```{-# LANGUAGE DeriveDataTypeable, FlexibleContexts, FlexibleInstances,
FunctionalDependencies, KindSignatures,
MultiParamTypeClasses, NoImplicitPrelude,
TypeOperators, UndecidableInstances  #-}
{-# OPTIONS -Wall #-}
-- | A tensor algebra library. Main ingredients are :
--
-- 'Vec' and ':~' are data constructors for rank-1 tensor.
-- This is essentially a touple of objects of the same type.
--
-- 'Vector' is a class for rank-1 tensor.
--
-- 'Axis' is an object for accessing the tensor components.

module Data.Tensor.TypeLevel
(
(:~)(..), Vec(..), Axis(..), (!),
Vector(..), VectorRing(..),
contract,
Vec0, Vec1, Vec2, Vec3, Vec4,
Vec5, Vec6, Vec7, Vec8, Vec9, Vec10,
vec0, vec1, vec2, vec3, vec4,
vec5, vec6, vec7, vec8, vec9, vec10
) where

import qualified Algebra.Ring as Ring
import           System.IO.Unsafe

import           Control.Applicative
(mapM_, sequence_, forM_, msum, mapM, sequence, forM)
import           Data.Foldable
import           Data.List (intercalate)
import           Data.Traversable
import           NumericPrelude hiding
(>>=), (>>), return, fail, fmap, mapM, mapM_, sequence, sequence_,
(=<<), foldl, foldl1, foldr, foldr1, and, or, any, all, sum, product,
concat, concatMap, maximum, minimum, elem, notElem)
import qualified Prelude as P98
import qualified Test.QuickCheck.Arbitrary as QC

infixl 9 !
-- | a component operator.
(!) :: Vector v => v a -> Axis v -> a
v ! i  = component i v

-- | data constructor for 0-dimensional tensor.
data Vec a
= Vec
deriving (Eq, Ord)

-- | data constructor for constructing n+1-dimensional tensor
-- from n-dimensional tensor.
data (n :: * -> * ) :~ a
= (n a) :~ a
deriving (Eq)
infixl 3 :~

instance Show (Vec a) where
show = const "()"

instance (Show a, Traversable ((:~) n)) => Show (n :~ a) where
show = ("("++) . (++")") .
intercalate "," . map show . toList

_ <- P.string "()"
return Vec

instance (Read a, Vector ((:~) n)) => Read (n :~ a) where
_ <- P.char '('
ret <- sequence \$ compose (\(Axis i) -> do
when (i>0) \$ (P.char ',' >> return ())
)
_ <- P.char ')'
return ret

instance QC.Arbitrary (Vec a) where
arbitrary = return Vec
shrink = const []

instance (QC.Arbitrary a, QC.Arbitrary (n a), Traversable n) => QC.Arbitrary (n :~ a) where
arbitrary = (:~) <\$> QC.arbitrary <*> QC.arbitrary
shrink = sequence . fmap QC.shrink

-- | the last component contributes the most to the ordering
instance (Ord (n a), Ord a) => Ord (n :~ a) where
compare (xs :~ x) (ys :~ y) = compare (x, xs) (y, ys)

instance Foldable Vec where
foldMap = foldMapDefault
instance Functor Vec where
fmap = fmapDefault
instance Traversable Vec where
traverse _ Vec = pure Vec
instance Applicative Vec where
pure _  = Vec
_ <*> _ = Vec

instance (Traversable n) => Foldable ((:~) n) where
foldMap = foldMapDefault
instance (Traversable n) => Functor ((:~) n) where
fmap = fmapDefault
instance (Traversable n) => Traversable ((:~) n) where
traverse f (x :~ y) = (:~) <\$> traverse f x <*> f y
instance (Applicative n, Traversable n) => Applicative ((:~) n) where
pure x = pure x :~ x
(vf :~ f) <*> (vx :~ x) = (vf <*> vx) :~ (f x)

-- | An coordinate 'Axis' , labeled by an integer.
-- Axis also carries v, the container type for its corresponding
-- vector. Therefore, An axis of one type can access only vectors
-- of a fixed dimension, but of arbitrary type.
newtype Axis (v :: * -> *) = Axis {axisIndex::Int} deriving (Eq,Ord,Show,Read)

-- | An object that allows component-wise access.
class (Traversable v) => Vector v where
-- | Get a component within f, a context which allows failure.
componentF :: (Alternative f) =>
Axis v -- ^the axis of the component you want
-> v a -- ^the target vector
-> f a -- ^the component, obtained within a failure

-- | Get a component. This computation may result in a runtime error,
-- though, as long as the 'Axis' is generated from library functions
-- such as 'compose', there will be no error.
component :: Axis v -> v a -> a
component axis vec = case componentF axis vec of
Just x  -> x
Nothing -> error \$ "axis out of bound: " ++ show axis
-- | The dimension of the vector.
dimension :: v a -> Int
-- | Create a 'Vector' from a function that maps
-- axis to components.
compose :: (Axis v -> a) -> v a

instance Vector Vec where
componentF axis Vec
= empty
dimension _ = 0
compose _ = Vec

instance (Vector v) => Vector ((:~) v) where
componentF (Axis i) vx@(v :~ x)
| i==dimension vx - 1 = pure x
| True                = componentF (Axis i) v
dimension (v :~ _) = 1 + dimension v
compose f = let
xs = compose (\(Axis i)->f (Axis i)) in xs :~ f (Axis (dimension xs))

zero = compose \$ const Additive.zero
x+y  = compose (\i ->  x!i +  y!i)
x-y  = compose (\i ->  x!i -  y!i)
negate x = compose (\i -> negate \$ x!i)

zero = compose \$ const Additive.zero
x+y  = compose (\i -> x!i + y!i)
x-y  = compose (\i -> x!i - y!i)
negate x = compose (\i -> negate \$ x!i)

-- | Tensor contraction. Create a 'Vector' from a function that maps
-- axis to component, then sums over the axis and returns @a@.
contract :: (Vector v, Additive.C a) => (Axis v -> a) -> a
contract f = foldl (+) Additive.zero (compose f)

-- | 'VectorRing' is a 'Vector' whose components belongs to 'Ring.C',
-- thus providing unit vectors.
class  (Vector v, Ring.C a) => VectorRing v a where
-- | A vector where 'Axis'th component is unity but others are zero.
unitVectorF :: (Alternative f) => Axis v -> f (v a)
-- | pure but unsafe version means of obtaining a 'unitVector'
unitVector :: Axis v -> v a
unitVector axis = case unitVectorF axis of
Just x  -> x
Nothing -> error \$ "axis out of bound: " ++ show axis

instance (Ring.C a) => VectorRing Vec a where
unitVectorF axis = empty

instance (Ring.C a, VectorRing v a, Additive.C (v a))
=> VectorRing ((:~) v) a where
unitVectorF axis@(Axis i) = ret
where
d = dimension z
ret
| i < 0 || i >= d   = empty
| i == d-1          = pure \$ Additive.zero :~ Ring.one
| 0 <= i && i < d-1 = fmap (:~ Additive.zero) \$ unitVectorF (Axis i)
| True              = pure z
-- this last guard never matches, but needed to infer the type of z.

instance (Vector v, P98.Num a) => P98.Num ((:~) v a) where
x+y  = compose (\i -> x!i P98.+ y!i)
x-y  = compose (\i -> x!i P98.- y!i)
negate x = compose (\i -> P98.negate \$ x!i)
(*) = error "P98 legacy instance"
fromInteger x = compose \$ const (P98.fromInteger x)
abs = error "P98 legacy instance"
signum = error "P98 legacy instance"

-- | Type synonyms
type Vec0 = Vec
type Vec1 = (:~) Vec0
type Vec2 = (:~) Vec1
type Vec3 = (:~) Vec2
type Vec4 = (:~) Vec3
type Vec5 = (:~) Vec4
type Vec6 = (:~) Vec5
type Vec7 = (:~) Vec6
type Vec8 = (:~) Vec7
type Vec9 = (:~) Vec8
type Vec10 = (:~) Vec9

-- | Utility functions
vec0 :: Vec0 a
vec0  = Vec
vec1 :: a -> Vec1 a
vec1 x0 = Vec :~ x0
vec2 :: a -> a -> Vec2 a
vec2 x0 x1 = Vec :~ x0 :~ x1
vec3 :: a -> a -> a -> Vec3 a
vec3 x0 x1 x2 = Vec :~ x0 :~ x1 :~ x2
vec4 :: a -> a -> a -> a -> Vec4 a
vec4 x0 x1 x2 x3 = Vec :~ x0 :~ x1 :~ x2 :~ x3
vec5 :: a -> a -> a -> a -> a -> Vec5 a
vec5 x0 x1 x2 x3 x4 = Vec :~ x0 :~ x1 :~ x2 :~ x3 :~ x4
vec6 :: a -> a -> a -> a -> a -> a -> Vec6 a
vec6 x0 x1 x2 x3 x4 x5 = Vec :~ x0 :~ x1 :~ x2 :~ x3 :~ x4 :~ x5
vec7 :: a -> a -> a -> a -> a -> a -> a -> Vec7 a
vec7 x0 x1 x2 x3 x4 x5 x6 = Vec :~ x0 :~ x1 :~ x2 :~ x3 :~ x4 :~ x5 :~ x6
vec8 :: a -> a -> a -> a -> a -> a -> a -> a -> Vec8 a
vec8 x0 x1 x2 x3 x4 x5 x6 x7 = Vec :~ x0 :~ x1 :~ x2 :~ x3 :~ x4 :~ x5 :~ x6 :~ x7
vec9 :: a -> a -> a -> a -> a -> a -> a -> a -> a -> Vec9 a
vec9 x0 x1 x2 x3 x4 x5 x6 x7 x8 = Vec :~ x0 :~ x1 :~ x2 :~ x3 :~ x4 :~ x5 :~ x6 :~ x7 :~ x8
vec10 :: a -> a -> a -> a -> a -> a -> a -> a -> a -> a -> Vec10 a
vec10 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 = Vec :~ x0 :~ x1 :~ x2 :~ x3 :~ x4 :~ x5 :~ x6 :~ x7 :~ x8 :~ x9
```