base-4.14.0.0: Basic libraries
Copyright(C) 2014 Herbert Valerio Riedel
(C) 2011 Edward Kmett
Licensesee libraries/base/LICENSE
Maintainerlibraries@haskell.org
Stabilityinternal
Portabilitynon-portable (GHC Extensions)
Safe HaskellNone
LanguageHaskell2010

GHC.Natural

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

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

Eq Natural #

Since: base-4.8.0.0

Instance details

Defined in GHC.Natural

Integral Natural #

Since: base-4.8.0.0

Instance details

Defined in GHC.Real

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

Ord Natural #

Since: base-4.8.0.0

Instance details

Defined in GHC.Natural

Read Natural #

Since: base-4.8.0.0

Instance details

Defined in GHC.Read

Real Natural #

Since: base-4.8.0.0

Instance details

Defined in GHC.Real

Show Natural #

Since: base-4.8.0.0

Instance details

Defined in GHC.Show

Ix Natural #

Since: base-4.8.0.0

Instance details

Defined in GHC.Ix

Bits Natural #

Since: base-4.8.0

Instance details

Defined in Data.Bits

PrintfArg Natural #

Since: base-4.8.0.0

Instance details

Defined in Text.Printf

mkNatural Source #

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 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

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

Natural subtraction. Returns Nothings for non-positive results.

Since: base-4.8.0.0

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

Compute greatest common divisor.

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

Compute least common multiple.

Bits

Conversions

naturalToInteger :: Natural -> Integer Source #

Since: base-4.12.0.0

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