Add {m,r,s}dot1Inner

10 files changed, 172 insertions, 73 deletions

diff --git a/src/Data/Array/Mixed/Internal/Arith.hs b/src/Data/Array/Mixed/Internal/Arith.hs
index 9f99c3b..fc26633 100644
--- a/src/Data/Array/Mixed/Internal/Arith.hs
+++ b/src/Data/Array/Mixed/Internal/Arith.hs
@@ -31,6 +31,7 @@ import System.IO.Unsafe
 
 import Data.Array.Mixed.Internal.Arith.Foreign
 import Data.Array.Mixed.Internal.Arith.Lists
+import Data.Array.Mixed.Types (fromSNat')
 
 
 -- TODO: need to sort strides for reduction-like functions so that the C inner-loop specialisation has some chance of working even after transposition
@@ -304,36 +305,44 @@ vectorExtremumOp ptrconv fextrem (RS.A (RG.A sh (OI.T strides offset vec)))
              . VS.toList
              <$> VS.unsafeFreeze outvR
 
-vectorDotprodOp :: (Num a, Storable a)
-                => (b -> a)
-                -> (Ptr a -> Ptr b)
-                -> (Int64 -> Ptr b -> Ptr Int64 -> Ptr Int64 -> Ptr b -> IO ())  -- ^ reduction kernel
-                -> (Int64 -> Ptr b -> Ptr b -> IO b)  -- ^ dotprod kernel
-                -> (Int64 -> Int64 -> Int64 -> Ptr b -> Int64 -> Int64 -> Ptr b -> IO b)  -- ^ strided dotprod kernel
-                -> RS.Array 1 a -> RS.Array 1 a -> a
-vectorDotprodOp valbackconv ptrconv fred fdot fdotstrided
-    (RS.A (RG.A [len1] (OI.T [stride1] offset1 vec1)))
-    (RS.A (RG.A [len2] (OI.T [stride2] offset2 vec2)))
-  | len1 /= len2 = error $ "vectorDotprodOp: lengths unequal: " ++ show len1 ++ " vs " ++ show len2
-  | len1 == 0 = 0  -- if the arrays are empty, just return zero
-  | otherwise = case (stride1, stride2) of
-      (0, 0) ->  -- replicated scalar * replicated scalar
-        fromIntegral len1 * (vec1 VS.! offset1) * (vec2 VS.! offset2)
-      (0, 1) ->  -- replicated scalar * dense
-        dotScalarVector len1 ptrconv fred (vec1 VS.! offset1) (VS.slice offset2 len1 vec2)
-      (0, -1) ->  -- replicated scalar * reversed dense
-        dotScalarVector len1 ptrconv fred (vec1 VS.! offset1) (VS.slice (offset2 - (len1 - 1)) len1 vec2)
-      (1, 0) ->  -- dense * replicated scalar
-        dotScalarVector len1 ptrconv fred (vec2 VS.! offset2) (VS.slice offset1 len1 vec1)
-      (-1, 0) ->  -- reversed dense * replicated scalar
-        dotScalarVector len1 ptrconv fred (vec2 VS.! offset2) (VS.slice (offset1 - (len1 - 1)) len1 vec1)
-      (1, 1) ->  -- dense * dense
-        dotVectorVector len1 valbackconv ptrconv fdot (VS.slice offset1 len1 vec1) (VS.slice offset2 len1 vec2)
-      (-1, -1) ->  -- reversed dense * reversed dense
-        dotVectorVector len1 valbackconv ptrconv fdot (VS.slice (offset1 - (len1 - 1)) len1 vec1) (VS.slice (offset2 - (len1 - 1)) len1 vec2)
-      (_, _) ->  -- fallback case
-        dotVectorVectorStrided len1 valbackconv ptrconv fdotstrided offset1 stride1 vec1 offset2 stride2 vec2
-vectorDotprodOp _ _ _ _ _ _ _ = error "vectorDotprodOp: not one-dimensional?"
+vectorDotprodInnerOp :: forall a b n. (Num a, Storable a)
+                     => SNat n
+                     -> (a -> b)
+                     -> (Ptr a -> Ptr b)
+                     -> (SNat n -> RS.Array n a -> RS.Array n a -> RS.Array n a)  -- ^ elementwise multiplication
+                     -> (Int64 -> Ptr b -> b -> Ptr b -> IO ())  -- ^ scale by constant
+                     -> (Int64 -> Ptr b -> Ptr Int64 -> Ptr Int64 -> Ptr b -> IO ())  -- ^ reduction kernel
+                     -> (Int64 -> Ptr Int64 -> Ptr b -> Ptr Int64 -> Ptr b -> Ptr Int64 -> Ptr b -> IO ())  -- ^ dotprod kernel
+                     -> RS.Array (n + 1) a -> RS.Array (n + 1) a -> RS.Array n a
+vectorDotprodInnerOp sn@SNat valconv ptrconv fmul fscale fred fdotinner
+    arr1@(RS.A (RG.A sh1 (OI.T strides1 offset1 vec1)))
+    arr2@(RS.A (RG.A sh2 (OI.T strides2 offset2 vec2)))
+  | null sh1 || null sh2 = error "unreachable"
+  | sh1 /= sh2 = error $ "vectorDotprodInnerOp: shapes unequal: " ++ show sh1 ++ " vs " ++ show sh2
+  | last sh1 <= 0 = RS.stretch (init sh1) (RS.fromList (1 <$ init sh1) [0])
+  | any (<= 0) (init sh1) = RS.A (RG.A (init sh1) (OI.T (0 <$ init strides1) 0 VS.empty))
+  -- now the input arrays are nonempty
+  | last sh1 == 1 = fmul sn (RS.reshape (init sh1) arr1) (RS.reshape (init sh1) arr2)
+  | last strides1 == 0 =
+      fmul sn
+        (RS.A (RG.A (init sh1) (OI.T (init strides1) offset1 vec1)))
+        (vectorRedInnerOp sn valconv ptrconv fscale fred arr2)
+  | last strides2 == 0 =
+      fmul sn
+        (vectorRedInnerOp sn valconv ptrconv fscale fred arr1)
+        (RS.A (RG.A (init sh2) (OI.T (init strides2) offset2 vec2)))
+  -- now there is useful dotprod work along the inner dimension
+  | otherwise = unsafePerformIO $ do
+      let inrank = fromSNat' sn + 1
+      outv <- VSM.unsafeNew (product (init sh1))
+      VSM.unsafeWith outv $ \poutv ->
+        VS.unsafeWith (VS.fromListN inrank (map fromIntegral sh1)) $ \psh ->
+        VS.unsafeWith (VS.fromListN inrank (map fromIntegral strides1)) $ \pstrides1 ->
+        VS.unsafeWith vec1 $ \pvec1 ->
+        VS.unsafeWith (VS.fromListN inrank (map fromIntegral strides2)) $ \pstrides2 ->
+        VS.unsafeWith vec2 $ \pvec2 ->
+          fdotinner (fromIntegral @Int @Int64 inrank) psh (ptrconv poutv) pstrides1 (ptrconv pvec1) pstrides2 (ptrconv pvec2)
+      RS.fromVector @_ @n (init sh1) <$> VS.unsafeFreeze outv
 
 {-# NOINLINE dotScalarVector #-}
 dotScalarVector :: forall a b. (Num a, Storable a)
@@ -461,13 +470,14 @@ $(fmap concat . forM typesList $ \arithtype ->
 
 $(fmap concat . forM typesList $ \arithtype -> do
     let ttyp = conT (atType arithtype)
-        name = mkName ("dotprodVector" ++ nameBase (atType arithtype))
-        c_op = varE (mkName ("c_dotprod_" ++ atCName arithtype))
-        c_op_strided = varE (mkName ("c_dotprod_" ++ atCName arithtype ++ "_strided"))
+        name = mkName ("dotprodinnerVector" ++ nameBase (atType arithtype))
+        c_op = varE (mkName ("c_dotprodinner_" ++ atCName arithtype))
+        mul_op = varE (mkName ("mulVector" ++ nameBase (atType arithtype)))
+        c_scale_op = varE (mkName ("c_binary_" ++ atCName arithtype ++ "_sv")) `appE` litE (integerL (fromIntegral (aboEnum BO_MUL)))
         c_red_op = varE (mkName ("c_reduce1_" ++ atCName arithtype)) `appE` litE (integerL (fromIntegral (aroEnum RO_SUM)))
     sequence [SigD name <$>
-                   [t| RS.Array 1 $ttyp -> RS.Array 1 $ttyp -> $ttyp |]
-             ,do body <- [| vectorDotprodOp id id $c_red_op $c_op $c_op_strided |]
+                   [t| forall n. SNat n -> RS.Array (n + 1) $ttyp -> RS.Array (n + 1) $ttyp -> RS.Array n $ttyp |]
+             ,do body <- [| \sn -> vectorDotprodInnerOp sn id id $mul_op $c_scale_op $c_red_op $c_op |]
                  return $ FunD name [Clause [] (NormalB body) []]])
 
 -- This branch is ostensibly a runtime branch, but will (hopefully) be
@@ -533,19 +543,19 @@ intWidBranchExtr fextr32 fextr64
   | finiteBitSize (undefined :: i) == 64 = vectorExtremumOp @i @Int64 castPtr fextr64
   | otherwise = error "Unsupported Int width"
 
-intWidBranchDotprod :: forall i. (FiniteBits i, Storable i, Integral i)
+intWidBranchDotprod :: forall i n. (FiniteBits i, Storable i, Integral i, NumElt i)
                     => -- int32
-                       (Int64 -> Ptr Int32 -> Ptr Int64 -> Ptr Int64 -> Ptr Int32 -> IO ())  -- ^ reduction kernel
-                    -> (Int64 -> Ptr Int32 -> Ptr Int32 -> IO Int32)  -- ^ dotprod kernel
-                    -> (Int64 -> Int64 -> Int64 -> Ptr Int32 -> Int64 -> Int64 -> Ptr Int32 -> IO Int32)  -- ^ strided dotprod kernel
+                       (Int64 -> Ptr Int32 -> Int32 -> Ptr Int32 -> IO ())  -- ^ scale by constant
+                    -> (Int64 -> Ptr Int32 -> Ptr Int64 -> Ptr Int64 -> Ptr Int32 -> IO ())  -- ^ reduction kernel
+                    -> (Int64 -> Ptr Int64 -> Ptr Int32 -> Ptr Int64 -> Ptr Int32 -> Ptr Int64 -> Ptr Int32 -> IO ())  -- ^ dotprod kernel
                        -- int64
+                    -> (Int64 -> Ptr Int64 -> Int64 -> Ptr Int64 -> IO ())  -- ^ scale by constant
                     -> (Int64 -> Ptr Int64 -> Ptr Int64 -> Ptr Int64 -> Ptr Int64 -> IO ())  -- ^ reduction kernel
-                    -> (Int64 -> Ptr Int64 -> Ptr Int64 -> IO Int64)  -- ^ dotprod kernel
-                    -> (Int64 -> Int64 -> Int64 -> Ptr Int64 -> Int64 -> Int64 -> Ptr Int64 -> IO Int64)  -- ^ strided dotprod kernel
-                    -> (RS.Array 1 i -> RS.Array 1 i -> i)
-intWidBranchDotprod fred32 fdot32 fdot32strided fred64 fdot64 fdot64strided
-  | finiteBitSize (undefined :: i) == 32 = vectorDotprodOp @i @Int32 fromIntegral castPtr fred32 fdot32 fdot32strided
-  | finiteBitSize (undefined :: i) == 64 = vectorDotprodOp @i @Int64 fromIntegral castPtr fred64 fdot64 fdot64strided
+                    -> (Int64 -> Ptr Int64 -> Ptr Int64 -> Ptr Int64 -> Ptr Int64 -> Ptr Int64 -> Ptr Int64 -> IO ())  -- ^ dotprod kernel
+                    -> (SNat n -> RS.Array (n + 1) i -> RS.Array (n + 1) i -> RS.Array n i)
+intWidBranchDotprod fsc32 fred32 fdot32 fsc64 fred64 fdot64 sn
+  | finiteBitSize (undefined :: i) == 32 = vectorDotprodInnerOp @i @Int32 sn fromIntegral castPtr numEltMul fsc32 fred32 fdot32
+  | finiteBitSize (undefined :: i) == 64 = vectorDotprodInnerOp @i @Int64 sn fromIntegral castPtr numEltMul fsc64 fred64 fdot64
   | otherwise = error "Unsupported Int width"
 
 class NumElt a where
@@ -561,7 +571,7 @@ class NumElt a where
   numEltProductFull :: SNat n -> RS.Array n a -> a
   numEltMinIndex :: RS.Array n a -> [Int]
   numEltMaxIndex :: RS.Array n a -> [Int]
-  numEltDotprod :: RS.Array 1 a -> RS.Array 1 a -> a
+  numEltDotprodInner :: SNat n -> RS.Array (n + 1) a -> RS.Array (n + 1) a -> RS.Array n a
 
 instance NumElt Int32 where
   numEltAdd = addVectorInt32
@@ -576,7 +586,7 @@ instance NumElt Int32 where
   numEltProductFull = productFullVectorInt32
   numEltMinIndex = minindexVectorInt32
   numEltMaxIndex = maxindexVectorInt32
-  numEltDotprod = dotprodVectorInt32
+  numEltDotprodInner = dotprodinnerVectorInt32
 
 instance NumElt Int64 where
   numEltAdd = addVectorInt64
@@ -591,7 +601,7 @@ instance NumElt Int64 where
   numEltProductFull = productFullVectorInt64
   numEltMinIndex = minindexVectorInt64
   numEltMaxIndex = maxindexVectorInt64
-  numEltDotprod = dotprodVectorInt64
+  numEltDotprodInner = dotprodinnerVectorInt64
 
 instance NumElt Float where
   numEltAdd = addVectorFloat
@@ -606,7 +616,7 @@ instance NumElt Float where
   numEltProductFull = productFullVectorFloat
   numEltMinIndex = minindexVectorFloat
   numEltMaxIndex = maxindexVectorFloat
-  numEltDotprod = dotprodVectorFloat
+  numEltDotprodInner = dotprodinnerVectorFloat
 
 instance NumElt Double where
   numEltAdd = addVectorDouble
@@ -621,7 +631,7 @@ instance NumElt Double where
   numEltProductFull = productFullVectorDouble
   numEltMinIndex = minindexVectorDouble
   numEltMaxIndex = maxindexVectorDouble
-  numEltDotprod = dotprodVectorDouble
+  numEltDotprodInner = dotprodinnerVectorDouble
 
 instance NumElt Int where
   numEltAdd = intWidBranch2 @Int (+)
@@ -646,8 +656,8 @@ instance NumElt Int where
   numEltProductFull = intWidBranchRedFull @Int (^) (c_reducefull_i32 (aroEnum RO_PRODUCT)) (c_reducefull_i64 (aroEnum RO_PRODUCT))
   numEltMinIndex = intWidBranchExtr @Int c_extremum_min_i32 c_extremum_min_i64
   numEltMaxIndex = intWidBranchExtr @Int c_extremum_max_i32 c_extremum_max_i64
-  numEltDotprod = intWidBranchDotprod @Int (c_reduce1_i32 (aroEnum RO_SUM)) c_dotprod_i32 c_dotprod_i32_strided
-                                           (c_reduce1_i64 (aroEnum RO_SUM)) c_dotprod_i64 c_dotprod_i64_strided
+  numEltDotprodInner = intWidBranchDotprod @Int (c_binary_i32_sv (aboEnum BO_MUL)) (c_reduce1_i32 (aroEnum RO_SUM)) c_dotprodinner_i32
+                                                (c_binary_i64_sv (aboEnum BO_MUL)) (c_reduce1_i64 (aroEnum RO_SUM)) c_dotprodinner_i64
 
 instance NumElt CInt where
   numEltAdd = intWidBranch2 @CInt (+)
@@ -672,8 +682,8 @@ instance NumElt CInt where
   numEltProductFull = intWidBranchRedFull @CInt (^) (c_reducefull_i32 (aroEnum RO_PRODUCT)) (c_reducefull_i64 (aroEnum RO_PRODUCT))
   numEltMinIndex = intWidBranchExtr @CInt c_extremum_min_i32 c_extremum_min_i64
   numEltMaxIndex = intWidBranchExtr @CInt c_extremum_max_i32 c_extremum_max_i64
-  numEltDotprod = intWidBranchDotprod @CInt (c_reduce1_i32 (aroEnum RO_SUM)) c_dotprod_i32 c_dotprod_i32_strided
-                                            (c_reduce1_i64 (aroEnum RO_SUM)) c_dotprod_i64 c_dotprod_i64_strided
+  numEltDotprodInner = intWidBranchDotprod @CInt (c_binary_i32_sv (aboEnum BO_MUL)) (c_reduce1_i32 (aroEnum RO_SUM)) c_dotprodinner_i32
+                                                 (c_binary_i64_sv (aboEnum BO_MUL)) (c_reduce1_i64 (aroEnum RO_SUM)) c_dotprodinner_i64
 
 class FloatElt a where
   floatEltDiv :: SNat n -> RS.Array n a -> RS.Array n a -> RS.Array n a
diff --git a/src/Data/Array/Mixed/Internal/Arith/Foreign.hs b/src/Data/Array/Mixed/Internal/Arith/Foreign.hs
index c1c0070..ade7ce1 100644
--- a/src/Data/Array/Mixed/Internal/Arith/Foreign.hs
+++ b/src/Data/Array/Mixed/Internal/Arith/Foreign.hs
@@ -22,6 +22,7 @@ $(do
         ,("extremum_max_" ++ tyn,          [t| Ptr Int64 -> Int64 -> Ptr Int64 -> Ptr Int64 -> Ptr $ttyp -> IO () |])
         ,("dotprod_" ++ tyn,               [t| Int64 -> Ptr $ttyp -> Ptr $ttyp -> IO $ttyp |])
         ,("dotprod_" ++ tyn ++ "_strided", [t| Int64 -> Int64 -> Int64 -> Ptr $ttyp -> Int64 -> Int64 -> Ptr $ttyp -> IO $ttyp |])
+        ,("dotprodinner_" ++ tyn,          [t| Int64 -> Ptr Int64 -> Ptr $ttyp -> Ptr Int64 -> Ptr $ttyp -> Ptr Int64 -> Ptr $ttyp -> IO () |])
         ]
 
   let importsFloat ttyp tyn =
diff --git a/src/Data/Array/Mixed/Lemmas.hs b/src/Data/Array/Mixed/Lemmas.hs
index 633c9c2..ec7e7bd 100644
--- a/src/Data/Array/Mixed/Lemmas.hs
+++ b/src/Data/Array/Mixed/Lemmas.hs
@@ -108,6 +108,9 @@ lemTakeLenApp _ _ _ = unsafeCoerceRefl
 lemInitApp :: Proxy l -> Proxy x -> Init (l ++ '[x]) :~: l
 lemInitApp _ _ = unsafeCoerceRefl
 
+lemLastApp :: Proxy l -> Proxy x -> Last (l ++ '[x]) :~: x
+lemLastApp _ _ = unsafeCoerceRefl
+
 
 -- ** KnownNat
 
diff --git a/src/Data/Array/Mixed/Shape.hs b/src/Data/Array/Mixed/Shape.hs
index a15e0a2..1285aa1 100644
--- a/src/Data/Array/Mixed/Shape.hs
+++ b/src/Data/Array/Mixed/Shape.hs
@@ -121,6 +121,10 @@ listxInit :: forall f n sh. ListX (n : sh) f -> ListX (Init (n : sh)) f
 listxInit (i ::% sh@(_ ::% _)) = i ::% listxInit sh
 listxInit (_ ::% ZX) = ZX
 
+listxLast :: forall f n sh. ListX (n : sh) f -> f (Last (n : sh))
+listxLast (_ ::% sh@(_ ::% _)) = listxLast sh
+listxLast (x ::% ZX) = x
+
 
 -- * Mixed indices
 
@@ -179,6 +183,9 @@ ixxDrop = coerce (listxDrop @(Const i) @(Const i))
 ixxInit :: forall n sh i. IxX (n : sh) i -> IxX (Init (n : sh)) i
 ixxInit = coerce (listxInit @(Const i))
 
+ixxLast :: forall n sh i. IxX (n : sh) i -> i
+ixxLast = coerce (listxLast @(Const i))
+
 ixxFromLinear :: IShX sh -> Int -> IIxX sh
 ixxFromLinear = \sh i -> case go sh i of
   (idx, 0) -> idx
@@ -330,6 +337,9 @@ shxDropSh = coerce (listxDrop @(SMayNat i SNat) @(SMayNat i SNat))
 shxInit :: forall n sh i. ShX (n : sh) i -> ShX (Init (n : sh)) i
 shxInit = coerce (listxInit @(SMayNat i SNat))
 
+shxLast :: forall n sh i. ShX (n : sh) i -> SMayNat i SNat (Last (n : sh))
+shxLast = coerce (listxLast @(SMayNat i SNat))
+
 shxTakeSSX :: forall sh sh' i. Proxy sh' -> ShX (sh ++ sh') i -> StaticShX sh -> ShX sh i
 shxTakeSSX _ = flip go
   where
@@ -404,6 +414,12 @@ ssxTail (_ :!% ssh) = ssh
 ssxDropIx :: forall sh sh' i. StaticShX (sh ++ sh') -> IxX sh i -> StaticShX sh'
 ssxDropIx = coerce (listxDrop @(SMayNat () SNat) @(Const i))
 
+ssxInit :: forall n sh. StaticShX (n : sh) -> StaticShX (Init (n : sh))
+ssxInit = coerce (listxInit @(SMayNat () SNat))
+
+ssxLast :: forall n sh. StaticShX (n : sh) -> SMayNat () SNat (Last (n : sh))
+ssxLast = coerce (listxLast @(SMayNat () SNat))
+
 -- | This may fail if @sh@ has @Nothing@s in it.
 ssxToShX' :: StaticShX sh -> Maybe (IShX sh)
 ssxToShX' ZKX = Just ZSX
diff --git a/src/Data/Array/Mixed/Types.hs b/src/Data/Array/Mixed/Types.hs
index 22d06e5..8e90a88 100644
--- a/src/Data/Array/Mixed/Types.hs
+++ b/src/Data/Array/Mixed/Types.hs
@@ -29,6 +29,7 @@ module Data.Array.Mixed.Types (
   MapJust,
   Tail,
   Init,
+  Last,
 
   -- * Unsafe
   unsafeCoerceRefl,
@@ -105,6 +106,10 @@ type family Init l where
   Init (x : y : xs) = x : Init (y : xs)
   Init '[x] = '[]
 
+type family Last l where
+  Last (x : y : xs) = Last (y : xs)
+  Last '[x] = x
+
 
 -- | This is just @'Unsafe.Coerce.unsafeCoerce' 'Refl'@, but specialised to
 -- only typecheck for actual type equalities. One cannot, e.g. accidentally
diff --git a/src/Data/Array/Nested.hs b/src/Data/Array/Nested.hs
index 53417bd..f37619f 100644
--- a/src/Data/Array/Nested.hs
+++ b/src/Data/Array/Nested.hs
@@ -12,7 +12,7 @@ module Data.Array.Nested (
   rreplicate, rreplicateScal, rfromListOuter, rfromList1, rfromList1Prim, rtoListOuter, rtoList1,
   rfromListLinear, rtoListLinear,
   rslice, rrev1, rreshape, rflatten, riota,
-  rminIndexPrim, rmaxIndexPrim, rdot, rdot1,
+  rminIndexPrim, rmaxIndexPrim, rdot1Inner, rdot,
   rnest, runNest,
   -- ** Lifting orthotope operations to 'Ranked' arrays
   rlift, rlift2,
@@ -33,7 +33,7 @@ module Data.Array.Nested (
   sreplicate, sreplicateScal, sfromListOuter, sfromList1, sfromList1Prim, stoListOuter, stoList1,
   sfromListLinear, stoListLinear,
   sslice, srev1, sreshape, sflatten, siota,
-  sminIndexPrim, smaxIndexPrim, sdot, sdot1,
+  sminIndexPrim, smaxIndexPrim, sdot1Inner, sdot,
   snest, sunNest,
   -- ** Lifting orthotope operations to 'Shaped' arrays
   slift, slift2,
@@ -54,7 +54,7 @@ module Data.Array.Nested (
   mreplicate, mreplicateScal, mfromListOuter, mfromList1, mfromList1Prim, mtoListOuter, mtoList1,
   mfromListLinear, mtoListLinear,
   mslice, mrev1, mreshape, mflatten, miota,
-  mminIndexPrim, mmaxIndexPrim, mdot, mdot1,
+  mminIndexPrim, mmaxIndexPrim, mdot1Inner, mdot,
   mnest, munNest,
   -- ** Lifting orthotope operations to 'Mixed' arrays
   mlift, mlift2,
diff --git a/src/Data/Array/Nested/Internal/Mixed.hs b/src/Data/Array/Nested/Internal/Mixed.hs
index 215313e..50202ba 100644
--- a/src/Data/Array/Nested/Internal/Mixed.hs
+++ b/src/Data/Array/Nested/Internal/Mixed.hs
@@ -104,7 +104,7 @@ newtype Primitive a = Primitive a
 
 -- | Element types that are primitive; arrays of these types are just a newtype
 -- wrapper over an array.
-class Storable a => PrimElt a where
+class (Storable a, Elt a) => PrimElt a where
   fromPrimitive :: Mixed sh (Primitive a) -> Mixed sh a
   toPrimitive :: Mixed sh a -> Mixed sh (Primitive a)
 
@@ -854,15 +854,26 @@ mmaxIndexPrim :: (PrimElt a, NumElt a) => Mixed sh a -> IIxX sh
 mmaxIndexPrim (toPrimitive -> M_Primitive sh (XArray arr)) =
   ixxFromList (ssxFromShape sh) (numEltMaxIndex arr)
 
-mdot1 :: (PrimElt a, NumElt a) => Mixed '[n] a -> Mixed '[n] a -> a
-mdot1 (toPrimitive -> M_Primitive _ (XArray arr1)) (toPrimitive -> M_Primitive _ (XArray arr2)) =
-  numEltDotprod arr1 arr2
+mdot1Inner :: forall sh n a. (PrimElt a, NumElt a)
+           => Proxy n -> Mixed (sh ++ '[n]) a -> Mixed (sh ++ '[n]) a -> Mixed sh a
+mdot1Inner _ (toPrimitive -> M_Primitive sh1 (XArray a)) (toPrimitive -> M_Primitive sh2 (XArray b))
+  | Refl <- lemInitApp (Proxy @sh) (Proxy @n)
+  , Refl <- lemLastApp (Proxy @sh) (Proxy @n)
+  = case sh1 of
+      _ :$% _
+        | sh1 == sh2
+        , Refl <- lemRankApp (ssxInit (ssxFromShape sh1)) (ssxLast (ssxFromShape sh1) :!% ZKX) ->
+            fromPrimitive $ M_Primitive (shxInit sh1) (XArray (numEltDotprodInner (shxRank (shxInit sh1)) a b))
+        | otherwise -> error "mdot1Inner: Unequal shapes"
+      ZSX -> error "unreachable"
 
 -- | This has a temporary, suboptimal implementation in terms of 'mflatten'.
--- Prefer 'mdot1' if applicable.
+-- Prefer 'mdot1Inner' if applicable.
 mdot :: (PrimElt a, NumElt a) => Mixed sh a -> Mixed sh a -> a
-mdot a b = mdot1 (fromPrimitive (mflatten (toPrimitive a)))
-                 (fromPrimitive (mflatten (toPrimitive b)))
+mdot a b =
+  munScalar $
+    mdot1Inner Proxy (fromPrimitive (mflatten (toPrimitive a)))
+                     (fromPrimitive (mflatten (toPrimitive b)))
 
 mtoXArrayPrimP :: Mixed sh (Primitive a) -> (IShX sh, XArray sh a)
 mtoXArrayPrimP (M_Primitive sh arr) = (sh, arr)
diff --git a/src/Data/Array/Nested/Internal/Ranked.hs b/src/Data/Array/Nested/Internal/Ranked.hs
index 74b2186..735d1a3 100644
--- a/src/Data/Array/Nested/Internal/Ranked.hs
+++ b/src/Data/Array/Nested/Internal/Ranked.hs
@@ -483,11 +483,14 @@ rmaxIndexPrim rarr@(Ranked arr)
   | Refl <- lemRankReplicate (rrank (rtoPrimitive rarr))
   = ixCvtXR (mmaxIndexPrim arr)
 
-rdot1 :: (PrimElt a, NumElt a) => Ranked 1 a -> Ranked 1 a -> a
-rdot1 = coerce mdot1
+rdot1Inner :: forall n a. (PrimElt a, NumElt a) => Ranked (n + 1) a -> Ranked (n + 1) a -> Ranked n a
+rdot1Inner arr1 arr2
+  | SNat <- rrank arr1
+  , Refl <- lemReplicatePlusApp (SNat @n) (Proxy @1) (Proxy @(Nothing @Nat))
+  = coerce (mdot1Inner (Proxy @(Nothing @Nat))) arr1 arr2
 
 -- | This has a temporary, suboptimal implementation in terms of 'mflatten'.
--- Prefer 'rdot1' if applicable.
+-- Prefer 'rdot1Inner' if applicable.
 rdot :: (PrimElt a, NumElt a) => Ranked n a -> Ranked n a -> a
 rdot = coerce mdot
 
diff --git a/src/Data/Array/Nested/Internal/Shape.hs b/src/Data/Array/Nested/Internal/Shape.hs
index ca04840..7077053 100644
--- a/src/Data/Array/Nested/Internal/Shape.hs
+++ b/src/Data/Array/Nested/Internal/Shape.hs
@@ -87,6 +87,16 @@ listrTail :: ListR (n + 1) i -> ListR n i
 listrTail (_ ::: sh) = sh
 listrTail ZR = error "unreachable"
 
+listrInit :: ListR (n + 1) i -> ListR n i
+listrInit (n ::: sh@(_ ::: _)) = n ::: listrInit sh
+listrInit (_ ::: ZR) = ZR
+listrInit ZR = error "unreachable"
+
+listrLast :: ListR (n + 1) i -> i
+listrLast (_ ::: sh@(_ ::: _)) = listrLast sh
+listrLast (n ::: ZR) = n
+listrLast ZR = error "unreachable"
+
 listrIndex :: forall k n i. (k + 1 <= n) => SNat k -> ListR n i -> i
 listrIndex SZ (x ::: _) = x
 listrIndex (SS i) (_ ::: xs) | Refl <- lemLeqSuccSucc (Proxy @k) (Proxy @n) = listrIndex i xs
@@ -166,6 +176,12 @@ ixrHead (IxR list) = listrHead list
 ixrTail :: IxR (n + 1) i -> IxR n i
 ixrTail (IxR list) = IxR (listrTail list)
 
+ixrInit :: IxR (n + 1) i -> IxR n i
+ixrInit (IxR list) = IxR (listrInit list)
+
+ixrLast :: IxR (n + 1) i -> i
+ixrLast (IxR list) = listrLast list
+
 ixrAppend :: forall n m i. IxR n i -> IxR m i -> IxR (n + m) i
 ixrAppend = coerce (listrAppend @_ @i)
 
@@ -235,6 +251,12 @@ shrHead (ShR list) = listrHead list
 shrTail :: ShR (n + 1) i -> ShR n i
 shrTail (ShR list) = ShR (listrTail list)
 
+shrInit :: ShR (n + 1) i -> ShR n i
+shrInit (ShR list) = ShR (listrInit list)
+
+shrLast :: ShR (n + 1) i -> i
+shrLast (ShR list) = listrLast list
+
 shrAppend :: forall n m i. ShR n i -> ShR m i -> ShR (n + m) i
 shrAppend = coerce (listrAppend @_ @i)
 
@@ -310,17 +332,25 @@ listsToList :: ListS sh (Const i) -> [i]
 listsToList ZS = []
 listsToList (Const i ::$ is) = i : listsToList is
 
-listsHead :: ListS (n : sh) i -> i n
+listsHead :: ListS (n : sh) f -> f n
 listsHead (i ::$ _) = i
 
-listsTail :: ListS (n : sh) i -> ListS sh i
+listsTail :: ListS (n : sh) f -> ListS sh f
 listsTail (_ ::$ sh) = sh
 
+listsInit :: ListS (n : sh) f -> ListS (Init (n : sh)) f
+listsInit (n ::$ sh@(_ ::$ _)) = n ::$ listsInit sh
+listsInit (_ ::$ ZS) = ZS
+
+listsLast :: ListS (n : sh) f -> f (Last (n : sh))
+listsLast (_ ::$ sh@(_ ::$ _)) = listsLast sh
+listsLast (n ::$ ZS) = n
+
 listsAppend :: ListS sh f -> ListS sh' f -> ListS (sh ++ sh') f
 listsAppend ZS idx' = idx'
 listsAppend (i ::$ idx) idx' = i ::$ listsAppend idx idx'
 
-listsRank :: ListS sh i -> SNat (Rank sh)
+listsRank :: ListS sh f -> SNat (Rank sh)
 listsRank ZS = SNat
 listsRank (_ ::$ sh) = snatSucc (listsRank sh)
 
@@ -403,6 +433,12 @@ ixsHead (IxS list) = getConst (listsHead list)
 ixsTail :: IxS (n : sh) i -> IxS sh i
 ixsTail (IxS list) = IxS (listsTail list)
 
+ixsInit :: IxS (n : sh) i -> IxS (Init (n : sh)) i
+ixsInit (IxS list) = IxS (listsInit list)
+
+ixsLast :: IxS (n : sh) i -> i
+ixsLast (IxS list) = getConst (listsLast list)
+
 ixsAppend :: forall sh sh' i. IxS sh i -> IxS sh' i -> IxS (sh ++ sh') i
 ixsAppend = coerce (listsAppend @_ @(Const i))
 
@@ -469,6 +505,12 @@ shsHead (ShS list) = listsHead list
 shsTail :: ShS (n : sh) -> ShS sh
 shsTail (ShS list) = ShS (listsTail list)
 
+shsInit :: ShS (n : sh) -> ShS (Init (n : sh))
+shsInit (ShS list) = ShS (listsInit list)
+
+shsLast :: ShS (n : sh) -> SNat (Last (n : sh))
+shsLast (ShS list) = listsLast list
+
 shsAppend :: forall sh sh'. ShS sh -> ShS sh' -> ShS (sh ++ sh')
 shsAppend = coerce (listsAppend @_ @SNat)
 
diff --git a/src/Data/Array/Nested/Internal/Shaped.hs b/src/Data/Array/Nested/Internal/Shaped.hs
index d013959..995507f 100644
--- a/src/Data/Array/Nested/Internal/Shaped.hs
+++ b/src/Data/Array/Nested/Internal/Shaped.hs
@@ -418,11 +418,19 @@ sminIndexPrim sarr@(Shaped arr) = ixCvtXS (sshape (stoPrimitive sarr)) (mminInde
 smaxIndexPrim :: (PrimElt a, NumElt a) => Shaped sh a -> IIxS sh
 smaxIndexPrim sarr@(Shaped arr) = ixCvtXS (sshape (stoPrimitive sarr)) (mmaxIndexPrim arr)
 
-sdot1 :: (PrimElt a, NumElt a) => Shaped '[n] a -> Shaped '[n] a -> a
-sdot1 = coerce mdot1
+sdot1Inner :: forall sh n a. (PrimElt a, NumElt a)
+           => Proxy n -> Shaped (sh ++ '[n]) a -> Shaped (sh ++ '[n]) a -> Shaped sh a
+sdot1Inner Proxy sarr1@(Shaped arr1) (Shaped arr2)
+  | Refl <- lemInitApp (Proxy @sh) (Proxy @n)
+  , Refl <- lemLastApp (Proxy @sh) (Proxy @n)
+  = case sshape sarr1 of
+      _ :$$ _
+        | Refl <- lemMapJustApp (shsInit (sshape sarr1)) (Proxy @'[n])
+        -> Shaped (mdot1Inner (Proxy @(Just n)) arr1 arr2)
+      _ -> error "unreachable"
 
 -- | This has a temporary, suboptimal implementation in terms of 'mflatten'.
--- Prefer 'sdot1' if applicable.
+-- Prefer 'sdot1Inner' if applicable.
 sdot :: (PrimElt a, NumElt a) => Shaped sh a -> Shaped sh a -> a
 sdot = coerce mdot