From 303c5e7bba583f08e59bc6c848be8e75c1155a3b Mon Sep 17 00:00:00 2001
From: Michael Orlitzky <michael@orlitzky.com>
Date: Wed, 20 Feb 2013 21:47:50 -0500
Subject: [PATCH] A huge pile of crap upon Matrix/Vector.

---
 src/Linear/Matrix.hs | 260 +++++++++++++++++++++++++++++++++-------
 src/Linear/Vector.hs | 278 +++++++++----------------------------------
 2 files changed, 270 insertions(+), 268 deletions(-)

diff --git a/src/Linear/Matrix.hs b/src/Linear/Matrix.hs
index 63c0348..39576dc 100644
--- a/src/Linear/Matrix.hs
+++ b/src/Linear/Matrix.hs
@@ -1,78 +1,146 @@
-{-# LANGUAGE ScopedTypeVariables #-}
+{-# LANGUAGE ExistentialQuantification #-}
 {-# LANGUAGE FlexibleContexts #-}
 {-# LANGUAGE FlexibleInstances #-}
 {-# LANGUAGE MultiParamTypeClasses #-}
+{-# LANGUAGE ScopedTypeVariables #-}
 {-# LANGUAGE TypeFamilies #-}
+{-# LANGUAGE RebindableSyntax #-}
 
 module Linear.Matrix
 where
 
+import Data.List (intercalate)
+
 import Data.Vector.Fixed (
   Dim,
   Vector
   )
 import qualified Data.Vector.Fixed as V (
+  Fun(..),
+  N1,
+  and,
+  eq,
+  foldl,
   fromList,
   length,
   map,
-  toList
+  maximum,
+  replicate,
+  toList,
+  zipWith
   )
-import Data.Vector.Fixed.Internal (arity)
-
+import Data.Vector.Fixed.Internal (Arity, arity, S, Dim)
 import Linear.Vector
+import Normed
+
+import NumericPrelude hiding (abs)
+import qualified Algebra.Algebraic as Algebraic
+import qualified Algebra.Absolute as Absolute
+import qualified Algebra.Additive as Additive
+import qualified Algebra.Ring as Ring
+import Algebra.Absolute (abs)
+import qualified Algebra.Field as Field
+import qualified Algebra.RealField as RealField
+import qualified Algebra.RealRing as RealRing
+import qualified Algebra.ToRational as ToRational
+import qualified Algebra.Transcendental as Transcendental
+import qualified Prelude as P
+
+data Mat v w a = (Vector v (w a), Vector w a) => Mat (v (w a))
+type Mat1 a = Mat D1 D1 a
+type Mat2 a = Mat D2 D2 a
+type Mat3 a = Mat D3 D3 a
+type Mat4 a = Mat D4 D4 a
+
+-- We can't just declare that all instances of Vector are instances of
+-- Eq unfortunately. We wind up with an overlapping instance for
+-- w (w a).
+instance (Eq a, Vector v Bool, Vector w Bool) => Eq (Mat v w a) where
+  -- | Compare a row at a time.
+  --
+  --   Examples:
+  --
+  --   >>> let m1 = fromList [[1,2],[3,4]] :: Mat2 Int
+  --   >>> let m2 = fromList [[1,2],[3,4]] :: Mat2 Int
+  --   >>> let m3 = fromList [[5,6],[7,8]] :: Mat2 Int
+  --   >>> m1 == m2
+  --   True
+  --   >>> m1 == m3
+  --   False
+  --
+  (Mat rows1) == (Mat rows2) =
+    V.and $ V.zipWith comp rows1 rows2
+    where
+      -- Compare a row, one column at a time.
+      comp row1 row2 = V.and (V.zipWith (==) row1 row2)
+
+
+instance (Show a, Vector v String, Vector w String) => Show (Mat v w a) where
+  -- | Display matrices and vectors as ordinary tuples. This is poor
+  --   practice, but these results are primarily displayed
+  --   interactively and convenience trumps correctness (said the guy
+  --   who insists his vector lengths be statically checked at
+  --   compile-time).
+  --
+  --   Examples:
+  --
+  --   >>> let m = fromList [[1,2],[3,4]] :: Mat2 Int
+  --   >>> show m
+  --   ((1,2),(3,4))
+  --
+  show (Mat rows) =
+    "(" ++ (intercalate "," (V.toList row_strings)) ++ ")"
+    where
+      row_strings = V.map show_vector rows
+      show_vector v1 =
+        "(" ++ (intercalate "," element_strings) ++ ")"
+        where
+          v1l = V.toList v1
+          element_strings = P.map show v1l
+
 
-type Mat v w a = Vn v (Vn w a)
-type Mat2 a = Mat Vec2D Vec2D a
-type Mat3 a = Mat Vec3D Vec3D a
-type Mat4 a = Mat Vec4D Vec4D a
 
 -- | Convert a matrix to a nested list.
-toList :: (Vector v (Vn w a), Vector w a) => Mat v w a -> [[a]]
-toList m = map V.toList (V.toList m)
+toList :: Mat v w a -> [[a]]
+toList (Mat rows) = map V.toList (V.toList rows)
 
 -- | Create a matrix from a nested list.
-fromList :: (Vector v (Vn w a), Vector w a) => [[a]] -> Mat v w a
-fromList vs = V.fromList $ map V.fromList vs
+fromList :: (Vector v (w a), Vector w a, Vector v a) => [[a]] -> Mat v w a
+fromList vs = Mat (V.fromList $ map V.fromList vs)
 
 
 -- | Unsafe indexing.
-(!!!) :: (Vector v (Vn w a), Vector w a) => Mat v w a -> (Int, Int) -> a
+(!!!) :: (Vector w a) => Mat v w a -> (Int, Int) -> a
 (!!!) m (i, j) = (row m i) ! j
 
 -- | Safe indexing.
-(!!?) :: (Vector v (Vn w a), Vector w a) => Mat v w a
-                                         -> (Int, Int)
-                                         -> Maybe a
-(!!?) m (i, j)
+(!!?) :: Mat v w a -> (Int, Int) -> Maybe a
+(!!?) m@(Mat rows) (i, j)
   | i < 0 || j < 0 = Nothing
-  | i > V.length m = Nothing
+  | i > V.length rows = Nothing
   | otherwise = if j > V.length (row m j)
                 then Nothing
                 else Just $ (row m j) ! j
 
 
 -- | The number of rows in the matrix.
-nrows :: forall v w a. (Vector v (Vn w a), Vector w a) => Mat v w a -> Int
-nrows = V.length
+nrows :: Mat v w a -> Int
+nrows (Mat rows) = V.length rows
 
 -- | The number of columns in the first row of the
 --   matrix. Implementation stolen from Data.Vector.Fixed.length.
-ncols :: forall v w a. (Vector v (Vn w a), Vector w a) => Mat v w a -> Int
-ncols _ = arity (undefined :: Dim w)
+ncols :: forall v w a. (Vector w a) => Mat v w a -> Int
+ncols _ = (arity (undefined :: Dim w))
 
 -- | Return the @i@th row of @m@. Unsafe.
-row :: (Vector v (Vn w a), Vector w a) => Mat v w a
-                                       -> Int
-                                       -> Vn w a
-row m i = m ! i
+row :: Mat v w a -> Int -> w a
+row (Mat rows) i = rows ! i
 
 
 -- | Return the @j@th column of @m@. Unsafe.
-column :: (Vector v a, Vector v (Vn w a), Vector w a) => Mat v w a
-                                                      -> Int
-                                                      -> Vn v a
-column m j =
-  V.map (element j) m
+column :: (Vector v a) => Mat v w a -> Int -> v a
+column (Mat rows) j =
+  V.map (element j) rows
   where
     element = flip (!)
 
@@ -89,16 +157,16 @@ column m j =
 --   >>> transpose m
 --   ((1,3),(2,4))
 --
-transpose :: (Vector v (Vn w a),
-              Vector w (Vn v a),
+transpose :: (Vector w (v a),
               Vector v a,
               Vector w a)
              => Mat v w a
              -> Mat w v a
-transpose m = V.fromList column_list
+transpose m = Mat $ V.fromList column_list
   where
     column_list = [ column m i | i <- [0..(ncols m)-1] ]
 
+
 -- | Is @m@ symmetric?
 --
 --   Examples:
@@ -111,7 +179,7 @@ transpose m = V.fromList column_list
 --   >>> symmetric m2
 --   False
 --
-symmetric :: (Vector v (Vn w a),
+symmetric :: (Vector v (w a),
               Vector w a,
               v ~ w,
               Vector w Bool,
@@ -136,11 +204,11 @@ symmetric m =
 --   ((0,1,2),(1,2,3),(2,3,4))
 --
 construct :: forall v w a.
-             (Vector v (Vn w a),
+             (Vector v (w a),
               Vector w a)
              => (Int -> Int -> a)
              -> Mat v w a
-construct lambda = rows
+construct lambda = Mat rows
   where
     -- The arity trick is used in Data.Vector.Fixed.length.
     imax = (arity (undefined :: Dim v)) - 1
@@ -161,19 +229,21 @@ construct lambda = rows
 --   ((20.000000000000004,-1.0),(-1.0,20.0))
 --
 cholesky :: forall a v w.
-            (RealFloat a,
-             Vector v (Vn w a),
-             Vector w a)
+            (Algebraic.C a,
+             Vector v (w a),
+             Vector w a,
+             Vector v a)
             => (Mat v w a)
             -> (Mat v w a)
 cholesky m = construct r
   where
     r :: Int -> Int -> a
-    r i j | i == j = sqrt(m !!! (i,j) - sum [(r k i)**2 | k <- [0..i-1]])
+    r i j | i == j = sqrt(m !!! (i,j) - sum [(r k i)^2 | k <- [0..i-1]])
           | i < j =
               (((m !!! (i,j)) - sum [(r k i)*(r k j) | k <- [0..i-1]]))/(r i i)
           | otherwise = 0
 
+
 -- | Matrix multiplication. Our 'Num' instance doesn't define one, and
 --   we need additional restrictions on the result type anyway.
 --
@@ -184,12 +254,11 @@ cholesky m = construct r
 --   >>> m1 `mult` m2
 --   ((22,28),(49,64))
 --
-mult :: (Num a,
-         Vector v (Vn w a),
+mult :: (Ring.C a,
+         Vector v a,
          Vector w a,
-         Vector w (Vn z a),
          Vector z a,
-         Vector v (Vn z a))
+         Vector v (z a))
         => Mat v w a
         -> Mat w z a
         -> Mat v z a
@@ -197,3 +266,104 @@ mult m1 m2 = construct lambda
   where
     lambda i j =
       sum [(m1 !!! (i,k)) * (m2 !!! (k,j)) | k <- [0..(ncols m1)-1] ]
+
+
+
+instance (Ring.C a,
+          Vector v (w a),
+          Vector w a)
+         => Additive.C (Mat v w a) where
+
+  (Mat rows1) + (Mat rows2) =
+    Mat $ V.zipWith (V.zipWith (+)) rows1 rows2
+
+  (Mat rows1) - (Mat rows2) =
+    Mat $ V.zipWith (V.zipWith (-)) rows1 rows2
+
+  zero = Mat (V.replicate $ V.replicate (fromInteger 0))
+
+
+instance (Ring.C a,
+          Vector v (w a),
+          Vector w a,
+          v ~ w)
+         => Ring.C (Mat v w a) where
+    one = Mat (V.replicate $ V.replicate (fromInteger 1))
+    m1 * m2 = m1 `mult` m2
+
+
+instance (Algebraic.C a,
+          ToRational.C a,
+          Vector v (w a),
+          Vector w a,
+          Vector v a,
+          Vector v [a])
+         => Normed (Mat v w a) where
+   -- Treat the matrix as a big vector.
+   norm_p p (Mat rows) =
+     sqrt $ sum [(fromRational' $ toRational x)^2 | x <- xs]
+     where
+       xs = concat $ V.toList $ V.map V.toList rows
+
+   norm_infty m@(Mat rows)
+     | nrows m == 0 || ncols m == 0 = 0
+     | otherwise =
+         fromRational' $ toRational $
+         P.maximum $ V.toList $ V.map (P.maximum . V.toList) rows
+
+
+
+
+
+-- Vector helpers. We want it to be easy to create low-dimension
+-- column vectors.
+type Vec a b = Mat a D1 b
+
+vec2d :: (a,a) -> Mat D2 D1 a
+vec2d (x,y) = Mat (D2 (D1 x) (D1 y))
+
+vec3d :: (a,a,a) -> Mat D3 D1 a
+vec3d (x,y,z) = Mat (D3 (D1 x) (D1 y) (D1 z))
+
+vec4d :: (a,a,a,a) -> Mat D4 D1 a
+vec4d (w,x,y,z) = Mat (D4 (D1 w) (D1 x) (D1 y) (D1 z))
+
+dot :: (RealRing.C a,
+        Dim w ~ V.N1,
+        Vector v a,
+        Vector w a,
+        Vector w (v a),
+        Vector w (w a))
+       => Mat v w a
+       -> Mat v w a
+       -> a
+v1 `dot` v2 = ((transpose v1) `mult` v2) !!! (0, 0)
+
+
+-- | The angle between @v1@ and @v2@ in Euclidean space.
+--
+--   Examples:
+--
+--   >>> let v1 = make2d (1.0, 0.0)
+--   >>> let v2 = make2d (0.0, 1.0)
+--   >>> angle v1 v2 == pi/2.0
+--   True
+--
+angle :: (Transcendental.C a,
+          RealRing.C a,
+          Dim w ~ V.N1,
+          Vector w (w a),
+          Vector v [a],
+          Vector v a,
+          Vector w a,
+          Vector v (w a),
+          Vector w (v a),
+          ToRational.C a)
+          => Mat v w a
+          -> Mat v w a
+          -> a
+angle v1 v2 =
+  acos theta
+  where
+   theta = (recip norms) * (v1 `dot` v2)
+   norms = (norm v1) * (norm v2)
diff --git a/src/Linear/Vector.hs b/src/Linear/Vector.hs
index 7cc5e00..9774dcd 100644
--- a/src/Linear/Vector.hs
+++ b/src/Linear/Vector.hs
@@ -11,6 +11,7 @@ import Data.List (intercalate)
 import Data.Vector.Fixed (
   Dim,
   Fun(..),
+  N1,
   N2,
   N3,
   N4,
@@ -20,119 +21,74 @@ import Data.Vector.Fixed (
   toList,
   )
 import qualified Data.Vector.Fixed as V (
-  eq,
-  foldl,
   length,
-  map,
-  replicate,
-  sum,
-  zipWith
   )
 
 import Normed
 
--- | The Vn newtype simply wraps (Vector v a) so that we avoid
---   undecidable instances.
-newtype Vn v a = Vn (v a)
 
--- | Declare the dimension of the wrapper to be the dimension of what
---   it contains.
-type instance Dim (Vn v) = Dim v
+-- * Low-dimension vector wrappers.
+--
+-- These wrappers are instances of 'Vector', so they inherit all of
+-- the userful instances defined above. But, they use fixed
+-- constructors, so you can pattern match out the individual
+-- components.
 
-instance (Vector v a) => Vector (Vn v) a where
-  -- | Fortunately, 'Fun' is an instance of 'Functor'. The
-  --   'construct' defined on our contained type will return a
-  --   'Fun', and we simply slap our constructor on top with fmap.
-  construct = fmap Vn construct
+data D1 a = D1 a
+type instance Dim D1 = N1
+instance Vector D1 a where
+  inspect (D1 x) (Fun f) = f x
+  construct = Fun D1
 
-  -- | Defer to the inspect defined on the contained type.
-  inspect (Vn v1) = inspect v1
+data D2 a = D2 a a
+type instance Dim D2 = N2
+instance Vector D2 a where
+  inspect (D2 x y) (Fun f) = f x y
+  construct = Fun D2
 
-instance (Show a, Vector v a) => Show (Vn v a) where
-  -- | Display vectors as ordinary tuples. This is poor practice, but
-  --   these results are primarily displayed interactively and
-  --   convenience trumps correctness (said the guy who insists his
-  --   vector lengths be statically checked at compile-time).
-  --
-  --   Examples:
-  --
-  --   >>> let v1 = make2d (1,2)
-  --   >>> show v1
-  --   (1,2)
-  --
-  show (Vn v1) =
-    "(" ++ (intercalate "," element_strings) ++ ")"
-    where
-      v1l = toList v1
-      element_strings = Prelude.map show v1l
+data D3 a = D3 a a a
+type instance Dim D3 = N3
+instance Vector D3 a where
+  inspect (D3 x y z) (Fun f) = f x y z
+  construct = Fun D3
+
+data D4 a = D4 a a a a
+type instance Dim D4 = N4
+instance Vector D4 a where
+  inspect (D4 w x y z) (Fun f) = f w x y z
+  construct = Fun D4
 
 
--- | We would really like to say, "anything that is a vector of
---   equatable things is itself equatable." The 'Vn' class
---   allows us to express this without a GHC battle.
+-- | Unsafe indexing.
 --
 --   Examples:
 --
---   >>> let v1 = make2d (1,2)
---   >>> let v2 = make2d (1,2)
---   >>> let v3 = make2d (3,4)
---   >>> v1 == v2
---   True
---   >>> v1 == v3
---   False
+--   >>> let v1 = Vec2D 1 2
+--   >>> v1 ! 1
+--   2
 --
-instance (Eq a, Vector v a) => Eq (Vn v a) where
-  (Vn v1) == (Vn v2) = v1 `V.eq` v2
-
-
--- | The use of 'Num' here is of course incorrect (otherwise, we
---   wouldn't have to throw errors). But it's really nice to be able
---   to use normal addition/subtraction.
-instance (Num a, Vector v a) => Num (Vn v a) where
-  -- | Componentwise addition.
-  --
-  --   Examples:
-  --
-  --   >>> let v1 = make2d (1,2)
-  --   >>> let v2 = make2d (3,4)
-  --   >>> v1 + v2
-  --   (4,6)
-  --
-  (Vn v1) + (Vn v2) = Vn $ V.zipWith (+) v1 v2
-
-  -- | Componentwise subtraction.
-  --
-  --   Examples:
-  --
-  --   >>> let v1 = make2d (1,2)
-  --   >>> let v2 = make2d (3,4)
-  --   >>> v1 - v2
-  --   (-2,-2)
-  --
-  (Vn v1) - (Vn v2) = Vn $ V.zipWith (-) v1 v2
+(!) :: (Vector v a) => v a -> Int -> a
+(!) v1 idx = (toList v1) !! idx
 
-  -- | Create an n-vector whose components are all equal to the given
-  --   integer. The result type must be specified since otherwise the
-  --   length n would be unknown.
-  --
-  --   Examples:
-  --
-  --   >>> let v1 = fromInteger 17 :: Vn Vec3 Int
-  --   (17,17,17)
-  --
-  fromInteger x = Vn $ V.replicate (fromInteger x)
-  (*) = error "multiplication of vectors is undefined"
-  abs = error "absolute value of vectors is undefined"
-  signum = error "signum of vectors is undefined"
+-- | Safe indexing.
+--
+--   Examples:
+--
+--   >>> let v1 = Vec3D 1 2 3
+--   >>> v1 !? 2
+--   Just 3
+--   >>> v1 !? 3
+--   Nothing
+--
+(!?) :: (Vector v a) => v a -> Int -> Maybe a
+(!?) v1 idx
+  | idx < 0 || idx >= V.length v1 = Nothing
+  | otherwise                     = Just $ v1 ! idx
 
 
--- | This is probably useless, since the vectors we usually contain
---   aren't functor instances.
-instance (Functor v) => Functor (Vn v) where
-  fmap f (Vn v1) = Vn (f `fmap` v1)
 
 
-instance (RealFloat a, Ord a, Vector v a) => Normed (Vn v a) where
+--instance (RealFloat a, Ord a, Vector v a) => Normed (Vn v a) where
   -- | The infinity norm. We don't use V.maximum here because it
   --   relies on a type constraint that the vector be non-empty and I
   --   don't know how to pattern match it away.
@@ -143,7 +99,7 @@ instance (RealFloat a, Ord a, Vector v a) => Normed (Vn v a) where
   --   >>> norm_infty v1
   --   5
   --
-  norm_infty (Vn v1) = realToFrac $ V.foldl max 0 v1
+--  norm_infty (Vn v1) = realToFrac $ V.foldl max 0 v1
 
   -- | Generic p-norms. The usual norm in R^n is (norm_p 2).
   --
@@ -155,78 +111,16 @@ instance (RealFloat a, Ord a, Vector v a) => Normed (Vn v a) where
   --   >>> norm_p 2 v1
   --   5.0
   --
-  norm_p p (Vn v1) =
-    realToFrac $ root $ V.sum $ V.map (exponentiate . abs) v1
-    where
-      exponentiate = (** (fromIntegral p))
-      root = (** (recip (fromIntegral p)))
-
--- | Dot (standard inner) product.
---
---   Examples:
---
---   >>> let v1 = make3d (1,2,3)
---   >>> let v2 = make3d (4,5,6)
---   >>> dot v1 v2
---   32
---
-dot :: (Num a, Vector v a) => Vn v a -> Vn v a -> a
-dot (Vn v1) (Vn v2) = V.sum $ V.zipWith (*) v1 v2
-
-
--- | The angle between @v1@ and @v2@ in Euclidean space.
---
---   Examples:
---
---   >>> let v1 = make2d (1.0, 0.0)
---   >>> let v2 = make2d (0.0, 1.0)
---   >>> angle v1 v2 == pi/2.0
---   True
---
-angle :: (RealFloat a, Vector v a) => Vn v a -> Vn v a -> a
-angle v1 v2 =
-  acos theta
-  where
-    theta = (v1 `dot` v2) / norms
-    norms = (norm v1) * (norm v2)
+--  norm_p p (Vn v1) =
+--    realToFrac $ root $ V.sum $ V.map (exponentiate . abs) v1
+--    where
+--      exponentiate = (** (fromIntegral p))
+--      root = (** (recip (fromIntegral p)))
 
 
--- | Unsafe indexing.
---
---   Examples:
---
---   >>> let v1 = make2d (1,2)
---   >>> v1 ! 1
---   2
---
-(!) :: (Vector v a) => v a -> Int -> a
-(!) v1 idx = (toList v1) !! idx
 
--- | Safe indexing.
---
---   Examples:
---
---   >>> let v1 = make3d (1,2,3)
---   >>> v1 !? 2
---   Just 3
---   >>> v1 !? 3
---   Nothing
---
-(!?) :: (Vector v a) => v a -> Int -> Maybe a
-(!?) v1 idx
-  | idx < 0 || idx >= V.length v1 = Nothing
-  | otherwise                     = Just $ v1 ! idx
 
 
-
-
--- * Low-dimension vector wrappers.
---
--- These wrappers are instances of 'Vector', so they inherit all of
--- the userful instances defined above. But, they use fixed
--- constructors, so you can pattern match out the individual
--- components.
-
 -- | Convenient constructor for 2D vectors.
 --
 --   Examples:
@@ -241,65 +135,3 @@ angle v1 v2 =
 --   >>> fixed_point g eps u0
 --   (1.0728549599342185,1.0820591495686167)
 --
-data Vec2D a = Vec2D a a
-type instance Dim Vec2D = N2
-instance Vector Vec2D a where
-  inspect (Vec2D x y) (Fun f) = f x y
-  construct = Fun Vec2D
-
-data Vec3D a = Vec3D a a a
-type instance Dim Vec3D = N3
-instance Vector Vec3D a where
-  inspect (Vec3D x y z) (Fun f) = f x y z
-  construct = Fun Vec3D
-
-data Vec4D a = Vec4D a a a a
-type instance Dim Vec4D = N4
-instance Vector Vec4D a where
-  inspect (Vec4D w x y z) (Fun f) = f w x y z
-  construct = Fun Vec4D
-
-
--- | Convenience function for creating 2d vectors.
---
---   Examples:
---
---   >>> let v1 = make2d (1,2)
---   >>> v1
---   (1,2)
---   >>> let Vn (Vec2D x y) = v1
---   >>> (x,y)
---   (1,2)
---
-make2d :: forall a. (a,a) -> Vn Vec2D a
-make2d (x,y) = Vn (Vec2D x y)
-
-
--- | Convenience function for creating 3d vectors.
---
---   Examples:
---
---   >>> let v1 = make3d (1,2,3)
---   >>> v1
---   (1,2,3)
---   >>> let Vn (Vec3D x y z) = v1
---   >>> (x,y,z)
---   (1,2,3)
---
-make3d :: forall a. (a,a,a) -> Vn Vec3D a
-make3d (x,y,z) = Vn (Vec3D x y z)
-
-
--- | Convenience function for creating 4d vectors.
---
---   Examples:
---
---   >>> let v1 = make4d (1,2,3,4)
---   >>> v1
---   (1,2,3,4)
---   >>> let Vn (Vec4D w x y z) = v1
---   >>> (w,x,y,z)
---   (1,2,3,4)
---
-make4d :: forall a. (a,a,a,a) -> Vn Vec4D a
-make4d (w,x,y,z) = Vn (Vec4D w x y z)
-- 
2.44.2