Copyright	(C) 2014 Herbert Valerio Riedel (C) 2011 Edward Kmett
License	see libraries/base/LICENSE
Maintainer	libraries@haskell.org
Stability	internal
Portability	non-portable (GHC Extensions)
Safe Haskell	None
Language	Haskell2010

GHC.Natural

Contents

The Natural number type
Arithmetic
Bits
Conversions
Modular arithmetic

Description

The arbitrary-precision Natural number type.

Note: This is an internal GHC module with an API subject to change. It's recommended use the Numeric.Natural module to import the Natural type.

Since: base-4.8.0.0

Synopsis

data Natural
- = NatS# GmpLimb#
- | NatJ# !BigNat
mkNatural :: [Word] -> Natural
isValidNatural :: Natural -> Bool
plusNatural :: Natural -> Natural -> Natural
minusNatural :: Natural -> Natural -> Natural
minusNaturalMaybe :: Natural -> Natural -> Maybe Natural
timesNatural :: Natural -> Natural -> Natural
negateNatural :: Natural -> Natural
signumNatural :: Natural -> Natural
quotRemNatural :: Natural -> Natural -> (Natural, Natural)
quotNatural :: Natural -> Natural -> Natural
remNatural :: Natural -> Natural -> Natural
gcdNatural :: Natural -> Natural -> Natural
lcmNatural :: Natural -> Natural -> Natural
andNatural :: Natural -> Natural -> Natural
orNatural :: Natural -> Natural -> Natural
xorNatural :: Natural -> Natural -> Natural
bitNatural :: Int# -> Natural
testBitNatural :: Natural -> Int -> Bool
popCountNatural :: Natural -> Int
shiftLNatural :: Natural -> Int -> Natural
shiftRNatural :: Natural -> Int -> Natural
naturalToInteger :: Natural -> Integer
naturalToWord :: Natural -> Word
naturalToInt :: Natural -> Int
naturalFromInteger :: Integer -> Natural
wordToNatural :: Word -> Natural
intToNatural :: Int -> Natural
naturalToWordMaybe :: Natural -> Maybe Word
wordToNatural# :: Word# -> Natural
wordToNaturalBase :: Word# -> Natural
powModNatural :: Natural -> Natural -> Natural -> Natural

The `Natural` number type

Warning: The internal implementation of Natural (i.e. which constructors are available) depends on the Integer backend used!

data Natural Source #

Type representing arbitrary-precision non-negative integers.

>>> 2^100 :: Natural
1267650600228229401496703205376

Operations whose result would be negative throw (Underflow :: ArithException ),

>>> -1 :: Natural
*** Exception: arithmetic underflow

Since: base-4.8.0.0

Constructors

NatS# GmpLimb#

in [0, maxBound::Word]

NatJ# !BigNat

in ]maxBound::Word, +inf[

Invariant: NatJ# is used iff value doesn't fit in NatS# constructor. NB: Order of constructors *must* coincide with Ord relation

Instances

Instances details

Enum Natural #

Since: base-4.8.0.0

Instance details

Defined in GHC.Enum

Methods

succ :: Natural -> Natural Source #

pred :: Natural -> Natural Source #

toEnum :: Int -> Natural Source #

fromEnum :: Natural -> Int Source #

enumFrom :: Natural -> [Natural] Source #

enumFromThen :: Natural -> Natural -> [Natural] Source #

enumFromTo :: Natural -> Natural -> [Natural] Source #

enumFromThenTo :: Natural -> Natural -> Natural -> [Natural] Source #

Eq Natural #

Since: base-4.8.0.0

Instance details

Defined in GHC.Natural

Methods

(==) :: Natural -> Natural -> Bool Source #

(/=) :: Natural -> Natural -> Bool Source #

Integral Natural #

Since: base-4.8.0.0

Instance details

Defined in GHC.Real

Methods

quot :: Natural -> Natural -> Natural Source #

rem :: Natural -> Natural -> Natural Source #

div :: Natural -> Natural -> Natural Source #

mod :: Natural -> Natural -> Natural Source #

quotRem :: Natural -> Natural -> (Natural, Natural) Source #

divMod :: Natural -> Natural -> (Natural, Natural) Source #

toInteger :: Natural -> Integer Source #

Data Natural #

Since: base-4.8.0.0

Instance details

Defined in Data.Data

Methods

gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> Natural -> c Natural Source #

gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c Natural Source #

toConstr :: Natural -> Constr Source #

dataTypeOf :: Natural -> DataType Source #

dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c Natural) Source #

dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c Natural) Source #

gmapT :: (forall b. Data b => b -> b) -> Natural -> Natural Source #

gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Natural -> r Source #

gmapQr :: forall r r'. (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Natural -> r Source #

gmapQ :: (forall d. Data d => d -> u) -> Natural -> [u] Source #

gmapQi :: Int -> (forall d. Data d => d -> u) -> Natural -> u Source #

gmapM :: Monad m => (forall d. Data d => d -> m d) -> Natural -> m Natural Source #

gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Natural -> m Natural Source #

gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Natural -> m Natural Source #

Num Natural #

Note that Natural 's Num instance isn't a ring: no element but 0 has an additive inverse. It is a semiring though.

Since: base-4.8.0.0

Instance details

Defined in GHC.Num

Methods

(+) :: Natural -> Natural -> Natural Source #

(-) :: Natural -> Natural -> Natural Source #

(*) :: Natural -> Natural -> Natural Source #

negate :: Natural -> Natural Source #

abs :: Natural -> Natural Source #

signum :: Natural -> Natural Source #

fromInteger :: Integer -> Natural Source #

Ord Natural #

Since: base-4.8.0.0

Instance details

Defined in GHC.Natural

Methods

compare :: Natural -> Natural -> Ordering Source #

(<) :: Natural -> Natural -> Bool Source #

(<=) :: Natural -> Natural -> Bool Source #

(>) :: Natural -> Natural -> Bool Source #

(>=) :: Natural -> Natural -> Bool Source #

max :: Natural -> Natural -> Natural Source #

min :: Natural -> Natural -> Natural Source #

Read Natural #

Since: base-4.8.0.0

Instance details

Defined in GHC.Read

Methods

readsPrec :: Int -> ReadS Natural Source #

readList :: ReadS [Natural] Source #

readPrec :: ReadPrec Natural Source #

readListPrec :: ReadPrec [Natural] Source #

Real Natural #

Since: base-4.8.0.0

Instance details

Defined in GHC.Real

Methods

toRational :: Natural -> Rational Source #

Show Natural #

Since: base-4.8.0.0

Instance details

Defined in GHC.Show

Methods

showsPrec :: Int -> Natural -> ShowS Source #

show :: Natural -> String Source #

showList :: [Natural] -> ShowS Source #

Ix Natural #

Since: base-4.8.0.0

Instance details

Defined in GHC.Ix

Methods

range :: (Natural, Natural) -> [Natural] Source #

index :: (Natural, Natural) -> Natural -> Int Source #

unsafeIndex :: (Natural, Natural) -> Natural -> Int Source #

inRange :: (Natural, Natural) -> Natural -> Bool Source #

rangeSize :: (Natural, Natural) -> Int Source #

unsafeRangeSize :: (Natural, Natural) -> Int Source #

Bits Natural #

Since: base-4.8.0

Instance details

Defined in Data.Bits

Methods

(.&.) :: Natural -> Natural -> Natural Source #

(.|.) :: Natural -> Natural -> Natural Source #

xor :: Natural -> Natural -> Natural Source #

complement :: Natural -> Natural Source #

shift :: Natural -> Int -> Natural Source #

rotate :: Natural -> Int -> Natural Source #

zeroBits :: Natural Source #

bit :: Int -> Natural Source #

setBit :: Natural -> Int -> Natural Source #

clearBit :: Natural -> Int -> Natural Source #

complementBit :: Natural -> Int -> Natural Source #

testBit :: Natural -> Int -> Bool Source #

bitSizeMaybe :: Natural -> Maybe Int Source #

bitSize :: Natural -> Int Source #

isSigned :: Natural -> Bool Source #

shiftL :: Natural -> Int -> Natural Source #

unsafeShiftL :: Natural -> Int -> Natural Source #

shiftR :: Natural -> Int -> Natural Source #

unsafeShiftR :: Natural -> Int -> Natural Source #

rotateL :: Natural -> Int -> Natural Source #

rotateR :: Natural -> Int -> Natural Source #

popCount :: Natural -> Int Source #

PrintfArg Natural #

Since: base-4.8.0.0

Instance details

Defined in Text.Printf

Methods

formatArg :: Natural -> FieldFormatter Source #

parseFormat :: Natural -> ModifierParser Source #

mkNatural Source #

Arguments

:: [Word]

value expressed in 32 bit chunks, least significant first

-> Natural

Construct Natural value from list of Word s.

This function is used by GHC for constructing Natural literals.

isValidNatural :: Natural -> Bool Source #

Test whether all internal invariants are satisfied by Natural value

This operation is mostly useful for test-suites and/or code which constructs Integer values directly.

Since: base-4.8.0.0

Arithmetic

plusNatural :: Natural -> Natural -> Natural Source #

Natural Addition

minusNatural :: Natural -> Natural -> Natural Source #

Natural subtraction. May throw Underflow.

minusNaturalMaybe :: Natural -> Natural -> Maybe Natural Source #

Natural subtraction. Returns Nothing s for non-positive results.

Since: base-4.8.0.0

timesNatural :: Natural -> Natural -> Natural Source #

Natural multiplication

negateNatural :: Natural -> Natural Source #

signumNatural :: Natural -> Natural Source #

quotRemNatural :: Natural -> Natural -> (Natural, Natural) Source #

quotNatural :: Natural -> Natural -> Natural Source #

remNatural :: Natural -> Natural -> Natural Source #

gcdNatural :: Natural -> Natural -> Natural Source #

Compute greatest common divisor.

lcmNatural :: Natural -> Natural -> Natural Source #

Compute least common multiple.

Bits

andNatural :: Natural -> Natural -> Natural Source #

orNatural :: Natural -> Natural -> Natural Source #

xorNatural :: Natural -> Natural -> Natural Source #

bitNatural :: Int# -> Natural Source #

testBitNatural :: Natural -> Int -> Bool Source #

popCountNatural :: Natural -> Int Source #

shiftLNatural :: Natural -> Int -> Natural Source #

shiftRNatural :: Natural -> Int -> Natural Source #

Conversions

naturalToInteger :: Natural -> Integer Source #

@since 4.X.0.0

naturalToWord :: Natural -> Word Source #

naturalToInt :: Natural -> Int Source #

naturalFromInteger :: Integer -> Natural Source #

Since: base-4.10.0.0

wordToNatural :: Word -> Natural Source #

Construct Natural from Word value.

Since: base-4.8.0.0

intToNatural :: Int -> Natural Source #

Convert Int to Natural . Throws Underflow when passed a negative Int .

naturalToWordMaybe :: Natural -> Maybe Word Source #

Try downcasting Natural to Word value. Returns Nothing if value doesn't fit in Word .

Since: base-4.8.0.0

wordToNatural# :: Word# -> Natural Source #

Convert a Word# into a Natural

Built-in rule ensures that applications of this function to literal Word# are lifted into Natural literals.

wordToNaturalBase :: Word# -> Natural Source #

Convert a Word# into a Natural

In base we can't use wordToNatural# as built-in rules transform some of them into Natural literals. Use this function instead.

Modular arithmetic

powModNatural :: Natural -> Natural -> Natural -> Natural Source #

"powModNatural b e m" computes base b raised to exponent e modulo m.

Since: base-4.8.0.0

The Natural number type

Instances

Arithmetic

Bits

Conversions

Modular arithmetic

The `Natural` number type