-- Hoogle documentation, generated by Haddock
-- See Hoogle, http://www.haskell.org/hoogle/


-- | Fast type-safe modular arithmetic
--   
--   <a>Modular arithmetic</a>, promoting moduli to the type level, with an
--   emphasis on performance. Originally part of <a>arithmoi</a> package.
@package mod
@version 0.1.0.0


-- | <a>Modular arithmetic</a>, promoting moduli to the type level, with an
--   emphasis on performance. Originally part of <a>arithmoi</a> package.
module Data.Mod

-- | This data type represents <a>integers modulo m</a>, equipped with
--   useful instances.
--   
--   For example, 3 :: <a>Mod</a> 10 stands for the class of integers
--   congruent to 3 modulo 10: …−17, −7, 3, 13, 23…
--   
--   <pre>
--   &gt;&gt;&gt; :set -XDataKinds
--   
--   &gt;&gt;&gt; 3 + 8 :: Mod 10
--   (1 `modulo` 10) -- because 3 + 8 = 11 ≡ 1 (mod 10)
--   </pre>
--   
--   <b>Warning:</b> division by residue, which is not <a>coprime</a> with
--   the modulo, throws <a>DivideByZero</a>. Consider using
--   <a>invertMod</a> for non-prime moduli.
data Mod (m :: Nat)

-- | The canonical representative of the residue class, always between 0
--   and m - 1 inclusively.
unMod :: Mod m -> Natural

-- | If an argument is <a>coprime</a> with the modulo, return its modular
--   inverse. Otherwise return <a>Nothing</a>.
--   
--   <pre>
--   &gt;&gt;&gt; :set -XDataKinds
--   
--   &gt;&gt;&gt; invertMod 3 :: Mod 10
--   Just (7 `modulo` 10) -- because 3 * 7 = 21 ≡ 1 (mod 10)
--   
--   &gt;&gt;&gt; invertMod 4 :: Mod 10
--   Nothing -- because 4 and 10 are not coprime
--   </pre>
invertMod :: KnownNat m => Mod m -> Maybe (Mod m)

-- | Drop-in replacement for <a>^</a> with much better performance.
--   Negative powers are allowed, but may throw <a>DivideByZero</a>, if an
--   argument is not <a>coprime</a> with the modulo.
--   
--   Building with <tt>-O</tt> triggers a rewrite rule <a>^</a> =
--   <a>^%</a>.
--   
--   <pre>
--   &gt;&gt;&gt; :set -XDataKinds
--   
--   &gt;&gt;&gt; 3 ^% 4 :: Mod 10
--   (1 `modulo` 10) -- because 3 ^ 4 = 81 ≡ 1 (mod 10)
--   
--   &gt;&gt;&gt; 3 ^% (-1) :: Mod 10
--   (7 `modulo` 10) -- because 3 * 7 = 21 ≡ 1 (mod 10)
--   
--   &gt;&gt;&gt; 4 ^% (-1) :: Mod 10
--   (*** Exception: divide by zero -- because 4 and 10 are not coprime
--   </pre>
(^%) :: (KnownNat m, Integral a) => Mod m -> a -> Mod m
infixr 8 ^%
instance GHC.Generics.Generic (Data.Mod.Mod m)
instance GHC.Classes.Ord (Data.Mod.Mod m)
instance GHC.Classes.Eq (Data.Mod.Mod m)
instance Control.DeepSeq.NFData (Data.Mod.Mod m)
instance GHC.TypeNats.KnownNat m => GHC.Show.Show (Data.Mod.Mod m)
instance GHC.TypeNats.KnownNat m => GHC.Enum.Enum (Data.Mod.Mod m)
instance GHC.TypeNats.KnownNat m => GHC.Enum.Bounded (Data.Mod.Mod m)
instance GHC.TypeNats.KnownNat m => GHC.Num.Num (Data.Mod.Mod m)
instance GHC.TypeNats.KnownNat m => Data.Semiring.Semiring (Data.Mod.Mod m)
instance GHC.TypeNats.KnownNat m => Data.Semiring.Ring (Data.Mod.Mod m)
instance GHC.TypeNats.KnownNat m => GHC.Real.Fractional (Data.Mod.Mod m)
instance GHC.TypeNats.KnownNat m => Data.Euclidean.GcdDomain (Data.Mod.Mod m)
instance GHC.TypeNats.KnownNat m => Data.Euclidean.Euclidean (Data.Mod.Mod m)
instance GHC.TypeNats.KnownNat m => Data.Euclidean.Field (Data.Mod.Mod m)