module Numeric.Coalgebra.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.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, cosh, sinh)
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
instance Semiring k => Algebra k HyperBasis where
mult f = f' where
fs = f Sinh Sinh
fc = f Cosh Cosh
f' Sinh = fs
f' Cosh = fc
instance Semiring k => UnitalAlgebra k HyperBasis where
unit = const
instance (Commutative k, Semiring k) => Coalgebra k HyperBasis where
comult f = f' where
fs = f Sinh
fc = f Cosh
f' Sinh Sinh = fc
f' Sinh Cosh = fs
f' Cosh Sinh = fs
f' Cosh Cosh = fc
instance (Commutative k, Semiring k) => CounitalCoalgebra k HyperBasis where
counit f = f Cosh
instance (Commutative 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, Group r, InvolutiveSemiring r) => InvolutiveMultiplication (Hyper r) where
adjoint (Hyper a b) = Hyper (adjoint a) (negate b)
instance (Commutative r, Group r, InvolutiveSemiring r) => InvolutiveSemiring (Hyper r)