{-# LANGUAGE FlexibleInstances
           , MultiParamTypeClasses
           , TypeFamilies
           , UndecidableInstances
           , DeriveDataTypeable
           , TypeOperators #-}
module Numeric.Algebra.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
import Data.Semigroup.Traversable
import Data.Semigroup.Foldable
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 quaternion algebra
instance (TriviallyInvolutive r, Rng r) => Algebra r QuaternionBasis where
  mult f = f' where
    fe = f E E - (f I I + f J J + f K K)
    fi = f E I + f I E + f J K - f K J
    fj = f E J + f J E + f K I - f I K
    fk = f E K + f K E + f I J - f J I
    f' E = fe
    f' I = fi
    f' J = fj
    f' K = fk
             
instance (TriviallyInvolutive r, Rng r) => UnitalAlgebra r QuaternionBasis where
  unit x E = x 
  unit _ _ = zero

-- | the trivial diagonal coalgebra
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
    f' E E = fe
    f' I I = fi
    f' J J = fj
    f' K K = fk
    f' _ _ = zero

instance (TriviallyInvolutive r, Rng r) => CounitalCoalgebra r QuaternionBasis where
  counit f = f E + f I + f J + f K

{-
-- 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, Rng r) => Multiplicative (Quaternion r) where
  (*) = mulRep

instance (TriviallyInvolutive r, Rng 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