.reshape (shxToList sh2) arr)
 
 -- | Throws if the given array and the target shape do not have the same number of elements.
 reshapePartial :: forall sh1 sh2 sh' a. Storable a => StaticShX sh1 -> StaticShX sh' -> IShX sh2 -> XArray (sh1 ++ sh') a -> XArray (sh2 ++ sh') a
 reshapePartial ssh1 ssh' sh2 (XArray arr)
   | Dict <- lemKnownNatRankSSX (ssxAppend ssh1 ssh')
-  , Dict <- lemKnownNatRankSSX (ssxAppend (staticShapeFrom sh2) ssh')
-  = XArray (S.reshape (shapeLshape sh2 ++ drop (lengthStaticShX ssh1) (S.shapeL arr)) arr)
+  , Dict <- lemKnownNatRankSSX (ssxAppend (ssxFromShape sh2) ssh')
+  = XArray (S.reshape (shxToList sh2 ++ drop (ssxLength ssh1) (S.shapeL arr)) arr)
 
 -- this was benchmarked to be (slightly) faster than S.iota, S.generate and S.fromVector(VS.enumFromTo).
 iota :: (Enum a, Storable a) => SNat n -> XArray '[Just n] a
diff --git a/src/Data/Array/Mixed/Internal/Arith.hs b/src/Data/Array/Mixed/Internal/Arith.hs
new file mode 100644
index 0000000..cf6820b
--- /dev/null
+++ b/src/Data/Array/Mixed/Internal/Arith.hs
@@ -0,0 +1,435 @@
+{-# LANGUAGE DataKinds #-}
+{-# LANGUAGE LambdaCase #-}
+{-# LANGUAGE ScopedTypeVariables #-}
+{-# LANGUAGE TemplateHaskell #-}
+{-# LANGUAGE TypeApplications #-}
+{-# LANGUAGE TypeOperators #-}
+{-# LANGUAGE ViewPatterns #-}
+{-# OPTIONS_GHC -fplugin GHC.TypeLits.KnownNat.Solver #-}
+module Data.Array.Mixed.Internal.Arith where
+
+import Control.Monad (forM, guard)
+import qualified Data.Array.Internal as OI
+import qualified Data.Array.Internal.RankedG as RG
+import qualified Data.Array.Internal.RankedS as RS
+import Data.Bits
+import Data.Int
+import Data.List (sort)
+import qualified Data.Vector.Storable as VS
+import qualified Data.Vector.Storable.Mutable as VSM
+import Foreign.C.Types
+import Foreign.Ptr
+import Foreign.Storable (Storable)
+import GHC.TypeLits
+import Language.Haskell.TH
+import System.IO.Unsafe
+
+import Data.Array.Mixed.Internal.Arith.Foreign
+import Data.Array.Mixed.Internal.Arith.Lists
+
+
+liftVEltwise1 :: Storable a
+              => SNat n
+              -> (VS.Vector a -> VS.Vector a)
+              -> RS.Array n a -> RS.Array n a
+liftVEltwise1 SNat f arr@(RS.A (RG.A sh (OI.T strides offset vec)))
+  | Just prefixSz <- stridesDense sh strides =
+      let vec' = f (VS.slice offset prefixSz vec)
+      in RS.A (RG.A sh (OI.T strides 0 vec'))
+  | otherwise = RS.fromVector sh (f (RS.toVector arr))
+
+liftVEltwise2 :: Storable a
+              => SNat n
+              -> (Either a (VS.Vector a) -> Either a (VS.Vector a) -> VS.Vector a)
+              -> RS.Array n a -> RS.Array n a -> RS.Array n a
+liftVEltwise2 SNat f
+    arr1@(RS.A (RG.A sh1 (OI.T strides1 offset1 vec1)))
+    arr2@(RS.A (RG.A sh2 (OI.T strides2 offset2 vec2)))
+  | sh1 /= sh2 = error $ "liftVEltwise2: shapes unequal: " ++ show sh1 ++ " vs " ++ show sh2
+  | product sh1 == 0 = arr1  -- if the arrays are empty, just return one of the empty inputs
+  | otherwise = case (stridesDense sh1 strides1, stridesDense sh2 strides2) of
+      (Just 1, Just 1) ->  -- both are a (potentially replicated) scalar; just apply f to the scalars
+        let vec' = f (Left (vec1 VS.! offset1)) (Left (vec2 VS.! offset2))
+        in RS.A (RG.A sh1 (OI.T strides1 0 vec'))
+      (Just 1, Just n) ->  -- scalar * dense
+        RS.fromVector sh1 (f (Left (vec1 VS.! offset1)) (Right (VS.slice offset2 n vec2)))
+      (Just n, Just 1) ->  -- dense * scalar
+        RS.fromVector sh1 (f (Right (VS.slice offset1 n vec1)) (Left (vec2 VS.! offset2)))
+      (_, _) ->  -- fallback case
+        RS.fromVector sh1 (f (Right (RS.toVector arr1)) (Right (RS.toVector arr2)))
+
+-- | Given the shape vector and the stride vector, return whether this vector
+-- of strides uses a dense prefix of its backing array. If so, the number of
+-- elements in this prefix is returned.
+-- This excludes any offset.
+stridesDense :: [Int] -> [Int] -> Maybe Int
+stridesDense sh _ | any (<= 0) sh = Just 0
+stridesDense sh str =
+  -- sort dimensions on their stride, ascending, dropping any zero strides
+  case dropWhile ((== 0) . fst) (sort (zip str sh)) of
+    [] -> Just 1
+    (1, n) : (unzip -> (str', sh')) -> checkCover n sh' str'
+    _ -> Nothing  -- if the smallest stride is not 1, it will never be dense
+  where
+    -- Given size of currently densely covered region at beginning of the
+    -- array, the remaining shape vector and the corresponding remaining stride
+    -- vector, return whether this all together covers a dense prefix of the
+    -- array. If it does, return the number of elements in this prefix.
+    checkCover :: Int -> [Int] -> [Int] -> Maybe Int
+    checkCover block [] [] = Just block
+    checkCover block (n : sh') (s : str') = guard (s <= block) >> checkCover (max block (n * s)) sh' str'
+    checkCover _ _ _ = error "Orthotope array's shape vector and stride vector have different lengths"
+
+{-# NOINLINE vectorOp1 #-}
+vectorOp1 :: forall a b. Storable a
+          => (Ptr a -> Ptr b)
+          -> (Int64 -> Ptr b -> Ptr b -> IO ())
+          -> VS.Vector a -> VS.Vector a
+vectorOp1 ptrconv f v = unsafePerformIO $ do
+  outv <- VSM.unsafeNew (VS.length v)
+  VSM.unsafeWith outv $ \poutv ->
+    VS.unsafeWith v $ \pv ->
+      f (fromIntegral (VS.length v)) (ptrconv poutv) (ptrconv pv)
+  VS.unsafeFreeze outv
+
+-- | If two vectors are given, assumes that they have the same length.
+{-# NOINLINE vectorOp2 #-}
+vectorOp2 :: forall a b. Storable a
+          => (a -> b)
+          -> (Ptr a -> Ptr b)
+          -> (a -> a -> a)
+          -> (Int64 -> Ptr b -> b -> Ptr b -> IO ())  -- sv
+          -> (Int64 -> Ptr b -> Ptr b -> b -> IO ())  -- vs
+          -> (Int64 -> Ptr b -> Ptr b -> Ptr b -> IO ())  -- vv
+          -> Either a (VS.Vector a) -> Either a (VS.Vector a) -> VS.Vector a
+vectorOp2 valconv ptrconv fss fsv fvs fvv = \cases
+  (Left x) (Left y) -> VS.singleton (fss x y)
+
+  (Left x) (Right vy) ->
+    unsafePerformIO $ do
+      outv <- VSM.unsafeNew (VS.length vy)
+      VSM.unsafeWith outv $ \poutv ->
+        VS.unsafeWith vy $ \pvy ->
+          fsv (fromIntegral (VS.length vy)) (ptrconv poutv) (valconv x) (ptrconv pvy)
+      VS.unsafeFreeze outv
+
+  (Right vx) (Left y) ->
+    unsafePerformIO $ do
+      outv <- VSM.unsafeNew (VS.length vx)
+      VSM.unsafeWith outv $ \poutv ->
+        VS.unsafeWith vx $ \pvx ->
+          fvs (fromIntegral (VS.length vx)) (ptrconv poutv) (ptrconv pvx) (valconv y)
+      VS.unsafeFreeze outv
+
+  (Right vx) (Right vy)
+    | VS.length vx == VS.length vy ->
+        unsafePerformIO $ do
+          outv <- VSM.unsafeNew (VS.length vx)
+          VSM.unsafeWith outv $ \poutv ->
+            VS.unsafeWith vx $ \pvx ->
+              VS.unsafeWith vy $ \pvy ->
+                fvv (fromIntegral (VS.length vx)) (ptrconv poutv) (ptrconv pvx) (ptrconv pvy)
+          VS.unsafeFreeze outv
+    | otherwise -> error $ "vectorOp: unequal lengths: " ++ show (VS.length vx) ++ " /= " ++ show (VS.length vy)
+
+-- TODO: test all the weird cases of this function
+-- | Reduce along the inner dimension
+{-# NOINLINE vectorRedInnerOp #-}
+vectorRedInnerOp :: forall a b n. (Num a, Storable a)
+                 => SNat n
+                 -> (a -> b)
+                 -> (Ptr a -> Ptr b)
+                 -> (Int64 -> Ptr b -> b -> Ptr b -> IO ())  -- ^ scale by constant
+                 -> (Int64 -> Ptr Int64 -> Ptr Int64 -> Ptr b -> Ptr b -> IO ())  -- ^ reduction kernel
+                 -> RS.Array (n + 1) a -> RS.Array n a
+vectorRedInnerOp sn@SNat valconv ptrconv fscale fred (RS.A (RG.A sh (OI.T strides offset vec)))
+  | null sh = error "unreachable"
+  | last sh <= 0 = RS.stretch (init sh) (RS.fromList (map (const 1) (init sh)) [0])
+  | any (<= 0) (init sh) = RS.A (RG.A (init sh) (OI.T (map (const 0) (init strides)) 0 VS.empty))
+  -- now the input array is nonempty
+  | last sh == 1 = RS.A (RG.A (init sh) (OI.T (init strides) offset vec))
+  | last strides == 0 =
+      liftVEltwise1 sn
+        (vectorOp1 id (\n pout px -> fscale n (ptrconv pout) (valconv (fromIntegral (last sh))) (ptrconv px)))
+        (RS.A (RG.A (init sh) (OI.T (init strides) offset vec)))
+  -- now there is useful work along the inner dimension
+  | otherwise =
+      let -- filter out zero-stride dimensions; the reduction kernel need not concern itself with those
+          (shF, stridesF) = unzip $ filter ((/= 0) . snd) (zip sh strides)
+          ndimsF = length shF
+      in unsafePerformIO $ do
+           outv <- VSM.unsafeNew (product (init shF))
+           VSM.unsafeWith outv $ \poutv ->
+             VS.unsafeWith (VS.fromListN ndimsF (map fromIntegral shF)) $ \pshF ->
+               VS.unsafeWith (VS.fromListN ndimsF (map fromIntegral stridesF)) $ \pstridesF ->
+                 VS.unsafeWith (VS.slice offset (VS.length vec - offset) vec) $ \pvec ->
+                   fred (fromIntegral ndimsF) pshF pstridesF (ptrconv poutv) (ptrconv pvec)
+           RS.fromVector (init sh) <$> VS.unsafeFreeze outv
+
+flipOp :: (Int64 -> Ptr a -> a -> Ptr a -> IO ())
+       ->  Int64 -> Ptr a -> Ptr a -> a -> IO ()
+flipOp f n out v s = f n out s v
+
+$(fmap concat . forM typesList $ \arithtype -> do
+    let ttyp = conT (atType arithtype)
+    fmap concat . forM [minBound..maxBound] $ \arithop -> do
+      let name = mkName (aboName arithop ++ "Vector" ++ nameBase (atType arithtype))
+          cnamebase = "c_binary_" ++ atCName arithtype
+          c_ss = varE (aboNumOp arithop)
+          c_sv = varE (mkName (cnamebase ++ "_sv")) `appE` litE (integerL (fromIntegral (aboEnum arithop)))
+          c_vs = varE (mkName (cnamebase ++ "_vs")) `appE` litE (integerL (fromIntegral (aboEnum arithop)))
+          c_vv = varE (mkName (cnamebase ++ "_vv")) `appE` litE (integerL (fromIntegral (aboEnum arithop)))
+      sequence [SigD name <$>
+                     [t| forall n. SNat n -> RS.Array n $ttyp -> RS.Array n $ttyp -> RS.Array n $ttyp |]
+               ,do body <- [| \sn -> liftVEltwise2 sn (vectorOp2 id id $c_ss $c_sv $c_vs $c_vv) |]
+                   return $ FunD name [Clause [] (NormalB body) []]])
+
+$(fmap concat . forM floatTypesList $ \arithtype -> do
+    let ttyp = conT (atType arithtype)
+    fmap concat . forM [minBound..maxBound] $ \arithop -> do
+      let name = mkName (afboName arithop ++ "Vector" ++ nameBase (atType arithtype))
+          cnamebase = "c_fbinary_" ++ atCName arithtype
+          c_ss = varE (afboNumOp arithop)
+          c_sv = varE (mkName (cnamebase ++ "_sv")) `appE` litE (integerL (fromIntegral (afboEnum arithop)))
+          c_vs = varE (mkName (cnamebase ++ "_vs")) `appE` litE (integerL (fromIntegral (afboEnum arithop)))
+          c_vv = varE (mkName (cnamebase ++ "_vv")) `appE` litE (integerL (fromIntegral (afboEnum arithop)))
+      sequence [SigD name <$>
+                     [t| forall n. SNat n -> RS.Array n $ttyp -> RS.Array n $ttyp -> RS.Array n $ttyp |]
+               ,do body <- [| \sn -> liftVEltwise2 sn (vectorOp2 id id $c_ss $c_sv $c_vs $c_vv) |]
+                   return $ FunD name [Clause [] (NormalB body) []]])
+
+$(fmap concat . forM typesList $ \arithtype -> do
+    let ttyp = conT (atType arithtype)
+    fmap concat . forM [minBound..maxBound] $ \arithop -> do
+      let name = mkName (auoName arithop ++ "Vector" ++ nameBase (atType arithtype))
+          c_op = varE (mkName ("c_unary_" ++ atCName arithtype)) `appE` litE (integerL (fromIntegral (auoEnum arithop)))
+      sequence [SigD name <$>
+                     [t| forall n. SNat n -> RS.Array n $ttyp -> RS.Array n $ttyp |]
+               ,do body <- [| \sn -> liftVEltwise1 sn (vectorOp1 id $c_op) |]
+                   return $ FunD name [Clause [] (NormalB body) []]])
+
+$(fmap concat . forM floatTypesList $ \arithtype -> do
+    let ttyp = conT (atType arithtype)
+    fmap concat . forM [minBound..maxBound] $ \arithop -> do
+      let name = mkName (afuoName arithop ++ "Vector" ++ nameBase (atType arithtype))
+          c_op = varE (mkName ("c_funary_" ++ atCName arithtype)) `appE` litE (integerL (fromIntegral (afuoEnum arithop)))
+      sequence [SigD name <$>
+                     [t| forall n. SNat n -> RS.Array n $ttyp -> RS.Array n $ttyp |]
+               ,do body <- [| \sn -> liftVEltwise1 sn (vectorOp1 id $c_op) |]
+                   return $ FunD name [Clause [] (NormalB body) []]])
+
+$(fmap concat . forM typesList $ \arithtype -> do
+    let ttyp = conT (atType arithtype)
+    fmap concat . forM [minBound..maxBound] $ \arithop -> do
+      let name = mkName (aroName arithop ++ "Vector" ++ nameBase (atType arithtype))
+          c_op = varE (mkName ("c_reduce_" ++ atCName arithtype)) `appE` litE (integerL (fromIntegral (aroEnum arithop)))
+          c_scale_op = varE (mkName ("c_binary_" ++ atCName arithtype ++ "_sv")) `appE` litE (integerL (fromIntegral (aboEnum BO_MUL)))
+      sequence [SigD name <$>
+                     [t| forall n. SNat n -> RS.Array (n + 1) $ttyp -> RS.Array n $ttyp |]
+               ,do body <- [| \sn -> vectorRedInnerOp sn id id $c_scale_op $c_op |]
+                   return $ FunD name [Clause [] (NormalB body) []]])
+
+-- This branch is ostensibly a runtime branch, but will (hopefully) be
+-- constant-folded away by GHC.
+intWidBranch1 :: forall i n. (FiniteBits i, Storable i)
+              => (Int64 -> Ptr Int32 -> Ptr Int32 -> IO ())
+              -> (Int64 -> Ptr Int64 -> Ptr Int64 -> IO ())
+              -> (SNat n -> RS.Array n i -> RS.Array n i)
+intWidBranch1 f32 f64 sn
+  | finiteBitSize (undefined :: i) == 32 = liftVEltwise1 sn (vectorOp1 @i @Int32 castPtr f32)
+  | finiteBitSize (undefined :: i) == 64 = liftVEltwise1 sn (vectorOp1 @i @Int64 castPtr f64)
+  | otherwise = error "Unsupported Int width"
+
+intWidBranch2 :: forall i n. (FiniteBits i, Storable i, Integral i)
+              => (i -> i -> i)  -- ss
+                 -- int32
+              -> (Int64 -> Ptr Int32 -> Int32 -> Ptr Int32 -> IO ())  -- sv
+              -> (Int64 -> Ptr Int32 -> Ptr Int32 -> Int32 -> IO ())  -- vs
+              -> (Int64 -> Ptr Int32 -> Ptr Int32 -> Ptr Int32 -> IO ())  -- vv
+                 -- int64
+              -> (Int64 -> Ptr Int64 -> Int64 -> Ptr Int64 -> IO ())  -- sv
+              -> (Int64 -> Ptr Int64 -> Ptr Int64 -> Int64 -> IO ())  -- vs
+              -> (Int64 -> Ptr Int64 -> Ptr Int64 -> Ptr Int64 -> IO ())  -- vv
+              -> (SNat n -> RS.Array n i -> RS.Array n i -> RS.Array n i)
+intWidBranch2 ss sv32 vs32 vv32 sv64 vs64 vv64 sn
+  | finiteBitSize (undefined :: i) == 32 = liftVEltwise2 sn (vectorOp2 @i @Int32 fromIntegral castPtr ss sv32 vs32 vv32)
+  | finiteBitSize (undefined :: i) == 64 = liftVEltwise2 sn (vectorOp2 @i @Int64 fromIntegral castPtr ss sv64 vs64 vv64)
+  | otherwise = error "Unsupported Int width"
+
+intWidBranchRed :: forall i n. (FiniteBits i, Storable i, Integral i)
+                => -- int32
+                   (Int64 -> Ptr Int32 -> Int32 -> Ptr Int32 -> IO ())  -- ^ scale by constant
+                -> (Int64 -> Ptr Int64 -> Ptr Int64 -> Ptr Int32 -> Ptr Int32 -> IO ())  -- ^ reduction kernel
+                   -- int64
+                -> (Int64 -> Ptr Int64 -> Int64 -> Ptr Int64 -> IO ())  -- ^ scale by constant
+                -> (Int64 -> Ptr Int64 -> Ptr Int64 -> Ptr Int64 -> Ptr Int64 -> IO ())  -- ^ reduction kernel
+                -> (SNat n -> RS.Array (n + 1) i -> RS.Array n i)
+intWidBranchRed fsc32 fred32 fsc64 fred64 sn
+  | finiteBitSize (undefined :: i) == 32 = vectorRedInnerOp @i @Int32 sn fromIntegral castPtr fsc32 fred32
+  | finiteBitSize (undefined :: i) == 64 = vectorRedInnerOp @i @Int64 sn fromIntegral castPtr fsc64 fred64
+  | otherwise = error "Unsupported Int width"
+
+class NumElt a where
+  numEltAdd :: SNat n -> RS.Array n a -> RS.Array n a -> RS.Array n a
+  numEltSub :: SNat n -> RS.Array n a -> RS.Array n a -> RS.Array n a
+  numEltMul :: SNat n -> RS.Array n a -> RS.Array n a -> RS.Array n a
+  numEltNeg :: SNat n -> RS.Array n a -> RS.Array n a
+  numEltAbs :: SNat n -> RS.Array n a -> RS.Array n a
+  numEltSignum :: SNat n -> RS.Array n a -> RS.Array n a
+  numEltSum1Inner :: SNat n -> RS.Array (n + 1) a -> RS.Array n a
+  numEltProduct1Inner :: SNat n -> RS.Array (n + 1) a -> RS.Array n a
+
+instance NumElt Int32 where
+  numEltAdd = addVectorInt32
+  numEltSub = subVectorInt32
+  numEltMul = mulVectorInt32
+  numEltNeg = negVectorInt32
+  numEltAbs = absVectorInt32
+  numEltSignum = signumVectorInt32
+  numEltSum1Inner = sum1VectorInt32
+  numEltProduct1Inner = product1VectorInt32
+
+instance NumElt Int64 where
+  numEltAdd = addVectorInt64
+  numEltSub = subVectorInt64
+  numEltMul = mulVectorInt64
+  numEltNeg = negVectorInt64
+  numEltAbs = absVectorInt64
+  numEltSignum = signumVectorInt64
+  numEltSum1Inner = sum1VectorInt64
+  numEltProduct1Inner = product1VectorInt64
+
+instance NumElt Float where
+  numEltAdd = addVectorFloat
+  numEltSub = subVectorFloat
+  numEltMul = mulVectorFloat
+  numEltNeg = negVectorFloat
+  numEltAbs = absVectorFloat
+  numEltSignum = signumVectorFloat
+  numEltSum1Inner = sum1VectorFloat
+  numEltProduct1Inner = product1VectorFloat
+
+instance NumElt Double where
+  numEltAdd = addVectorDouble
+  numEltSub = subVectorDouble
+  numEltMul = mulVectorDouble
+  numEltNeg = negVectorDouble
+  numEltAbs = absVectorDouble
+  numEltSignum = signumVectorDouble
+  numEltSum1Inner = sum1VectorDouble
+  numEltProduct1Inner = product1VectorDouble
+
+instance NumElt Int where
+  numEltAdd = intWidBranch2 @Int (+)
+                (c_binary_i32_sv (aboEnum BO_ADD)) (flipOp (c_binary_i32_sv (aboEnum BO_ADD))) (c_binary_i32_vv (aboEnum BO_ADD))
+                (c_binary_i64_sv (aboEnum BO_ADD)) (flipOp (c_binary_i64_sv (aboEnum BO_ADD))) (c_binary_i64_vv (aboEnum BO_ADD))
+  numEltSub = intWidBranch2 @Int (-)
+                (c_binary_i32_sv (aboEnum BO_SUB)) (flipOp (c_binary_i32_sv (aboEnum BO_SUB))) (c_binary_i32_vv (aboEnum BO_SUB))
+                (c_binary_i64_sv (aboEnum BO_SUB)) (flipOp (c_binary_i64_sv (aboEnum BO_SUB))) (c_binary_i64_vv (aboEnum BO_SUB))
+  numEltMul = intWidBranch2 @Int (*)
+                (c_binary_i32_sv (aboEnum BO_MUL)) (flipOp (c_binary_i32_sv (aboEnum BO_MUL))) (c_binary_i32_vv (aboEnum BO_MUL))
+                (c_binary_i64_sv (aboEnum BO_MUL)) (flipOp (c_binary_i64_sv (aboEnum BO_MUL))) (c_binary_i64_vv (aboEnum BO_MUL))
+  numEltNeg = intWidBranch1 @Int (c_unary_i32 (auoEnum UO_NEG)) (c_unary_i64 (auoEnum UO_NEG))
+  numEltAbs = intWidBranch1 @Int (c_unary_i32 (auoEnum UO_ABS)) (c_unary_i64 (auoEnum UO_ABS))
+  numEltSignum = intWidBranch1 @Int (c_unary_i32 (auoEnum UO_SIGNUM)) (c_unary_i64 (auoEnum UO_SIGNUM))
+  numEltSum1Inner = intWidBranchRed @Int
+                      (c_binary_i32_sv (aboEnum BO_MUL)) (c_reduce_i32 (aroEnum RO_SUM1))
+                      (c_binary_i64_sv (aboEnum BO_MUL)) (c_reduce_i64 (aroEnum RO_SUM1))
+  numEltProduct1Inner = intWidBranchRed @Int
+                          (c_binary_i32_sv (aboEnum BO_MUL)) (c_reduce_i32 (aroEnum RO_PRODUCT1))
+                          (c_binary_i64_sv (aboEnum BO_MUL)) (c_reduce_i64 (aroEnum RO_PRODUCT1))
+
+instance NumElt CInt where
+  numEltAdd = intWidBranch2 @CInt (+)
+                (c_binary_i32_sv (aboEnum BO_ADD)) (flipOp (c_binary_i32_sv (aboEnum BO_ADD))) (c_binary_i32_vv (aboEnum BO_ADD))
+                (c_binary_i64_sv (aboEnum BO_ADD)) (flipOp (c_binary_i64_sv (aboEnum BO_ADD))) (c_binary_i64_vv (aboEnum BO_ADD))
+  numEltSub = intWidBranch2 @CInt (-)
+                (c_binary_i32_sv (aboEnum BO_SUB)) (flipOp (c_binary_i32_sv (aboEnum BO_SUB))) (c_binary_i32_vv (aboEnum BO_SUB))
+                (c_binary_i64_sv (aboEnum BO_SUB)) (flipOp (c_binary_i64_sv (aboEnum BO_SUB))) (c_binary_i64_vv (aboEnum BO_SUB))
+  numEltMul = intWidBranch2 @CInt (*)
+                (c_binary_i32_sv (aboEnum BO_MUL)) (flipOp (c_binary_i32_sv (aboEnum BO_MUL))) (c_binary_i32_vv (aboEnum BO_MUL))
+                (c_binary_i64_sv (aboEnum BO_MUL)) (flipOp (c_binary_i64_sv (aboEnum BO_MUL))) (c_binary_i64_vv (aboEnum BO_MUL))
+  numEltNeg = intWidBranch1 @CInt (c_unary_i32 (auoEnum UO_NEG)) (c_unary_i64 (auoEnum UO_NEG))
+  numEltAbs = intWidBranch1 @CInt (c_unary_i32 (auoEnum UO_ABS)) (c_unary_i64 (auoEnum UO_ABS))
+  numEltSignum = intWidBranch1 @CInt (c_unary_i32 (auoEnum UO_SIGNUM)) (c_unary_i64 (auoEnum UO_SIGNUM))
+  numEltSum1Inner = intWidBranchRed @CInt
+                      (c_binary_i32_sv (aboEnum BO_MUL)) (c_reduce_i32 (aroEnum RO_SUM1))
+                      (c_binary_i64_sv (aboEnum BO_MUL)) (c_reduce_i64 (aroEnum RO_SUM1))
+  numEltProduct1Inner = intWidBranchRed @CInt
+                          (c_binary_i32_sv (aboEnum BO_MUL)) (c_reduce_i32 (aroEnum RO_PRODUCT1))
+                          (c_binary_i64_sv (aboEnum BO_MUL)) (c_reduce_i64 (aroEnum RO_PRODUCT1))
+
+class FloatElt a where
+  floatEltDiv :: SNat n -> RS.Array n a -> RS.Array n a -> RS.Array n a
+  floatEltPow :: SNat n -> RS.Array n a -> RS.Array n a -> RS.Array n a
+  floatEltLogbase :: SNat n -> RS.Array n a -> RS.Array n a -> RS.Array n a
+  floatEltRecip :: SNat n -> RS.Array n a -> RS.Array n a
+  floatEltExp :: SNat n -> RS.Array n a -> RS.Array n a
+  floatEltLog :: SNat n -> RS.Array n a -> RS.Array n a
+  floatEltSqrt :: SNat n -> RS.Array n a -> RS.Array n a
+  floatEltSin :: SNat n -> RS.Array n a -> RS.Array n a
+  floatEltCos :: SNat n -> RS.Array n a -> RS.Array n a
+  floatEltTan :: SNat n -> RS.Array n a -> RS.Array n a
+  floatEltAsin :: SNat n -> RS.Array n a -> RS.Array n a
+  floatEltAcos :: SNat n -> RS.Array n a -> RS.Array n a
+  floatEltAtan :: SNat n -> RS.Array n a -> RS.Array n a
+  floatEltSinh :: SNat n -> RS.Array n a -> RS.Array n a
+  floatEltCosh :: SNat n -> RS.Array n a -> RS.Array n a
+  floatEltTanh :: SNat n -> RS.Array n a -> RS.Array n a
+  floatEltAsinh :: SNat n -> RS.Array n a -> RS.Array n a
+  floatEltAcosh :: SNat n -> RS.Array n a -> RS.Array n a
+  floatEltAtanh :: SNat n -> RS.Array n a -> RS.Array n a
+  floatEltLog1p :: SNat n -> RS.Array n a -> RS.Array n a
+  floatEltExpm1 :: SNat n -> RS.Array n a -> RS.Array n a
+  floatEltLog1pexp :: SNat n -> RS.Array n a -> RS.Array n a
+  floatEltLog1mexp :: SNat n -> RS.Array n a -> RS.Array n a
+
+instance FloatElt Float where
+  floatEltDiv = divVectorFloat
+  floatEltPow = powVectorFloat
+  floatEltLogbase = logbaseVectorFloat
+  floatEltRecip = recipVectorFloat
+  floatEltExp = expVectorFloat
+  floatEltLog = logVectorFloat
+  floatEltSqrt = sqrtVectorFloat
+  floatEltSin = sinVectorFloat
+  floatEltCos = cosVectorFloat
+  floatEltTan = tanVectorFloat
+  floatEltAsin = asinVectorFloat
+  floatEltAcos = acosVectorFloat
+  floatEltAtan = atanVectorFloat
+  floatEltSinh = sinhVectorFloat
+  floatEltCosh = coshVectorFloat
+  floatEltTanh = tanhVectorFloat
+  floatEltAsinh = asinhVectorFloat
+  floatEltAcosh = acoshVectorFloat
+  floatEltAtanh = atanhVectorFloat
+  floatEltLog1p = log1pVectorFloat
+  floatEltExpm1 = expm1VectorFloat
+  floatEltLog1pexp = log1pexpVectorFloat
+  floatEltLog1mexp = log1mexpVectorFloat
+
+instance FloatElt Double where
+  floatEltDiv = divVectorDouble
+  floatEltPow = powVectorDouble
+  floatEltLogbase = logbaseVectorDouble
+  floatEltRecip = recipVectorDouble
+  floatEltExp = expVectorDouble
+  floatEltLog = logVectorDouble
+  floatEltSqrt = sqrtVectorDouble
+  floatEltSin = sinVectorDouble
+  floatEltCos = cosVectorDouble
+  floatEltTan = tanVectorDouble
+  floatEltAsin = asinVectorDouble
+  floatEltAcos = acosVectorDouble
+  floatEltAtan = atanVectorDouble
+  floatEltSinh = sinhVectorDouble
+  floatEltCosh = coshVectorDouble
+  floatEltTanh = tanhVectorDouble
+  floatEltAsinh = asinhVectorDouble
+  floatEltAcosh = acoshVectorDouble
+  floatEltAtanh = atanhVectorDouble
+  floatEltLog1p = log1pVectorDouble
+  floatEltExpm1 = expm1VectorDouble
+  floatEltLog1pexp = log1pexpVectorDouble
+  floatEltLog1mexp = log1mexpVectorDouble
diff --git a/src/Data/Array/Mixed/Internal/Arith/Foreign.hs b/src/Data/Array/Mixed/Internal/Arith/Foreign.hs
new file mode 100644
index 0000000..6fc7229
--- /dev/null
+++ b/src/Data/Array/Mixed/Internal/Arith/Foreign.hs
@@ -0,0 +1,55 @@
+{-# LANGUAGE ForeignFunctionInterface #-}
+{-# LANGUAGE TemplateHaskell #-}
+module Data.Array.Mixed.Internal.Arith.Foreign where
+
+import Control.Monad
+import Data.Int
+import Data.Maybe
+import Foreign.C.Types
+import Foreign.Ptr
+import Language.Haskell.TH
+
+import Data.Array.Mixed.Internal.Arith.Lists
+
+
+$(fmap concat . forM typesList $ \arithtype -> do
+    let ttyp = conT (atType arithtype)
+    let base = "binary_" ++ atCName arithtype
+    sequence $ catMaybes
+      [Just (ForeignD . ImportF CCall Unsafe ("oxarop_" ++ base ++ "_sv") (mkName ("c_" ++ base ++ "_sv")) <$>
+               [t| CInt -> Int64 -> Ptr $ttyp -> $ttyp -> Ptr $ttyp -> IO () |])
+      ,Just (ForeignD . ImportF CCall Unsafe ("oxarop_" ++ base ++ "_vv") (mkName ("c_" ++ base ++ "_vv")) <$>
+               [t| CInt -> Int64 -> Ptr $ttyp -> Ptr $ttyp -> Ptr $ttyp -> IO () |])
+      ,Just (ForeignD . ImportF CCall Unsafe ("oxarop_" ++ base ++ "_vs") (mkName ("c_" ++ base ++ "_vs")) <$>
+               [t| CInt -> Int64 -> Ptr $ttyp -> Ptr $ttyp -> $ttyp -> IO () |])
+      ])
+
+$(fmap concat . forM floatTypesList $ \arithtype -> do
+    let ttyp = conT (atType arithtype)
+    let base = "fbinary_" ++ atCName arithtype
+    sequence $ catMaybes
+      [Just (ForeignD . ImportF CCall Unsafe ("oxarop_" ++ base ++ "_sv") (mkName ("c_" ++ base ++ "_sv")) <$>
+               [t| CInt -> Int64 -> Ptr $ttyp -> $ttyp -> Ptr $ttyp -> IO () |])
+      ,Just (ForeignD . ImportF CCall Unsafe ("oxarop_" ++ base ++ "_vv") (mkName ("c_" ++ base ++ "_vv")) <$>
+               [t| CInt -> Int64 -> Ptr $ttyp -> Ptr $ttyp -> Ptr $ttyp -> IO () |])
+      ,Just (ForeignD . ImportF CCall Unsafe ("oxarop_" ++ base ++ "_vs") (mkName ("c_" ++ base ++ "_vs")) <$>
+               [t| CInt -> Int64 -> Ptr $ttyp -> Ptr $ttyp -> $ttyp -> IO () |])
+      ])
+
+$(fmap concat . forM typesList $ \arithtype -> do
+    let ttyp = conT (atType arithtype)
+    let base = "unary_" ++ atCName arithtype
+    pure . ForeignD . ImportF CCall Unsafe ("oxarop_" ++ base) (mkName ("c_" ++ base)) <$>
+      [t| CInt -> Int64 -> Ptr $ttyp -> Ptr $ttyp -> IO () |])
+
+$(fmap concat . forM floatTypesList $ \arithtype -> do
+    let ttyp = conT (atType arithtype)
+    let base = "funary_" ++ atCName arithtype
+    pure . ForeignD . ImportF CCall Unsafe ("oxarop_" ++ base) (mkName ("c_" ++ base)) <$>
+      [t| CInt -> Int64 -> Ptr $ttyp -> Ptr $ttyp -> IO () |])
+
+$(fmap concat . forM typesList $ \arithtype -> do
+    let ttyp = conT (atType arithtype)
+    let base = "reduce_" ++ atCName arithtype
+    pure . ForeignD . ImportF CCall Unsafe ("oxarop_" ++ base) (mkName ("c_" ++ base)) <$>
+      [t| CInt -> Int64 -> Ptr Int64 -> Ptr Int64 -> Ptr $ttyp -> Ptr $ttyp -> IO () |])
diff --git a/src/Data/Array/Mixed/Internal/Arith/Lists.hs b/src/Data/Array/Mixed/Internal/Arith/Lists.hs
new file mode 100644
index 0000000..a284bc1
--- /dev/null
+++ b/src/Data/Array/Mixed/Internal/Arith/Lists.hs
@@ -0,0 +1,78 @@
+{-# LANGUAGE LambdaCase #-}
+{-# LANGUAGE TemplateHaskell #-}
+module Data.Array.Mixed.Internal.Arith.Lists where
+
+import Data.Char
+import Data.Int
+import Language.Haskell.TH
+
+import Data.Array.Mixed.Internal.Arith.Lists.TH
+
+
+data ArithType = ArithType
+  { atType :: Name  -- ''Int32
+  , atCName :: String  -- "i32"
+  }
+
+floatTypesList :: [ArithType]
+floatTypesList =
+  [ArithType ''Float "float"
+  ,ArithType ''Double "double"
+  ]
+
+typesList :: [ArithType]
+typesList =
+  [ArithType ''Int32 "i32"
+  ,ArithType ''Int64 "i64"
+  ]
+  ++ floatTypesList
+
+-- data ArithBOp = BO_ADD | BO_SUB | BO_MUL deriving (Show, Enum, Bounded)
+$(genArithDataType Binop "ArithBOp")
+
+$(genArithNameFun Binop ''ArithBOp "aboName" (map toLower . drop 3))
+$(genArithEnumFun Binop ''ArithBOp "aboEnum")
+
+$(do clauses <- readArithLists Binop
+                  (\name _num hsop -> return (Clause [ConP (mkName name) [] []]
+                                                     (NormalB (VarE 'mkName `AppE` LitE (StringL hsop)))
+                                                     []))
+                  return
+     sequence [SigD (mkName "aboNumOp") <$> [t| ArithBOp -> Name |]
+              ,return $ FunD (mkName "aboNumOp") clauses])
+
+
+-- data ArithFBOp = FB_DIV deriving (Show, Enum, Bounded)
+$(genArithDataType FBinop "ArithFBOp")
+
+$(genArithNameFun FBinop ''ArithFBOp "afboName" (map toLower . drop 3))
+$(genArithEnumFun FBinop ''ArithFBOp "afboEnum")
+
+$(do clauses <- readArithLists FBinop
+                  (\name _num hsop -> return (Clause [ConP (mkName name) [] []]
+                                                     (NormalB (VarE 'mkName `AppE` LitE (StringL hsop)))
+                                                     []))
+                  return
+     sequence [SigD (mkName "afboNumOp") <$> [t| ArithFBOp -> Name |]
+              ,return $ FunD (mkName "afboNumOp") clauses])
+
+
+-- data ArithUOp = UO_NEG | UO_ABS | UO_SIGNUM | ... deriving (Show, Enum, Bounded)
+$(genArithDataType Unop "ArithUOp")
+
+$(genArithNameFun Unop ''ArithUOp "auoName" (map toLower . drop 3))
+$(genArithEnumFun Unop ''ArithUOp "auoEnum")
+
+
+-- data ArithFUOp = FU_RECIP | ... deriving (Show, Enum, Bounded)
+$(genArithDataType FUnop "ArithFUOp")
+
+$(genArithNameFun FUnop ''ArithFUOp "afuoName" (map toLower . drop 3))
+$(genArithEnumFun FUnop ''ArithFUOp "afuoEnum")
+
+
+-- data ArithRedOp = RO_SUM1 | RO_PRODUCT1 deriving (Show, Enum, Bounded)
+$(genArithDataType Redop "ArithRedOp")
+
+$(genArithNameFun Redop ''ArithRedOp "aroName" (map toLower . drop 3))
+$(genArithEnumFun Redop ''ArithRedOp "aroEnum")
diff --git a/src/Data/Array/Mixed/Internal/Arith/Lists/TH.hs b/src/Data/Array/Mixed/Internal/Arith/Lists/TH.hs
new file mode 100644
index 0000000..8b7d05f
--- /dev/null
+++ b/src/Data/Array/Mixed/Internal/Arith/Lists/TH.hs
@@ -0,0 +1,82 @@
+{-# LANGUAGE TemplateHaskellQuotes #-}
+module Data.Array.Mixed.Internal.Arith.Lists.TH where
+
+import Control.Monad
+import Control.Monad.IO.Class
+import Data.Maybe
+import Foreign.C.Types
+import Language.Haskell.TH
+import Language.Haskell.TH.Syntax
+import Text.Read
+
+
+data OpKind = Binop | FBinop | Unop | FUnop | Redop
+  deriving (Show, Eq)
+
+readArithLists :: OpKind
+               -> (String -> Int -> String -> Q a)
+               -> ([a] -> Q r)
+               -> Q r
+readArithLists targetkind fop fcombine = do
+  addDependentFile "cbits/arith_lists.h"
+  lns <- liftIO $ lines <$> readFile "cbits/arith_lists.h"
+
+  mvals <- forM lns $ \line -> do
+    if null (dropWhile (== ' ') line)
+      then return Nothing
+      else do let (kind, name, num, aux) = parseLine line
+              if kind == targetkind
+                then Just <$> fop name num aux
+                else return Nothing
+
+  fcombine (catMaybes mvals)
+  where
+    parseLine s0
+      | ("LIST_", s1) <- splitAt 5 s0
+      , (kindstr, '(' : s2) <- break (== '(') s1
+      , (f1, ',' : s3) <- parseField s2
+      , (f2, ',' : s4) <- parseField s3
+      , (f3, ')' : _) <- parseField s4
+      , Just kind <- parseKind kindstr
+      , let name = f1
+      , Just num <- readMaybe f2
+      , let aux = f3
+      = (kind, name, num, aux)
+      | otherwise
+      = error $ "readArithLists: unrecognised line in cbits/arith_lists.h: " ++ show s0
+
+    parseField s = break (`elem` ",)") (dropWhile (== ' ') s)
+
+    parseKind "BINOP" = Just Binop
+    parseKind "FBINOP" = Just FBinop
+    parseKind "UNOP" = Just Unop
+    parseKind "FUNOP" = Just FUnop
+    parseKind "REDOP" = Just Redop
+    parseKind _ = Nothing
+
+genArithDataType :: OpKind -> String -> Q [Dec]
+genArithDataType kind dtname = do
+  cons <- readArithLists kind
+            (\name _num _ -> return $ NormalC (mkName name) [])
+            return
+  return [DataD [] (mkName dtname) [] Nothing cons [DerivClause Nothing [ConT ''Show, ConT ''Enum, ConT ''Bounded]]]
+
+genArithNameFun :: OpKind -> Name -> String -> (String -> String) -> Q [Dec]
+genArithNameFun kind dtname funname nametrans = do
+  clauses <- readArithLists kind
+               (\name _num _ -> return (Clause [ConP (mkName name) [] []]
+                                               (NormalB (LitE (StringL (nametrans name))))
+                                               []))
+               return
+  return [SigD (mkName funname) (ArrowT `AppT` ConT dtname `AppT` ConT ''String)
+         ,FunD (mkName funname) clauses]
+
+genArithEnumFun :: OpKind -> Name -> String -> Q [Dec]
+genArithEnumFun kind dtname funname = do
+  clauses <- readArithLists kind
+               (\name num _ -> return (Clause [ConP (mkName name) [] []]
+                                              (NormalB (LitE (IntegerL (fromIntegral num))))
+                                              []))
+               return
+  return [SigD (mkName funname) (ArrowT `AppT` ConT dtname `AppT` ConT ''CInt)
+         ,FunD (mkName funname) clauses]
diff --git a/src/Data/Array/Mixed/Lemmas.hs b/src/Data/Array/Mixed/Lemmas.hs
new file mode 100644
index 0000000..30ec9c0
--- /dev/null
+++ b/src/Data/Array/Mixed/Lemmas.hs
@@ -0,0 +1,47 @@
+{-# LANGUAGE GADTs #-}
+{-# LANGUAGE ScopedTypeVariables #-}
+{-# LANGUAGE TypeOperators #-}
+{-# OPTIONS_GHC -fplugin GHC.TypeLits.Normalise #-}
+{-# OPTIONS_GHC -fplugin GHC.TypeLits.KnownNat.Solver #-}
+{-# LANGUAGE TypeApplications #-}
+{-# LANGUAGE DataKinds #-}
+module Data.Array.Mixed.Lemmas where
+
+import Data.Proxy
+import Data.Type.Equality
+import GHC.TypeLits
+
+import Data.Array.Mixed.Shape
+import Data.Array.Mixed.Types
+
+
+lemRankApp :: forall sh1 sh2.
+              StaticShX sh1 -> StaticShX sh2
+           -> Rank (sh1 ++ sh2) :~: Rank sh1 + Rank sh2
+lemRankApp ZKX _ = Refl
+lemRankApp (_ :!% (ssh1 :: StaticShX sh1T)) ssh2
+  = lem2 (Proxy @(Rank sh1T)) Proxy Proxy $
+      lem (Proxy @(Rank sh2)) (Proxy @(Rank sh1T)) (Proxy @(Rank (sh1T ++ sh2))) $
+        lemRankApp ssh1 ssh2
+  where
+    lem :: proxy a -> proxy b -> proxy c
+        -> c :~: b + a
+        -> b + a :~: c
+    lem _ _ _ Refl = Refl
+
+    lem2 :: proxy a -> proxy b -> proxy c
+         -> (a + b :~: c)
+         -> c + 1 :~: (a + 1 + b)
+    lem2 _ _ _ Refl = Refl
+
+lemRankAppComm :: StaticShX sh1 -> StaticShX sh2
+               -> Rank (sh1 ++ sh2) :~: Rank (sh2 ++ sh1)
+lemRankAppComm _ _ = unsafeCoerceRefl  -- TODO improve this
+
+lemKnownNatRank :: IShX sh -> Dict KnownNat (Rank sh)
+lemKnownNatRank ZSX = Dict
+lemKnownNatRank (_ :$% sh) | Dict <- lemKnownNatRank sh = Dict
+
+lemKnownNatRankSSX :: StaticShX sh -> Dict KnownNat (Rank sh)
+lemKnownNatRankSSX ZKX = Dict
+lemKnownNatRankSSX (_ :!% ssh) | Dict <- lemKnownNatRankSSX ssh = Dict
diff --git a/src/Data/Array/Mixed/Permutation.hs b/src/Data/Array/Mixed/Permutation.hs
new file mode 100644
index 0000000..2710018
--- /dev/null
+++ b/src/Data/Array/Mixed/Permutation.hs
@@ -0,0 +1,252 @@
+{-# LANGUAGE ConstraintKinds #-}
+{-# LANGUAGE DataKinds #-}
+{-# LANGUAGE GADTs #-}
+{-# LANGUAGE LambdaCase #-}
+{-# LANGUAGE PolyKinds #-}
+{-# LANGUAGE QuantifiedConstraints #-}
+{-# LANGUAGE RankNTypes #-}
+{-# LANGUAGE ScopedTypeVariables #-}
+{-# LANGUAGE StandaloneDeriving #-}
+{-# LANGUAGE TypeApplications #-}
+{-# LANGUAGE TypeFamilies #-}
+{-# LANGUAGE TypeOperators #-}
+{-# LANGUAGE UndecidableInstances #-}
+{-# OPTIONS_GHC -fplugin GHC.TypeLits.Normalise #-}
+{-# OPTIONS_GHC -fplugin GHC.TypeLits.KnownNat.Solver #-}
+module Data.Array.Mixed.Permutation where
+
+import Data.Coerce (coerce)
+import Data.Functor.Const
+import Data.List (sort)
+import Data.Proxy
+import Data.Type.Bool
+import Data.Type.Equality
+import Data.Type.Ord
+import GHC.TypeError
+import GHC.TypeLits
+import qualified GHC.TypeNats as TN
+
+import Data.Array.Mixed.Shape
+import Data.Array.Mixed.Types
+
+
+-- * Permutations
+
+-- | A "backward" permutation of a dimension list. The operation on the
+-- dimension list is most similar to 'Data.Vector.backpermute'; see 'Permute'
+-- for code that implements this.
+data Perm list where
+  PNil :: Perm '[]
+  PCons :: SNat a -> Perm l -> Perm (a : l)
+infixr 5 `PCons`
+deriving instance Show (Perm list)
+deriving instance Eq (Perm list)
+
+permLengthSNat :: Perm list -> SNat (Rank list)
+permLengthSNat PNil = SNat
+permLengthSNat (_ `PCons` l) | SNat <- permLengthSNat l = SNat
+
+permFromList :: [Int] -> (forall list. Perm list -> r) -> r
+permFromList [] k = k PNil
+permFromList (x : xs) k = withSomeSNat (fromIntegral x) $ \case
+  Just sn -> permFromList xs $ \list -> k (sn `PCons` list)
+  Nothing -> error $ "Data.Array.Mixed.permFromList: negative number in list: " ++ show x
+
+permToList :: Perm list -> [Natural]
+permToList PNil = mempty
+permToList (x `PCons` l) = TN.fromSNat x : permToList l
+
+permToList' :: Perm list -> [Int]
+permToList' = map fromIntegral . permToList
+
+
+-- ** Applying permutations
+
+type family Elem x l where
+  Elem x '[] = 'False
+  Elem x (x : _) = 'True
+  Elem x (_ : ys) = Elem x ys
+
+type family AllElem' as bs where
+  AllElem' '[] bs = 'True
+  AllElem' (a : as) bs = Elem a bs && AllElem' as bs
+
+type AllElem as bs = Assert (AllElem' as bs)
+  (TypeError (Text "The elements of " :<>: ShowType as :<>: Text " are not all in " :<>: ShowType bs))
+
+type family Count i n where
+  Count n n = '[]
+  Count i n = i : Count (i + 1) n
+
+type IsPermutation as = (AllElem as (Count 0 (Rank as)), AllElem (Count 0 (Rank as)) as)
+
+type family Index i sh where
+  Index 0 (n : sh) = n
+  Index i (_ : sh) = Index (i - 1) sh
+
+type family Permute is sh where
+  Permute '[] sh = '[]
+  Permute (i : is) sh = Index i sh : Permute is sh
+
+type PermutePrefix is sh = Permute is (TakeLen is sh) ++ DropLen is sh
+
+type family TakeLen ref l where
+  TakeLen '[] l = '[]
+  TakeLen (_ : ref) (x : xs) = x : TakeLen ref xs
+
+type family DropLen ref l where
+  DropLen '[] l = l
+  DropLen (_ : ref) (_ : xs) = DropLen ref xs
+
+listxTakeLen :: forall f is sh. Perm is -> ListX sh f -> ListX (TakeLen is sh) f
+listxTakeLen PNil _ = ZX
+listxTakeLen (_ `PCons` is) (n ::% sh) = n ::% listxTakeLen is sh
+listxTakeLen (_ `PCons` _) ZX = error "IsPermutation longer than shape"
+
+listxDropLen :: forall f is sh. Perm is -> ListX sh f -> ListX (DropLen is sh) f
+listxDropLen PNil sh = sh
+listxDropLen (_ `PCons` is) (_ ::% sh) = listxDropLen is sh
+listxDropLen (_ `PCons` _) ZX = error "IsPermutation longer than shape"
+
+listxPermute :: forall f is sh. Perm is -> ListX sh f -> ListX (Permute is sh) f
+listxPermute PNil _ = ZX
+listxPermute (i `PCons` (is :: Perm is')) (sh :: ListX sh f) =
+  listxIndex (Proxy @is') (Proxy @sh) i sh (listxPermute is sh)
+
+listxIndex :: forall f is shT i sh. Proxy is -> Proxy shT -> SNat i -> ListX sh f -> ListX (Permute is shT) f -> ListX (Index i sh : Permute is shT) f
+listxIndex _ _ SZ (n ::% _) rest = n ::% rest
+listxIndex p pT (SS (i :: SNat i')) ((_ :: f n) ::% (sh :: ListX sh' f)) rest
+  | Refl <- lemIndexSucc (Proxy @i') (Proxy @n) (Proxy @sh')
+  = listxIndex p pT i sh rest
+listxIndex _ _ _ ZX _ = error "Index into empty shape"
+
+listxPermutePrefix :: forall f is sh. Perm is -> ListX sh f -> ListX (PermutePrefix is sh) f
+listxPermutePrefix perm sh = listxAppend (listxPermute perm (listxTakeLen perm sh)) (listxDropLen perm sh)
+
+ixxPermutePrefix :: forall i is sh. Perm is -> IxX sh i -> IxX (PermutePrefix is sh) i
+ixxPermutePrefix = coerce (listxPermutePrefix @(Const i))
+
+ssxTakeLen :: forall is sh. Perm is -> StaticShX sh -> StaticShX (TakeLen is sh)
+ssxTakeLen = coerce (listxTakeLen @(SMayNat () SNat))
+
+ssxDropLen :: Perm is -> StaticShX sh -> StaticShX (DropLen is sh)
+ssxDropLen = coerce (listxDropLen @(SMayNat () SNat))
+
+ssxPermute :: Perm is -> StaticShX sh -> StaticShX (Permute is sh)
+ssxPermute = coerce (listxPermute @(SMayNat () SNat))
+
+ssxIndex :: Proxy is -> Proxy shT -> SNat i -> StaticShX sh -> StaticShX (Permute is shT) -> StaticShX (Index i sh : Permute is shT)
+ssxIndex p1 p2 = coerce (listxIndex @(SMayNat () SNat) p1 p2)
+
+ssxPermutePrefix :: Perm is -> StaticShX sh -> StaticShX (PermutePrefix is sh)
+ssxPermutePrefix = coerce (listxPermutePrefix @(SMayNat () SNat))
+
+shxPermutePrefix :: Perm is -> IShX sh -> IShX (PermutePrefix is sh)
+shxPermutePrefix = coerce (listxPermutePrefix @(SMayNat Int SNat))
+
+
+-- * Operations on permutations
+
+-- TODO: test this thing more properly
+permInverse :: Perm is
+            -> (forall is'.
+                     IsPermutation is'
+                  => Perm is'
+                  -> (forall sh. Rank sh ~ Rank is => StaticShX sh -> Permute is' (Permute is sh) :~: sh)
+                  -> r)
+            -> r
+permInverse = \perm k ->
+  genPerm perm $ \(invperm :: Perm is') ->
+    let sn = permLengthSNat invperm
+    in case (provePerm1 (Proxy @is') sn invperm, provePerm2 (SNat @0) sn invperm) of
+         (Just Refl, Just Refl) ->
+           k invperm
+             (\ssh -> case provePermInverse perm invperm ssh of
+                        Just eq -> eq
+                        Nothing -> error $ "permInverse: did not generate inverse? perm = " ++ show perm
+                                           ++ " ; invperm = " ++ show invperm)
+         _ -> error $ "permInverse: did not generate permutation? perm = " ++ show perm
+                      ++ " ; invperm = " ++ show invperm
+  where
+    genPerm :: Perm is -> (forall is'. Perm is' -> r) -> r
+    genPerm perm =
+      let permList = permToList' perm
+      in toHList $ map snd (sort (zip permList [0..]))
+      where
+        toHList :: [Natural] -> (forall is'. Perm is' -> r) -> r
+        toHList [] k = k PNil
+        toHList (n : ns) k = toHList ns $ \l -> TN.withSomeSNat n $ \sn -> k (PCons sn l)
+
+    lemElemCount :: (0 <= n, Compare n m ~ LT) => proxy n -> proxy m -> Elem n (Count 0 m) :~: True
+    lemElemCount _ _ = unsafeCoerceRefl
+
+    lemCount :: (OrdCond (Compare i n) True False True ~ True) => proxy i -> proxy n -> Count i n :~: i : Count (i + 1) n
+    lemCount _ _ = unsafeCoerceRefl
+
+    lemElem :: Elem x ys ~ True => proxy x -> proxy' (y : ys) -> Elem x (y : ys) :~: True
+    lemElem _ _ = unsafeCoerceRefl
+
+    provePerm1 :: Proxy isTop -> SNat (Rank isTop) -> Perm is'
+               -> Maybe (AllElem' is' (Count 0 (Rank isTop)) :~: True)
+    provePerm1 _ _ PNil = Just (Refl)
+    provePerm1 p rtop@SNat (PCons sn@SNat perm)
+      | Just Refl <- provePerm1 p rtop perm
+      = case (cmpNat (SNat @0) sn, cmpNat sn rtop) of
+          (LTI, LTI) | Refl <- lemElemCount sn rtop -> Just Refl
+          (EQI, LTI) | Refl <- lemElemCount sn rtop -> Just Refl
+          _ -> Nothing
+      | otherwise
+      = Nothing
+
+    provePerm2 :: SNat i -> SNat n -> Perm is'
+               -> Maybe (AllElem' (Count i n) is' :~: True)
+    provePerm2 = \i@(SNat :: SNat i) n@SNat perm ->
+      case cmpNat i n of
+        EQI -> Just Refl
+        LTI | Refl <- lemCount i n
+            , Just Refl <- provePerm2 (SNat @(i + 1)) n perm
+            -> checkElem i perm
+            | otherwise -> Nothing
+        GTI -> error "unreachable"
+      where
+        checkElem :: SNat i -> Perm is' -> Maybe (Elem i is' :~: True)
+        checkElem _ PNil = Nothing
+        checkElem i@SNat (PCons k@SNat perm :: Perm is') =
+          case sameNat i k of
+            Just Refl -> Just Refl
+            Nothing | Just Refl <- checkElem i perm, Refl <- lemElem i (Proxy @is') -> Just Refl
+                    | otherwise -> Nothing
+
+    provePermInverse :: Perm is -> Perm is' -> StaticShX sh
+                     -> Maybe (Permute is' (Permute is sh) :~: sh)
+    provePermInverse perm perminv ssh =
+      ssxGeq (ssxPermute perminv (ssxPermute perm ssh)) ssh
+
+type family MapSucc is where
+  MapSucc '[] = '[]
+  MapSucc (i : is) = i + 1 : MapSucc is
+
+permShift1 :: Perm l -> Perm (0 : MapSucc l)
+permShift1 = (SNat @0 `PCons`) . permMapSucc
+  where
+    permMapSucc :: Perm l -> Perm (MapSucc l)
+    permMapSucc PNil = PNil
+    permMapSucc ((SNat :: SNat i) `PCons` ns) = SNat @(i + 1) `PCons` permMapSucc ns
+
+
+-- * Lemmas
+
+lemRankPermute :: Proxy sh -> Perm is -> Rank (Permute is sh) :~: Rank is
+lemRankPermute _ PNil = Refl
+lemRankPermute p (_ `PCons` is) | Refl <- lemRankPermute p is = Refl
+
+lemRankDropLen :: forall is sh. (Rank is <= Rank sh)
+               => StaticShX sh -> Perm is -> Rank (DropLen is sh) :~: Rank sh - Rank is
+lemRankDropLen ZKX PNil = Refl
+lemRankDropLen (_ :!% sh) (_ `PCons` is) | Refl <- lemRankDropLen sh is = Refl
+lemRankDropLen (_ :!% _) PNil = Refl
+lemRankDropLen ZKX (_ `PCons` _) = error "1 <= 0"
+
+lemIndexSucc :: Proxy i -> Proxy a -> Proxy l
+             -> Index (i + 1) (a : l) :~: Index i l
+lemIndexSucc _ _ _ = unsafeCoerceRefl
diff --git a/src/Data/Array/Mixed/Shape.hs b/src/Data/Array/Mixed/Shape.hs
new file mode 100644
index 0000000..a16da76
--- /dev/null
+++ b/src/Data/Array/Mixed/Shape.hs
@@ -0,0 +1,455 @@
+{-# LANGUAGE DataKinds #-}
+{-# LANGUAGE DeriveGeneric #-}
+{-# LANGUAGE FlexibleInstances #-}
+{-# LANGUAGE GADTs #-}
+{-# LANGUAGE NoStarIsType #-}
+{-# LANGUAGE PatternSynonyms #-}
+{-# LANGUAGE PolyKinds #-}
+{-# LANGUAGE QuantifiedConstraints #-}
+{-# LANGUAGE RankNTypes #-}
+{-# LANGUAGE RoleAnnotations #-}
+{-# LANGUAGE ScopedTypeVariables #-}
+{-# LANGUAGE StandaloneDeriving #-}
+{-# LANGUAGE StandaloneKindSignatures #-}
+{-# LANGUAGE TypeApplications #-}
+{-# LANGUAGE TypeFamilies #-}
+{-# LANGUAGE TypeOperators #-}
+{-# LANGUAGE UndecidableInstances #-}
+{-# LANGUAGE ViewPatterns #-}
+{-# OPTIONS_GHC -fplugin GHC.TypeLits.Normalise #-}
+{-# OPTIONS_GHC -fplugin GHC.TypeLits.KnownNat.Solver #-}
+module Data.Array.Mixed.Shape where
+
+import Control.DeepSeq (NFData(..))
+import qualified Data.Foldable as Foldable
+import Data.Functor.Const
+import Data.Kind (Type, Constraint)
+import Data.Monoid (Sum(..))
+import Data.Proxy
+import Data.Type.Equality
+import GHC.Generics (Generic)
+import GHC.IsList (IsList)
+import qualified GHC.IsList as IsList
+import GHC.TypeLits
+
+import Data.Array.Mixed.Types
+import Data.Coerce
+import Data.Bifunctor (first)
+
+
+-- | The length of a type-level list. If the argument is a shape, then the
+-- result is the rank of that shape.
+type family Rank sh where
+  Rank '[] = 0
+  Rank (_ : sh) = Rank sh + 1
+
+
+-- * Mixed lists
+
+type role ListX nominal representational
+type ListX :: [Maybe Nat] -> (Maybe Nat -> Type) -> Type
+data ListX sh f where
+  ZX :: ListX '[] f
+  (::%) :: f n -> ListX sh f -> ListX (n : sh) f
+deriving instance (forall n. Eq (f n)) => Eq (ListX sh f)
+deriving instance (forall n. Ord (f n)) => Ord (ListX sh f)
+infixr 3 ::%
+
+instance (forall n. Show (f n)) => Show (ListX sh f) where
+  showsPrec _ = listxShow shows
+
+instance (forall n. NFData (f n)) => NFData (ListX sh f) where
+  rnf ZX = ()
+  rnf (x ::% l) = rnf x `seq` rnf l
+
+data UnconsListXRes f sh1 =
+  forall n sh. (n : sh ~ sh1) => UnconsListXRes (ListX sh f) (f n)
+listxUncons :: ListX sh1 f -> Maybe (UnconsListXRes f sh1)
+listxUncons (i ::% shl') = Just (UnconsListXRes shl' i)
+listxUncons ZX = Nothing
+
+listxFmap :: (forall n. f n -> g n) -> ListX sh f -> ListX sh g
+listxFmap _ ZX = ZX
+listxFmap f (x ::% xs) = f x ::% listxFmap f xs
+
+listxFold :: Monoid m => (forall n. f n -> m) -> ListX sh f -> m
+listxFold _ ZX = mempty
+listxFold f (x ::% xs) = f x <> listxFold f xs
+
+listxLength :: ListX sh f -> Int
+listxLength = getSum . listxFold (\_ -> Sum 1)
+
+listxLengthSNat :: ListX sh f -> SNat (Rank sh)
+listxLengthSNat ZX = SNat
+listxLengthSNat (_ ::% l) | SNat <- listxLengthSNat l = SNat
+
+listxShow :: forall sh f. (forall n. f n -> ShowS) -> ListX sh f -> ShowS
+listxShow f l = showString "[" . go "" l . showString "]"
+  where
+    go :: String -> ListX sh' f -> ShowS
+    go _ ZX = id
+    go prefix (x ::% xs) = showString prefix . f x . go "," xs
+
+listxToList :: ListX sh' (Const i) -> [i]
+listxToList ZX = []
+listxToList (Const i ::% is) = i : listxToList is
+
+listxAppend :: ListX sh f -> ListX sh' f -> ListX (sh ++ sh') f
+listxAppend ZX idx' = idx'
+listxAppend (i ::% idx) idx' = i ::% listxAppend idx idx'
+
+listxDrop :: forall f g sh sh'. ListX (sh ++ sh') f -> ListX sh g -> ListX sh' f
+listxDrop long ZX = long
+listxDrop long (_ ::% short) = case long of _ ::% long' -> listxDrop long' short
+
+
+-- * Mixed indices
+
+-- | This is a newtype over 'ListX'.
+type role IxX nominal representational
+type IxX :: [Maybe Nat] -> Type -> Type
+newtype IxX sh i = IxX (ListX sh (Const i))
+  deriving (Eq, Ord, Generic)
+
+pattern ZIX :: forall sh i. () => sh ~ '[] => IxX sh i
+pattern ZIX = IxX ZX
+
+pattern (:.%)
+  :: forall {sh1} {i}.
+     forall n sh. (n : sh ~ sh1)
+  => i -> IxX sh i -> IxX sh1 i
+pattern i :.% shl <- IxX (listxUncons -> Just (UnconsListXRes (IxX -> shl) (getConst -> i)))
+  where i :.% IxX shl = IxX (Const i ::% shl)
+infixr 3 :.%
+
+{-# COMPLETE ZIX, (:.%) #-}
+
+type IIxX sh = IxX sh Int
+
+instance Show i => Show (IxX sh i) where
+  showsPrec _ (IxX l) = listxShow (\(Const i) -> shows i) l
+
+instance Functor (IxX sh) where
+  fmap f (IxX l) = IxX (listxFmap (Const . f . getConst) l)
+
+instance Foldable (IxX sh) where
+  foldMap f (IxX l) = listxFold (f . getConst) l
+
+instance NFData i => NFData (IxX sh i)
+
+ixxZero :: StaticShX sh -> IIxX sh
+ixxZero ZKX = ZIX
+ixxZero (_ :!% ssh) = 0 :.% ixxZero ssh
+
+ixxZero' :: IShX sh -> IIxX sh
+ixxZero' ZSX = ZIX
+ixxZero' (_ :$% sh) = 0 :.% ixxZero' sh
+
+ixxAppend :: forall sh sh' i. IxX sh i -> IxX sh' i -> IxX (sh ++ sh') i
+ixxAppend = coerce (listxAppend @_ @(Const i))
+
+ixxDrop :: forall sh sh' i. IxX (sh ++ sh') i -> IxX sh i -> IxX sh' i
+ixxDrop = coerce (listxDrop @(Const i) @(Const i))
+
+ixxFromLinear :: IShX sh -> Int -> IIxX sh
+ixxFromLinear = \sh i -> case go sh i of
+  (idx, 0) -> idx
+  _ -> error $ "ixxFromLinear: out of range (" ++ show i ++
+               " in array of shape " ++ show sh ++ ")"
+  where
+    -- returns (index in subarray, remaining index in enclosing array)
+    go :: IShX sh -> Int -> (IIxX sh, Int)
+    go ZSX i = (ZIX, i)
+    go (n :$% sh) i =
+      let (idx, i') = go sh i
+          (upi, locali) = i' `quotRem` fromSMayNat' n
+      in (locali :.% idx, upi)
+
+ixxToLinear :: IShX sh -> IIxX sh -> Int
+ixxToLinear = \sh i -> fst (go sh i)
+  where
+    -- returns (index in subarray, size of subarray)
+    go :: IShX sh -> IIxX sh -> (Int, Int)
+    go ZSX ZIX = (0, 1)
+    go (n :$% sh) (i :.% ix) =
+      let (lidx, sz) = go sh ix
+      in (sz * i + lidx, fromSMayNat' n * sz)
+
+
+-- * Mixed shapes
+
+data SMayNat i f n where
+  SUnknown :: i -> SMayNat i f Nothing
+  SKnown :: f n -> SMayNat i f (Just n)
+deriving instance (Show i, forall m. Show (f m)) => Show (SMayNat i f n)
+deriving instance (Eq i, forall m. Eq (f m)) => Eq (SMayNat i f n)
+deriving instance (Ord i, forall m. Ord (f m)) => Ord (SMayNat i f n)
+
+instance (NFData i, forall m. NFData (f m)) => NFData (SMayNat i f n) where
+  rnf (SUnknown i) = rnf i
+  rnf (SKnown x) = rnf x
+
+fromSMayNat :: (n ~ Nothing => i -> r)
+            -> (forall m. n ~ Just m => f m -> r)
+            -> SMayNat i f n -> r
+fromSMayNat f _ (SUnknown i) = f i
+fromSMayNat _ g (SKnown s) = g s
+
+fromSMayNat' :: SMayNat Int SNat n -> Int
+fromSMayNat' = fromSMayNat id fromSNat'
+
+type family AddMaybe n m where
+  AddMaybe Nothing _ = Nothing
+  AddMaybe (Just _) Nothing = Nothing
+  AddMaybe (Just n) (Just m) = Just (n + m)
+
+smnAddMaybe :: SMayNat Int SNat n -> SMayNat Int SNat m -> SMayNat Int SNat (AddMaybe n m)
+smnAddMaybe (SUnknown n) m = SUnknown (n + fromSMayNat' m)
+smnAddMaybe (SKnown n) (SUnknown m) = SUnknown (fromSNat' n + m)
+smnAddMaybe (SKnown n) (SKnown m) = SKnown (snatPlus n m)
+
+
+-- | This is a newtype over 'ListX'.
+type role ShX nominal representational
+type ShX :: [Maybe Nat] -> Type -> Type
+newtype ShX sh i = ShX (ListX sh (SMayNat i SNat))
+  deriving (Eq, Ord, Generic)
+
+pattern ZSX :: forall sh i. () => sh ~ '[] => ShX sh i
+pattern ZSX = ShX ZX
+
+pattern (:$%)
+  :: forall {sh1} {i}.
+     forall n sh. (n : sh ~ sh1)
+  => SMayNat i SNat n -> ShX sh i -> ShX sh1 i
+pattern i :$% shl <- ShX (listxUncons -> Just (UnconsListXRes (ShX -> shl) i))
+  where i :$% ShX shl = ShX (i ::% shl)
+infixr 3 :$%
+
+{-# COMPLETE ZSX, (:$%) #-}
+
+type IShX sh = ShX sh Int
+
+instance Show i => Show (ShX sh i) where
+  showsPrec _ (ShX l) = listxShow (fromSMayNat shows (shows . fromSNat)) l
+
+instance Functor (ShX sh) where
+  fmap f (ShX l) = ShX (listxFmap (fromSMayNat (SUnknown . f) SKnown) l)
+
+instance NFData i => NFData (ShX sh i) where
+  rnf (ShX ZX) = ()
+  rnf (ShX (SUnknown i ::% l)) = rnf i `seq` rnf (ShX l)
+  rnf (ShX (SKnown SNat ::% l)) = rnf (ShX l)
+
+shxLength :: ShX sh i -> Int
+shxLength (ShX l) = listxLength l
+
+-- | This is more than @geq@: it also checks that the integers (the unknown
+-- dimensions) are the same.
+shxEqual :: Eq i => ShX sh i -> ShX sh' i -> Maybe (sh :~: sh')
+shxEqual ZSX ZSX = Just Refl
+shxEqual (SKnown n@SNat :$% sh) (SKnown m@SNat :$% sh')
+  | Just Refl <- sameNat n m
+  , Just Refl <- shxEqual sh sh'
+  = Just Refl
+shxEqual (SUnknown i :$% sh) (SUnknown j :$% sh')
+  | i == j
+  , Just Refl <- shxEqual sh sh'
+  = Just Refl
+shxEqual _ _ = Nothing
+
+-- | The number of elements in an array described by this shape.
+shxSize :: IShX sh -> Int
+shxSize ZSX = 1
+shxSize (n :$% sh) = fromSMayNat' n * shxSize sh
+
+shxToList :: IShX sh -> [Int]
+shxToList ZSX = []
+shxToList (smn :$% sh) = fromSMayNat' smn : shxToList sh
+
+shxAppend :: forall sh sh' i. ShX sh i -> ShX sh' i -> ShX (sh ++ sh') i
+shxAppend = coerce (listxAppend @_ @(SMayNat i SNat))
+
+shxTail :: ShX (n : sh) i -> ShX sh i
+shxTail (_ :$% sh) = sh
+
+shxDropSSX :: forall sh sh' i. ShX (sh ++ sh') i -> StaticShX sh -> ShX sh' i
+shxDropSSX = coerce (listxDrop @(SMayNat i SNat) @(SMayNat () SNat))
+
+shxDropIx :: forall sh sh' i j. ShX (sh ++ sh') i -> IxX sh j -> ShX sh' i
+shxDropIx = coerce (listxDrop @(SMayNat i SNat) @(Const j))
+
+shxDropSh :: forall sh sh' i. ShX (sh ++ sh') i -> ShX sh i -> ShX sh' i
+shxDropSh = coerce (listxDrop @(SMayNat i SNat) @(SMayNat i SNat))
+
+shxTakeSSX :: forall sh sh' i. Proxy sh' -> ShX (sh ++ sh') i -> StaticShX sh -> ShX sh i
+shxTakeSSX _ = flip go
+  where
+    go :: StaticShX sh1 -> ShX (sh1 ++ sh') i -> ShX sh1 i
+    go ZKX _ = ZSX
+    go (_ :!% ssh1) (n :$% sh) = n :$% go ssh1 sh
+
+-- This is a weird operation, so it has a long name
+shxCompleteZeros :: StaticShX sh -> IShX sh
+shxCompleteZeros ZKX = ZSX
+shxCompleteZeros (SUnknown () :!% ssh) = SUnknown 0 :$% shxCompleteZeros ssh
+shxCompleteZeros (SKnown n :!% ssh) = SKnown n :$% shxCompleteZeros ssh
+
+shxSplitApp :: Proxy sh' -> StaticShX sh -> ShX (sh ++ sh') i -> (ShX sh i, ShX sh' i)
+shxSplitApp _ ZKX idx = (ZSX, idx)
+shxSplitApp p (_ :!% ssh) (i :$% idx) = first (i :$%) (shxSplitApp p ssh idx)
+
+shxEnum :: IShX sh -> [IIxX sh]
+shxEnum = \sh -> go sh id []
+  where
+    go :: IShX sh -> (IIxX sh -> a) -> [a] -> [a]
+    go ZSX f = (f ZIX :)
+    go (n :$% sh) f = foldr (.) id [go sh (f . (i :.%)) | i <- [0 .. fromSMayNat' n - 1]]
+
+
+-- * Static mixed shapes
+
+-- | The part of a shape that is statically known. (A newtype over 'ListX'.)
+type StaticShX :: [Maybe Nat] -> Type
+newtype StaticShX sh = StaticShX (ListX sh (SMayNat () SNat))
+  deriving (Eq, Ord)
+
+pattern ZKX :: forall sh. () => sh ~ '[] => StaticShX sh
+pattern ZKX = StaticShX ZX
+
+pattern (:!%)
+  :: forall {sh1}.
+     forall n sh. (n : sh ~ sh1)
+  => SMayNat () SNat n -> StaticShX sh -> StaticShX sh1
+pattern i :!% shl <- StaticShX (listxUncons -> Just (UnconsListXRes (StaticShX -> shl) i))
+  where i :!% StaticShX shl = StaticShX (i ::% shl)
+infixr 3 :!%
+
+{-# COMPLETE ZKX, (:!%) #-}
+
+instance Show (StaticShX sh) where
+  showsPrec _ (StaticShX l) = listxShow (fromSMayNat shows (shows . fromSNat)) l
+
+ssxLength :: StaticShX sh -> Int
+ssxLength (StaticShX l) = listxLength l
+
+-- | This suffices as an implementation of @geq@ in the @Data.GADT.Compare@
+-- class of the @some@ package.
+ssxGeq :: StaticShX sh -> StaticShX sh' -> Maybe (sh :~: sh')
+ssxGeq ZKX ZKX = Just Refl
+ssxGeq (SKnown n@SNat :!% sh) (SKnown m@SNat :!% sh')
+  | Just Refl <- sameNat n m
+  , Just Refl <- ssxGeq sh sh'
+  = Just Refl
+ssxGeq (SUnknown () :!% sh) (SUnknown () :!% sh')
+  | Just Refl <- ssxGeq sh sh'
+  = Just Refl
+ssxGeq _ _ = Nothing
+
+ssxAppend :: StaticShX sh -> StaticShX sh' -> StaticShX (sh ++ sh')
+ssxAppend ZKX sh' = sh'
+ssxAppend (n :!% sh) sh' = n :!% ssxAppend sh sh'
+
+ssxTail :: StaticShX (n : sh) -> StaticShX sh
+ssxTail (_ :!% ssh) = ssh
+
+ssxDropIx :: forall sh sh' i. StaticShX (sh ++ sh') -> IxX sh i -> StaticShX sh'
+ssxDropIx = coerce (listxDrop @(SMayNat () SNat) @(Const i))
+
+-- | This may fail if @sh@ has @Nothing@s in it.
+ssxToShX' :: StaticShX sh -> Maybe (IShX sh)
+ssxToShX' ZKX = Just ZSX
+ssxToShX' (SKnown n :!% sh) = (SKnown n :$%) <$> ssxToShX' sh
+ssxToShX' (SUnknown _ :!% _) = Nothing
+
+ssxReplicate :: SNat n -> StaticShX (Replicate n Nothing)
+ssxReplicate SZ = ZKX
+ssxReplicate (SS (n :: SNat n'))
+  | Refl <- lemReplicateSucc @(Nothing @Nat) @n'
+  = SUnknown () :!% ssxReplicate n
+
+ssxIotaFrom :: Int -> StaticShX sh -> [Int]
+ssxIotaFrom _ ZKX = []
+ssxIotaFrom i (_ :!% ssh) = i : ssxIotaFrom (i+1) ssh
+
+ssxFromShape :: IShX sh -> StaticShX sh
+ssxFromShape ZSX = ZKX
+ssxFromShape (n :$% sh) = fromSMayNat (\_ -> SUnknown ()) SKnown n :!% ssxFromShape sh
+
+
+-- | Evidence for the static part of a shape. This pops up only when you are
+-- polymorphic in the element type of an array.
+type KnownShX :: [Maybe Nat] -> Constraint
+class KnownShX sh where knownShX :: StaticShX sh
+instance KnownShX '[] where knownShX = ZKX
+instance (KnownNat n, KnownShX sh) => KnownShX (Just n : sh) where knownShX = SKnown natSing :!% knownShX
+instance KnownShX sh => KnownShX (Nothing : sh) where knownShX = SUnknown () :!% knownShX
+
+
+-- * Flattening
+
+type Flatten sh = Flatten' 1 sh
+
+type family Flatten' acc sh where
+  Flatten' acc '[] = Just acc
+  Flatten' acc (Nothing : sh) = Nothing
+  Flatten' acc (Just n : sh) = Flatten' (acc * n) sh
+
+-- This function is currently unused
+ssxFlatten :: StaticShX sh -> SMayNat () SNat (Flatten sh)
+ssxFlatten = go (SNat @1)
+  where
+    go :: SNat acc -> StaticShX sh -> SMayNat () SNat (Flatten' acc sh)
+    go acc ZKX = SKnown acc
+    go _ (SUnknown () :!% _) = SUnknown ()
+    go acc (SKnown sn :!% sh) = go (snatMul acc sn) sh
+
+shxFlatten :: IShX sh -> SMayNat Int SNat (Flatten sh)
+shxFlatten = go (SNat @1)
+  where
+    go :: SNat acc -> IShX sh -> SMayNat Int SNat (Flatten' acc sh)
+    go acc ZSX = SKnown acc
+    go acc (SUnknown n :$% sh) = SUnknown (goUnknown (fromSNat' acc * n) sh)
+    go acc (SKnown sn :$% sh) = go (snatMul acc sn) sh
+
+    goUnknown :: Int -> IShX sh -> Int
+    goUnknown acc ZSX = acc
+    goUnknown acc (SUnknown n :$% sh) = goUnknown (acc * n) sh
+    goUnknown acc (SKnown sn :$% sh) = goUnknown (acc * fromSNat' sn) sh
+
+
+-- | Very untyped: only length is checked (at runtime).
+instance KnownShX sh => IsList (ListX sh (Const i)) where
+  type Item (ListX sh (Const i)) = i
+  fromList topl = go (knownShX @sh) topl
+    where
+      go :: StaticShX sh' -> [i] -> ListX sh' (Const i)
+      go ZKX [] = ZX
+      go (_ :!% sh) (i : is) = Const i ::% go sh is
+      go _ _ = error $ "IsList(ListX): Mismatched list length (type says "
+                         ++ show (ssxLength (knownShX @sh)) ++ ", list has length "
+                         ++ show (length topl) ++ ")"
+  toList = listxToList
+
+-- | Very untyped: only length is checked (at runtime), index bounds are __not checked__.
+instance KnownShX sh => IsList (IxX sh i) where
+  type Item (IxX sh i) = i
+  fromList = IxX . IsList.fromList
+  toList = Foldable.toList
+
+-- | Untyped: length and known dimensions are checked (at runtime).
+instance KnownShX sh => IsList (ShX sh Int) where
+  type Item (ShX sh Int) = Int
+  fromList topl = ShX (go (knownShX @sh) topl)
+    where
+      go :: StaticShX sh' -> [Int] -> ListX sh' (SMayNat Int SNat)
+      go ZKX [] = ZX
+      go (SKnown sn :!% sh) (i : is)
+        | i == fromSNat' sn = SKnown sn ::% go sh is
+        | otherwise = error $ "IsList(ShX): Value does not match typing (type says "
+                                ++ show (fromSNat' sn) ++ ", list contains " ++ show i ++ ")"
+      go (SUnknown () :!% sh) (i : is) = SUnknown i ::% go sh is
+      go _ _ = error $ "IsList(ShX): Mismatched list length (type says "
+                         ++ show (ssxLength (knownShX @sh)) ++ ", list has length "
+                         ++ show (length topl) ++ ")"
+  toList = shxToList
diff --git a/src/Data/Array/Mixed/Types.hs b/src/Data/Array/Mixed/Types.hs
new file mode 100644
index 0000000..d77513f
--- /dev/null
+++ b/src/Data/Array/Mixed/Types.hs
@@ -0,0 +1,110 @@
+{-# LANGUAGE DataKinds #-}
+{-# LANGUAGE GADTs #-}
+{-# LANGUAGE NoStarIsType #-}
+{-# LANGUAGE PatternSynonyms #-}
+{-# LANGUAGE PolyKinds #-}
+{-# LANGUAGE ScopedTypeVariables #-}
+{-# LANGUAGE TypeApplications #-}
+{-# LANGUAGE TypeFamilies #-}
+{-# LANGUAGE TypeOperators #-}
+{-# LANGUAGE UndecidableInstances #-}
+{-# LANGUAGE ViewPatterns #-}
+{-# OPTIONS_GHC -fplugin GHC.TypeLits.Normalise #-}
+{-# OPTIONS_GHC -fplugin GHC.TypeLits.KnownNat.Solver #-}
+module Data.Array.Mixed.Types (
+  -- * Reified evidence of a type class
+  Dict(..),
+
+  -- * Type-level naturals
+  pattern SZ, pattern SS,
+  fromSNat',
+  snatPlus, snatMul,
+
+  -- * Type-level lists
+  type (++),
+  lemAppNil,
+  lemAppAssoc,
+  Replicate,
+  lemReplicateSucc,
+
+  -- * Unsafe
+  unsafeCoerceRefl,
+) where
+
+import Data.Type.Equality
+import Data.Proxy
+import GHC.TypeLits
+import qualified GHC.TypeNats as TN
+import qualified Unsafe.Coerce
+
+
+-- | Evidence for the constraint @c a@.
+data Dict c a where
+  Dict :: c a => Dict c a
+
+fromSNat' :: SNat n -> Int
+fromSNat' = fromIntegral . fromSNat
+
+pattern SZ :: () => (n ~ 0) => SNat n
+pattern SZ <- ((\sn -> testEquality sn (SNat @0)) -> Just Refl)
+  where SZ = SNat
+
+pattern SS :: forall np1. () => forall n. (n + 1 ~ np1) => SNat n -> SNat np1
+pattern SS sn <- (snatPred -> Just (SNatPredResult sn Refl))
+  where SS = snatSucc
+
+{-# COMPLETE SZ, SS #-}
+
+snatSucc :: SNat n -> SNat (n + 1)
+snatSucc SNat = SNat
+
+data SNatPredResult np1 = forall n. SNatPredResult (SNat n) (n + 1 :~: np1)
+snatPred :: forall np1. SNat np1 -> Maybe (SNatPredResult np1)
+snatPred snp1 =
+  withKnownNat snp1 $
+    case cmpNat (Proxy @1) (Proxy @np1) of
+      LTI -> Just (SNatPredResult (SNat @(np1 - 1)) Refl)
+      EQI -> Just (SNatPredResult (SNat @(np1 - 1)) Refl)
+      GTI -> Nothing
+
+-- This should be a function in base
+snatPlus :: SNat n -> SNat m -> SNat (n + m)
+snatPlus n m = TN.withSomeSNat (TN.fromSNat n + TN.fromSNat m) Unsafe.Coerce.unsafeCoerce
+
+-- This should be a function in base
+snatMul :: SNat n -> SNat m -> SNat (n * m)
+snatMul n m = TN.withSomeSNat (TN.fromSNat n * TN.fromSNat m) Unsafe.Coerce.unsafeCoerce
+
+
+-- | This is just @'Unsafe.Coerce.unsafeCoerce' 'Refl'@, but specialised to
+-- only typecheck for actual type equalities. One cannot, e.g. accidentally
+-- write this:
+--
+-- @
+-- foo :: Proxy a -> Proxy b -> a :~: b
+-- foo = unsafeCoerceRefl
+-- @
+--
+-- which would have been permitted with normal 'Unsafe.Coerce.unsafeCoerce',
+-- but would have resulted in interesting memory errors at runtime.
+unsafeCoerceRefl :: a :~: b
+unsafeCoerceRefl = Unsafe.Coerce.unsafeCoerce Refl
+
+
+-- | Type-level list append.
+type family l1 ++ l2 where
+  '[] ++ l2 = l2
+  (x : xs) ++ l2 = x : xs ++ l2
+
+lemAppNil :: l ++ '[] :~: l
+lemAppNil = unsafeCoerceRefl
+
+lemAppAssoc :: Proxy a -> Proxy b -> Proxy c -> (a ++ b) ++ c :~: a ++ (b ++ c)
+lemAppAssoc _ _ _ = unsafeCoerceRefl
+
+type family Replicate n a where
+  Replicate 0 a = '[]
+  Replicate n a = a : Replicate (n - 1) a
+
+lemReplicateSucc :: (a : Replicate n a) :~: Replicate (n + 1) a
+lemReplicateSucc = unsafeCoerceRefl
diff --git a/src/Data/Array/Nested.hs b/src/Data/Array/Nested.hs
index e3af0ee..1a4e094 100644
--- a/src/Data/Array/Nested.hs
+++ b/src/Data/Array/Nested.hs
@@ -58,8 +58,9 @@ module Data.Array.Nested (
   type (++),
   Storable,
   SNat, pattern SNat,
-  HList,
-  Permutation,
+  pattern SZ, pattern SS,
+  Perm(..),
+  IsPermutation,
   KnownNatList(..),
   listSToList,
   shSToList,
@@ -69,7 +70,10 @@ module Data.Array.Nested (
 import Prelude hiding (mappend)
 
 import Data.Array.Mixed
+import Data.Array.Mixed.Internal.Arith
+import Data.Array.Mixed.Permutation
+import Data.Array.Mixed.Shape
+import Data.Array.Mixed.Types
 import Data.Array.Nested.Internal
-import Data.Array.Nested.Internal.Arith
 import Foreign.Storable
 import GHC.TypeLits
diff --git a/src/Data/Array/Nested/Internal.hs b/src/Data/Array/Nested/Internal.hs
index 712c5f1..0870789 100644
--- a/src/Data/Array/Nested/Internal.hs
+++ b/src/Data/Array/Nested/Internal.hs
@@ -60,7 +60,11 @@ import Unsafe.Coerce
 
 import Data.Array.Mixed
 import qualified Data.Array.Mixed as X
-import Data.Array.Nested.Internal.Arith
+import Data.Array.Mixed.Lemmas
+import Data.Array.Mixed.Permutation
+import Data.Array.Mixed.Shape
+import Data.Array.Mixed.Internal.Arith
+import Data.Array.Mixed.Types
 
 
 -- Invariant in the API
@@ -123,19 +127,19 @@ lemKnownShX (SUnknown () :!% ssh) | Dict <- lemKnownShX ssh = Dict
 
 ssxFromSNat :: SNat n -> StaticShX (Replicate n Nothing)
 ssxFromSNat SZ = ZKX
-ssxFromSNat (SS (n :: SNat nm1)) | Refl <- X.lemReplicateSucc @(Nothing @Nat) @nm1 = SUnknown () :!% ssxFromSNat n
+ssxFromSNat (SS (n :: SNat nm1)) | Refl <- lemReplicateSucc @(Nothing @Nat) @nm1 = SUnknown () :!% ssxFromSNat n
 
 lemKnownReplicate :: SNat n -> Dict KnownShX (Replicate n Nothing)
 lemKnownReplicate sn = lemKnownShX (ssxFromSNat sn)
 
-lemRankReplicate :: SNat n -> X.Rank (Replicate n (Nothing @Nat)) :~: n
+lemRankReplicate :: SNat n -> Rank (Replicate n (Nothing @Nat)) :~: n
 lemRankReplicate SZ = Refl
 lemRankReplicate (SS (n :: SNat nm1))
-  | Refl <- X.lemReplicateSucc @(Nothing @Nat) @nm1
+  | Refl <- lemReplicateSucc @(Nothing @Nat) @nm1
   , Refl <- lemRankReplicate n
   = Refl
 
-lemRankMapJust :: forall sh. ShS sh -> X.Rank (MapJust sh) :~: X.Rank sh
+lemRankMapJust :: forall sh. ShS sh -> Rank (MapJust sh) :~: Rank sh
 lemRankMapJust ZSS = Refl
 lemRankMapJust (_ :$$ sh') | Refl <- lemRankMapJust sh' = Refl
 
@@ -146,9 +150,9 @@ lemReplicatePlusApp sn _ _ = go sn
     go :: SNat n' -> Replicate (n' + m) a :~: Replicate n' a ++ Replicate m a
     go SZ = Refl
     go (SS (n :: SNat n'm1))
-      | Refl <- X.lemReplicateSucc @a @n'm1
+      | Refl <- lemReplicateSucc @a @n'm1
       , Refl <- go n
-      = sym (X.lemReplicateSucc @a @(n'm1 + m))
+      = sym (lemReplicateSucc @a @(n'm1 + m))
 
 lemLeqPlus :: n <= m => Proxy n -> Proxy m -> Proxy k -> (n <=? (m + k)) :~: 'True
 lemLeqPlus _ _ _ = Refl
@@ -156,17 +160,17 @@ lemLeqPlus _ _ _ = Refl
 lemLeqSuccSucc :: (k + 1 <= n) => Proxy k -> Proxy n -> (k <=? n - 1) :~: True
 lemLeqSuccSucc _ _ = unsafeCoerce Refl
 
-lemDropLenApp :: X.Rank l1 <= X.Rank l2
+lemDropLenApp :: Rank l1 <= Rank l2
               => Proxy l1 -> Proxy l2 -> Proxy rest
-              -> X.DropLen l1 l2 ++ rest :~: X.DropLen l1 (l2 ++ rest)
+              -> DropLen l1 l2 ++ rest :~: DropLen l1 (l2 ++ rest)
 lemDropLenApp _ _ _ = unsafeCoerce Refl
 
-lemTakeLenApp :: X.Rank l1 <= X.Rank l2
+lemTakeLenApp :: Rank l1 <= Rank l2
               => Proxy l1 -> Proxy l2 -> Proxy rest
-              -> X.TakeLen l1 l2 :~: X.TakeLen l1 (l2 ++ rest)
+              -> TakeLen l1 l2 :~: TakeLen l1 (l2 ++ rest)
 lemTakeLenApp _ _ _ = unsafeCoerce Refl
 
-srankSh :: ShX sh f -> SNat (X.Rank sh)
+srankSh :: ShX sh f -> SNat (Rank sh)
 srankSh ZSX = SNat
 srankSh (_ :$% sh) | SNat <- srankSh sh = SNat
 
@@ -585,11 +589,11 @@ class Elt a where
          -> (forall sh' b. Storable b => StaticShX sh' -> XArray (sh1 ++ sh') b -> XArray (sh2 ++ sh') b -> XArray (sh3 ++ sh') b)
          -> Mixed sh1 a -> Mixed sh2 a -> Mixed sh3 a
 
-  mcast :: forall sh1 sh2 sh'. X.Rank sh1 ~ X.Rank sh2
+  mcast :: forall sh1 sh2 sh'. Rank sh1 ~ Rank sh2
         => StaticShX sh1 -> IShX sh2 -> Proxy sh' -> Mixed (sh1 ++ sh') a -> Mixed (sh2 ++ sh') a
 
-  mtranspose :: forall is sh. (X.Permutation is, X.Rank is <= X.Rank sh)
-             => HList SNat is -> Mixed sh a -> Mixed (X.PermutePrefix is sh) a
+  mtranspose :: forall is sh. (IsPermutation is, Rank is <= Rank sh)
+             => Perm is -> Mixed sh a -> Mixed (PermutePrefix is sh) a
 
   -- ====== PRIVATE METHODS ====== --
 
@@ -635,20 +639,20 @@ class Elt a => KnownElt a where
 instance Storable a => Elt (Primitive a) where
   mshape (M_Primitive sh _) = sh
   mindex (M_Primitive _ a) i = Primitive (X.index a i)
-  mindexPartial (M_Primitive sh a) i = M_Primitive (X.shDropIx sh i) (X.indexPartial a i)
+  mindexPartial (M_Primitive sh a) i = M_Primitive (shxDropIx sh i) (X.indexPartial a i)
   mscalar (Primitive x) = M_Primitive ZSX (X.scalar x)
   mfromListOuter l@(arr1 :| _) =
     let sh = SUnknown (length l) :$% mshape arr1
-    in M_Primitive sh (X.fromListOuter (X.staticShapeFrom sh) (map (\(M_Primitive _ a) -> a) (toList l)))
-  mtoListOuter (M_Primitive sh arr) = map (M_Primitive (X.shTail sh)) (X.toListOuter arr)
+    in M_Primitive sh (X.fromListOuter (ssxFromShape sh) (map (\(M_Primitive _ a) -> a) (toList l)))
+  mtoListOuter (M_Primitive sh arr) = map (M_Primitive (shxTail sh)) (X.toListOuter arr)
 
   mlift :: forall sh1 sh2.
            StaticShX sh2
         -> (StaticShX '[] -> XArray (sh1 ++ '[]) a -> XArray (sh2 ++ '[]) a)
         -> Mixed sh1 (Primitive a) -> Mixed sh2 (Primitive a)
   mlift ssh2 f (M_Primitive _ a)
-    | Refl <- X.lemAppNil @sh1
-    , Refl <- X.lemAppNil @sh2
+    | Refl <- lemAppNil @sh1
+    , Refl <- lemAppNil @sh2
     , let result = f ZKX a
     = M_Primitive (X.shape ssh2 result) result
 
@@ -657,36 +661,36 @@ instance Storable a => Elt (Primitive a) where
          -> (StaticShX '[] -> XArray (sh1 ++ '[]) a -> XArray (sh2 ++ '[]) a -> XArray (sh3 ++ '[]) a)
          -> Mixed sh1 (Primitive a) -> Mixed sh2 (Primitive a) -> Mixed sh3 (Primitive a)
   mlift2 ssh3 f (M_Primitive _ a) (M_Primitive _ b)
-    | Refl <- X.lemAppNil @sh1
-    , Refl <- X.lemAppNil @sh2
-    , Refl <- X.lemAppNil @sh3
+    | Refl <- lemAppNil @sh1
+    , Refl <- lemAppNil @sh2
+    , Refl <- lemAppNil @sh3
     , let result = f ZKX a b
     = M_Primitive (X.shape ssh3 result) result
 
-  mcast :: forall sh1 sh2 sh'. X.Rank sh1 ~ X.Rank sh2
+  mcast :: forall sh1 sh2 sh'. Rank sh1 ~ Rank sh2
         => StaticShX sh1 -> IShX sh2 -> Proxy sh' -> Mixed (sh1 ++ sh') (Primitive a) -> Mixed (sh2 ++ sh') (Primitive a)
   mcast ssh1 sh2 _ (M_Primitive sh1' arr) =
-    let (_, sh') = shAppSplit (Proxy @sh') ssh1 sh1'
-    in M_Primitive (shAppend sh2 sh') (X.cast ssh1 sh2 (X.staticShapeFrom sh') arr)
+    let (_, sh') = shxSplitApp (Proxy @sh') ssh1 sh1'
+    in M_Primitive (shxAppend sh2 sh') (X.cast ssh1 sh2 (ssxFromShape sh') arr)
 
   mtranspose perm (M_Primitive sh arr) =
-    M_Primitive (X.shPermutePrefix perm sh)
-                (X.transpose (X.staticShapeFrom sh) perm arr)
+    M_Primitive (shxPermutePrefix perm sh)
+                (X.transpose (ssxFromShape sh) perm arr)
 
   mshapeTree _ = ()
   mshapeTreeEq _ () () = True
   mshapeTreeEmpty _ () = False
   mshowShapeTree _ () = "()"
-  mvecsWrite sh i (Primitive x) (MV_Primitive v) = VSM.write v (X.toLinearIdx sh i) x
+  mvecsWrite sh i (Primitive x) (MV_Primitive v) = VSM.write v (ixxToLinear sh i) x
 
   -- TODO: this use of toVector is suboptimal
   mvecsWritePartial
     :: forall sh' sh s.
        IShX (sh ++ sh') -> IIxX sh -> Mixed sh' (Primitive a) -> MixedVecs s (sh ++ sh') (Primitive a) -> ST s ()
   mvecsWritePartial sh i (M_Primitive sh' arr) (MV_Primitive v) = do
-    let arrsh = X.shape (X.staticShapeFrom sh') arr
-        offset = X.toLinearIdx sh (X.ixAppend i (X.zeroIxX' arrsh))
-    VS.copy (VSM.slice offset (X.shapeSize arrsh) v) (X.toVector arr)
+    let arrsh = X.shape (ssxFromShape sh') arr
+        offset = ixxToLinear sh (ixxAppend i (ixxZero' arrsh))
+    VS.copy (VSM.slice offset (shxSize arrsh) v) (X.toVector arr)
 
   mvecsFreeze sh (MV_Primitive v) = M_Primitive sh . X.fromVector sh <$> VS.freeze v
 
@@ -701,7 +705,7 @@ deriving via Primitive () instance Elt ()
 
 instance Storable a => KnownElt (Primitive a) where
   memptyArray sh = M_Primitive sh (X.empty sh)
-  mvecsUnsafeNew sh _ = MV_Primitive <$> VSM.unsafeNew (X.shapeSize sh)
+  mvecsUnsafeNew sh _ = MV_Primitive <$> VSM.unsafeNew (shxSize sh)
   mvecsNewEmpty _ = MV_Primitive <$> VSM.unsafeNew 0
 
 -- [PRIMITIVE ELEMENT TYPES LIST]
@@ -755,7 +759,7 @@ instance Elt a => Elt (Mixed sh' a) where
   -- moverlongShape method, a prefix of which is mshape.
   mshape :: forall sh. Mixed sh (Mixed sh' a) -> IShX sh
   mshape (M_Nest sh arr)
-    = fst (shAppSplit (Proxy @sh') (X.staticShapeFrom sh) (mshape arr))
+    = fst (shxSplitApp (Proxy @sh') (ssxFromShape sh) (mshape arr))
 
   mindex :: Mixed sh (Mixed sh' a) -> IIxX sh -> Mixed sh' a
   mindex (M_Nest _ arr) i = mindexPartial arr i
@@ -763,8 +767,8 @@ instance Elt a => Elt (Mixed sh' a) where
   mindexPartial :: forall sh1 sh2.
                    Mixed (sh1 ++ sh2) (Mixed sh' a) -> IIxX sh1 -> Mixed sh2 (Mixed sh' a)
   mindexPartial (M_Nest sh arr) i
-    | Refl <- X.lemAppAssoc (Proxy @sh1) (Proxy @sh2) (Proxy @sh')
-    = M_Nest (X.shDropIx sh i) (mindexPartial @a @sh1 @(sh2 ++ sh') arr i)
+    | Refl <- lemAppAssoc (Proxy @sh1) (Proxy @sh2) (Proxy @sh')
+    = M_Nest (shxDropIx sh i) (mindexPartial @a @sh1 @(sh2 ++ sh') arr i)
 
   mscalar = M_Nest ZSX
 
@@ -773,95 +777,95 @@ instance Elt a => Elt (Mixed sh' a) where
     M_Nest (SUnknown (length l) :$% mshape arr)
            (mfromListOuter ((\(M_Nest _ a) -> a) <$> l))
 
-  mtoListOuter (M_Nest sh arr) = map (M_Nest (X.shTail sh)) (mtoListOuter arr)
+  mtoListOuter (M_Nest sh arr) = map (M_Nest (shxTail sh)) (mtoListOuter arr)
 
   mlift :: forall sh1 sh2.
            StaticShX sh2
         -> (forall shT b. Storable b => StaticShX shT -> XArray (sh1 ++ shT) b -> XArray (sh2 ++ shT) b)
         -> Mixed sh1 (Mixed sh' a) -> Mixed sh2 (Mixed sh' a)
   mlift ssh2 f (M_Nest sh1 arr) =
-    let result = mlift (X.ssxAppend ssh2 ssh') f' arr
-        (sh2, _) = shAppSplit (Proxy @sh') ssh2 (mshape result)
+    let result = mlift (ssxAppend ssh2 ssh') f' arr
+        (sh2, _) = shxSplitApp (Proxy @sh') ssh2 (mshape result)
     in M_Nest sh2 result
     where
-      ssh' = X.staticShapeFrom (snd (shAppSplit (Proxy @sh') (X.staticShapeFrom sh1) (mshape arr)))
+      ssh' = ssxFromShape (snd (shxSplitApp (Proxy @sh') (ssxFromShape sh1) (mshape arr)))
 
       f' :: forall shT b. Storable b => StaticShX shT -> XArray ((sh1 ++ sh') ++ shT) b -> XArray ((sh2 ++ sh') ++ shT) b
       f' sshT
-        | Refl <- X.lemAppAssoc (Proxy @sh1) (Proxy @sh') (Proxy @shT)
-        , Refl <- X.lemAppAssoc (Proxy @sh2) (Proxy @sh') (Proxy @shT)
-        = f (X.ssxAppend ssh' sshT)
+        | Refl <- lemAppAssoc (Proxy @sh1) (Proxy @sh') (Proxy @shT)
+        , Refl <- lemAppAssoc (Proxy @sh2) (Proxy @sh') (Proxy @shT)
+        = f (ssxAppend ssh' sshT)
 
   mlift2 :: forall sh1 sh2 sh3.
             StaticShX sh3
          -> (forall shT b. Storable b => StaticShX shT -> XArray (sh1 ++ shT) b -> XArray (sh2 ++ shT) b -> XArray (sh3 ++ shT) b)
          -> Mixed sh1 (Mixed sh' a) -> Mixed sh2 (Mixed sh' a) -> Mixed sh3 (Mixed sh' a)
   mlift2 ssh3 f (M_Nest sh1 arr1) (M_Nest _ arr2) =
-    let result = mlift2 (X.ssxAppend ssh3 ssh') f' arr1 arr2
-        (sh3, _) = shAppSplit (Proxy @sh') ssh3 (mshape result)
+    let result = mlift2 (ssxAppend ssh3 ssh') f' arr1 arr2
+        (sh3, _) = shxSplitApp (Proxy @sh') ssh3 (mshape result)
     in M_Nest sh3 result
     where
-      ssh' = X.staticShapeFrom (snd (shAppSplit (Proxy @sh') (X.staticShapeFrom sh1) (mshape arr1)))
+      ssh' = ssxFromShape (snd (shxSplitApp (Proxy @sh') (ssxFromShape sh1) (mshape arr1)))
 
       f' :: forall shT b. Storable b => StaticShX shT -> XArray ((sh1 ++ sh') ++ shT) b -> XArray ((sh2 ++ sh') ++ shT) b -> XArray ((sh3 ++ sh') ++ shT) b
       f' sshT
-        | Refl <- X.lemAppAssoc (Proxy @sh1) (Proxy @sh') (Proxy @shT)
-        , Refl <- X.lemAppAssoc (Proxy @sh2) (Proxy @sh') (Proxy @shT)
-        , Refl <- X.lemAppAssoc (Proxy @sh3) (Proxy @sh') (Proxy @shT)
-        = f (X.ssxAppend ssh' sshT)
+        | Refl <- lemAppAssoc (Proxy @sh1) (Proxy @sh') (Proxy @shT)
+        , Refl <- lemAppAssoc (Proxy @sh2) (Proxy @sh') (Proxy @shT)
+        , Refl <- lemAppAssoc (Proxy @sh3) (Proxy @sh') (Proxy @shT)
+        = f (ssxAppend ssh' sshT)
 
-  mcast :: forall sh1 sh2 shT. X.Rank sh1 ~ X.Rank sh2
+  mcast :: forall sh1 sh2 shT. Rank sh1 ~ Rank sh2
         => StaticShX sh1 -> IShX sh2 -> Proxy shT -> Mixed (sh1 ++ shT) (Mixed sh' a) -> Mixed (sh2 ++ shT) (Mixed sh' a)
   mcast ssh1 sh2 _ (M_Nest sh1T arr)
-    | Refl <- X.lemAppAssoc (Proxy @sh1) (Proxy @shT) (Proxy @sh')
-    , Refl <- X.lemAppAssoc (Proxy @sh2) (Proxy @shT) (Proxy @sh')
-    = let (_, shT) = shAppSplit (Proxy @shT) ssh1 sh1T
-      in M_Nest (shAppend sh2 shT) (mcast ssh1 sh2 (Proxy @(shT ++ sh')) arr)
-
-  mtranspose :: forall is sh. (X.Permutation is, X.Rank is <= X.Rank sh)
-             => HList SNat is -> Mixed sh (Mixed sh' a)
-             -> Mixed (X.PermutePrefix is sh) (Mixed sh' a)
+    | Refl <- lemAppAssoc (Proxy @sh1) (Proxy @shT) (Proxy @sh')
+    , Refl <- lemAppAssoc (Proxy @sh2) (Proxy @shT) (Proxy @sh')
+    = let (_, shT) = shxSplitApp (Proxy @shT) ssh1 sh1T
+      in M_Nest (shxAppend sh2 shT) (mcast ssh1 sh2 (Proxy @(shT ++ sh')) arr)
+
+  mtranspose :: forall is sh. (IsPermutation is, Rank is <= Rank sh)
+             => Perm is -> Mixed sh (Mixed sh' a)
+             -> Mixed (PermutePrefix is sh) (Mixed sh' a)
   mtranspose perm (M_Nest sh arr)
-    | let sh' = X.shDropSh @sh @sh' (mshape arr) sh
-    , Refl <- X.lemRankApp (X.staticShapeFrom sh) (X.staticShapeFrom sh')
-    , Refl <- lemLeqPlus (Proxy @(X.Rank is)) (Proxy @(X.Rank sh)) (Proxy @(X.Rank sh'))
-    , Refl <- X.lemAppAssoc (Proxy @(Permute is (TakeLen is (sh ++ sh')))) (Proxy @(DropLen is sh)) (Proxy @sh')
+    | let sh' = shxDropSh @sh @sh' (mshape arr) sh
+    , Refl <- lemRankApp (ssxFromShape sh) (ssxFromShape sh')
+    , Refl <- lemLeqPlus (Proxy @(Rank is)) (Proxy @(Rank sh)) (Proxy @(Rank sh'))
+    , Refl <- lemAppAssoc (Proxy @(Permute is (TakeLen is (sh ++ sh')))) (Proxy @(DropLen is sh)) (Proxy @sh')
     , Refl <- lemDropLenApp (Proxy @is) (Proxy @sh) (Proxy @sh')
     , Refl <- lemTakeLenApp (Proxy @is) (Proxy @sh) (Proxy @sh')
-    = M_Nest (X.shPermutePrefix perm sh)
+    = M_Nest (shxPermutePrefix perm sh)
              (mtranspose perm arr)
 
   mshapeTree :: Mixed sh' a -> ShapeTree (Mixed sh' a)
-  mshapeTree arr = (mshape arr, mshapeTree (mindex arr (X.zeroIxX (X.staticShapeFrom (mshape arr)))))
+  mshapeTree arr = (mshape arr, mshapeTree (mindex arr (ixxZero (ssxFromShape (mshape arr)))))
 
   mshapeTreeEq _ (sh1, t1) (sh2, t2) = sh1 == sh2 && mshapeTreeEq (Proxy @a) t1 t2
 
-  mshapeTreeEmpty _ (sh, t) = X.shapeSize sh == 0 && mshapeTreeEmpty (Proxy @a) t
+  mshapeTreeEmpty _ (sh, t) = shxSize sh == 0 && mshapeTreeEmpty (Proxy @a) t
 
   mshowShapeTree _ (sh, t) = "(" ++ show sh ++ ", " ++ mshowShapeTree (Proxy @a) t ++ ")"
 
-  mvecsWrite sh idx val (MV_Nest sh' vecs) = mvecsWritePartial (X.shAppend sh sh') idx val vecs
+  mvecsWrite sh idx val (MV_Nest sh' vecs) = mvecsWritePartial (shxAppend sh sh') idx val vecs
 
   mvecsWritePartial :: forall sh1 sh2 s.
                        IShX (sh1 ++ sh2) -> IIxX 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)
-    | Refl <- X.lemAppAssoc (Proxy @sh1) (Proxy @sh2) (Proxy @sh')
-    = mvecsWritePartial (X.shAppend sh12 sh') idx arr vecs
+    | Refl <- lemAppAssoc (Proxy @sh1) (Proxy @sh2) (Proxy @sh')
+    = mvecsWritePartial (shxAppend sh12 sh') idx arr vecs
 
-  mvecsFreeze sh (MV_Nest sh' vecs) = M_Nest sh <$> mvecsFreeze (X.shAppend sh sh') vecs
+  mvecsFreeze sh (MV_Nest sh' vecs) = M_Nest sh <$> mvecsFreeze (shxAppend sh sh') vecs
 
 instance (KnownShX sh', KnownElt a) => KnownElt (Mixed sh' a) where
-  memptyArray sh = M_Nest sh (memptyArray (X.shAppend sh (X.completeShXzeros (knownShX @sh'))))
+  memptyArray sh = M_Nest sh (memptyArray (shxAppend sh (shxCompleteZeros (knownShX @sh'))))
 
   mvecsUnsafeNew sh example
-    | X.shapeSize sh' == 0 = mvecsNewEmpty (Proxy @(Mixed sh' a))
-    | otherwise = MV_Nest sh' <$> mvecsUnsafeNew (X.shAppend sh sh') (mindex example (X.zeroIxX (X.staticShapeFrom sh')))
+    | shxSize sh' == 0 = mvecsNewEmpty (Proxy @(Mixed sh' a))
+    | otherwise = MV_Nest sh' <$> mvecsUnsafeNew (shxAppend sh sh') (mindex example (ixxZero (ssxFromShape sh')))
     where
       sh' = mshape example
 
-  mvecsNewEmpty _ = MV_Nest (X.completeShXzeros (knownShX @sh')) <$> mvecsNewEmpty (Proxy @a)
+  mvecsNewEmpty _ = MV_Nest (shxCompleteZeros (knownShX @sh')) <$> mvecsNewEmpty (Proxy @a)
 
 
 -- | Create an array given a size and a function that computes the element at a
@@ -882,10 +886,10 @@ instance (KnownShX sh', KnownElt a) => KnownElt (Mixed sh' a) where
 -- array. The type of 'mgenerate' allows this requirement to be broken very
 -- easily, hence the runtime check.
 mgenerate :: forall sh a. KnownElt a => IShX sh -> (IIxX sh -> a) -> Mixed sh a
-mgenerate sh f = case X.enumShape sh of
+mgenerate sh f = case shxEnum sh of
   [] -> memptyArray sh
   firstidx : restidxs ->
-    let firstelem = f (X.zeroIxX' sh)
+    let firstelem = f (ixxZero' sh)
         shapetree = mshapeTree firstelem
     in if mshapeTreeEmpty (Proxy @a) shapetree
          then memptyArray sh
@@ -905,28 +909,28 @@ msumOuter1P :: forall sh n a. (Storable a, NumElt a)
             => Mixed (n : sh) (Primitive a) -> Mixed sh (Primitive a)
 msumOuter1P (M_Primitive (n :$% sh) arr) =
   let nssh = fromSMayNat (\_ -> SUnknown ()) SKnown n :!% ZKX
-  in M_Primitive sh (X.sumOuter nssh (X.staticShapeFrom sh) arr)
+  in M_Primitive sh (X.sumOuter nssh (ssxFromShape sh) arr)
 
 msumOuter1 :: forall sh n a. (NumElt a, PrimElt a)
            => Mixed (n : sh) a -> Mixed sh a
 msumOuter1 = fromPrimitive . msumOuter1P @sh @n @a . toPrimitive
 
 mappend :: forall n m sh a. Elt a
-        => Mixed (n : sh) a -> Mixed (m : sh) a -> Mixed (X.AddMaybe n m : sh) a
+        => Mixed (n : sh) a -> Mixed (m : sh) a -> Mixed (AddMaybe n m : sh) a
 mappend arr1 arr2 = mlift2 (snm :!% ssh) f arr1 arr2
   where
     sn :$% sh = mshape arr1
     sm :$% _ = mshape arr2
-    ssh = X.staticShapeFrom sh
-    snm :: SMayNat () SNat (X.AddMaybe n m)
+    ssh = ssxFromShape sh
+    snm :: SMayNat () SNat (AddMaybe n m)
     snm = case (sn, sm) of
             (SUnknown{}, _) -> SUnknown ()
             (SKnown{}, SUnknown{}) -> SUnknown ()
-            (SKnown n, SKnown m) -> SKnown (X.plusSNat n m)
+            (SKnown n, SKnown m) -> SKnown (snatPlus n m)
 
     f :: forall sh' b. Storable b
-      => StaticShX sh' -> XArray (n : sh ++ sh') b -> XArray (m : sh ++ sh') b -> XArray (X.AddMaybe n m : sh ++ sh') b
-    f ssh' = X.append (X.ssxAppend ssh ssh')
+      => StaticShX sh' -> XArray (n : sh ++ sh') b -> XArray (m : sh ++ sh') b -> XArray (AddMaybe n m : sh ++ sh') b
+    f ssh' = X.append (ssxAppend ssh ssh')
 
 mfromVectorP :: forall sh a. Storable a => IShX sh -> VS.Vector a -> Mixed sh (Primitive a)
 mfromVectorP sh v = M_Primitive sh (X.fromVector sh v)
@@ -971,9 +975,9 @@ mrerankP :: forall sh1 sh2 sh a b. (Storable a, Storable b)
          -> (Mixed sh1 (Primitive a) -> Mixed sh2 (Primitive b))
          -> Mixed (sh ++ sh1) (Primitive a) -> Mixed (sh ++ sh2) (Primitive b)
 mrerankP ssh sh2 f (M_Primitive sh arr) =
-  let sh1 = shDropSSX sh ssh
-  in M_Primitive (X.shAppend (shTakeSSX (Proxy @sh1) sh ssh) sh2)
-                 (X.rerank ssh (X.staticShapeFrom sh1) (X.staticShapeFrom sh2)
+  let sh1 = shxDropSSX sh ssh
+  in M_Primitive (shxAppend (shxTakeSSX (Proxy @sh1) sh ssh) sh2)
+                 (X.rerank ssh (ssxFromShape sh1) (ssxFromShape sh2)
                            (\a -> let M_Primitive _ r = f (M_Primitive sh1 a) in r)
                            arr)
 
@@ -988,10 +992,10 @@ mrerank ssh sh2 f (toPrimitive -> arr) =
 mreplicate :: forall sh sh' a. Elt a
            => IShX sh -> Mixed sh' a -> Mixed (sh ++ sh') a
 mreplicate sh arr =
-  let ssh' = X.staticShapeFrom (mshape arr)
-  in mlift (X.ssxAppend (X.staticShapeFrom sh) ssh')
+  let ssh' = ssxFromShape (mshape arr)
+  in mlift (ssxAppend (ssxFromShape sh) ssh')
            (\(sshT :: StaticShX shT) ->
-              case X.lemAppAssoc (Proxy @sh) (Proxy @sh') (Proxy @shT) of
+              case lemAppAssoc (Proxy @sh) (Proxy @sh') (Proxy @shT) of
                 Refl -> X.replicate sh (ssxAppend ssh' sshT))
            arr
 
@@ -1005,18 +1009,18 @@ mreplicateScal sh x = fromPrimitive (mreplicateScalP sh x)
 mslice :: Elt a => SNat i -> SNat n -> Mixed (Just (i + n + k) : sh) a -> Mixed (Just n : sh) a
 mslice i n arr =
   let _ :$% sh = mshape arr
-  in mlift (SKnown n :!% X.staticShapeFrom sh) (\_ -> X.slice i n) arr
+  in mlift (SKnown n :!% ssxFromShape sh) (\_ -> X.slice i n) arr
 
 msliceU :: Elt a => Int -> Int -> Mixed (Nothing : sh) a -> Mixed (Nothing : sh) a
-msliceU i n arr = mlift (X.staticShapeFrom (mshape arr)) (\_ -> X.sliceU i n) arr
+msliceU i n arr = mlift (ssxFromShape (mshape arr)) (\_ -> X.sliceU i n) arr
 
 mrev1 :: Elt a => Mixed (n : sh) a -> Mixed (n : sh) a
-mrev1 arr = mlift (X.staticShapeFrom (mshape arr)) (\_ -> X.rev1) arr
+mrev1 arr = mlift (ssxFromShape (mshape arr)) (\_ -> X.rev1) arr
 
 mreshape :: forall sh sh' a. Elt a => IShX sh' -> Mixed sh a -> Mixed sh' a
 mreshape sh' arr =
-  mlift (X.staticShapeFrom sh')
-        (\sshIn -> X.reshapePartial (X.staticShapeFrom (mshape arr)) sshIn sh')
+  mlift (ssxFromShape sh')
+        (\sshIn -> X.reshapePartial (ssxFromShape (mshape arr)) sshIn sh')
         arr
 
 miota :: (Enum a, PrimElt a) => SNat n -> Mixed '[Just n] a
@@ -1095,26 +1099,26 @@ instance (FloatElt a, NumElt a, PrimElt a) => Floating (Mixed sh a) where
   log1pexp = mliftNumElt1 floatEltLog1pexp
   log1mexp = mliftNumElt1 floatEltLog1mexp
 
-mtoRanked :: forall sh a. Elt a => Mixed sh a -> Ranked (X.Rank sh) a
+mtoRanked :: forall sh a. Elt a => Mixed sh a -> Ranked (Rank sh) a
 mtoRanked arr
-  | Refl <- X.lemAppNil @sh
-  , Refl <- X.lemAppNil @(Replicate (X.Rank sh) (Nothing @Nat))
+  | Refl <- lemAppNil @sh
+  , Refl <- lemAppNil @(Replicate (Rank sh) (Nothing @Nat))
   , Refl <- lemRankReplicate (srankSh (mshape arr))
-  = Ranked (mcast (X.staticShapeFrom (mshape arr)) (convSh (mshape arr)) (Proxy @'[]) arr)
+  = Ranked (mcast (ssxFromShape (mshape arr)) (convSh (mshape arr)) (Proxy @'[]) arr)
   where
-    convSh :: IShX sh' -> IShX (Replicate (X.Rank sh') Nothing)
+    convSh :: IShX sh' -> IShX (Replicate (Rank sh') Nothing)
     convSh ZSX = ZSX
     convSh (smn :$% (sh :: IShX sh'T))
-      | Refl <- X.lemReplicateSucc @(Nothing @Nat) @(X.Rank sh'T)
+      | Refl <- lemReplicateSucc @(Nothing @Nat) @(Rank sh'T)
       = SUnknown (fromSMayNat' smn) :$% convSh sh
 
-mcastToShaped :: forall sh sh' a. (Elt a, X.Rank sh ~ X.Rank sh')
+mcastToShaped :: forall sh sh' a. (Elt a, Rank sh ~ Rank sh')
               => Mixed sh a -> ShS sh' -> Shaped sh' a
 mcastToShaped arr targetsh
-  | Refl <- X.lemAppNil @sh
-  , Refl <- X.lemAppNil @(MapJust sh')
+  | Refl <- lemAppNil @sh
+  , Refl <- lemAppNil @(MapJust sh')
   , Refl <- lemRankMapJust targetsh
-  = Shaped (mcast (X.staticShapeFrom (mshape arr)) (shCvtSX targetsh) (Proxy @'[]) arr)
+  = Shaped (mcast (ssxFromShape (mshape arr)) (shCvtSX targetsh) (Proxy @'[]) arr)
 
 
 -- | A rank-typed array: the number of dimensions of the array (its /rank/) is
@@ -1418,7 +1422,7 @@ zeroIxR :: SNat n -> IIxR n
 zeroIxR SZ = ZIR
 zeroIxR (SS n) = 0 :.: zeroIxR n
 
-ixCvtXR :: IIxX sh -> IIxR (X.Rank sh)
+ixCvtXR :: IIxX sh -> IIxR (Rank sh)
 ixCvtXR ZIX = ZIR
 ixCvtXR (n :.% idx) = n :.: ixCvtXR idx
 
@@ -1429,7 +1433,7 @@ shCvtXR' ZSX =
 shCvtXR' (n :$% (idx :: IShX sh))
   | Refl <- lemReplicateSucc @(Nothing @Nat) @(n - 1) =
   castWith (subst2 (lem1 @sh Refl))
-    (X.fromSMayNat' n :$: shCvtXR' (castWith (subst2 (lem2 Refl)) idx))
+    (fromSMayNat' n :$: shCvtXR' (castWith (subst2 (lem2 Refl)) idx))
   where
     lem1 :: forall sh' n' k.
             k : sh' :~: Replicate n' Nothing
@@ -1443,13 +1447,13 @@ shCvtXR' (n :$% (idx :: IShX sh))
 ixCvtRX :: IIxR n -> IIxX (Replicate n Nothing)
 ixCvtRX ZIR = ZIX
 ixCvtRX (n :.: (idx :: IxR m Int)) =
-  castWith (subst2 @IxX @Int (X.lemReplicateSucc @(Nothing @Nat) @m))
+  castWith (subst2 @IxX @Int (lemReplicateSucc @(Nothing @Nat) @m))
     (n :.% ixCvtRX idx)
 
 shCvtRX :: IShR n -> IShX (Replicate n Nothing)
 shCvtRX ZSR = ZSX
 shCvtRX (n :$: (idx :: ShR m Int)) =
-  castWith (subst2 @ShX @Int (X.lemReplicateSucc @(Nothing @Nat) @m))
+  castWith (subst2 @ShX @Int (lemReplicateSucc @(Nothing @Nat) @m))
     (SUnknown n :$% shCvtRX idx)
 
 shapeSizeR :: IShR n -> Int
@@ -1506,7 +1510,7 @@ rsumOuter1P :: forall n a.
                (Storable a, NumElt a)
             => Ranked (n + 1) (Primitive a) -> Ranked n (Primitive a)
 rsumOuter1P (Ranked arr)
-  | Refl <- X.lemReplicateSucc @(Nothing @Nat) @n
+  | Refl <- lemReplicateSucc @(Nothing @Nat) @n
   = Ranked (msumOuter1P arr)
 
 rsumOuter1 :: forall n a. (NumElt a, PrimElt a)
@@ -1559,7 +1563,7 @@ rappend :: forall n a. Elt a
 rappend arr1 arr2
   | sn@SNat <- snatFromShR (rshape arr1)
   , Dict <- lemKnownReplicate sn
-  , Refl <- X.lemReplicateSucc @(Nothing @Nat) @n
+  , Refl <- lemReplicateSucc @(Nothing @Nat) @n
   = coerce (mappend @Nothing @Nothing @(Replicate n Nothing))
       arr1 arr2
 
@@ -1582,7 +1586,7 @@ rtoVector = coerce mtoVector
 
 rfromListOuter :: forall n a. Elt a => NonEmpty (Ranked n a) -> Ranked (n + 1) a
 rfromListOuter l
-  | Refl <- X.lemReplicateSucc @(Nothing @Nat) @n
+  | Refl <- lemReplicateSucc @(Nothing @Nat) @n
   = Ranked (mfromListOuter (coerce l :: NonEmpty (Mixed (Replicate n Nothing) a)))
 
 rfromList1 :: Elt a => NonEmpty a -> Ranked 1 a
@@ -1593,7 +1597,7 @@ rfromList1Prim l = Ranked (mfromList1Prim l)
 
 rtoListOuter :: forall n a. Elt a => Ranked (n + 1) a -> [Ranked n a]
 rtoListOuter (Ranked arr)
-  | Refl <- X.lemReplicateSucc @(Nothing @Nat) @n
+  | Refl <- lemReplicateSucc @(Nothing @Nat) @n
   = coerce (mtoListOuter @a @Nothing @(Replicate n Nothing) arr)
 
 rtoList1 :: Elt a => Ranked 1 a -> [a]
@@ -1677,7 +1681,7 @@ rreplicateScal sh x = rfromPrimitive (rreplicateScalP sh x)
 
 rslice :: forall n a. Elt a => Int -> Int -> Ranked (n + 1) a -> Ranked (n + 1) a
 rslice i n arr
-  | Refl <- X.lemReplicateSucc @(Nothing @Nat) @n
+  | Refl <- lemReplicateSucc @(Nothing @Nat) @n
   = rlift (snatFromShR (rshape arr))
           (\_ -> X.sliceU i n)
           arr
@@ -1686,7 +1690,7 @@ rrev1 :: forall n a. Elt a => Ranked (n + 1) a -> Ranked (n + 1) a
 rrev1 arr =
   rlift (snatFromShR (rshape arr))
         (\(_ :: StaticShX sh') ->
-          case X.lemReplicateSucc @(Nothing @Nat) @n of
+          case lemReplicateSucc @(Nothing @Nat) @n of
             Refl -> X.rev1 @Nothing @(Replicate n Nothing ++ sh'))
         arr
 
@@ -1707,12 +1711,12 @@ rasXArrayPrim :: PrimElt a => Ranked n a -> (IShR n, XArray (Replicate n Nothing
 rasXArrayPrim (Ranked arr) = first shCvtXR' (masXArrayPrim arr)
 
 rfromXArrayPrimP :: SNat n -> XArray (Replicate n Nothing) a -> Ranked n (Primitive a)
-rfromXArrayPrimP sn arr = Ranked (mfromXArrayPrimP (X.staticShapeFrom (X.shape (ssxFromSNat sn) arr)) arr)
+rfromXArrayPrimP sn arr = Ranked (mfromXArrayPrimP (ssxFromShape (X.shape (ssxFromSNat sn) arr)) arr)
 
 rfromXArrayPrim :: PrimElt a => SNat n -> XArray (Replicate n Nothing) a -> Ranked n a
-rfromXArrayPrim sn arr = Ranked (mfromXArrayPrim (X.staticShapeFrom (X.shape (ssxFromSNat sn) arr)) arr)
+rfromXArrayPrim sn arr = Ranked (mfromXArrayPrim (ssxFromShape (X.shape (ssxFromSNat sn) arr)) arr)
 
-rcastToShaped :: Elt a => Ranked (X.Rank sh) a -> ShS sh -> Shaped sh a
+rcastToShaped :: Elt a => Ranked (Rank sh) a -> ShS sh -> Shaped sh a
 rcastToShaped (Ranked arr) targetsh
   | Refl <- lemRankReplicate (srankSh (shCvtSX targetsh))
   , Refl <- lemRankMapJust targetsh
@@ -1809,7 +1813,7 @@ shCvtSX (n :$$ sh) = SKnown n :$% shCvtSX sh
 
 shapeSizeS :: ShS sh -> Int
 shapeSizeS ZSS = 1
-shapeSizeS (n :$$ sh) = X.fromSNat' n * shapeSizeS sh
+shapeSizeS (n :$$ sh) = fromSNat' n * shapeSizeS sh
 
 
 sshape :: forall sh a. Elt a => Shaped sh a -> ShS sh
@@ -1838,14 +1842,14 @@ slift :: forall sh1 sh2 a. Elt a
       => ShS sh2
       -> (forall sh' b. Storable b => StaticShX sh' -> XArray (MapJust sh1 ++ sh') b -> XArray (MapJust sh2 ++ sh') b)
       -> Shaped sh1 a -> Shaped sh2 a
-slift sh2 f (Shaped arr) = Shaped (mlift (X.staticShapeFrom (shCvtSX sh2)) f arr)
+slift sh2 f (Shaped arr) = Shaped (mlift (ssxFromShape (shCvtSX sh2)) f arr)
 
 -- | See the documentation of 'mlift'.
 slift2 :: forall sh1 sh2 sh3 a. Elt a
        => ShS sh3
        -> (forall sh' b. Storable b => StaticShX sh' -> XArray (MapJust sh1 ++ sh') b -> XArray (MapJust sh2 ++ sh') b -> XArray (MapJust sh3 ++ sh') b)
        -> Shaped sh1 a -> Shaped sh2 a -> Shaped sh3 a
-slift2 sh3 f (Shaped arr1) (Shaped arr2) = Shaped (mlift2 (X.staticShapeFrom (shCvtSX sh3)) f arr1 arr2)
+slift2 sh3 f (Shaped arr1) (Shaped arr2) = Shaped (mlift2 (ssxFromShape (shCvtSX sh3)) f arr1 arr2)
 
 ssumOuter1P :: forall sh n a. (Storable a, NumElt a)
             => Shaped (n : sh) (Primitive a) -> Shaped sh (Primitive a)
@@ -1855,28 +1859,28 @@ ssumOuter1 :: forall sh n a. (NumElt a, PrimElt a)
            => Shaped (n : sh) a -> Shaped sh a
 ssumOuter1 = sfromPrimitive . ssumOuter1P . stoPrimitive
 
-lemCommMapJustTakeLen :: HList SNat is -> ShS sh -> X.TakeLen is (MapJust sh) :~: MapJust (X.TakeLen is sh)
-lemCommMapJustTakeLen HNil _ = Refl
-lemCommMapJustTakeLen (_ `HCons` is) (_ :$$ sh) | Refl <- lemCommMapJustTakeLen is sh = Refl
-lemCommMapJustTakeLen (_ `HCons` _) ZSS = error "TakeLen of empty"
+lemCommMapJustTakeLen :: Perm is -> ShS sh -> TakeLen is (MapJust sh) :~: MapJust (TakeLen is sh)
+lemCommMapJustTakeLen PNil _ = Refl
+lemCommMapJustTakeLen (_ `PCons` is) (_ :$$ sh) | Refl <- lemCommMapJustTakeLen is sh = Refl
+lemCommMapJustTakeLen (_ `PCons` _) ZSS = error "TakeLen of empty"
 
-lemCommMapJustDropLen :: HList SNat is -> ShS sh -> X.DropLen is (MapJust sh) :~: MapJust (X.DropLen is sh)
-lemCommMapJustDropLen HNil _ = Refl
-lemCommMapJustDropLen (_ `HCons` is) (_ :$$ sh) | Refl <- lemCommMapJustDropLen is sh = Refl
-lemCommMapJustDropLen (_ `HCons` _) ZSS = error "DropLen of empty"
+lemCommMapJustDropLen :: Perm is -> ShS sh -> DropLen is (MapJust sh) :~: MapJust (DropLen is sh)
+lemCommMapJustDropLen PNil _ = Refl
+lemCommMapJustDropLen (_ `PCons` is) (_ :$$ sh) | Refl <- lemCommMapJustDropLen is sh = Refl
+lemCommMapJustDropLen (_ `PCons` _) ZSS = error "DropLen of empty"
 
-lemCommMapJustIndex :: SNat i -> ShS sh -> X.Index i (MapJust sh) :~: Just (X.Index i sh)
+lemCommMapJustIndex :: SNat i -> ShS sh -> Index i (MapJust sh) :~: Just (Index i sh)
 lemCommMapJustIndex SZ (_ :$$ _) = Refl
 lemCommMapJustIndex (SS (i :: SNat i')) ((_ :: SNat n) :$$ (sh :: ShS sh'))
   | Refl <- lemCommMapJustIndex i sh
-  , Refl <- X.lemIndexSucc (Proxy @i') (Proxy @(Just n)) (Proxy @(MapJust sh'))
-  , Refl <- X.lemIndexSucc (Proxy @i') (Proxy @n) (Proxy @sh')
+  , Refl <- lemIndexSucc (Proxy @i') (Proxy @(Just n)) (Proxy @(MapJust sh'))
+  , Refl <- lemIndexSucc (Proxy @i') (Proxy @n) (Proxy @sh')
   = Refl
 lemCommMapJustIndex _ ZSS = error "Index of empty"
 
-lemCommMapJustPermute :: HList SNat is -> ShS sh -> X.Permute is (MapJust sh) :~: MapJust (X.Permute is sh)
-lemCommMapJustPermute HNil _ = Refl
-lemCommMapJustPermute (i `HCons` is) sh
+lemCommMapJustPermute :: Perm is -> ShS sh -> Permute is (MapJust sh) :~: MapJust (Permute is sh)
+lemCommMapJustPermute PNil _ = Refl
+lemCommMapJustPermute (i `PCons` is) sh
   | Refl <- lemCommMapJustPermute is sh
   , Refl <- lemCommMapJustIndex i sh
   = Refl
@@ -1885,53 +1889,53 @@ listsAppend :: ListS sh f -> ListS sh' f -> ListS (sh ++ sh') f
 listsAppend ZS idx' = idx'
 listsAppend (i ::$ idx) idx' = i ::$ listsAppend idx idx'
 
-listsTakeLen :: forall f is sh. HList SNat is -> ListS sh f -> ListS (X.TakeLen is sh) f
-listsTakeLen HNil _ = ZS
-listsTakeLen (_ `HCons` is) (n ::$ sh) = n ::$ listsTakeLen is sh
-listsTakeLen (_ `HCons` _) ZS = error "Permutation longer than shape"
+listsTakeLen :: forall f is sh. Perm is -> ListS sh f -> ListS (TakeLen is sh) f
+listsTakeLen PNil _ = ZS
+listsTakeLen (_ `PCons` is) (n ::$ sh) = n ::$ listsTakeLen is sh
+listsTakeLen (_ `PCons` _) ZS = error "Permutation longer than shape"
 
-listsDropLen :: forall f is sh. HList SNat is -> ListS sh f -> ListS (DropLen is sh) f
-listsDropLen HNil sh = sh
-listsDropLen (_ `HCons` is) (_ ::$ sh) = listsDropLen is sh
-listsDropLen (_ `HCons` _) ZS = error "Permutation longer than shape"
+listsDropLen :: forall f is sh. Perm is -> ListS sh f -> ListS (DropLen is sh) f
+listsDropLen PNil sh = sh
+listsDropLen (_ `PCons` is) (_ ::$ sh) = listsDropLen is sh
+listsDropLen (_ `PCons` _) ZS = error "Permutation longer than shape"
 
-listsPermute :: forall f is sh. HList SNat is -> ListS sh f -> ListS (X.Permute is sh) f
-listsPermute HNil _ = ZS
-listsPermute (i `HCons` (is :: HList SNat is')) (sh :: ListS sh f) = listsIndex (Proxy @is') (Proxy @sh) i sh (listsPermute is sh)
+listsPermute :: forall f is sh. Perm is -> ListS sh f -> ListS (Permute is sh) f
+listsPermute PNil _ = ZS
+listsPermute (i `PCons` (is :: Perm is')) (sh :: ListS sh f) = listsIndex (Proxy @is') (Proxy @sh) i sh (listsPermute is sh)
 
-listsIndex :: forall f i is sh shT. Proxy is -> Proxy shT -> SNat i -> ListS sh f -> ListS (X.Permute is shT) f -> ListS (X.Index i sh : X.Permute is shT) f
+listsIndex :: forall f i is sh shT. Proxy is -> Proxy shT -> SNat i -> ListS sh f -> ListS (Permute is shT) f -> ListS (Index i sh : Permute is shT) f
 listsIndex _ _ SZ (n ::$ _) rest = n ::$ rest
 listsIndex p pT (SS (i :: SNat i')) ((_ :: f n) ::$ (sh :: ListS sh' f)) rest
-  | Refl <- X.lemIndexSucc (Proxy @i') (Proxy @n) (Proxy @sh')
+  | Refl <- lemIndexSucc (Proxy @i') (Proxy @n) (Proxy @sh')
   = listsIndex p pT i sh rest
 listsIndex _ _ _ ZS _ = error "Index into empty shape"
 
-shsTakeLen :: HList SNat is -> ShS sh -> ShS (X.TakeLen is sh)
+shsTakeLen :: Perm is -> ShS sh -> ShS (TakeLen is sh)
 shsTakeLen = coerce (listsTakeLen @SNat)
 
-shsPermute :: HList SNat is -> ShS sh -> ShS (X.Permute is sh)
+shsPermute :: Perm is -> ShS sh -> ShS (Permute is sh)
 shsPermute = coerce (listsPermute @SNat)
 
-shsIndex :: Proxy is -> Proxy shT -> SNat i -> ShS sh -> ShS (X.Permute is shT) -> ShS (X.Index i sh : X.Permute is shT)
+shsIndex :: Proxy is -> Proxy shT -> SNat i -> ShS sh -> ShS (Permute is shT) -> ShS (Index i sh : Permute is shT)
 shsIndex pis pshT = coerce (listsIndex @SNat pis pshT)
 
-applyPermS :: forall f is sh. HList SNat is -> ListS sh f -> ListS (PermutePrefix is sh) f
+applyPermS :: forall f is sh. Perm is -> ListS sh f -> ListS (PermutePrefix is sh) f
 applyPermS perm sh = listsAppend (listsPermute perm (listsTakeLen perm sh)) (listsDropLen perm sh)
 
-applyPermIxS :: forall i is sh. HList SNat is -> IxS sh i -> IxS (PermutePrefix is sh) i
+applyPermIxS :: forall i is sh. Perm is -> IxS sh i -> IxS (PermutePrefix is sh) i
 applyPermIxS = coerce (applyPermS @(Const i))
 
-applyPermShS :: forall is sh. HList SNat is -> ShS sh -> ShS (PermutePrefix is sh)
+applyPermShS :: forall is sh. Perm is -> ShS sh -> ShS (PermutePrefix is sh)
 applyPermShS = coerce (applyPermS @SNat)
 
-stranspose :: forall is sh a. (X.Permutation is, X.Rank is <= X.Rank sh, Elt a)
-           => HList SNat is -> Shaped sh a -> Shaped (X.PermutePrefix is sh) a
+stranspose :: forall is sh a. (IsPermutation is, Rank is <= Rank sh, Elt a)
+           => Perm is -> Shaped sh a -> Shaped (PermutePrefix is sh) a
 stranspose perm sarr@(Shaped arr)
   | Refl <- lemRankMapJust (sshape sarr)
   , Refl <- lemCommMapJustTakeLen perm (sshape sarr)
   , Refl <- lemCommMapJustDropLen perm (sshape sarr)
   , Refl <- lemCommMapJustPermute perm (shsTakeLen perm (sshape sarr))
-  , Refl <- lemCommMapJustApp (shsPermute perm (shsTakeLen perm (sshape sarr))) (Proxy @(X.DropLen is sh))
+  , Refl <- lemCommMapJustApp (shsPermute perm (shsTakeLen perm (sshape sarr))) (Proxy @(DropLen is sh))
   = Shaped (mtranspose perm arr)
 
 sappend :: Elt a => Shaped (n : sh) a -> Shaped (m : sh) a -> Shaped (n + m : sh) a
@@ -1969,7 +1973,7 @@ stoList1 = map sunScalar . stoListOuter
 
 sfromListPrim :: forall n a. PrimElt a => SNat n -> [a] -> Shaped '[n] a
 sfromListPrim sn l
-  | Refl <- X.lemAppNil @'[Just n]
+  | Refl <- lemAppNil @'[Just n]
   = let ssh = SUnknown () :!% ZKX
         xarr = X.cast ssh (SKnown sn :$% ZSX) ZKX (X.fromList1 ssh l)
     in Shaped $ fromPrimitive $ M_Primitive (X.shape (SKnown sn :!% ZKX) xarr) xarr
@@ -1989,7 +1993,7 @@ srerankP :: forall sh1 sh2 sh a b. (Storable a, Storable b)
 srerankP sh sh2 f sarr@(Shaped arr)
   | Refl <- lemCommMapJustApp sh (Proxy @sh1)
   , Refl <- lemCommMapJustApp sh (Proxy @sh2)
-  = Shaped (mrerankP (X.staticShapeFrom (shTakeSSX (Proxy @(MapJust sh1)) (shCvtSX (sshape sarr)) (X.staticShapeFrom (shCvtSX sh))))
+  = Shaped (mrerankP (ssxFromShape (shxTakeSSX (Proxy @(MapJust sh1)) (shCvtSX (sshape sarr)) (ssxFromShape (shCvtSX sh))))
                      (shCvtSX sh2)
                      (\a -> let Shaped r = f (Shaped a) in r)
                      arr)
@@ -2033,12 +2037,12 @@ sasXArrayPrim :: PrimElt a => Shaped sh a -> (ShS sh, XArray (MapJust sh) a)
 sasXArrayPrim (Shaped arr) = first shCvtXS' (masXArrayPrim arr)
 
 sfromXArrayPrimP :: ShS sh -> XArray (MapJust sh) a -> Shaped sh (Primitive a)
-sfromXArrayPrimP sh arr = Shaped (mfromXArrayPrimP (X.staticShapeFrom (shCvtSX sh)) arr)
+sfromXArrayPrimP sh arr = Shaped (mfromXArrayPrimP (ssxFromShape (shCvtSX sh)) arr)
 
 sfromXArrayPrim :: PrimElt a => ShS sh -> XArray (MapJust sh) a -> Shaped sh a
-sfromXArrayPrim sh arr = Shaped (mfromXArrayPrim (X.staticShapeFrom (shCvtSX sh)) arr)
+sfromXArrayPrim sh arr = Shaped (mfromXArrayPrim (ssxFromShape (shCvtSX sh)) arr)
 
-stoRanked :: Elt a => Shaped sh a -> Ranked (X.Rank sh) a
+stoRanked :: Elt a => Shaped sh a -> Ranked (Rank sh) a
 stoRanked sarr@(Shaped arr)
   | Refl <- lemRankMapJust (sshape sarr)
   = mtoRanked arr
diff --git a/src/Data/Array/Nested/Internal/Arith.hs b/src/Data/Array/Nested/Internal/Arith.hs
deleted file mode 100644
index 95fcfcf..0000000
--- a/src/Data/Array/Nested/Internal/Arith.hs
+++ /dev/null
@@ -1,435 +0,0 @@
-{-# LANGUAGE DataKinds #-}
-{-# LANGUAGE LambdaCase #-}
-{-# LANGUAGE ScopedTypeVariables #-}
-{-# LANGUAGE TemplateHaskell #-}
-{-# LANGUAGE TypeApplications #-}
-{-# LANGUAGE TypeOperators #-}
-{-# LANGUAGE ViewPatterns #-}
-{-# OPTIONS_GHC -fplugin GHC.TypeLits.KnownNat.Solver #-}
-module Data.Array.Nested.Internal.Arith where
-
-import Control.Monad (forM, guard)
-import qualified Data.Array.Internal as OI
-import qualified Data.Array.Internal.RankedG as RG
-import qualified Data.Array.Internal.RankedS as RS
-import Data.Bits
-import Data.Int
-import Data.List (sort)
-import qualified Data.Vector.Storable as VS
-import qualified Data.Vector.Storable.Mutable as VSM
-import Foreign.C.Types
-import Foreign.Ptr
-import Foreign.Storable (Storable)
-import GHC.TypeLits
-import Language.Haskell.TH
-import System.IO.Unsafe
-
-import Data.Array.Nested.Internal.Arith.Foreign
-import Data.Array.Nested.Internal.Arith.Lists
-
-
-liftVEltwise1 :: Storable a
-              => SNat n
-              -> (VS.Vector a -> VS.Vector a)
-              -> RS.Array n a -> RS.Array n a
-liftVEltwise1 SNat f arr@(RS.A (RG.A sh (OI.T strides offset vec)))
-  | Just prefixSz <- stridesDense sh strides =
-      let vec' = f (VS.slice offset prefixSz vec)
-      in RS.A (RG.A sh (OI.T strides 0 vec'))
-  | otherwise = RS.fromVector sh (f (RS.toVector arr))
-
-liftVEltwise2 :: Storable a
-              => SNat n
-              -> (Either a (VS.Vector a) -> Either a (VS.Vector a) -> VS.Vector a)
-              -> RS.Array n a -> RS.Array n a -> RS.Array n a
-liftVEltwise2 SNat f
-    arr1@(RS.A (RG.A sh1 (OI.T strides1 offset1 vec1)))
-    arr2@(RS.A (RG.A sh2 (OI.T strides2 offset2 vec2)))
-  | sh1 /= sh2 = error $ "liftVEltwise2: shapes unequal: " ++ show sh1 ++ " vs " ++ show sh2
-  | product sh1 == 0 = arr1  -- if the arrays are empty, just return one of the empty inputs
-  | otherwise = case (stridesDense sh1 strides1, stridesDense sh2 strides2) of
-      (Just 1, Just 1) ->  -- both are a (potentially replicated) scalar; just apply f to the scalars
-        let vec' = f (Left (vec1 VS.! offset1)) (Left (vec2 VS.! offset2))
-        in RS.A (RG.A sh1 (OI.T strides1 0 vec'))
-      (Just 1, Just n) ->  -- scalar * dense
-        RS.fromVector sh1 (f (Left (vec1 VS.! offset1)) (Right (VS.slice offset2 n vec2)))
-      (Just n, Just 1) ->  -- dense * scalar
-        RS.fromVector sh1 (f (Right (VS.slice offset1 n vec1)) (Left (vec2 VS.! offset2)))
-      (_, _) ->  -- fallback case
-        RS.fromVector sh1 (f (Right (RS.toVector arr1)) (Right (RS.toVector arr2)))
-
--- | Given the shape vector and the stride vector, return whether this vector
--- of strides uses a dense prefix of its backing array. If so, the number of
--- elements in this prefix is returned.
--- This excludes any offset.
-stridesDense :: [Int] -> [Int] -> Maybe Int
-stridesDense sh _ | any (<= 0) sh = Just 0
-stridesDense sh str =
-  -- sort dimensions on their stride, ascending, dropping any zero strides
-  case dropWhile ((== 0) . fst) (sort (zip str sh)) of
-    [] -> Just 1
-    (1, n) : (unzip -> (str', sh')) -> checkCover n sh' str'
-    _ -> Nothing  -- if the smallest stride is not 1, it will never be dense
-  where
-    -- Given size of currently densely covered region at beginning of the
-    -- array, the remaining shape vector and the corresponding remaining stride
-    -- vector, return whether this all together covers a dense prefix of the
-    -- array. If it does, return the number of elements in this prefix.
-    checkCover :: Int -> [Int] -> [Int] -> Maybe Int
-    checkCover block [] [] = Just block
-    checkCover block (n : sh') (s : str') = guard (s <= block) >> checkCover (max block (n * s)) sh' str'
-    checkCover _ _ _ = error "Orthotope array's shape vector and stride vector have different lengths"
-
-{-# NOINLINE vectorOp1 #-}
-vectorOp1 :: forall a b. Storable a
-          => (Ptr a -> Ptr b)
-          -> (Int64 -> Ptr b -> Ptr b -> IO ())
-          -> VS.Vector a -> VS.Vector a
-vectorOp1 ptrconv f v = unsafePerformIO $ do
-  outv <- VSM.unsafeNew (VS.length v)
-  VSM.unsafeWith outv $ \poutv ->
-    VS.unsafeWith v $ \pv ->
-      f (fromIntegral (VS.length v)) (ptrconv poutv) (ptrconv pv)
-  VS.unsafeFreeze outv
-
--- | If two vectors are given, assumes that they have the same length.
-{-# NOINLINE vectorOp2 #-}
-vectorOp2 :: forall a b. Storable a
-          => (a -> b)
-          -> (Ptr a -> Ptr b)
-          -> (a -> a -> a)
-          -> (Int64 -> Ptr b -> b -> Ptr b -> IO ())  -- sv
-          -> (Int64 -> Ptr b -> Ptr b -> b -> IO ())  -- vs
-          -> (Int64 -> Ptr b -> Ptr b -> Ptr b -> IO ())  -- vv
-          -> Either a (VS.Vector a) -> Either a (VS.Vector a) -> VS.Vector a
-vectorOp2 valconv ptrconv fss fsv fvs fvv = \cases
-  (Left x) (Left y) -> VS.singleton (fss x y)
-
-  (Left x) (Right vy) ->
-    unsafePerformIO $ do
-      outv <- VSM.unsafeNew (VS.length vy)
-      VSM.unsafeWith outv $ \poutv ->
-        VS.unsafeWith vy $ \pvy ->
-          fsv (fromIntegral (VS.length vy)) (ptrconv poutv) (valconv x) (ptrconv pvy)
-      VS.unsafeFreeze outv
-
-  (Right vx) (Left y) ->
-    unsafePerformIO $ do
-      outv <- VSM.unsafeNew (VS.length vx)
-      VSM.unsafeWith outv $ \poutv ->
-        VS.unsafeWith vx $ \pvx ->
-          fvs (fromIntegral (VS.length vx)) (ptrconv poutv) (ptrconv pvx) (valconv y)
-      VS.unsafeFreeze outv
-
-  (Right vx) (Right vy)
-    | VS.length vx == VS.length vy ->
-        unsafePerformIO $ do
-          outv <- VSM.unsafeNew (VS.length vx)
-          VSM.unsafeWith outv $ \poutv ->
-            VS.unsafeWith vx $ \pvx ->
-              VS.unsafeWith vy $ \pvy ->
-                fvv (fromIntegral (VS.length vx)) (ptrconv poutv) (ptrconv pvx) (ptrconv pvy)
-          VS.unsafeFreeze outv
-    | otherwise -> error $ "vectorOp: unequal lengths: " ++ show (VS.length vx) ++ " /= " ++ show (VS.length vy)
-
--- TODO: test all the weird cases of this function
--- | Reduce along the inner dimension
-{-# NOINLINE vectorRedInnerOp #-}
-vectorRedInnerOp :: forall a b n. (Num a, Storable a)
-                 => SNat n
-                 -> (a -> b)
-                 -> (Ptr a -> Ptr b)
-                 -> (Int64 -> Ptr b -> b -> Ptr b -> IO ())  -- ^ scale by constant
-                 -> (Int64 -> Ptr Int64 -> Ptr Int64 -> Ptr b -> Ptr b -> IO ())  -- ^ reduction kernel
-                 -> RS.Array (n + 1) a -> RS.Array n a
-vectorRedInnerOp sn@SNat valconv ptrconv fscale fred (RS.A (RG.A sh (OI.T strides offset vec)))
-  | null sh = error "unreachable"
-  | last sh <= 0 = RS.stretch (init sh) (RS.fromList (map (const 1) (init sh)) [0])
-  | any (<= 0) (init sh) = RS.A (RG.A (init sh) (OI.T (map (const 0) (init strides)) 0 VS.empty))
-  -- now the input array is nonempty
-  | last sh == 1 = RS.A (RG.A (init sh) (OI.T (init strides) offset vec))
-  | last strides == 0 =
-      liftVEltwise1 sn
-        (vectorOp1 id (\n pout px -> fscale n (ptrconv pout) (valconv (fromIntegral (last sh))) (ptrconv px)))
-        (RS.A (RG.A (init sh) (OI.T (init strides) offset vec)))
-  -- now there is useful work along the inner dimension
-  | otherwise =
-      let -- filter out zero-stride dimensions; the reduction kernel need not concern itself with those
-          (shF, stridesF) = unzip $ filter ((/= 0) . snd) (zip sh strides)
-          ndimsF = length shF
-      in unsafePerformIO $ do
-           outv <- VSM.unsafeNew (product (init shF))
-           VSM.unsafeWith outv $ \poutv ->
-             VS.unsafeWith (VS.fromListN ndimsF (map fromIntegral shF)) $ \pshF ->
-               VS.unsafeWith (VS.fromListN ndimsF (map fromIntegral stridesF)) $ \pstridesF ->
-                 VS.unsafeWith (VS.slice offset (VS.length vec - offset) vec) $ \pvec ->
-                   fred (fromIntegral ndimsF) pshF pstridesF (ptrconv poutv) (ptrconv pvec)
-           RS.fromVector (init sh) <$> VS.unsafeFreeze outv
-
-flipOp :: (Int64 -> Ptr a -> a -> Ptr a -> IO ())
-       ->  Int64 -> Ptr a -> Ptr a -> a -> IO ()
-flipOp f n out v s = f n out s v
-
-$(fmap concat . forM typesList $ \arithtype -> do
-    let ttyp = conT (atType arithtype)
-    fmap concat . forM [minBound..maxBound] $ \arithop -> do
-      let name = mkName (aboName arithop ++ "Vector" ++ nameBase (atType arithtype))
-          cnamebase = "c_binary_" ++ atCName arithtype
-          c_ss = varE (aboNumOp arithop)
-          c_sv = varE (mkName (cnamebase ++ "_sv")) `appE` litE (integerL (fromIntegral (aboEnum arithop)))
-          c_vs = varE (mkName (cnamebase ++ "_vs")) `appE` litE (integerL (fromIntegral (aboEnum arithop)))
-          c_vv = varE (mkName (cnamebase ++ "_vv")) `appE` litE (integerL (fromIntegral (aboEnum arithop)))
-      sequence [SigD name <$>
-                     [t| forall n. SNat n -> RS.Array n $ttyp -> RS.Array n $ttyp -> RS.Array n $ttyp |]
-               ,do body <- [| \sn -> liftVEltwise2 sn (vectorOp2 id id $c_ss $c_sv $c_vs $c_vv) |]
-                   return $ FunD name [Clause [] (NormalB body) []]])
-
-$(fmap concat . forM floatTypesList $ \arithtype -> do
-    let ttyp = conT (atType arithtype)
-    fmap concat . forM [minBound..maxBound] $ \arithop -> do
-      let name = mkName (afboName arithop ++ "Vector" ++ nameBase (atType arithtype))
-          cnamebase = "c_fbinary_" ++ atCName arithtype
-          c_ss = varE (afboNumOp arithop)
-          c_sv = varE (mkName (cnamebase ++ "_sv")) `appE` litE (integerL (fromIntegral (afboEnum arithop)))
-          c_vs = varE (mkName (cnamebase ++ "_vs")) `appE` litE (integerL (fromIntegral (afboEnum arithop)))
-          c_vv = varE (mkName (cnamebase ++ "_vv")) `appE` litE (integerL (fromIntegral (afboEnum arithop)))
-      sequence [SigD name <$>
-                     [t| forall n. SNat n -> RS.Array n $ttyp -> RS.Array n $ttyp -> RS.Array n $ttyp |]
-               ,do body <- [| \sn -> liftVEltwise2 sn (vectorOp2 id id $c_ss $c_sv $c_vs $c_vv) |]
-                   return $ FunD name [Clause [] (NormalB body) []]])
-
-$(fmap concat . forM typesList $ \arithtype -> do
-    let ttyp = conT (atType arithtype)
-    fmap concat . forM [minBound..maxBound] $ \arithop -> do
-      let name = mkName (auoName arithop ++ "Vector" ++ nameBase (atType arithtype))
-          c_op = varE (mkName ("c_unary_" ++ atCName arithtype)) `appE` litE (integerL (fromIntegral (auoEnum arithop)))
-      sequence [SigD name <$>
-                     [t| forall n. SNat n -> RS.Array n $ttyp -> RS.Array n $ttyp |]
-               ,do body <- [| \sn -> liftVEltwise1 sn (vectorOp1 id $c_op) |]
-                   return $ FunD name [Clause [] (NormalB body) []]])
-
-$(fmap concat . forM floatTypesList $ \arithtype -> do
-    let ttyp = conT (atType arithtype)
-    fmap concat . forM [minBound..maxBound] $ \arithop -> do
-      let name = mkName (afuoName arithop ++ "Vector" ++ nameBase (atType arithtype))
-          c_op = varE (mkName ("c_funary_" ++ atCName arithtype)) `appE` litE (integerL (fromIntegral (afuoEnum arithop)))
-      sequence [SigD name <$>
-                     [t| forall n. SNat n -> RS.Array n $ttyp -> RS.Array n $ttyp |]
-               ,do body <- [| \sn -> liftVEltwise1 sn (vectorOp1 id $c_op) |]
-                   return $ FunD name [Clause [] (NormalB body) []]])
-
-$(fmap concat . forM typesList $ \arithtype -> do
-    let ttyp = conT (atType arithtype)
-    fmap concat . forM [minBound..maxBound] $ \arithop -> do
-      let name = mkName (aroName arithop ++ "Vector" ++ nameBase (atType arithtype))
-          c_op = varE (mkName ("c_reduce_" ++ atCName arithtype)) `appE` litE (integerL (fromIntegral (aroEnum arithop)))
-          c_scale_op = varE (mkName ("c_binary_" ++ atCName arithtype ++ "_sv")) `appE` litE (integerL (fromIntegral (aboEnum BO_MUL)))
-      sequence [SigD name <$>
-                     [t| forall n. SNat n -> RS.Array (n + 1) $ttyp -> RS.Array n $ttyp |]
-               ,do body <- [| \sn -> vectorRedInnerOp sn id id $c_scale_op $c_op |]
-                   return $ FunD name [Clause [] (NormalB body) []]])
-
--- This branch is ostensibly a runtime branch, but will (hopefully) be
--- constant-folded away by GHC.
-intWidBranch1 :: forall i n. (FiniteBits i, Storable i)
-              => (Int64 -> Ptr Int32 -> Ptr Int32 -> IO ())
-              -> (Int64 -> Ptr Int64 -> Ptr Int64 -> IO ())
-              -> (SNat n -> RS.Array n i -> RS.Array n i)
-intWidBranch1 f32 f64 sn
-  | finiteBitSize (undefined :: i) == 32 = liftVEltwise1 sn (vectorOp1 @i @Int32 castPtr f32)
-  | finiteBitSize (undefined :: i) == 64 = liftVEltwise1 sn (vectorOp1 @i @Int64 castPtr f64)
-  | otherwise = error "Unsupported Int width"
-
-intWidBranch2 :: forall i n. (FiniteBits i, Storable i, Integral i)
-              => (i -> i -> i)  -- ss
-                 -- int32
-              -> (Int64 -> Ptr Int32 -> Int32 -> Ptr Int32 -> IO ())  -- sv
-              -> (Int64 -> Ptr Int32 -> Ptr Int32 -> Int32 -> IO ())  -- vs
-              -> (Int64 -> Ptr Int32 -> Ptr Int32 -> Ptr Int32 -> IO ())  -- vv
-                 -- int64
-              -> (Int64 -> Ptr Int64 -> Int64 -> Ptr Int64 -> IO ())  -- sv
-              -> (Int64 -> Ptr Int64 -> Ptr Int64 -> Int64 -> IO ())  -- vs
-              -> (Int64 -> Ptr Int64 -> Ptr Int64 -> Ptr Int64 -> IO ())  -- vv
-              -> (SNat n -> RS.Array n i -> RS.Array n i -> RS.Array n i)
-intWidBranch2 ss sv32 vs32 vv32 sv64 vs64 vv64 sn
-  | finiteBitSize (undefined :: i) == 32 = liftVEltwise2 sn (vectorOp2 @i @Int32 fromIntegral castPtr ss sv32 vs32 vv32)
-  | finiteBitSize (undefined :: i) == 64 = liftVEltwise2 sn (vectorOp2 @i @Int64 fromIntegral castPtr ss sv64 vs64 vv64)
-  | otherwise = error "Unsupported Int width"
-
-intWidBranchRed :: forall i n. (FiniteBits i, Storable i, Integral i)
-                => -- int32
-                   (Int64 -> Ptr Int32 -> Int32 -> Ptr Int32 -> IO ())  -- ^ scale by constant
-                -> (Int64 -> Ptr Int64 -> Ptr Int64 -> Ptr Int32 -> Ptr Int32 -> IO ())  -- ^ reduction kernel
-                   -- int64
-                -> (Int64 -> Ptr Int64 -> Int64 -> Ptr Int64 -> IO ())  -- ^ scale by constant
-                -> (Int64 -> Ptr Int64 -> Ptr Int64 -> Ptr Int64 -> Ptr Int64 -> IO ())  -- ^ reduction kernel
-                -> (SNat n -> RS.Array (n + 1) i -> RS.Array n i)
-intWidBranchRed fsc32 fred32 fsc64 fred64 sn
-  | finiteBitSize (undefined :: i) == 32 = vectorRedInnerOp @i @Int32 sn fromIntegral castPtr fsc32 fred32
-  | finiteBitSize (undefined :: i) == 64 = vectorRedInnerOp @i @Int64 sn fromIntegral castPtr fsc64 fred64
-  | otherwise = error "Unsupported Int width"
-
-class NumElt a where
-  numEltAdd :: SNat n -> RS.Array n a -> RS.Array n a -> RS.Array n a
-  numEltSub :: SNat n -> RS.Array n a -> RS.Array n a -> RS.Array n a
-  numEltMul :: SNat n -> RS.Array n a -> RS.Array n a -> RS.Array n a
-  numEltNeg :: SNat n -> RS.Array n a -> RS.Array n a
-  numEltAbs :: SNat n -> RS.Array n a -> RS.Array n a
-  numEltSignum :: SNat n -> RS.Array n a -> RS.Array n a
-  numEltSum1Inner :: SNat n -> RS.Array (n + 1) a -> RS.Array n a
-  numEltProduct1Inner :: SNat n -> RS.Array (n + 1) a -> RS.Array n a
-
-instance NumElt Int32 where
-  numEltAdd = addVectorInt32
-  numEltSub = subVectorInt32
-  numEltMul = mulVectorInt32
-  numEltNeg = negVectorInt32
-  numEltAbs = absVectorInt32
-  numEltSignum = signumVectorInt32
-  numEltSum1Inner = sum1VectorInt32
-  numEltProduct1Inner = product1VectorInt32
-
-instance NumElt Int64 where
-  numEltAdd = addVectorInt64
-  numEltSub = subVectorInt64
-  numEltMul = mulVectorInt64
-  numEltNeg = negVectorInt64
-  numEltAbs = absVectorInt64
-  numEltSignum = signumVectorInt64
-  numEltSum1Inner = sum1VectorInt64
-  numEltProduct1Inner = product1VectorInt64
-
-instance NumElt Float where
-  numEltAdd = addVectorFloat
-  numEltSub = subVectorFloat
-  numEltMul = mulVectorFloat
-  numEltNeg = negVectorFloat
-  numEltAbs = absVectorFloat
-  numEltSignum = signumVectorFloat
-  numEltSum1Inner = sum1VectorFloat
-  numEltProduct1Inner = product1VectorFloat
-
-instance NumElt Double where
-  numEltAdd = addVectorDouble
-  numEltSub = subVectorDouble
-  numEltMul = mulVectorDouble
-  numEltNeg = negVectorDouble
-  numEltAbs = absVectorDouble
-  numEltSignum = signumVectorDouble
-  numEltSum1Inner = sum1VectorDouble
-  numEltProduct1Inner = product1VectorDouble
-
-instance NumElt Int where
-  numEltAdd = intWidBranch2 @Int (+)
-                (c_binary_i32_sv (aboEnum BO_ADD)) (flipOp (c_binary_i32_sv (aboEnum BO_ADD))) (c_binary_i32_vv (aboEnum BO_ADD))
-                (c_binary_i64_sv (aboEnum BO_ADD)) (flipOp (c_binary_i64_sv (aboEnum BO_ADD))) (c_binary_i64_vv (aboEnum BO_ADD))
-  numEltSub = intWidBranch2 @Int (-)
-                (c_binary_i32_sv (aboEnum BO_SUB)) (flipOp (c_binary_i32_sv (aboEnum BO_SUB))) (c_binary_i32_vv (aboEnum BO_SUB))
-                (c_binary_i64_sv (aboEnum BO_SUB)) (flipOp (c_binary_i64_sv (aboEnum BO_SUB))) (c_binary_i64_vv (aboEnum BO_SUB))
-  numEltMul = intWidBranch2 @Int (*)
-                (c_binary_i32_sv (aboEnum BO_MUL)) (flipOp (c_binary_i32_sv (aboEnum BO_MUL))) (c_binary_i32_vv (aboEnum BO_MUL))
-                (c_binary_i64_sv (aboEnum BO_MUL)) (flipOp (c_binary_i64_sv (aboEnum BO_MUL))) (c_binary_i64_vv (aboEnum BO_MUL))
-  numEltNeg = intWidBranch1 @Int (c_unary_i32 (auoEnum UO_NEG)) (c_unary_i64 (auoEnum UO_NEG))
-  numEltAbs = intWidBranch1 @Int (c_unary_i32 (auoEnum UO_ABS)) (c_unary_i64 (auoEnum UO_ABS))
-  numEltSignum = intWidBranch1 @Int (c_unary_i32 (auoEnum UO_SIGNUM)) (c_unary_i64 (auoEnum UO_SIGNUM))
-  numEltSum1Inner = intWidBranchRed @Int
-                      (c_binary_i32_sv (aboEnum BO_MUL)) (c_reduce_i32 (aroEnum RO_SUM1))
-                      (c_binary_i64_sv (aboEnum BO_MUL)) (c_reduce_i64 (aroEnum RO_SUM1))
-  numEltProduct1Inner = intWidBranchRed @Int
-                          (c_binary_i32_sv (aboEnum BO_MUL)) (c_reduce_i32 (aroEnum RO_PRODUCT1))
-                          (c_binary_i64_sv (aboEnum BO_MUL)) (c_reduce_i64 (aroEnum RO_PRODUCT1))
-
-instance NumElt CInt where
-  numEltAdd = intWidBranch2 @CInt (+)
-                (c_binary_i32_sv (aboEnum BO_ADD)) (flipOp (c_binary_i32_sv (aboEnum BO_ADD))) (c_binary_i32_vv (aboEnum BO_ADD))
-                (c_binary_i64_sv (aboEnum BO_ADD)) (flipOp (c_binary_i64_sv (aboEnum BO_ADD))) (c_binary_i64_vv (aboEnum BO_ADD))
-  numEltSub = intWidBranch2 @CInt (-)
-                (c_binary_i32_sv (aboEnum BO_SUB)) (flipOp (c_binary_i32_sv (aboEnum BO_SUB))) (c_binary_i32_vv (aboEnum BO_SUB))
-                (c_binary_i64_sv (aboEnum BO_SUB)) (flipOp (c_binary_i64_sv (aboEnum BO_SUB))) (c_binary_i64_vv (aboEnum BO_SUB))
-  numEltMul = intWidBranch2 @CInt (*)
-                (c_binary_i32_sv (aboEnum BO_MUL)) (flipOp (c_binary_i32_sv (aboEnum BO_MUL))) (c_binary_i32_vv (aboEnum BO_MUL))
-                (c_binary_i64_sv (aboEnum BO_MUL)) (flipOp (c_binary_i64_sv (aboEnum BO_MUL))) (c_binary_i64_vv (aboEnum BO_MUL))
-  numEltNeg = intWidBranch1 @CInt (c_unary_i32 (auoEnum UO_NEG)) (c_unary_i64 (auoEnum UO_NEG))
-  numEltAbs = intWidBranch1 @CInt (c_unary_i32 (auoEnum UO_ABS)) (c_unary_i64 (auoEnum UO_ABS))
-  numEltSignum = intWidBranch1 @CInt (c_unary_i32 (auoEnum UO_SIGNUM)) (c_unary_i64 (auoEnum UO_SIGNUM))
-  numEltSum1Inner = intWidBranchRed @CInt
-                      (c_binary_i32_sv (aboEnum BO_MUL)) (c_reduce_i32 (aroEnum RO_SUM1))
-                      (c_binary_i64_sv (aboEnum BO_MUL)) (c_reduce_i64 (aroEnum RO_SUM1))
-  numEltProduct1Inner = intWidBranchRed @CInt
-                          (c_binary_i32_sv (aboEnum BO_MUL)) (c_reduce_i32 (aroEnum RO_PRODUCT1))
-                          (c_binary_i64_sv (aboEnum BO_MUL)) (c_reduce_i64 (aroEnum RO_PRODUCT1))
-
-class FloatElt a where
-  floatEltDiv :: SNat n -> RS.Array n a -> RS.Array n a -> RS.Array n a
-  floatEltPow :: SNat n -> RS.Array n a -> RS.Array n a -> RS.Array n a
-  floatEltLogbase :: SNat n -> RS.Array n a -> RS.Array n a -> RS.Array n a
-  floatEltRecip :: SNat n -> RS.Array n a -> RS.Array n a
-  floatEltExp :: SNat n -> RS.Array n a -> RS.Array n a
-  floatEltLog :: SNat n -> RS.Array n a -> RS.Array n a
-  floatEltSqrt :: SNat n -> RS.Array n a -> RS.Array n a
-  floatEltSin :: SNat n -> RS.Array n a -> RS.Array n a
-  floatEltCos :: SNat n -> RS.Array n a -> RS.Array n a
-  floatEltTan :: SNat n -> RS.Array n a -> RS.Array n a
-  floatEltAsin :: SNat n -> RS.Array n a -> RS.Array n a
-  floatEltAcos :: SNat n -> RS.Array n a -> RS.Array n a
-  floatEltAtan :: SNat n -> RS.Array n a -> RS.Array n a
-  floatEltSinh :: SNat n -> RS.Array n a -> RS.Array n a
-  floatEltCosh :: SNat n -> RS.Array n a -> RS.Array n a
-  floatEltTanh :: SNat n -> RS.Array n a -> RS.Array n a
-  floatEltAsinh :: SNat n -> RS.Array n a -> RS.Array n a
-  floatEltAcosh :: SNat n -> RS.Array n a -> RS.Array n a
-  floatEltAtanh :: SNat n -> RS.Array n a -> RS.Array n a
-  floatEltLog1p :: SNat n -> RS.Array n a -> RS.Array n a
-  floatEltExpm1 :: SNat n -> RS.Array n a -> RS.Array n a
-  floatEltLog1pexp :: SNat n -> RS.Array n a -> RS.Array n a
-  floatEltLog1mexp :: SNat n -> RS.Array n a -> RS.Array n a
-
-instance FloatElt Float where
-  floatEltDiv = divVectorFloat
-  floatEltPow = powVectorFloat
-  floatEltLogbase = logbaseVectorFloat
-  floatEltRecip = recipVectorFloat
-  floatEltExp = expVectorFloat
-  floatEltLog = logVectorFloat
-  floatEltSqrt = sqrtVectorFloat
-  floatEltSin = sinVectorFloat
-  floatEltCos = cosVectorFloat
-  floatEltTan = tanVectorFloat
-  floatEltAsin = asinVectorFloat
-  floatEltAcos = acosVectorFloat
-  floatEltAtan = atanVectorFloat
-  floatEltSinh = sinhVectorFloat
-  floatEltCosh = coshVectorFloat
-  floatEltTanh = tanhVectorFloat
-  floatEltAsinh = asinhVectorFloat
-  floatEltAcosh = acoshVectorFloat
-  floatEltAtanh = atanhVectorFloat
-  floatEltLog1p = log1pVectorFloat
-  floatEltExpm1 = expm1VectorFloat
-  floatEltLog1pexp = log1pexpVectorFloat
-  floatEltLog1mexp = log1mexpVectorFloat
-
-instance FloatElt Double where
-  floatEltDiv = divVectorDouble
-  floatEltPow = powVectorDouble
-  floatEltLogbase = logbaseVectorDouble
-  floatEltRecip = recipVectorDouble
-  floatEltExp = expVectorDouble
-  floatEltLog = logVectorDouble
-  floatEltSqrt = sqrtVectorDouble
-  floatEltSin = sinVectorDouble
-  floatEltCos = cosVectorDouble
-  floatEltTan = tanVectorDouble
-  floatEltAsin = asinVectorDouble
-  floatEltAcos = acosVectorDouble
-  floatEltAtan = atanVectorDouble
-  floatEltSinh = sinhVectorDouble
-  floatEltCosh = coshVectorDouble
-  floatEltTanh = tanhVectorDouble
-  floatEltAsinh = asinhVectorDouble
-  floatEltAcosh = acoshVectorDouble
-  floatEltAtanh = atanhVectorDouble
-  floatEltLog1p = log1pVectorDouble
-  floatEltExpm1 = expm1VectorDouble
-  floatEltLog1pexp = log1pexpVectorDouble
-  floatEltLog1mexp = log1mexpVectorDouble
diff --git a/src/Data/Array/Nested/Internal/Arith/Foreign.hs b/src/Data/Array/Nested/Internal/Arith/Foreign.hs
deleted file mode 100644
index ac83188..0000000
--- a/src/Data/Array/Nested/Internal/Arith/Foreign.hs
+++ /dev/null
@@ -1,55 +0,0 @@
-{-# LANGUAGE ForeignFunctionInterface #-}
-{-# LANGUAGE TemplateHaskell #-}
-module Data.Array.Nested.Internal.Arith.Foreign where
-
-import Control.Monad
-import Data.Int
-import Data.Maybe
-import Foreign.C.Types
-import Foreign.Ptr
-import Language.Haskell.TH
-
-import Data.Array.Nested.Internal.Arith.Lists
-
-
-$(fmap concat . forM typesList $ \arithtype -> do
-    let ttyp = conT (atType arithtype)
-    let base = "binary_" ++ atCName arithtype
-    sequence $ catMaybes
-      [Just (ForeignD . ImportF CCall Unsafe ("oxarop_" ++ base ++ "_sv") (mkName ("c_" ++ base ++ "_sv")) <$>
-               [t| CInt -> Int64 -> Ptr $ttyp -> $ttyp -> Ptr $ttyp -> IO () |])
-      ,Just (ForeignD . ImportF CCall Unsafe ("oxarop_" ++ base ++ "_vv") (mkName ("c_" ++ base ++ "_vv")) <$>
-               [t| CInt -> Int64 -> Ptr $ttyp -> Ptr $ttyp -> Ptr $ttyp -> IO () |])
-      ,Just (ForeignD . ImportF CCall Unsafe ("oxarop_" ++ base ++ "_vs") (mkName ("c_" ++ base ++ "_vs")) <$>
-               [t| CInt -> Int64 -> Ptr $ttyp -> Ptr $ttyp -> $ttyp -> IO () |])
-      ])
-
-$(fmap concat . forM floatTypesList $ \arithtype -> do
-    let ttyp = conT (atType arithtype)
-    let base = "fbinary_" ++ atCName arithtype
-    sequence $ catMaybes
-      [Just (ForeignD . ImportF CCall Unsafe ("oxarop_" ++ base ++ "_sv") (mkName ("c_" ++ base ++ "_sv")) <$>
-               [t| CInt -> Int64 -> Ptr $ttyp -> $ttyp -> Ptr $ttyp -> IO () |])
-      ,Just (ForeignD . ImportF CCall Unsafe ("oxarop_" ++ base ++ "_vv") (mkName ("c_" ++ base ++ "_vv")) <$>
-               [t| CInt -> Int64 -> Ptr $ttyp -> Ptr $ttyp -> Ptr $ttyp -> IO () |])
-      ,Just (ForeignD . ImportF CCall Unsafe ("oxarop_" ++ base ++ "_vs") (mkName ("c_" ++ base ++ "_vs")) <$>
-               [t| CInt -> Int64 -> Ptr $ttyp -> Ptr $ttyp -> $ttyp -> IO () |])
-      ])
-
-$(fmap concat . forM typesList $ \arithtype -> do
-    let ttyp = conT (atType arithtype)
-    let base = "unary_" ++ atCName arithtype
-    pure . ForeignD . ImportF CCall Unsafe ("oxarop_" ++ base) (mkName ("c_" ++ base)) <$>
-      [t| CInt -> Int64 -> Ptr $ttyp -> Ptr $ttyp -> IO () |])
-
-$(fmap concat . forM floatTypesList $ \arithtype -> do
-    let ttyp = conT (atType arithtype)
-    let base = "funary_" ++ atCName arithtype
-    pure . ForeignD . ImportF CCall Unsafe ("oxarop_" ++ base) (mkName ("c_" ++ base)) <$>
-      [t| CInt -> Int64 -> Ptr $ttyp -> Ptr $ttyp -> IO () |])
-
-$(fmap concat . forM typesList $ \arithtype -> do
-    let ttyp = conT (atType arithtype)
-    let base = "reduce_" ++ atCName arithtype
-    pure . ForeignD . ImportF CCall Unsafe ("oxarop_" ++ base) (mkName ("c_" ++ base)) <$>
-      [t| CInt -> Int64 -> Ptr Int64 -> Ptr Int64 -> Ptr $ttyp -> Ptr $ttyp -> IO () |])
diff --git a/src/Data/Array/Nested/Internal/Arith/Lists.hs b/src/Data/Array/Nested/Internal/Arith/Lists.hs
deleted file mode 100644
index ce2836d..0000000
--- a/src/Data/Array/Nested/Internal/Arith/Lists.hs
+++ /dev/null
@@ -1,78 +0,0 @@
-{-# LANGUAGE LambdaCase #-}
-{-# LANGUAGE TemplateHaskell #-}
-module Data.Array.Nested.Internal.Arith.Lists where
-
-import Data.Char
-import Data.Int
-import Language.Haskell.TH
-
-import Data.Array.Nested.Internal.Arith.Lists.TH
-
-
-data ArithType = ArithType
-  { atType :: Name  -- ''Int32
-  , atCName :: String  -- "i32"
-  }
-
-floatTypesList :: [ArithType]
-floatTypesList =
-  [ArithType ''Float "float"
-  ,ArithType ''Double "double"
-  ]
-
-typesList :: [ArithType]
-typesList =
-  [ArithType ''Int32 "i32"
-  ,ArithType ''Int64 "i64"
-  ]
-  ++ floatTypesList
-
--- data ArithBOp = BO_ADD | BO_SUB | BO_MUL deriving (Show, Enum, Bounded)
-$(genArithDataType Binop "ArithBOp")
-
-$(genArithNameFun Binop ''ArithBOp "aboName" (map toLower . drop 3))
-$(genArithEnumFun Binop ''ArithBOp "aboEnum")
-
-$(do clauses <- readArithLists Binop
-                  (\name _num hsop -> return (Clause [ConP (mkName name) [] []]
-                                                     (NormalB (VarE 'mkName `AppE` LitE (StringL hsop)))
-                                                     []))
-                  return
-     sequence [SigD (mkName "aboNumOp") <$> [t| ArithBOp -> Name |]
-              ,return $ FunD (mkName "aboNumOp") clauses])
-
-
--- data ArithFBOp = FB_DIV deriving (Show, Enum, Bounded)
-$(genArithDataType FBinop "ArithFBOp")
-
-$(genArithNameFun FBinop ''ArithFBOp "afboName" (map toLower . drop 3))
-$(genArithEnumFun FBinop ''ArithFBOp "afboEnum")
-
-$(do clauses <- readArithLists FBinop
-                  (\name _num hsop -> return (Clause [ConP (mkName name) [] []]
-                                                     (NormalB (VarE 'mkName `AppE` LitE (StringL hsop)))
-                                                     []))
-                  return
-     sequence [SigD (mkName "afboNumOp") <$> [t| ArithFBOp -> Name |]
-              ,return $ FunD (mkName "afboNumOp") clauses])
-
-
--- data ArithUOp = UO_NEG | UO_ABS | UO_SIGNUM | ... deriving (Show, Enum, Bounded)
-$(genArithDataType Unop "ArithUOp")
-
-$(genArithNameFun Unop ''ArithUOp "auoName" (map toLower . drop 3))
-$(genArithEnumFun Unop ''ArithUOp "auoEnum")
-
-
--- data ArithFUOp = FU_RECIP | ... deriving (Show, Enum, Bounded)
-$(genArithDataType FUnop "ArithFUOp")
-
-$(genArithNameFun FUnop ''ArithFUOp "afuoName" (map toLower . drop 3))
-$(genArithEnumFun FUnop ''ArithFUOp "afuoEnum")
-
-
--- data ArithRedOp = RO_SUM1 | RO_PRODUCT1 deriving (Show, Enum, Bounded)
-$(genArithDataType Redop "ArithRedOp")
-
-$(genArithNameFun Redop ''ArithRedOp "aroName" (map toLower . drop 3))
-$(genArithEnumFun Redop ''ArithRedOp "aroEnum")
diff --git a/src/Data/Array/Nested/Internal/Arith/Lists/TH.hs b/src/Data/Array/Nested/Internal/Arith/Lists/TH.hs
deleted file mode 100644
index 7142dfa..0000000
--- a/src/Data/Array/Nested/Internal/Arith/Lists/TH.hs
+++ /dev/null
@@ -1,82 +0,0 @@
-{-# LANGUAGE TemplateHaskellQuotes #-}
-module Data.Array.Nested.Internal.Arith.Lists.TH where
-
-import Control.Monad
-import Control.Monad.IO.Class
-import Data.Maybe
-import Foreign.C.Types
-import Language.Haskell.TH
-import Language.Haskell.TH.Syntax
-import Text.Read
-
-
-data OpKind = Binop | FBinop | Unop | FUnop | Redop
-  deriving (Show, Eq)
-
-readArithLists :: OpKind
-               -> (String -> Int -> String -> Q a)
-               -> ([a] -> Q r)
-               -> Q r
-readArithLists targetkind fop fcombine = do
-  addDependentFile "cbits/arith_lists.h"
-  lns <- liftIO $ lines <$> readFile "cbits/arith_lists.h"
-
-  mvals <- forM lns $ \line -> do
-    if null (dropWhile (== ' ') line)
-      then return Nothing
-      else do let (kind, name, num, aux) = parseLine line
-              if kind == targetkind
-                then Just <$> fop name num aux
-                else return Nothing
-
-  fcombine (catMaybes mvals)
-  where
-    parseLine s0
-      | ("LIST_", s1) <- splitAt 5 s0
-      , (kindstr, '(' : s2) <- break (== '(') s1
-      , (f1, ',' : s3) <- parseField s2
-      , (f2, ',' : s4) <- parseField s3
-      , (f3, ')' : _) <- parseField s4
-      , Just kind <- parseKind kindstr
-      , let name = f1
-      , Just num <- readMaybe f2
-      , let aux = f3
-      = (kind, name, num, aux)
-      | otherwise
-      = error $ "readArithLists: unrecognised line in cbits/arith_lists.h: " ++ show s0
-
-    parseField s = break (`elem` ",)") (dropWhile (== ' ') s)
-
-    parseKind "BINOP" = Just Binop
-    parseKind "FBINOP" = Just FBinop
-    parseKind "UNOP" = Just Unop
-    parseKind "FUNOP" = Just FUnop
-    parseKind "REDOP" = Just Redop
-    parseKind _ = Nothing
-
-genArithDataType :: OpKind -> String -> Q [Dec]
-genArithDataType kind dtname = do
-  cons <- readArithLists kind
-            (\name _num _ -> return $ NormalC (mkName name) [])
-            return
-  return [DataD [] (mkName dtname) [] Nothing cons [DerivClause Nothing [ConT ''Show, ConT ''Enum, ConT ''Bounded]]]
-
-genArithNameFun :: OpKind -> Name -> String -> (String -> String) -> Q [Dec]
-genArithNameFun kind dtname funname nametrans = do
-  clauses <- readArithLists kind
-               (\name _num _ -> return (Clause [ConP (mkName name) [] []]
-                                               (NormalB (LitE (StringL (nametrans name))))
-                                               []))
-               return
-  return [SigD (mkName funname) (ArrowT `AppT` ConT dtname `AppT` ConT ''String)
-         ,FunD (mkName funname) clauses]
-
-genArithEnumFun :: OpKind -> Name -> String -> Q [Dec]
-genArithEnumFun kind dtname funname = do
-  clauses <- readArithLists kind
-               (\name num _ -> return (Clause [ConP (mkName name) [] []]
-                                              (NormalB (LitE (IntegerL (fromIntegral num))))
-                                              []))
-               return
-  return [SigD (mkName funname) (ArrowT `AppT` ConT dtname `AppT` ConT ''CInt)
-         ,FunD (mkName funname) clauses]
-- 
cgit v1.2.3-70-g09d2