{-# LANGUAGE MultiParamTypeClasses, FlexibleInstances, TypeFamilies, UndecidableInstances, DeriveDataTypeable #-}
module Numeric.Algebra.Hyperbolic
  ( Hyperbolic(..)
  , HyperBasis'(..)
  , Hyper'(..)
  ) where

import Control.Applicative
import Control.Monad.Reader.Class
import Data.Data
import Data.Distributive
import Data.Functor.Bind
import Data.Functor.Representable
import Data.Functor.Representable.Trie
import Data.Foldable
import Data.Ix
import Data.Key
import Data.Monoid
import Data.Semigroup.Traversable
import Data.Semigroup.Foldable
import Data.Semigroup
import Data.Traversable
import Numeric.Algebra
import Numeric.Coalgebra.Hyperbolic.Class
import Prelude hiding ((-),(+),(*),negate,subtract, fromInteger)

-- the dual hyperbolic basis
data HyperBasis' = Cosh' | Sinh' deriving (Eq,Ord,Show,Read,Enum,Ix,Bounded,Data,Typeable)
data Hyper' a = Hyper' a a deriving (Eq,Show,Read,Data,Typeable)

instance Hyperbolic HyperBasis' where
  cosh = Cosh'
  sinh = Sinh'

instance Rig r => Hyperbolic (Hyper' r) where
  cosh = Hyper' one zero
  sinh = Hyper' zero one
  
instance Rig r => Hyperbolic (HyperBasis' -> r) where
  cosh Sinh' = zero
  cosh Cosh' = one
  sinh Sinh' = one
  sinh Cosh' = zero

type instance Key Hyper' = HyperBasis'

instance Representable Hyper' where
  tabulate f = Hyper' (f Cosh') (f Sinh')

instance Indexable Hyper' where
  index (Hyper' a _ ) Cosh' = a
  index (Hyper' _ b ) Sinh' = b

instance Lookup Hyper' where
  lookup = lookupDefault

instance Adjustable Hyper' where
  adjust f Cosh' (Hyper' a b) = Hyper' (f a) b
  adjust f Sinh' (Hyper' a b) = Hyper' a (f b)

instance Distributive Hyper' where
  distribute = distributeRep 

instance Functor Hyper' where
  fmap f (Hyper' a b) = Hyper' (f a) (f b)

instance Zip Hyper' where
  zipWith f (Hyper' a1 b1) (Hyper' a2 b2) = Hyper' (f a1 a2) (f b1 b2)

instance ZipWithKey Hyper' where
  zipWithKey f (Hyper' a1 b1) (Hyper' a2 b2) = Hyper' (f Cosh' a1 a2) (f Sinh' b1 b2)

instance Keyed Hyper' where
  mapWithKey = mapWithKeyRep

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

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

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

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

instance MonadReader HyperBasis' Hyper' where
  ask = askRep
  local = localRep

instance Foldable Hyper' where
  foldMap f (Hyper' a b) = f a `mappend` f b

instance FoldableWithKey Hyper' where
  foldMapWithKey f (Hyper' a b) = f Cosh' a `mappend` f Sinh' b

instance Traversable Hyper' where
  traverse f (Hyper' a b) = Hyper' <$> f a <*> f b

instance TraversableWithKey Hyper' where
  traverseWithKey f (Hyper' a b) = Hyper' <$> f Cosh' a <*> f Sinh' b

instance Foldable1 Hyper' where
  foldMap1 f (Hyper' a b) = f a <> f b

instance FoldableWithKey1 Hyper' where
  foldMapWithKey1 f (Hyper' a b) = f Cosh' a <> f Sinh' b

instance Traversable1 Hyper' where
  traverse1 f (Hyper' a b) = Hyper' <$> f a <.> f b

instance TraversableWithKey1 Hyper' where
  traverseWithKey1 f (Hyper' a b) = Hyper' <$> f Cosh' a <.> f Sinh' b

instance HasTrie HyperBasis' where
  type BaseTrie HyperBasis' = Hyper'
  embedKey = id
  projectKey = id

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

instance LeftModule r s => LeftModule r (Hyper' s) where
  r .* Hyper' a b = Hyper' (r .* a) (r .* b)

instance RightModule r s => RightModule r (Hyper' s) where
  Hyper' a b *. r = Hyper' (a *. r) (b *. r)

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

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

instance Abelian r => Abelian (Hyper' r)

instance Idempotent r => Idempotent (Hyper' r)

instance Partitionable r => Partitionable (Hyper' r) where
  partitionWith f (Hyper' a b) = id =<<
    partitionWith (\a1 a2 -> 
    partitionWith (\b1 b2 -> f (Hyper' a1 b1) (Hyper' a2 b2)) b) a

-- the dual hyperbolic trigonometric algebra
instance (Commutative k, Semiring k) => Algebra k HyperBasis' where
  mult f = f' where
    fs = f Sinh' Cosh' + f Cosh' Sinh'
    fc = f Cosh' Cosh' + f Sinh' Sinh'
    f' Sinh' = fs
    f' Cosh' = fc

instance (Commutative k, Monoidal k, Semiring k) => UnitalAlgebra k HyperBasis' where
  unit _ Sinh' = zero
  unit x Cosh' = x

-- the diagonal coalgebra
instance (Commutative k, Monoidal k, Semiring k) => Coalgebra k HyperBasis' where
  comult f = f' where
     fs = f Sinh'
     fc = f Cosh'
     f' Sinh' Sinh' = fs
     f' Sinh' Cosh' = zero
     f' Cosh' Sinh' = zero
     f' Cosh' Cosh' = fc

instance (Commutative k, Monoidal k, Semiring k) => CounitalCoalgebra k HyperBasis' where
  counit f = f Cosh' + f Sinh'

instance (Commutative k, Monoidal k, Semiring k) => Bialgebra k HyperBasis'

instance (Commutative k, Group k, InvolutiveSemiring k) => InvolutiveAlgebra k HyperBasis' where
  inv f = f' where
    afc = adjoint (f Cosh')
    nfs = negate (f Sinh')
    f' Cosh' = afc
    f' Sinh' = nfs

instance (Commutative k, Group k, InvolutiveSemiring k) => InvolutiveCoalgebra k HyperBasis' where
  coinv = inv

instance (Commutative k, Group k, InvolutiveSemiring k) => HopfAlgebra k HyperBasis' where
  antipode = inv

instance (Commutative k, Semiring k) => Multiplicative (Hyper' k) where
  (*) = mulRep

instance (Commutative k, Semiring k) => Commutative (Hyper' k)

instance (Commutative k, Semiring k) => Semiring (Hyper' k)

instance (Commutative k, Rig k) => Unital (Hyper' k) where
  one = Hyper' one zero

instance (Commutative r, Rig r) => Rig (Hyper' r) where
  fromNatural n = Hyper' (fromNatural n) zero

instance (Commutative r, Ring r) => Ring (Hyper' r) where
  fromInteger n = Hyper' (fromInteger n) zero

instance (Commutative r, Semiring r) => LeftModule (Hyper' r) (Hyper' r) where (.*) = (*)
instance (Commutative r, Semiring r) => RightModule (Hyper' r) (Hyper' r) where (*.) = (*)

instance (Commutative r, InvolutiveSemiring r, Rng r) => InvolutiveMultiplication (Hyper' r) where
  adjoint (Hyper' a b) = Hyper' (adjoint a) (negate b)

instance (Commutative r, InvolutiveSemiring r, Rng r) => InvolutiveSemiring (Hyper' r)

instance (Commutative r, InvolutiveSemiring r, Rng r) => Quadrance r (Hyper' r) where
  quadrance n = case adjoint n * n of
    Hyper' a _ -> a

instance (Commutative r, InvolutiveSemiring r, DivisionRing r) => Division (Hyper' r) where
  recip q@(Hyper' a b) = Hyper' (qq \\ a) (qq \\ b)
    where qq = quadrance q