Data.Modular

This module provides types for working with integers modulo some constant.

This module uses some new Haskell features introduced in 7.6. In particular, it needs DataKinds and type literals (GHC.TypeLits). The TypeOperators extension is needed for the nice infix syntax.

These types are created with the type constructor Mod (or its synonym /). To work with integers mod 7, you could write:

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

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

All the usual typeclasses are defined for these types. You can also get the constant using bound or extract the underlying value using unMod.

Here is a quick example:

*Data.Modular> (10 :: ℤ/7) * (11 :: ℤ/7)
5

It also works correctly with negative numeric literals:

*Data.Modular> (-10 :: ℤ/7) * (11 :: ℤ/7)
2

Documentation

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

Wraps the underlying type into the modular type, wrapping as appropriate.

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

Wraps an integral number to a mod, converting between integral types.

data Mod i n

The actual type, wrapping an underlying Integeral type i in a newtype annotated with the bound.

Instances

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

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

Eq i => Eq (Mod i n)

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

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

Ord i => Ord (Mod i n)

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

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

Show i => Show (Mod i n)

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.