-- 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 2.0.0.3


-- | 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)

-- | The modulus has to be a non-zero type-level natural number.
type Modulus n = (KnownNat n, 1 <= n)

-- | Extract the underlying integral value from a modular type.
--   
--   <pre>
--   &gt;&gt;&gt; unMod (10 :: ℤ/4)
--   2
--   </pre>
unMod :: (i `Mod` n) -> i

-- | Injects a value of the underlying type into the modulus type, wrapping
--   as appropriate.
--   
--   If <tt>n</tt> is ambiguous, you can specify it with
--   <tt>TypeApplications</tt>:
--   
--   <pre>
--   &gt;&gt;&gt; toMod @6 10
--   4
--   </pre>
--   
--   Note that <tt>n</tt> cannot be 0.
toMod :: forall n i. (Integral i, Modulus n) => i -> i `Mod` n

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

-- | The modular inverse.
--   
--   <pre>
--   &gt;&gt;&gt; inv 3 :: Maybe (ℤ/7)
--   Just 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>
--   &gt;&gt;&gt; inv 6 :: Maybe (ℤ/15)
--   Nothing
--   </pre>
inv :: forall n i. (Modulus n, Integral i) => (i / n) -> Maybe (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
--   with the type-level modulus <tt>n</tt> specified with a proxy
--   argument.
--   
--   This lets you use <a>toMod</a> without <tt>TypeApplications</tt> in
--   contexts where <tt>n</tt> is ambiguous.
modVal :: forall i proxy n. (Integral i, Modulus n) => i -> proxy n -> Mod i n

-- | A modular number with an unknown modulus.
--   
--   Conceptually <tt>SomeMod i = ∃n. i/n</tt>.
data SomeMod i

-- | Convert an integral number <tt>i</tt> into <tt><a>SomeMod</a></tt>
--   with the modulus given at runtime.
--   
--   That is, given <tt>i :: ℤ</tt>, <tt>someModVal i modulus</tt> is
--   equivalent to <tt>i :: ℤ/modulus</tt> except we don't know
--   <tt>modulus</tt> statically.
--   
--   <pre>
--   &gt;&gt;&gt; someModVal 10 4
--   Just (someModVal 2 4)
--   </pre>
--   
--   Will return <a>Nothing</a> if the modulus is 0 or negative:
--   
--   <pre>
--   &gt;&gt;&gt; someModVal 10 (-10)
--   Nothing
--   </pre>
--   
--   <pre>
--   &gt;&gt;&gt; someModVal 10 0
--   Nothing
--   </pre>
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.Read.Read i, GHC.Real.Integral i, Data.Modular.Modulus n) => GHC.Read.Read (Data.Modular.Mod i n)
instance (GHC.Real.Integral i, Data.Modular.Modulus n) => GHC.Num.Num (Data.Modular.Mod i n)
instance (GHC.Real.Integral i, Data.Modular.Modulus n) => GHC.Enum.Enum (Data.Modular.Mod i n)
instance (GHC.Real.Integral i, Data.Modular.Modulus n) => GHC.Enum.Bounded (Data.Modular.Mod i n)
instance (GHC.Real.Integral i, Data.Modular.Modulus n) => GHC.Real.Real (Data.Modular.Mod i n)
instance (GHC.Real.Integral i, Data.Modular.Modulus n) => GHC.Real.Fractional (Data.Modular.Mod i n)
instance GHC.Show.Show i => GHC.Show.Show (Data.Modular.Mod i n)