{-# OPTIONS_GHC -Wall #-}
{-# OPTIONS_GHC -Wno-unused-imports #-}

-- | Multiplicative classes
module NumHask.Algebra.Multiplicative
  ( Multiplicative (..),
    product,
    accproduct,
    Divisive (..),
  )
where

import Data.Int (Int16, Int32, Int64, Int8)
import Data.Traversable (mapAccumL)
import Data.Word (Word, Word16, Word32, Word64, Word8)
import GHC.Natural (Natural (..))
import Prelude (Double, Float, Int, Integer, fromInteger, fromRational)
import qualified Prelude as P

-- $setup
--
-- >>> :set -XRebindableSyntax
-- >>> import NumHask.Prelude

-- | or [Multiplication](https://en.wikipedia.org/wiki/Multiplication)
--
-- For practical reasons, we begin the class tree with 'NumHask.Algebra.Additive.Additive' and 'Multiplicative'.  Starting with  'NumHask.Algebra.Group.Associative' and 'NumHask.Algebra.Group.Unital', or using 'Data.Semigroup.Semigroup' and 'Data.Monoid.Monoid' from base tends to confuse the interface once you start having to disinguish between (say) monoidal addition and monoidal multiplication.
--
--
-- prop> \a -> one * a == a
-- prop> \a -> a * one == a
-- prop> \a b c -> (a * b) * c == a * (b * c)
--
-- By convention, (*) is regarded as not necessarily commutative, but this is not universal, and the introduction of another symbol which means commutative multiplication seems a bit dogmatic.
--
-- >>> one * 2
-- 2
--
-- >>> 2 * 3
-- 6
class Multiplicative a where
  infixl 7 *
  (*) :: a -> a -> a

  one :: a

-- | Compute the product of a 'Data.Foldable.Foldable'.
--
-- >>> product [1..5]
-- 120
product :: (Multiplicative a, P.Foldable f) => f a -> a
product :: forall a (f :: * -> *). (Multiplicative a, Foldable f) => f a -> a
product = forall (t :: * -> *) a b.
Foldable t =>
(a -> b -> b) -> b -> t a -> b
P.foldr forall a. Multiplicative a => a -> a -> a
(*) forall a. Multiplicative a => a
one

-- | Compute the accumulating product of a 'Data.Traversable.Traversable'.
--
-- >>> accproduct [1..5]
-- [1,2,6,24,120]
accproduct :: (Multiplicative a, P.Traversable f) => f a -> f a
accproduct :: forall a (f :: * -> *).
(Multiplicative a, Traversable f) =>
f a -> f a
accproduct = forall a b. (a, b) -> b
P.snd forall b c a. (b -> c) -> (a -> b) -> a -> c
P.. forall (t :: * -> *) s a b.
Traversable t =>
(s -> a -> (s, b)) -> s -> t a -> (s, t b)
mapAccumL (\a
a a
b -> (a
a forall a. Multiplicative a => a -> a -> a
* a
b, a
a forall a. Multiplicative a => a -> a -> a
* a
b)) forall a. Multiplicative a => a
one

-- | or [Division](https://en.wikipedia.org/wiki/Division_(mathematics\))
--
-- Though unusual, the term Divisive usefully fits in with the grammer of other classes and avoids name clashes that occur with some popular libraries.
--
-- prop> \(a :: Double) -> a / a ~= one || a == zero
-- prop> \(a :: Double) -> recip a ~= one / a || a == zero
-- prop> \(a :: Double) -> recip a * a ~= one || a == zero
-- prop> \(a :: Double) -> a * recip a ~= one || a == zero
--
-- >>> recip 2.0
-- 0.5
--
-- >>> 1 / 2
-- 0.5
class (Multiplicative a) => Divisive a where
  recip :: a -> a
  recip a
a = forall a. Multiplicative a => a
one forall a. Divisive a => a -> a -> a
/ a
a

  infixl 7 /

  (/) :: a -> a -> a
  (/) a
a a
b = a
a forall a. Multiplicative a => a -> a -> a
* forall a. Divisive a => a -> a
recip a
b

instance Multiplicative Double where
  * :: Double -> Double -> Double
(*) = forall a. Num a => a -> a -> a
(P.*)
  one :: Double
one = Double
1.0

instance Divisive Double where
  recip :: Double -> Double
recip = forall a. Fractional a => a -> a
P.recip

instance Multiplicative Float where
  * :: Float -> Float -> Float
(*) = forall a. Num a => a -> a -> a
(P.*)
  one :: Float
one = Float
1.0

instance Divisive Float where
  recip :: Float -> Float
recip = forall a. Fractional a => a -> a
P.recip

instance Multiplicative Int where
  * :: Int -> Int -> Int
(*) = forall a. Num a => a -> a -> a
(P.*)
  one :: Int
one = Int
1

instance Multiplicative Integer where
  * :: Integer -> Integer -> Integer
(*) = forall a. Num a => a -> a -> a
(P.*)
  one :: Integer
one = Integer
1

instance Multiplicative P.Bool where
  * :: Bool -> Bool -> Bool
(*) = Bool -> Bool -> Bool
(P.&&)
  one :: Bool
one = Bool
P.True

instance Multiplicative Natural where
  * :: Natural -> Natural -> Natural
(*) = forall a. Num a => a -> a -> a
(P.*)
  one :: Natural
one = Natural
1

instance Multiplicative Int8 where
  * :: Int8 -> Int8 -> Int8
(*) = forall a. Num a => a -> a -> a
(P.*)
  one :: Int8
one = Int8
1

instance Multiplicative Int16 where
  * :: Int16 -> Int16 -> Int16
(*) = forall a. Num a => a -> a -> a
(P.*)
  one :: Int16
one = Int16
1

instance Multiplicative Int32 where
  * :: Int32 -> Int32 -> Int32
(*) = forall a. Num a => a -> a -> a
(P.*)
  one :: Int32
one = Int32
1

instance Multiplicative Int64 where
  * :: Int64 -> Int64 -> Int64
(*) = forall a. Num a => a -> a -> a
(P.*)
  one :: Int64
one = Int64
1

instance Multiplicative Word where
  * :: Word -> Word -> Word
(*) = forall a. Num a => a -> a -> a
(P.*)
  one :: Word
one = Word
1

instance Multiplicative Word8 where
  * :: Word8 -> Word8 -> Word8
(*) = forall a. Num a => a -> a -> a
(P.*)
  one :: Word8
one = Word8
1

instance Multiplicative Word16 where
  * :: Word16 -> Word16 -> Word16
(*) = forall a. Num a => a -> a -> a
(P.*)
  one :: Word16
one = Word16
1

instance Multiplicative Word32 where
  * :: Word32 -> Word32 -> Word32
(*) = forall a. Num a => a -> a -> a
(P.*)
  one :: Word32
one = Word32
1

instance Multiplicative Word64 where
  * :: Word64 -> Word64 -> Word64
(*) = forall a. Num a => a -> a -> a
(P.*)
  one :: Word64
one = Word64
1

instance (Multiplicative b) => Multiplicative (a -> b) where
  a -> b
f * :: (a -> b) -> (a -> b) -> a -> b
* a -> b
f' = \a
a -> a -> b
f a
a forall a. Multiplicative a => a -> a -> a
* a -> b
f' a
a
  one :: a -> b
one a
_ = forall a. Multiplicative a => a
one

instance (Divisive b) => Divisive (a -> b) where
  recip :: (a -> b) -> a -> b
recip a -> b
f = forall a. Divisive a => a -> a
recip forall b c a. (b -> c) -> (a -> b) -> a -> c
P.. a -> b
f