Multiplicative.hs

{-# OPTIONS_GHC -Wall #-}

-- | A magma heirarchy for multiplication. The basic magma structure is repeated and prefixed with 'Multiplicative-'.
module NumHask.Algebra.Multiplicative
  ( MultiplicativeMagma(..)
  , MultiplicativeUnital(..)
  , MultiplicativeAssociative
  , MultiplicativeCommutative
  , MultiplicativeInvertible(..)
  , product
  , Multiplicative(..)
  , MultiplicativeRightCancellative(..)
  , MultiplicativeLeftCancellative(..)
  , MultiplicativeGroup(..)
  , MultiplicativeIdempotent
  ) where

import Data.Complex (Complex(..))
import Data.Int (Int8, Int16, Int32, Int64)
import Data.Word (Word, Word8, Word16, Word32, Word64)
import GHC.Natural (Natural(..))
import NumHask.Algebra.Additive
import qualified Prelude as P
import Prelude (Bool(..), Double, Float, Int, Integer)

-- | 'times' is used as the operator for the multiplicative magam to distinguish from '*' which, by convention, implies commutativity
--
-- > ∀ a,b ∈ A: a `times` b ∈ A
--
-- law is true by construction in Haskell
class MultiplicativeMagma a where
  times :: a -> a -> a

instance MultiplicativeMagma Double where
  times = (P.*)

instance MultiplicativeMagma Float where
  times = (P.*)

instance MultiplicativeMagma Int where
  times = (P.*)

instance MultiplicativeMagma Integer where
  times = (P.*)

instance MultiplicativeMagma Bool where
  times = (P.&&)

instance (MultiplicativeMagma a, AdditiveGroup a) =>
         MultiplicativeMagma (Complex a) where
  (rx :+ ix) `times` (ry :+ iy) =
    (rx `times` ry - ix `times` iy) :+ (ix `times` ry + iy `times` rx)

instance MultiplicativeMagma Natural where
  times = (P.*)

instance MultiplicativeMagma Int8 where
  times = (P.*)

instance MultiplicativeMagma Int16 where
  times = (P.*)

instance MultiplicativeMagma Int32 where
  times = (P.*)

instance MultiplicativeMagma Int64 where
  times = (P.*)

instance MultiplicativeMagma Word where
  times = (P.*)

instance MultiplicativeMagma Word8 where
  times = (P.*)

instance MultiplicativeMagma Word16 where
  times = (P.*)

instance MultiplicativeMagma Word32 where
  times = (P.*)

instance MultiplicativeMagma Word64 where
  times = (P.*)

-- | Unital magma for multiplication.
--
-- > one `times` a == a
-- > a `times` one == a
class MultiplicativeMagma a =>
      MultiplicativeUnital a where
  one :: a

instance MultiplicativeUnital Double where
  one = 1

instance MultiplicativeUnital Float where
  one = 1

instance MultiplicativeUnital Int where
  one = 1

instance MultiplicativeUnital Integer where
  one = 1

instance MultiplicativeUnital Bool where
  one = True

instance (AdditiveUnital a, AdditiveGroup a, MultiplicativeUnital a) =>
         MultiplicativeUnital (Complex a) where
  one = one :+ zero

instance MultiplicativeUnital Natural where
  one = 1

instance MultiplicativeUnital Int8 where
  one = 1

instance MultiplicativeUnital Int16 where
  one = 1

instance MultiplicativeUnital Int32 where
  one = 1

instance MultiplicativeUnital Int64 where
  one = 1

instance MultiplicativeUnital Word where
  one = 1

instance MultiplicativeUnital Word8 where
  one = 1

instance MultiplicativeUnital Word16 where
  one = 1

instance MultiplicativeUnital Word32 where
  one = 1

instance MultiplicativeUnital Word64 where
  one = 1

-- | Associative magma for multiplication.
--
-- > (a `times` b) `times` c == a `times` (b `times` c)
class MultiplicativeMagma a =>
      MultiplicativeAssociative a

instance MultiplicativeAssociative Double

instance MultiplicativeAssociative Float

instance MultiplicativeAssociative Int

instance MultiplicativeAssociative Integer

instance MultiplicativeAssociative Bool

instance (AdditiveGroup a, MultiplicativeAssociative a) =>
         MultiplicativeAssociative (Complex a)

instance MultiplicativeAssociative Natural

instance MultiplicativeAssociative Int8

instance MultiplicativeAssociative Int16

instance MultiplicativeAssociative Int32

instance MultiplicativeAssociative Int64

instance MultiplicativeAssociative Word

instance MultiplicativeAssociative Word8

instance MultiplicativeAssociative Word16

instance MultiplicativeAssociative Word32

instance MultiplicativeAssociative Word64

-- | Commutative magma for multiplication.
--
-- > a `times` b == b `times` a
class MultiplicativeMagma a =>
      MultiplicativeCommutative a

instance MultiplicativeCommutative Double

instance MultiplicativeCommutative Float

instance MultiplicativeCommutative Int

instance MultiplicativeCommutative Integer

instance MultiplicativeCommutative Bool

instance (AdditiveGroup a, MultiplicativeCommutative a) =>
         MultiplicativeCommutative (Complex a)

instance MultiplicativeCommutative Natural

instance MultiplicativeCommutative Int8

instance MultiplicativeCommutative Int16

instance MultiplicativeCommutative Int32

instance MultiplicativeCommutative Int64

instance MultiplicativeCommutative Word

instance MultiplicativeCommutative Word8

instance MultiplicativeCommutative Word16

instance MultiplicativeCommutative Word32

instance MultiplicativeCommutative Word64

-- | Invertible magma for multiplication.
--
-- > ∀ a ∈ A: recip a ∈ A
--
-- law is true by construction in Haskell
class MultiplicativeMagma a =>
      MultiplicativeInvertible a where
  recip :: a -> a

instance MultiplicativeInvertible Double where
  recip = P.recip

instance MultiplicativeInvertible Float where
  recip = P.recip

instance (AdditiveGroup a, MultiplicativeInvertible a) =>
         MultiplicativeInvertible (Complex a) where
  recip (rx :+ ix) = (rx `times` d) :+ (negate ix `times` d)
    where
      d = recip ((rx `times` rx) `plus` (ix `times` ix))

-- | Idempotent magma for multiplication.
--
-- > a `times` a == a
class MultiplicativeMagma a =>
      MultiplicativeIdempotent a

instance MultiplicativeIdempotent Bool

-- | product definition avoiding a clash with the Product monoid in base
-- fixme: fit in with Product in base
--
product :: (Multiplicative a, P.Foldable f) => f a -> a
product = P.foldr (*) one

-- | Multiplicative is commutative, associative and unital under multiplication
--
-- > one * a == a
-- > a * one == a
-- > (a * b) * c == a * (b * c)
-- > a * b == b * a
class ( MultiplicativeCommutative a
      , MultiplicativeUnital a
      , MultiplicativeAssociative a
      ) =>
      Multiplicative a where
  infixl 7 *
  (*) :: a -> a -> a
  a * b = times a b

instance Multiplicative Double

instance Multiplicative Float

instance Multiplicative Int

instance Multiplicative Integer

instance Multiplicative Bool

instance (AdditiveGroup a, Multiplicative a) => Multiplicative (Complex a)

instance Multiplicative Natural

instance Multiplicative Int8

instance Multiplicative Int16

instance Multiplicative Int32

instance Multiplicative Int64

instance Multiplicative Word

instance Multiplicative Word8

instance Multiplicative Word16

instance Multiplicative Word32

instance Multiplicative Word64

-- | Non-commutative left divide
--
-- > recip a `times` a = one
class ( MultiplicativeUnital a
      , MultiplicativeAssociative a
      , MultiplicativeInvertible a
      ) =>
      MultiplicativeLeftCancellative a where
  infixl 7 ~/
  (~/) :: a -> a -> a
  a ~/ b = recip b `times` a

-- | Non-commutative right divide
--
-- > a `times` recip a = one
class ( MultiplicativeUnital a
      , MultiplicativeAssociative a
      , MultiplicativeInvertible a
      ) =>
      MultiplicativeRightCancellative a where
  infixl 7 /~
  (/~) :: a -> a -> a
  a /~ b = a `times` recip b

-- | Divide ('/') is reserved for where both the left and right cancellative laws hold.  This then implies that the MultiplicativeGroup is also Abelian.
--
-- > a / a = one
-- > recip a = one / a
-- > recip a * a = one
-- > a * recip a = one
class (Multiplicative a, MultiplicativeInvertible a) =>
      MultiplicativeGroup a where
  infixl 7 /
  (/) :: a -> a -> a
  (/) a b = a `times` recip b

instance MultiplicativeGroup Double

instance MultiplicativeGroup Float

instance (AdditiveGroup a, MultiplicativeGroup a) =>
         MultiplicativeGroup (Complex a)