Data.Modular

Types for working with integers modulo some constant.

Documentation

Mod and its synonym / let you wrap arbitrary numeric types in a modulus. To work with integers (mod 7) backed by Integer, you could use one of the following equivalent types:

Mod Integer 7
Integer `Mod` 7
Integer/7
ℤ/7

('ℤ' is a synonym for Integer provided by this library. In Emacs, you can use the TeX input mode to type it with \Bbb{Z}.)

The usual numeric typeclasses are defined for these types. You can always extract the underlying value with unMod.

Here is a quick example:

>>> 10 * 11 :: ℤ/7
5

It also works correctly with negative numeric literals:

>>> (-10) * 11 :: ℤ/7
2

Modular division is an inverse of modular multiplication. It is defined when divisor is coprime to modulus:

>>> 7 `div` 3 :: ℤ/16
13
>>> 3 * 13 :: ℤ/16
7

To use type level numeric literals you need to enable the DataKinds extension and to use infix syntax for Mod or the / synonym, you need TypeOperators.

Preliminaries

To use type level numeric literals you need to enable the DataKinds extension:

>>> :set -XDataKinds

To use infix syntax for Mod or the / synonym, enable TypeOperators:

>>> :set -XTypeOperators

Modular arithmetic

data Mod i n

Wraps an underlying Integeral type i in a newtype annotated with the bound n.

Instances

(Integral i, KnownNat n) => Bounded (Mod i n)

Methods

minBound :: Mod i n

maxBound :: Mod i n

(Integral i, KnownNat n) => Enum (Mod i n)

Methods

succ :: Mod i n -> Mod i n

pred :: Mod i n -> Mod i n

toEnum :: Int -> Mod i n

fromEnum :: Mod i n -> Int

enumFrom :: Mod i n -> [Mod i n]

enumFromThen :: Mod i n -> Mod i n -> [Mod i n]

enumFromTo :: Mod i n -> Mod i n -> [Mod i n]

enumFromThenTo :: Mod i n -> Mod i n -> Mod i n -> [Mod i n]

Eq i => Eq (Mod i n)

Methods

(==) :: Mod i n -> Mod i n -> Bool

(/=) :: Mod i n -> Mod i n -> Bool

(Integral i, KnownNat n) => Integral (Mod i n)

Integer division uses modular inverse inv, so it is possible to divide only by numbers coprime to n and the remainder is always 0.

Methods

quot :: Mod i n -> Mod i n -> Mod i n

rem :: Mod i n -> Mod i n -> Mod i n

div :: Mod i n -> Mod i n -> Mod i n

mod :: Mod i n -> Mod i n -> Mod i n

quotRem :: Mod i n -> Mod i n -> (Mod i n, Mod i n)

divMod :: Mod i n -> Mod i n -> (Mod i n, Mod i n)

toInteger :: Mod i n -> Integer

(Integral i, KnownNat n) => Num (Mod i n)

Methods

(+) :: Mod i n -> Mod i n -> Mod i n

(-) :: Mod i n -> Mod i n -> Mod i n

(*) :: Mod i n -> Mod i n -> Mod i n

negate :: Mod i n -> Mod i n

abs :: Mod i n -> Mod i n

signum :: Mod i n -> Mod i n

fromInteger :: Integer -> Mod i n

Ord i => Ord (Mod i n)

Methods

compare :: Mod i n -> Mod i n -> Ordering

(<) :: Mod i n -> Mod i n -> Bool

(<=) :: Mod i n -> Mod i n -> Bool

(>) :: Mod i n -> Mod i n -> Bool

(>=) :: Mod i n -> Mod i n -> Bool

max :: Mod i n -> Mod i n -> Mod i n

min :: Mod i n -> Mod i n -> Mod i n

(Read i, Integral i, KnownNat n) => Read (Mod i n)

Methods

readsPrec :: Int -> ReadS (Mod i n)

readList :: ReadS [Mod i n]

readPrec :: ReadPrec (Mod i n)

readListPrec :: ReadPrec [Mod i n]

(Integral i, KnownNat n) => Real (Mod i n)

Methods

toRational :: Mod i n -> Rational

Show i => Show (Mod i n)

Methods

showsPrec :: Int -> Mod i n -> ShowS

show :: Mod i n -> String

showList :: [Mod i n] -> ShowS

unMod :: (i `Mod` n) -> i

Extract the underlying integral value from a modular type.

toMod :: forall n i. (Integral i, KnownNat n) => i -> i `Mod` n

Injects a value of the underlying type into the modulus type, wrapping as appropriate.

toMod' :: forall n i j. (Integral i, Integral j, KnownNat n) => i -> j `Mod` n

Wraps an integral number, converting between integral types.

inv :: forall n i. (KnownNat n, Integral i) => Mod i n -> Mod i n

The modular inverse.

>>> inv 3 :: ℤ/7
5
>>> 3 * 5 :: ℤ/7
1

Note that only numbers coprime to n have an inverse modulo n:

inv 6 :: ℤ/15

• ** Exception: divide by 6 (mod 15), non-coprime to modulus

type (/) = Mod

A synonym for Mod, inspired by the ℤ/n syntax from mathematics.

type ℤ = Integer

A synonym for Integer, also inspired by the ℤ/n syntax.

modVal :: forall i proxy n. (Integral i, KnownNat n) => i -> proxy n -> Mod i n

Convert an integral number i into a Mod value given modular bound n at type level.

data SomeMod i

A modular number with an unknown bound.

Instances

Show i => Show (SomeMod i)

Methods

showsPrec :: Int -> SomeMod i -> ShowS

show :: SomeMod i -> String

showList :: [SomeMod i] -> ShowS

someModVal :: Integral i => i -> Integer -> Maybe (SomeMod i)

Convert an integral number i into a Mod value with an unknown modulus.