# LANGUAGE FlexibleInstances 
            , MultiParamTypeClasses 
            , TypeFamilies 
            , UndecidableInstances 
            , DeriveDataTypeable 
            , TypeOperators #
           , MultiParamTypeClasses
           , TypeFamilies
           , UndecidableInstances
           , DeriveDataTypeable
           , TypeOperators #-}
module Numeric.Coalgebra.Quaternion
  ( Distinguished(..)
  , Complicated(..)
  , Hamiltonian(..)
  , QuaternionBasis'(..)
  , Quaternion'(..)
  , complicate'
  , vectorPart'
  , scalarPart'
  ) where

import Control.Applicative
import Control.Monad.Reader.Class
import Data.Ix hiding (index)
import Data.Data
import Data.Distributive
import Data.Functor.Bind
import Data.Functor.Rep
import Data.Foldable
import Data.Traversable
import Data.Semigroup.Traversable
import Data.Semigroup.Foldable
import Data.Semigroup
import Numeric.Algebra
import Numeric.Algebra.Distinguished.Class
import Numeric.Algebra.Complex.Class
import Numeric.Algebra.Quaternion.Class
import Prelude hiding ((-),(+),(*),negate,subtract, fromInteger)

instance Distinguished QuaternionBasis' where
  e = E'

instance Complicated QuaternionBasis' where
  i = I'

instance Hamiltonian QuaternionBasis' where
  j = J'
  k = K'

instance Rig r => Distinguished (Quaternion' r) where
  e = Quaternion' one zero zero zero

instance Rig r => Complicated (Quaternion' r) where
  i = Quaternion' zero one zero zero

instance Rig r => Hamiltonian (Quaternion' r) where
  j = Quaternion' zero zero one zero
  k = Quaternion' one zero zero one

instance Rig r => Distinguished (QuaternionBasis' -> r) where
  e E' = one
  e _ = zero

instance Rig r => Complicated (QuaternionBasis' -> r) where
  i I' = one
  i _ = zero

instance Rig r => Hamiltonian (QuaternionBasis' -> r) where
  j J' = one
  j _ = zero

  k K' = one
  k _ = zero

-- quaternion basis
data QuaternionBasis' = E' | I' | J' | K' deriving (Eq,Ord,Enum,Read,Show,Bounded,Ix,Data,Typeable)

data Quaternion' a = Quaternion' a a a a deriving (Eq,Show,Read,Data,Typeable)

instance Representable Quaternion' where
  type Rep Quaternion' = QuaternionBasis'
  tabulate f = Quaternion' (f E') (f I') (f J') (f K')
  index (Quaternion' a _ _ _) E' = a
  index (Quaternion' _ b _ _) I' = b
  index (Quaternion' _ _ c _) J' = c
  index (Quaternion' _ _ _ d) K' = d

instance Distributive Quaternion' where
  distribute = distributeRep

instance Functor Quaternion' where
  fmap = fmapRep

instance Apply Quaternion' where
  (<.>) = apRep

instance Applicative Quaternion' where
  pure = pureRep
  (<*>) = apRep

instance Bind Quaternion' where
  (>>-) = bindRep

instance Monad Quaternion' where
  return = pureRep
  (>>=) = bindRep

instance MonadReader QuaternionBasis' Quaternion' where
  ask = askRep
  local = localRep

instance Foldable Quaternion' where
  foldMap f (Quaternion' a b c d) =
    f a `mappend` f b `mappend` f c `mappend` f d

instance Traversable Quaternion' where
  traverse f (Quaternion' a b c d) =
    Quaternion' <$> f a <*> f b <*> f c <*> f d

instance Foldable1 Quaternion' where
  foldMap1 f (Quaternion' a b c d) =
    f a <> f b <> f c <> f d

instance Traversable1 Quaternion' where
  traverse1 f (Quaternion' a b c d) =
    Quaternion' <$> f a <.> f b <.> f c <.> f d

instance Additive r => Additive (Quaternion' r) where
  (+) = addRep
  sinnum1p = sinnum1pRep

instance LeftModule r s => LeftModule r (Quaternion' s) where
  r .* Quaternion' a b c d =
    Quaternion' (r .* a) (r .* b) (r .* c) (r .* d)

instance RightModule r s => RightModule r (Quaternion' s) where
  Quaternion' a b c d *. r =
    Quaternion' (a *. r) (b *. r) (c *. r) (d *. r)

instance Monoidal r => Monoidal (Quaternion' r) where
  zero = zeroRep
  sinnum = sinnumRep

instance Group r => Group (Quaternion' r) where
  (-) = minusRep
  negate = negateRep
  subtract = subtractRep
  times = timesRep

instance Abelian r => Abelian (Quaternion' r)

instance Idempotent r => Idempotent (Quaternion' r)

instance Partitionable r => Partitionable (Quaternion' r) where
  partitionWith f (Quaternion' a b c d) = id =<<
    partitionWith (\a1 a2 -> id =<<
    partitionWith (\b1 b2 -> id =<<
    partitionWith (\c1 c2 ->
    partitionWith (\d1 d2 -> f (Quaternion' a1 b1 c1 d1)
                               (Quaternion' a2 b2 c2 d2)
                  ) d) c) b) a

-- | the trivial diagonal algebra
instance (TriviallyInvolutive r, Semiring r) => Algebra r QuaternionBasis' where
  mult f = f' where
    fe = f E' E'
    fi = f I' I'
    fj = f J' J'
    fk = f K' K'
    f' E' = fe
    f' I' = fi
    f' J' = fj
    f' K' = fk

instance (TriviallyInvolutive r, Semiring r) => UnitalAlgebra r QuaternionBasis' where
  unit = const


-- | dual quaternion comultiplication
instance (TriviallyInvolutive r, Rng r) => Coalgebra r QuaternionBasis' where
  comult f = f' where
    fe = f E'
    fi = f I'
    fj = f J'
    fk = f K'
    fe' = negate fe
    fi' = negate fi
    fj' = negate fj
    fk' = negate fk
    f' E' E' = fe
    f' E' I' = fi
    f' E' J' = fj
    f' E' K' = fk
    f' I' E' = fi
    f' I' I' = fe'
    f' I' J' = fk
    f' I' K' = fj'
    f' J' E' = fj
    f' J' I' = fk'
    f' J' J' = fe'
    f' J' K' = fi
    f' K' E' = fk
    f' K' I' = fj
    f' K' J' = fi'
    f' K' K' = fe'

instance (TriviallyInvolutive r, Rng r) => CounitalCoalgebra r QuaternionBasis' where
  counit f = f E'

instance (TriviallyInvolutive r, Rng r)  => Bialgebra r QuaternionBasis'

instance (TriviallyInvolutive r, InvolutiveSemiring r, Rng r)  => InvolutiveAlgebra r QuaternionBasis' where
  inv f E' = f E'
  inv f b = negate (f b)

instance (TriviallyInvolutive r, InvolutiveSemiring r, Rng r) => InvolutiveCoalgebra r QuaternionBasis' where
  coinv = inv

instance (TriviallyInvolutive r, InvolutiveSemiring r, Rng r) => HopfAlgebra r QuaternionBasis' where
  antipode = inv

instance (TriviallyInvolutive r, Semiring r) => Multiplicative (Quaternion' r) where
  (*) = mulRep

instance (TriviallyInvolutive r, Semiring r) => Semiring (Quaternion' r)

instance (TriviallyInvolutive r, Ring r) => Unital (Quaternion' r) where
  one = oneRep

instance (TriviallyInvolutive r, Ring r) => Rig (Quaternion' r) where
  fromNatural n = Quaternion' (fromNatural n) zero zero zero

instance (TriviallyInvolutive r, Ring r) => Ring (Quaternion' r) where
  fromInteger n = Quaternion' (fromInteger n) zero zero zero

instance ( TriviallyInvolutive r, Rng r) => LeftModule (Quaternion' r) (Quaternion' r) where
  (.*) = (*)
instance (TriviallyInvolutive r, Rng r) => RightModule (Quaternion' r) (Quaternion' r) where
  (*.) = (*)

instance (TriviallyInvolutive r, Rng r) => InvolutiveMultiplication (Quaternion' r) where
  -- without trivial involution, multiplication fails associativity, and we'd need to
  -- support weaker multiplicative properties like Alternative and PowerAssociative
  adjoint (Quaternion' a b c d) = Quaternion' a (negate b) (negate c) (negate d)

  | Cayley - Dickson quaternion isomorphism ( one way )
complicate' :: Complicated c => QuaternionBasis' -> (c , c)
complicate' E' = (e, e)
complicate' I' = (i, e)
complicate' J' = (e, i)
complicate' K' = (i, i)

scalarPart' :: (Representable f, Rep f ~ QuaternionBasis') => f r -> r
scalarPart' f = index f E'

vectorPart' :: (Representable f, Rep f ~ QuaternionBasis') => f r -> (r,r,r)
vectorPart' f = (index f I', index f J', index f K')

instance (TriviallyInvolutive r, Rng r) => Quadrance r (Quaternion' r) where
  quadrance n = scalarPart' (adjoint n * n)

instance (TriviallyInvolutive r, Ring r, Division r) => Division (Quaternion' r) where
  recip q@(Quaternion' a b c d) = Quaternion' (qq \\ a) (qq \\ b) (qq \\ c) (qq \\ d)
    where qq = quadrance q

 quaternion basis
 | the trivial diagonal algebra
 | dual quaternion comultiplication
 without trivial involution, multiplication fails associativity, and we'd need to
 support weaker multiplicative properties like Alternative and PowerAssociative
# LANGUAGE FlexibleInstances 
            , MultiParamTypeClasses 
            , TypeFamilies 
            , UndecidableInstances 
            , DeriveDataTypeable 
            , TypeOperators #
           , MultiParamTypeClasses
           , TypeFamilies
           , UndecidableInstances
           , DeriveDataTypeable
           , TypeOperators #-}
module Numeric.Coalgebra.Quaternion
  ( Distinguished(..)
  , Complicated(..)
  , Hamiltonian(..)
  , QuaternionBasis'(..)
  , Quaternion'(..)
  , complicate'
  , vectorPart'
  , scalarPart'
  ) where

import Control.Applicative
import Control.Monad.Reader.Class
import Data.Ix hiding (index)
import Data.Data
import Data.Distributive
import Data.Functor.Bind
import Data.Functor.Rep
import Data.Foldable
import Data.Traversable
import Data.Semigroup.Traversable
import Data.Semigroup.Foldable
import Data.Semigroup
import Numeric.Algebra
import Numeric.Algebra.Distinguished.Class
import Numeric.Algebra.Complex.Class
import Numeric.Algebra.Quaternion.Class
import Prelude hiding ((-),(+),(*),negate,subtract, fromInteger)

instance Distinguished QuaternionBasis' where
  e = E'

instance Complicated QuaternionBasis' where
  i = I'

instance Hamiltonian QuaternionBasis' where
  j = J'
  k = K'

instance Rig r => Distinguished (Quaternion' r) where
  e = Quaternion' one zero zero zero

instance Rig r => Complicated (Quaternion' r) where
  i = Quaternion' zero one zero zero

instance Rig r => Hamiltonian (Quaternion' r) where
  j = Quaternion' zero zero one zero
  k = Quaternion' one zero zero one

instance Rig r => Distinguished (QuaternionBasis' -> r) where
  e E' = one
  e _ = zero

instance Rig r => Complicated (QuaternionBasis' -> r) where
  i I' = one
  i _ = zero

instance Rig r => Hamiltonian (QuaternionBasis' -> r) where
  j J' = one
  j _ = zero

  k K' = one
  k _ = zero

data QuaternionBasis' = E' | I' | J' | K' deriving (Eq,Ord,Enum,Read,Show,Bounded,Ix,Data,Typeable)

data Quaternion' a = Quaternion' a a a a deriving (Eq,Show,Read,Data,Typeable)

instance Representable Quaternion' where
  type Rep Quaternion' = QuaternionBasis'
  tabulate f = Quaternion' (f E') (f I') (f J') (f K')
  index (Quaternion' a _ _ _) E' = a
  index (Quaternion' _ b _ _) I' = b
  index (Quaternion' _ _ c _) J' = c
  index (Quaternion' _ _ _ d) K' = d

instance Distributive Quaternion' where
  distribute = distributeRep

instance Functor Quaternion' where
  fmap = fmapRep

instance Apply Quaternion' where
  (<.>) = apRep

instance Applicative Quaternion' where
  pure = pureRep
  (<*>) = apRep

instance Bind Quaternion' where
  (>>-) = bindRep

instance Monad Quaternion' where
  return = pureRep
  (>>=) = bindRep

instance MonadReader QuaternionBasis' Quaternion' where
  ask = askRep
  local = localRep

instance Foldable Quaternion' where
  foldMap f (Quaternion' a b c d) =
    f a `mappend` f b `mappend` f c `mappend` f d

instance Traversable Quaternion' where
  traverse f (Quaternion' a b c d) =
    Quaternion' <$> f a <*> f b <*> f c <*> f d

instance Foldable1 Quaternion' where
  foldMap1 f (Quaternion' a b c d) =
    f a <> f b <> f c <> f d

instance Traversable1 Quaternion' where
  traverse1 f (Quaternion' a b c d) =
    Quaternion' <$> f a <.> f b <.> f c <.> f d

instance Additive r => Additive (Quaternion' r) where
  (+) = addRep
  sinnum1p = sinnum1pRep

instance LeftModule r s => LeftModule r (Quaternion' s) where
  r .* Quaternion' a b c d =
    Quaternion' (r .* a) (r .* b) (r .* c) (r .* d)

instance RightModule r s => RightModule r (Quaternion' s) where
  Quaternion' a b c d *. r =
    Quaternion' (a *. r) (b *. r) (c *. r) (d *. r)

instance Monoidal r => Monoidal (Quaternion' r) where
  zero = zeroRep
  sinnum = sinnumRep

instance Group r => Group (Quaternion' r) where
  (-) = minusRep
  negate = negateRep
  subtract = subtractRep
  times = timesRep

instance Abelian r => Abelian (Quaternion' r)

instance Idempotent r => Idempotent (Quaternion' r)

instance Partitionable r => Partitionable (Quaternion' r) where
  partitionWith f (Quaternion' a b c d) = id =<<
    partitionWith (\a1 a2 -> id =<<
    partitionWith (\b1 b2 -> id =<<
    partitionWith (\c1 c2 ->
    partitionWith (\d1 d2 -> f (Quaternion' a1 b1 c1 d1)
                               (Quaternion' a2 b2 c2 d2)
                  ) d) c) b) a

instance (TriviallyInvolutive r, Semiring r) => Algebra r QuaternionBasis' where
  mult f = f' where
    fe = f E' E'
    fi = f I' I'
    fj = f J' J'
    fk = f K' K'
    f' E' = fe
    f' I' = fi
    f' J' = fj
    f' K' = fk

instance (TriviallyInvolutive r, Semiring r) => UnitalAlgebra r QuaternionBasis' where
  unit = const


instance (TriviallyInvolutive r, Rng r) => Coalgebra r QuaternionBasis' where
  comult f = f' where
    fe = f E'
    fi = f I'
    fj = f J'
    fk = f K'
    fe' = negate fe
    fi' = negate fi
    fj' = negate fj
    fk' = negate fk
    f' E' E' = fe
    f' E' I' = fi
    f' E' J' = fj
    f' E' K' = fk
    f' I' E' = fi
    f' I' I' = fe'
    f' I' J' = fk
    f' I' K' = fj'
    f' J' E' = fj
    f' J' I' = fk'
    f' J' J' = fe'
    f' J' K' = fi
    f' K' E' = fk
    f' K' I' = fj
    f' K' J' = fi'
    f' K' K' = fe'

instance (TriviallyInvolutive r, Rng r) => CounitalCoalgebra r QuaternionBasis' where
  counit f = f E'

instance (TriviallyInvolutive r, Rng r)  => Bialgebra r QuaternionBasis'

instance (TriviallyInvolutive r, InvolutiveSemiring r, Rng r)  => InvolutiveAlgebra r QuaternionBasis' where
  inv f E' = f E'
  inv f b = negate (f b)

instance (TriviallyInvolutive r, InvolutiveSemiring r, Rng r) => InvolutiveCoalgebra r QuaternionBasis' where
  coinv = inv

instance (TriviallyInvolutive r, InvolutiveSemiring r, Rng r) => HopfAlgebra r QuaternionBasis' where
  antipode = inv

instance (TriviallyInvolutive r, Semiring r) => Multiplicative (Quaternion' r) where
  (*) = mulRep

instance (TriviallyInvolutive r, Semiring r) => Semiring (Quaternion' r)

instance (TriviallyInvolutive r, Ring r) => Unital (Quaternion' r) where
  one = oneRep

instance (TriviallyInvolutive r, Ring r) => Rig (Quaternion' r) where
  fromNatural n = Quaternion' (fromNatural n) zero zero zero

instance (TriviallyInvolutive r, Ring r) => Ring (Quaternion' r) where
  fromInteger n = Quaternion' (fromInteger n) zero zero zero

instance ( TriviallyInvolutive r, Rng r) => LeftModule (Quaternion' r) (Quaternion' r) where
  (.*) = (*)
instance (TriviallyInvolutive r, Rng r) => RightModule (Quaternion' r) (Quaternion' r) where
  (*.) = (*)

instance (TriviallyInvolutive r, Rng r) => InvolutiveMultiplication (Quaternion' r) where
  adjoint (Quaternion' a b c d) = Quaternion' a (negate b) (negate c) (negate d)

  | Cayley - Dickson quaternion isomorphism ( one way )
complicate' :: Complicated c => QuaternionBasis' -> (c , c)
complicate' E' = (e, e)
complicate' I' = (i, e)
complicate' J' = (e, i)
complicate' K' = (i, i)

scalarPart' :: (Representable f, Rep f ~ QuaternionBasis') => f r -> r
scalarPart' f = index f E'

vectorPart' :: (Representable f, Rep f ~ QuaternionBasis') => f r -> (r,r,r)
vectorPart' f = (index f I', index f J', index f K')

instance (TriviallyInvolutive r, Rng r) => Quadrance r (Quaternion' r) where
  quadrance n = scalarPart' (adjoint n * n)

instance (TriviallyInvolutive r, Ring r, Division r) => Division (Quaternion' r) where
  recip q@(Quaternion' a b c d) = Quaternion' (qq \\ a) (qq \\ b) (qq \\ c) (qq \\ d)
    where qq = quadrance q
