Copyright	(C) 2014 Herbert Valerio Riedel (C) 2011 Edward Kmett
License	see libraries/base/LICENSE
Maintainer	[email protected]
Stability	internal
Portability	non-portable (GHC Extensions)
Safe Haskell	None
Language	Haskell2010

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

mkNatural Source

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

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 Nothings 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 multiplier.

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