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


-- | A type for integers modulo some constant.
--   
--   A convenient type for working with integers modulo some constant. It
--   saves you from manually wrapping numeric operations all over the place
--   and prevents a range of simple mistakes. <tt>Integer <a>Mod</a> 7</tt>
--   is the type of integers (mod 7) backed by <tt>Integer</tt>. We also
--   have some cute syntax for these types like <tt>ℤ/7</tt> for integers
--   modulo 7.
@package modular-arithmetic
@version 1.2.1.5


-- | Types for working with integers modulo some constant.
module Data.Modular

-- | Wraps an underlying <tt>Integeral</tt> type <tt>i</tt> in a newtype
--   annotated with the bound <tt>n</tt>.
data i `Mod` (n :: Nat)

-- | Extract the underlying integral value from a modular type.
unMod :: (i `Mod` n) -> i

-- | Injects a value of the underlying type into the modulus type, wrapping
--   as appropriate.
toMod :: forall n i. (Integral i, KnownNat n) => i -> i `Mod` n

-- | Wraps an integral number, converting between integral types.
toMod' :: forall n i j. (Integral i, Integral j, KnownNat n) => i -> j `Mod` n

-- | The modular inverse.
--   
--   <pre>
--   &gt;&gt;&gt; inv 3 :: ℤ/7
--   5
--   
--   &gt;&gt;&gt; 3 * 5 :: ℤ/7
--   1
--   </pre>
--   
--   Note that only numbers coprime to <tt>n</tt> have an inverse modulo
--   <tt>n</tt>:
--   
--   <pre>
--   inv 6 :: ℤ/15
--   </pre>
--   
--   <ul>
--   <li>** Exception: divide by 6 (mod 15), non-coprime to modulus</li>
--   </ul>
inv :: forall n i. (KnownNat n, Integral i) => Mod i n -> Mod i n

-- | A synonym for <tt>Mod</tt>, inspired by the ℤ/n syntax from
--   mathematics.
type (/) = Mod

-- | A synonym for Integer, also inspired by the ℤ/n syntax.
type ℤ = Integer

-- | Convert an integral number <tt>i</tt> into a <tt><a>Mod</a></tt> value
--   given modular bound <tt>n</tt> at type level.
modVal :: forall i proxy n. (Integral i, KnownNat n) => i -> proxy n -> Mod i n

-- | A modular number with an unknown bound.
data SomeMod i

-- | Convert an integral number <tt>i</tt> into a <tt><a>Mod</a></tt> value
--   with an unknown modulus.
someModVal :: Integral i => i -> Integer -> Maybe (SomeMod i)
instance GHC.Classes.Ord i => GHC.Classes.Ord (Data.Modular.Mod i n)
instance GHC.Classes.Eq i => GHC.Classes.Eq (Data.Modular.Mod i n)
instance GHC.Show.Show i => GHC.Show.Show (Data.Modular.SomeMod i)
instance GHC.Show.Show i => GHC.Show.Show (Data.Modular.Mod i n)
instance (GHC.Read.Read i, GHC.Real.Integral i, GHC.TypeNats.KnownNat n) => GHC.Read.Read (Data.Modular.Mod i n)
instance (GHC.Real.Integral i, GHC.TypeNats.KnownNat n) => GHC.Num.Num (Data.Modular.Mod i n)
instance (GHC.Real.Integral i, GHC.TypeNats.KnownNat n) => GHC.Enum.Enum (Data.Modular.Mod i n)
instance (GHC.Real.Integral i, GHC.TypeNats.KnownNat n) => GHC.Enum.Bounded (Data.Modular.Mod i n)
instance (GHC.Real.Integral i, GHC.TypeNats.KnownNat n) => GHC.Real.Real (Data.Modular.Mod i n)
instance (GHC.Real.Integral i, GHC.TypeNats.KnownNat n) => GHC.Real.Integral (Data.Modular.Mod i n)