GHC.Natural

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: 4.8.0.0

The Natural number type

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

data Natural

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: 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

Enum Natural

Since: 4.8.0.0

Defined in GHC.Enum

Methods

succ :: Natural -> Natural

pred :: Natural -> Natural

toEnum :: Int -> Natural

fromEnum :: Natural -> Int

enumFrom :: Natural -> [Natural]

enumFromThen :: Natural -> Natural -> [Natural]

enumFromTo :: Natural -> Natural -> [Natural]

enumFromThenTo :: Natural -> Natural -> Natural -> [Natural]

Eq Natural

Since: 4.8.0.0

Defined in GHC.Natural

Methods

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

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

Integral Natural

Since: 4.8.0.0

Defined in GHC.Real

Methods

quot :: Natural -> Natural -> Natural

rem :: Natural -> Natural -> Natural

div :: Natural -> Natural -> Natural

mod :: Natural -> Natural -> Natural

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

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

toInteger :: Natural -> Integer

Data Natural

Since: 4.8.0.0

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

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

toConstr :: Natural -> Constr

dataTypeOf :: Natural -> DataType

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

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

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

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

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

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

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

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

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

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

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: 4.8.0.0

Defined in GHC.Num

Methods

(+) :: Natural -> Natural -> Natural

(-) :: Natural -> Natural -> Natural

(*) :: Natural -> Natural -> Natural

negate :: Natural -> Natural

abs :: Natural -> Natural

signum :: Natural -> Natural

fromInteger :: Integer -> Natural

Ord Natural

Since: 4.8.0.0

Defined in GHC.Natural

Methods

compare :: Natural -> Natural -> Ordering

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

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

(>) :: Natural -> Natural -> Bool

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

max :: Natural -> Natural -> Natural

min :: Natural -> Natural -> Natural

Read Natural

Since: 4.8.0.0

Defined in GHC.Read

Methods

readsPrec :: Int -> ReadS Natural

readList :: ReadS [Natural]

readPrec :: ReadPrec Natural

readListPrec :: ReadPrec [Natural]

Real Natural

Since: 4.8.0.0

Defined in GHC.Real

Methods

toRational :: Natural -> Rational

Show Natural

Since: 4.8.0.0

Defined in GHC.Show

Methods

showsPrec :: Int -> Natural -> ShowS

show :: Natural -> String

showList :: [Natural] -> ShowS

Ix Natural

Since: 4.8.0.0

Defined in GHC.Arr

Methods

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

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

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

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

rangeSize :: (Natural, Natural) -> Int

unsafeRangeSize :: (Natural, Natural) -> Int

Bits Natural

Since: 4.8.0

Defined in Data.Bits

Methods

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

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

xor :: Natural -> Natural -> Natural

complement :: Natural -> Natural

shift :: Natural -> Int -> Natural

rotate :: Natural -> Int -> Natural

zeroBits :: Natural

bit :: Int -> Natural

setBit :: Natural -> Int -> Natural

clearBit :: Natural -> Int -> Natural

complementBit :: Natural -> Int -> Natural

testBit :: Natural -> Int -> Bool

bitSizeMaybe :: Natural -> Maybe Int

bitSize :: Natural -> Int

isSigned :: Natural -> Bool

shiftL :: Natural -> Int -> Natural

unsafeShiftL :: Natural -> Int -> Natural

shiftR :: Natural -> Int -> Natural

unsafeShiftR :: Natural -> Int -> Natural

rotateL :: Natural -> Int -> Natural

rotateR :: Natural -> Int -> Natural

popCount :: Natural -> Int

PrintfArg Natural

Since: 4.8.0.0

Defined in Text.Printf

Methods

formatArg :: Natural -> FieldFormatter

parseFormat :: Natural -> ModifierParser

mkNatural

Arguments

:: [Word]

value expressed in 32 bit chunks, least significant first

-> Natural

Construct Natural value from list of Words.

This function is used by GHC for constructing Natural literals.

isValidNatural :: Natural -> Bool

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: 4.8.0.0

Arithmetic

plusNatural :: Natural -> Natural -> Natural

Natural Addition

minusNatural :: Natural -> Natural -> Natural

Natural subtraction. May throw Underflow.

minusNaturalMaybe :: Natural -> Natural -> Maybe Natural

Natural subtraction. Returns Nothings for non-positive results.

Since: 4.8.0.0

timesNatural :: Natural -> Natural -> Natural

Natural multiplication

negateNatural :: Natural -> Natural

signumNatural :: Natural -> Natural

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

quotNatural :: Natural -> Natural -> Natural

remNatural :: Natural -> Natural -> Natural

gcdNatural :: Natural -> Natural -> Natural

Compute greatest common divisor.

lcmNatural :: Natural -> Natural -> Natural

compute least common multiplier.

Bits

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

Conversions

naturalToInteger :: Natural -> Integer

Since: 4.12.0.0

naturalToWord :: Natural -> Word

naturalToInt :: Natural -> Int

naturalFromInteger :: Integer -> Natural

Since: 4.10.0.0

wordToNatural :: Word -> Natural

Construct Natural from Word value.

Since: 4.8.0.0

intToNatural :: Int -> Natural

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

naturalToWordMaybe :: Natural -> Maybe Word

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

Since: 4.8.0.0

wordToNatural# :: Word# -> Natural

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

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

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

Since: 4.8.0.0