Wrapper for residues modulo m.

Mod 3 :: Mod 10 stands for the class of integers, congruent to 3 modulo 10 (…−17, −7, 3, 13, 23…). The modulo is stored on type level, so it is impossible, for example, to add up by mistake residues with different moduli.

>>> :set -XDataKinds
>>> (3 :: Mod 10) + (4 :: Mod 12)
error: Couldn't match type ‘12’ with ‘10’...
>>> (3 :: Mod 10) + 8
(1 `modulo` 10)

Note that modulo cannot be negative.

The canonical representative of the residue class, always between 0 and m-1 inclusively.

The canonical representative of the residue class, always between 0 and m-1 inclusively.

Linking type and value levels: extract modulo m as a value.

Linking type and value levels: extract modulo m as a value.

Computes the modular inverse, if the residue is coprime with the modulo.

>>> :set -XDataKinds
>>> invertMod (3 :: Mod 10)
Just (7 `modulo` 10) -- because 3 * 7 = 1 :: Mod 10
>>> invertMod (4 :: Mod 10)
Nothing

Drop-in replacement for ^, with much better performance.

>>> :set -XDataKinds
>>> powMod (3 :: Mod 10) 4
(1 `modulo` 10)

Infix synonym of powMod.

This type represents elements of the multiplicative group mod m, i.e. those elements which are coprime to m. Use toMultElement to construct.

Unwrap a residue.

Attempt to construct a multiplicative group element.

For elements of the multiplicative group, we can safely perform the inverse without needing to worry about failure.

This type represents residues with unknown modulo and rational numbers. One can freely combine them in arithmetic expressions, but each operation will spend time on modulo's recalculation:

>>> 2 `modulo` 10 + 4 `modulo` 15
(1 `modulo` 5)
>>> (2 `modulo` 10) * (4 `modulo` 15)
(3 `modulo` 5)
>>> 2 `modulo` 10 + fromRational (3 % 7)
(1 `modulo` 10)
>>> 2 `modulo` 10 * fromRational (3 % 7)
(8 `modulo` 10)

If performance is crucial, it is recommended to extract Mod m for further processing by pattern matching. E. g.,

case modulo n m of
  SomeMod k -> process k -- Here k has type Mod m
  InfMod{}  -> error "impossible"

Create modular value by representative of residue class and modulo. One can use the result either directly (via functions from Num and Fractional), or deconstruct it by pattern matching. Note that modulo never returns InfMod.

Computes the inverse value, if it exists.

>>> invertSomeMod (3 `modulo` 10)
Just (7 `modulo` 10) -- because 3 * 7 = 1 :: Mod 10
>>> invertSomeMod (4 `modulo` 10)
Nothing
>>> invertSomeMod (fromRational (2 % 5))
Just 5 % 2

Drop-in replacement for ^, with much better performance. When -O is enabled, there is a rewrite rule, which specialises ^ to powSomeMod.

>>> powSomeMod (3 `modulo` 10) 4
(1 `modulo` 10)

This class gives the integer associated with a type-level natural. There are instances of the class for every concrete literal: 0, 1, 2, etc.

Since: base-4.7.0.0