{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE TypeSynonymInstances #-}
{-# LANGUAGE DefaultSignatures #-}
module Basement.Numerical.Multiplicative
( Multiplicative(..)
, IDivisible(..)
, Divisible(..)
, recip
) where
import Basement.Compat.Base
import Basement.Compat.C.Types
import Basement.Compat.Natural
import Basement.Compat.NumLiteral
import Basement.Numerical.Number
import Basement.Numerical.Additive
import Basement.Types.Word128 (Word128)
import Basement.Types.Word256 (Word256)
import qualified Basement.Types.Word128 as Word128
import qualified Basement.Types.Word256 as Word256
import qualified Prelude
class Multiplicative a where
{-# MINIMAL midentity, (*) #-}
midentity :: a
(*) :: a -> a -> a
(^) :: (IsNatural n, Enum n, IDivisible n) => a -> n -> a
(^) = a -> n -> a
forall n a.
(Enum n, IsNatural n, IDivisible n, Multiplicative a) =>
a -> n -> a
power
class (Additive a, Multiplicative a) => IDivisible a where
{-# MINIMAL (div, mod) | divMod #-}
div :: a -> a -> a
div a
a a
b = (a, a) -> a
forall a b. (a, b) -> a
fst ((a, a) -> a) -> (a, a) -> a
forall a b. (a -> b) -> a -> b
$ a -> a -> (a, a)
forall a. IDivisible a => a -> a -> (a, a)
divMod a
a a
b
mod :: a -> a -> a
mod a
a a
b = (a, a) -> a
forall a b. (a, b) -> b
snd ((a, a) -> a) -> (a, a) -> a
forall a b. (a -> b) -> a -> b
$ a -> a -> (a, a)
forall a. IDivisible a => a -> a -> (a, a)
divMod a
a a
b
divMod :: a -> a -> (a, a)
divMod a
a a
b = (a -> a -> a
forall a. IDivisible a => a -> a -> a
div a
a a
b, a -> a -> a
forall a. IDivisible a => a -> a -> a
mod a
a a
b)
class Multiplicative a => Divisible a where
{-# MINIMAL (/) #-}
(/) :: a -> a -> a
infixl 7 *, /
infixr 8 ^
instance Multiplicative Integer where
midentity :: Integer
midentity = Integer
1
* :: Integer -> Integer -> Integer
(*) = Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
(Prelude.*)
instance Multiplicative Int where
midentity :: Int
midentity = Int
1
* :: Int -> Int -> Int
(*) = Int -> Int -> Int
forall a. Num a => a -> a -> a
(Prelude.*)
instance Multiplicative Int8 where
midentity :: Int8
midentity = Int8
1
* :: Int8 -> Int8 -> Int8
(*) = Int8 -> Int8 -> Int8
forall a. Num a => a -> a -> a
(Prelude.*)
instance Multiplicative Int16 where
midentity :: Int16
midentity = Int16
1
* :: Int16 -> Int16 -> Int16
(*) = Int16 -> Int16 -> Int16
forall a. Num a => a -> a -> a
(Prelude.*)
instance Multiplicative Int32 where
midentity :: Int32
midentity = Int32
1
* :: Int32 -> Int32 -> Int32
(*) = Int32 -> Int32 -> Int32
forall a. Num a => a -> a -> a
(Prelude.*)
instance Multiplicative Int64 where
midentity :: Int64
midentity = Int64
1
* :: Int64 -> Int64 -> Int64
(*) = Int64 -> Int64 -> Int64
forall a. Num a => a -> a -> a
(Prelude.*)
instance Multiplicative Natural where
midentity :: Natural
midentity = Natural
1
* :: Natural -> Natural -> Natural
(*) = Natural -> Natural -> Natural
forall a. Num a => a -> a -> a
(Prelude.*)
instance Multiplicative Word where
midentity :: Word
midentity = Word
1
* :: Word -> Word -> Word
(*) = Word -> Word -> Word
forall a. Num a => a -> a -> a
(Prelude.*)
instance Multiplicative Word8 where
midentity :: Word8
midentity = Word8
1
* :: Word8 -> Word8 -> Word8
(*) = Word8 -> Word8 -> Word8
forall a. Num a => a -> a -> a
(Prelude.*)
instance Multiplicative Word16 where
midentity :: Word16
midentity = Word16
1
* :: Word16 -> Word16 -> Word16
(*) = Word16 -> Word16 -> Word16
forall a. Num a => a -> a -> a
(Prelude.*)
instance Multiplicative Word32 where
midentity :: Word32
midentity = Word32
1
* :: Word32 -> Word32 -> Word32
(*) = Word32 -> Word32 -> Word32
forall a. Num a => a -> a -> a
(Prelude.*)
instance Multiplicative Word64 where
midentity :: Word64
midentity = Word64
1
* :: Word64 -> Word64 -> Word64
(*) = Word64 -> Word64 -> Word64
forall a. Num a => a -> a -> a
(Prelude.*)
instance Multiplicative Word128 where
midentity :: Word128
midentity = Word128
1
* :: Word128 -> Word128 -> Word128
(*) = Word128 -> Word128 -> Word128
(Word128.*)
instance Multiplicative Word256 where
midentity :: Word256
midentity = Word256
1
* :: Word256 -> Word256 -> Word256
(*) = Word256 -> Word256 -> Word256
(Word256.*)
instance Multiplicative Prelude.Float where
midentity :: Float
midentity = Float
1.0
* :: Float -> Float -> Float
(*) = Float -> Float -> Float
forall a. Num a => a -> a -> a
(Prelude.*)
instance Multiplicative Prelude.Double where
midentity :: Double
midentity = Double
1.0
* :: Double -> Double -> Double
(*) = Double -> Double -> Double
forall a. Num a => a -> a -> a
(Prelude.*)
instance Multiplicative Prelude.Rational where
midentity :: Rational
midentity = Rational
1.0
* :: Rational -> Rational -> Rational
(*) = Rational -> Rational -> Rational
forall a. Num a => a -> a -> a
(Prelude.*)
instance Multiplicative CChar where
midentity :: CChar
midentity = CChar
1
* :: CChar -> CChar -> CChar
(*) = CChar -> CChar -> CChar
forall a. Num a => a -> a -> a
(Prelude.*)
instance Multiplicative CSChar where
midentity :: CSChar
midentity = CSChar
1
* :: CSChar -> CSChar -> CSChar
(*) = CSChar -> CSChar -> CSChar
forall a. Num a => a -> a -> a
(Prelude.*)
instance Multiplicative CUChar where
midentity :: CUChar
midentity = CUChar
1
* :: CUChar -> CUChar -> CUChar
(*) = CUChar -> CUChar -> CUChar
forall a. Num a => a -> a -> a
(Prelude.*)
instance Multiplicative CShort where
midentity :: CShort
midentity = CShort
1
* :: CShort -> CShort -> CShort
(*) = CShort -> CShort -> CShort
forall a. Num a => a -> a -> a
(Prelude.*)
instance Multiplicative CUShort where
midentity :: CUShort
midentity = CUShort
1
* :: CUShort -> CUShort -> CUShort
(*) = CUShort -> CUShort -> CUShort
forall a. Num a => a -> a -> a
(Prelude.*)
instance Multiplicative CInt where
midentity :: CInt
midentity = CInt
1
* :: CInt -> CInt -> CInt
(*) = CInt -> CInt -> CInt
forall a. Num a => a -> a -> a
(Prelude.*)
instance Multiplicative CUInt where
midentity :: CUInt
midentity = CUInt
1
* :: CUInt -> CUInt -> CUInt
(*) = CUInt -> CUInt -> CUInt
forall a. Num a => a -> a -> a
(Prelude.*)
instance Multiplicative CLong where
midentity :: CLong
midentity = CLong
1
* :: CLong -> CLong -> CLong
(*) = CLong -> CLong -> CLong
forall a. Num a => a -> a -> a
(Prelude.*)
instance Multiplicative CULong where
midentity :: CULong
midentity = CULong
1
* :: CULong -> CULong -> CULong
(*) = CULong -> CULong -> CULong
forall a. Num a => a -> a -> a
(Prelude.*)
instance Multiplicative CPtrdiff where
midentity :: CPtrdiff
midentity = CPtrdiff
1
* :: CPtrdiff -> CPtrdiff -> CPtrdiff
(*) = CPtrdiff -> CPtrdiff -> CPtrdiff
forall a. Num a => a -> a -> a
(Prelude.*)
instance Multiplicative CSize where
midentity :: CSize
midentity = CSize
1
* :: CSize -> CSize -> CSize
(*) = CSize -> CSize -> CSize
forall a. Num a => a -> a -> a
(Prelude.*)
instance Multiplicative CWchar where
midentity :: CWchar
midentity = CWchar
1
* :: CWchar -> CWchar -> CWchar
(*) = CWchar -> CWchar -> CWchar
forall a. Num a => a -> a -> a
(Prelude.*)
instance Multiplicative CSigAtomic where
midentity :: CSigAtomic
midentity = CSigAtomic
1
* :: CSigAtomic -> CSigAtomic -> CSigAtomic
(*) = CSigAtomic -> CSigAtomic -> CSigAtomic
forall a. Num a => a -> a -> a
(Prelude.*)
instance Multiplicative CLLong where
midentity :: CLLong
midentity = CLLong
1
* :: CLLong -> CLLong -> CLLong
(*) = CLLong -> CLLong -> CLLong
forall a. Num a => a -> a -> a
(Prelude.*)
instance Multiplicative CULLong where
midentity :: CULLong
midentity = CULLong
1
* :: CULLong -> CULLong -> CULLong
(*) = CULLong -> CULLong -> CULLong
forall a. Num a => a -> a -> a
(Prelude.*)
instance Multiplicative CIntPtr where
midentity :: CIntPtr
midentity = CIntPtr
1
* :: CIntPtr -> CIntPtr -> CIntPtr
(*) = CIntPtr -> CIntPtr -> CIntPtr
forall a. Num a => a -> a -> a
(Prelude.*)
instance Multiplicative CUIntPtr where
midentity :: CUIntPtr
midentity = CUIntPtr
1
* :: CUIntPtr -> CUIntPtr -> CUIntPtr
(*) = CUIntPtr -> CUIntPtr -> CUIntPtr
forall a. Num a => a -> a -> a
(Prelude.*)
instance Multiplicative CIntMax where
midentity :: CIntMax
midentity = CIntMax
1
* :: CIntMax -> CIntMax -> CIntMax
(*) = CIntMax -> CIntMax -> CIntMax
forall a. Num a => a -> a -> a
(Prelude.*)
instance Multiplicative CUIntMax where
midentity :: CUIntMax
midentity = CUIntMax
1
* :: CUIntMax -> CUIntMax -> CUIntMax
(*) = CUIntMax -> CUIntMax -> CUIntMax
forall a. Num a => a -> a -> a
(Prelude.*)
instance Multiplicative CClock where
midentity :: CClock
midentity = CClock
1
* :: CClock -> CClock -> CClock
(*) = CClock -> CClock -> CClock
forall a. Num a => a -> a -> a
(Prelude.*)
instance Multiplicative CTime where
midentity :: CTime
midentity = CTime
1
* :: CTime -> CTime -> CTime
(*) = CTime -> CTime -> CTime
forall a. Num a => a -> a -> a
(Prelude.*)
instance Multiplicative CUSeconds where
midentity :: CUSeconds
midentity = CUSeconds
1
* :: CUSeconds -> CUSeconds -> CUSeconds
(*) = CUSeconds -> CUSeconds -> CUSeconds
forall a. Num a => a -> a -> a
(Prelude.*)
instance Multiplicative CSUSeconds where
midentity :: CSUSeconds
midentity = CSUSeconds
1
* :: CSUSeconds -> CSUSeconds -> CSUSeconds
(*) = CSUSeconds -> CSUSeconds -> CSUSeconds
forall a. Num a => a -> a -> a
(Prelude.*)
instance Multiplicative COff where
midentity :: COff
midentity = COff
1
* :: COff -> COff -> COff
(*) = COff -> COff -> COff
forall a. Num a => a -> a -> a
(Prelude.*)
instance Multiplicative CFloat where
midentity :: CFloat
midentity = CFloat
1.0
* :: CFloat -> CFloat -> CFloat
(*) = CFloat -> CFloat -> CFloat
forall a. Num a => a -> a -> a
(Prelude.*)
instance Multiplicative CDouble where
midentity :: CDouble
midentity = CDouble
1.0
* :: CDouble -> CDouble -> CDouble
(*) = CDouble -> CDouble -> CDouble
forall a. Num a => a -> a -> a
(Prelude.*)
instance IDivisible Integer where
div :: Integer -> Integer -> Integer
div = Integer -> Integer -> Integer
forall a. Integral a => a -> a -> a
Prelude.div
mod :: Integer -> Integer -> Integer
mod = Integer -> Integer -> Integer
forall a. Integral a => a -> a -> a
Prelude.mod
instance IDivisible Int where
div :: Int -> Int -> Int
div = Int -> Int -> Int
forall a. Integral a => a -> a -> a
Prelude.div
mod :: Int -> Int -> Int
mod = Int -> Int -> Int
forall a. Integral a => a -> a -> a
Prelude.mod
instance IDivisible Int8 where
div :: Int8 -> Int8 -> Int8
div = Int8 -> Int8 -> Int8
forall a. Integral a => a -> a -> a
Prelude.div
mod :: Int8 -> Int8 -> Int8
mod = Int8 -> Int8 -> Int8
forall a. Integral a => a -> a -> a
Prelude.mod
instance IDivisible Int16 where
div :: Int16 -> Int16 -> Int16
div = Int16 -> Int16 -> Int16
forall a. Integral a => a -> a -> a
Prelude.div
mod :: Int16 -> Int16 -> Int16
mod = Int16 -> Int16 -> Int16
forall a. Integral a => a -> a -> a
Prelude.mod
instance IDivisible Int32 where
div :: Int32 -> Int32 -> Int32
div = Int32 -> Int32 -> Int32
forall a. Integral a => a -> a -> a
Prelude.div
mod :: Int32 -> Int32 -> Int32
mod = Int32 -> Int32 -> Int32
forall a. Integral a => a -> a -> a
Prelude.mod
instance IDivisible Int64 where
div :: Int64 -> Int64 -> Int64
div = Int64 -> Int64 -> Int64
forall a. Integral a => a -> a -> a
Prelude.div
mod :: Int64 -> Int64 -> Int64
mod = Int64 -> Int64 -> Int64
forall a. Integral a => a -> a -> a
Prelude.mod
instance IDivisible Natural where
div :: Natural -> Natural -> Natural
div = Natural -> Natural -> Natural
forall a. Integral a => a -> a -> a
Prelude.quot
mod :: Natural -> Natural -> Natural
mod = Natural -> Natural -> Natural
forall a. Integral a => a -> a -> a
Prelude.rem
instance IDivisible Word where
div :: Word -> Word -> Word
div = Word -> Word -> Word
forall a. Integral a => a -> a -> a
Prelude.quot
mod :: Word -> Word -> Word
mod = Word -> Word -> Word
forall a. Integral a => a -> a -> a
Prelude.rem
instance IDivisible Word8 where
div :: Word8 -> Word8 -> Word8
div = Word8 -> Word8 -> Word8
forall a. Integral a => a -> a -> a
Prelude.quot
mod :: Word8 -> Word8 -> Word8
mod = Word8 -> Word8 -> Word8
forall a. Integral a => a -> a -> a
Prelude.rem
instance IDivisible Word16 where
div :: Word16 -> Word16 -> Word16
div = Word16 -> Word16 -> Word16
forall a. Integral a => a -> a -> a
Prelude.quot
mod :: Word16 -> Word16 -> Word16
mod = Word16 -> Word16 -> Word16
forall a. Integral a => a -> a -> a
Prelude.rem
instance IDivisible Word32 where
div :: Word32 -> Word32 -> Word32
div = Word32 -> Word32 -> Word32
forall a. Integral a => a -> a -> a
Prelude.quot
mod :: Word32 -> Word32 -> Word32
mod = Word32 -> Word32 -> Word32
forall a. Integral a => a -> a -> a
Prelude.rem
instance IDivisible Word64 where
div :: Word64 -> Word64 -> Word64
div = Word64 -> Word64 -> Word64
forall a. Integral a => a -> a -> a
Prelude.quot
mod :: Word64 -> Word64 -> Word64
mod = Word64 -> Word64 -> Word64
forall a. Integral a => a -> a -> a
Prelude.rem
instance IDivisible Word128 where
div :: Word128 -> Word128 -> Word128
div = Word128 -> Word128 -> Word128
Word128.quot
mod :: Word128 -> Word128 -> Word128
mod = Word128 -> Word128 -> Word128
Word128.rem
instance IDivisible Word256 where
div :: Word256 -> Word256 -> Word256
div = Word256 -> Word256 -> Word256
Word256.quot
mod :: Word256 -> Word256 -> Word256
mod = Word256 -> Word256 -> Word256
Word256.rem
instance IDivisible CChar where
div :: CChar -> CChar -> CChar
div = CChar -> CChar -> CChar
forall a. Integral a => a -> a -> a
Prelude.quot
mod :: CChar -> CChar -> CChar
mod = CChar -> CChar -> CChar
forall a. Integral a => a -> a -> a
Prelude.rem
instance IDivisible CSChar where
div :: CSChar -> CSChar -> CSChar
div = CSChar -> CSChar -> CSChar
forall a. Integral a => a -> a -> a
Prelude.quot
mod :: CSChar -> CSChar -> CSChar
mod = CSChar -> CSChar -> CSChar
forall a. Integral a => a -> a -> a
Prelude.rem
instance IDivisible CUChar where
div :: CUChar -> CUChar -> CUChar
div = CUChar -> CUChar -> CUChar
forall a. Integral a => a -> a -> a
Prelude.quot
mod :: CUChar -> CUChar -> CUChar
mod = CUChar -> CUChar -> CUChar
forall a. Integral a => a -> a -> a
Prelude.rem
instance IDivisible CShort where
div :: CShort -> CShort -> CShort
div = CShort -> CShort -> CShort
forall a. Integral a => a -> a -> a
Prelude.quot
mod :: CShort -> CShort -> CShort
mod = CShort -> CShort -> CShort
forall a. Integral a => a -> a -> a
Prelude.rem
instance IDivisible CUShort where
div :: CUShort -> CUShort -> CUShort
div = CUShort -> CUShort -> CUShort
forall a. Integral a => a -> a -> a
Prelude.quot
mod :: CUShort -> CUShort -> CUShort
mod = CUShort -> CUShort -> CUShort
forall a. Integral a => a -> a -> a
Prelude.rem
instance IDivisible CInt where
div :: CInt -> CInt -> CInt
div = CInt -> CInt -> CInt
forall a. Integral a => a -> a -> a
Prelude.quot
mod :: CInt -> CInt -> CInt
mod = CInt -> CInt -> CInt
forall a. Integral a => a -> a -> a
Prelude.rem
instance IDivisible CUInt where
div :: CUInt -> CUInt -> CUInt
div = CUInt -> CUInt -> CUInt
forall a. Integral a => a -> a -> a
Prelude.quot
mod :: CUInt -> CUInt -> CUInt
mod = CUInt -> CUInt -> CUInt
forall a. Integral a => a -> a -> a
Prelude.rem
instance IDivisible CLong where
div :: CLong -> CLong -> CLong
div = CLong -> CLong -> CLong
forall a. Integral a => a -> a -> a
Prelude.quot
mod :: CLong -> CLong -> CLong
mod = CLong -> CLong -> CLong
forall a. Integral a => a -> a -> a
Prelude.rem
instance IDivisible CULong where
div :: CULong -> CULong -> CULong
div = CULong -> CULong -> CULong
forall a. Integral a => a -> a -> a
Prelude.quot
mod :: CULong -> CULong -> CULong
mod = CULong -> CULong -> CULong
forall a. Integral a => a -> a -> a
Prelude.rem
instance IDivisible CPtrdiff where
div :: CPtrdiff -> CPtrdiff -> CPtrdiff
div = CPtrdiff -> CPtrdiff -> CPtrdiff
forall a. Integral a => a -> a -> a
Prelude.quot
mod :: CPtrdiff -> CPtrdiff -> CPtrdiff
mod = CPtrdiff -> CPtrdiff -> CPtrdiff
forall a. Integral a => a -> a -> a
Prelude.rem
instance IDivisible CSize where
div :: CSize -> CSize -> CSize
div = CSize -> CSize -> CSize
forall a. Integral a => a -> a -> a
Prelude.quot
mod :: CSize -> CSize -> CSize
mod = CSize -> CSize -> CSize
forall a. Integral a => a -> a -> a
Prelude.rem
instance IDivisible CWchar where
div :: CWchar -> CWchar -> CWchar
div = CWchar -> CWchar -> CWchar
forall a. Integral a => a -> a -> a
Prelude.quot
mod :: CWchar -> CWchar -> CWchar
mod = CWchar -> CWchar -> CWchar
forall a. Integral a => a -> a -> a
Prelude.rem
instance IDivisible CSigAtomic where
div :: CSigAtomic -> CSigAtomic -> CSigAtomic
div = CSigAtomic -> CSigAtomic -> CSigAtomic
forall a. Integral a => a -> a -> a
Prelude.quot
mod :: CSigAtomic -> CSigAtomic -> CSigAtomic
mod = CSigAtomic -> CSigAtomic -> CSigAtomic
forall a. Integral a => a -> a -> a
Prelude.rem
instance IDivisible CLLong where
div :: CLLong -> CLLong -> CLLong
div = CLLong -> CLLong -> CLLong
forall a. Integral a => a -> a -> a
Prelude.quot
mod :: CLLong -> CLLong -> CLLong
mod = CLLong -> CLLong -> CLLong
forall a. Integral a => a -> a -> a
Prelude.rem
instance IDivisible CULLong where
div :: CULLong -> CULLong -> CULLong
div = CULLong -> CULLong -> CULLong
forall a. Integral a => a -> a -> a
Prelude.quot
mod :: CULLong -> CULLong -> CULLong
mod = CULLong -> CULLong -> CULLong
forall a. Integral a => a -> a -> a
Prelude.rem
instance IDivisible CIntPtr where
div :: CIntPtr -> CIntPtr -> CIntPtr
div = CIntPtr -> CIntPtr -> CIntPtr
forall a. Integral a => a -> a -> a
Prelude.quot
mod :: CIntPtr -> CIntPtr -> CIntPtr
mod = CIntPtr -> CIntPtr -> CIntPtr
forall a. Integral a => a -> a -> a
Prelude.rem
instance IDivisible CUIntPtr where
div :: CUIntPtr -> CUIntPtr -> CUIntPtr
div = CUIntPtr -> CUIntPtr -> CUIntPtr
forall a. Integral a => a -> a -> a
Prelude.quot
mod :: CUIntPtr -> CUIntPtr -> CUIntPtr
mod = CUIntPtr -> CUIntPtr -> CUIntPtr
forall a. Integral a => a -> a -> a
Prelude.rem
instance IDivisible CIntMax where
div :: CIntMax -> CIntMax -> CIntMax
div = CIntMax -> CIntMax -> CIntMax
forall a. Integral a => a -> a -> a
Prelude.quot
mod :: CIntMax -> CIntMax -> CIntMax
mod = CIntMax -> CIntMax -> CIntMax
forall a. Integral a => a -> a -> a
Prelude.rem
instance IDivisible CUIntMax where
div :: CUIntMax -> CUIntMax -> CUIntMax
div = CUIntMax -> CUIntMax -> CUIntMax
forall a. Integral a => a -> a -> a
Prelude.quot
mod :: CUIntMax -> CUIntMax -> CUIntMax
mod = CUIntMax -> CUIntMax -> CUIntMax
forall a. Integral a => a -> a -> a
Prelude.rem
instance Divisible Prelude.Rational where
/ :: Rational -> Rational -> Rational
(/) = Rational -> Rational -> Rational
forall a. Fractional a => a -> a -> a
(Prelude./)
instance Divisible Float where
/ :: Float -> Float -> Float
(/) = Float -> Float -> Float
forall a. Fractional a => a -> a -> a
(Prelude./)
instance Divisible Double where
/ :: Double -> Double -> Double
(/) = Double -> Double -> Double
forall a. Fractional a => a -> a -> a
(Prelude./)
instance Divisible CFloat where
/ :: CFloat -> CFloat -> CFloat
(/) = CFloat -> CFloat -> CFloat
forall a. Fractional a => a -> a -> a
(Prelude./)
instance Divisible CDouble where
/ :: CDouble -> CDouble -> CDouble
(/) = CDouble -> CDouble -> CDouble
forall a. Fractional a => a -> a -> a
(Prelude./)
recip :: Divisible a => a -> a
recip :: a -> a
recip a
x = a
forall a. Multiplicative a => a
midentity a -> a -> a
forall a. Divisible a => a -> a -> a
/ a
x
power :: (Enum n, IsNatural n, IDivisible n, Multiplicative a) => a -> n -> a
power :: a -> n -> a
power a
a n
n
| n
n n -> n -> Bool
forall a. Eq a => a -> a -> Bool
== n
0 = a
forall a. Multiplicative a => a
midentity
| Bool
otherwise = a -> a -> n -> a
forall a a.
(IDivisible a, IsIntegral a, Enum a, Multiplicative a) =>
a -> a -> a -> a
squaring a
forall a. Multiplicative a => a
midentity a
a n
n
where
squaring :: a -> a -> a -> a
squaring a
y a
x a
i
| a
i a -> a -> Bool
forall a. Eq a => a -> a -> Bool
== a
0 = a
y
| a
i a -> a -> Bool
forall a. Eq a => a -> a -> Bool
== a
1 = a
x a -> a -> a
forall a. Multiplicative a => a -> a -> a
* a
y
| a -> Bool
forall n. (IDivisible n, IsIntegral n) => n -> Bool
even a
i = a -> a -> a -> a
squaring a
y (a
xa -> a -> a
forall a. Multiplicative a => a -> a -> a
*a
x) (a
ia -> a -> a
forall a. IDivisible a => a -> a -> a
`div`a
2)
| Bool
otherwise = a -> a -> a -> a
squaring (a
xa -> a -> a
forall a. Multiplicative a => a -> a -> a
*a
y) (a
xa -> a -> a
forall a. Multiplicative a => a -> a -> a
*a
x) (a -> a
forall a. Enum a => a -> a
pred a
ia -> a -> a
forall a. IDivisible a => a -> a -> a
`div` a
2)
even :: (IDivisible n, IsIntegral n) => n -> Bool
even :: n -> Bool
even n
n = (n
n n -> n -> n
forall a. IDivisible a => a -> a -> a
`mod` n
2) n -> n -> Bool
forall a. Eq a => a -> a -> Bool
== n
0