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.Traversable
import Numeric.Algebra
import Prelude hiding ((),(+),(*),negate,subtract, fromInteger)
data HyperBasis = S | C deriving (Eq,Ord,Show,Read,Enum,Ix,Bounded,Data,Typeable)
data Hyper a = Hyper a a deriving (Eq,Show,Read,Data,Typeable)
class Hyperbolic r where
s :: r
c :: r
instance Hyperbolic HyperBasis where
s = S
c = C
instance Rig r => Hyperbolic (Hyper r) where
s = Hyper one zero
c = Hyper zero one
instance Rig r => Hyperbolic (HyperBasis -> r) where
s S = one
s C = zero
c S = zero
c C = one
instance Hyperbolic a => Hyperbolic (Covector r a) where
s = return s
c = return c
type instance Key Hyper = HyperBasis
instance Representable Hyper where
tabulate f = Hyper (f S) (f C)
instance Indexable Hyper where
index (Hyper a _ ) S = a
index (Hyper _ b ) C = b
instance Lookup Hyper where
lookup = lookupDefault
instance Adjustable Hyper where
adjust f S (Hyper a b) = Hyper (f a) b
adjust f C (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 S a1 a2) (f C 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 S a `mappend` f C 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 S a <*> f C b
instance Foldable1 Hyper where
foldMap1 f (Hyper a b) = f a <> f b
instance FoldableWithKey1 Hyper where
foldMapWithKey1 f (Hyper a b) = f S a <> f C 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 S a <.> f C b
instance HasTrie HyperBasis where
type BaseTrie HyperBasis = Hyper
embedKey = id
projectKey = id
instance Additive r => Additive (Hyper r) where
(+) = addRep
replicate1p = replicate1pRep
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
replicate = replicateRep
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 (Commutative k, Semiring k) => Algebra k HyperBasis where
mult f = f' where
fs = f S C + f C S
fc = f C C + f S S
f' S = fs
f' C = fc
instance (Commutative k, Monoidal k, Semiring k) => UnitalAlgebra k HyperBasis where
unit _ S = zero
unit x C = x
instance (Commutative k, Semiring k) => Coalgebra k HyperBasis where
comult f = f' where
fs = f S
fc = f C
f' S S = fc
f' S C = fs
f' C S = fs
f' C C = fc
instance (Commutative k, Semiring k) => CounitalCoalgebra k HyperBasis where
counit f = f C
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 zero one
instance (Commutative r, Rig r) => Rig (Hyper r) where
fromNatural n = Hyper zero (fromNatural n)
instance (Commutative r, Ring r) => Ring (Hyper r) where
fromInteger n = Hyper zero (fromInteger n)
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, Semiring r, InvolutiveMultiplication r) => InvolutiveMultiplication (Hyper r) where
adjoint (Hyper a b) = Hyper (adjoint a) (adjoint b)
instance (Commutative r, InvolutiveSemiring r) => InvolutiveSemiring (Hyper r)