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


-- | Coya monoids
--   
--   Take some log semiring R. Then, for any two x,y :: R, the following
--   holds:
--   
--   x ^ log y == y ^ log x == e ^ (log x * log y)
--   
--   A Coya monoid is some commutative monoid (R, y = x ^ log y. The
--   following laws hold:
--   
--   e # x = x (Left Identity)
--   
--   x # e = x (Right Identity)
--   
--   (x z == x z) (Associativity)
--   
--   x x (Commutativity)
--   
--   If the R is a poset where all elements in R are greater than one, then
--   R also forms a group:
--   
--   x # (e ^ (1 / log (x))) == x
@package coya
@version 0.1


-- | Consider some log-semiring R. Then, for any two x,y :: R, the
--   following holds:
--   
--   <pre>
--   x ^ log y == y ^ log x == e ^ (log x * log y)
--   </pre>
--   
--   Coya is a commutative monoid (R, y = x ^ log y.
--   
--   The following laws hold:
--   
--   <ul>
--   <li><i><b>Left Identity</b></i> <tt>e # x == x</tt></li>
--   <li><i><b>Right Identity</b></i> <tt>x # e == x</tt></li>
--   <li><i><b>Associativity</b></i> <tt>(x # y) # z == x # (y #
--   z)</tt></li>
--   <li><i><b>Commutativity</b></i> <tt>x # y == y # x</tt></li>
--   </ul>
--   
--   If R is a poset where all elements in R are greater than one, then R
--   also forms a group:
--   
--   <pre>
--   x # (exp (1 / log x)) = x
--   </pre>
module Coya

-- | The <a>Coya</a> monoid. Its <tt>semigroup</tt> instance is a binary
--   operation that distributes over multiplication, i.e:
--   
--   <pre>
--   <a>Coya</a> x <a>&lt;&gt;</a> (<a>Coya</a> y <tt>*</tt> <a>Coya</a> z) <a>==</a> (<a>Coya</a> x &lt;&gt; <a>Coya</a> y) <tt>*</tt> (<a>Coya</a> x &lt;&gt; <a>Coya</a> z)
--   </pre>
--   
--   The <a>Semiring</a> and <a>Num</a> instances simply lift the
--   underlying type's.
newtype Coya a
Coya :: a -> Coya a
[getCoya] :: Coya a -> a

-- | The <a>Coya</a> monoid constrained to numbers which are greater than
--   1. This ensures that the group property of inversion holds:
--   
--   <pre>
--   x <a>&lt;&gt;</a> (<a>exp</a> (1 <a>/</a> <a>log</a> x)) <a>==</a> x
--   </pre>
newtype CoyaGroup a
CoyaGroup :: Refined (From 1) (Coya a) -> CoyaGroup a
[getCoyaGroup] :: CoyaGroup a -> Refined (From 1) (Coya a)

-- | A smart constructor for <a>CoyaGroup</a>.
coyaGroup :: forall a. (Ord a, Num a) => a -> Maybe (CoyaGroup a)
instance GHC.Read.Read a => GHC.Read.Read (Coya.Coya a)
instance GHC.Show.Show a => GHC.Show.Show (Coya.Coya a)
instance Data.Primitive.Types.Prim a => Data.Primitive.Types.Prim (Coya.Coya a)
instance Foreign.Storable.Storable a => Foreign.Storable.Storable (Coya.Coya a)
instance GHC.Real.RealFrac a => GHC.Real.RealFrac (Coya.Coya a)
instance GHC.Float.RealFloat a => GHC.Float.RealFloat (Coya.Coya a)
instance GHC.Real.Real a => GHC.Real.Real (Coya.Coya a)
instance GHC.Real.Fractional a => GHC.Real.Fractional (Coya.Coya a)
instance GHC.Float.Floating a => GHC.Float.Floating (Coya.Coya a)
instance GHC.Num.Num a => GHC.Num.Num (Coya.Coya a)
instance Data.Semiring.Ring a => Data.Semiring.Ring (Coya.Coya a)
instance Data.Semiring.Semiring a => Data.Semiring.Semiring (Coya.Coya a)
instance GHC.Classes.Ord a => GHC.Classes.Ord (Coya.Coya a)
instance GHC.Classes.Eq a => GHC.Classes.Eq (Coya.Coya a)
instance (GHC.Float.Floating a, GHC.Classes.Ord a) => GHC.Base.Semigroup (Coya.CoyaGroup a)
instance (GHC.Float.Floating a, GHC.Classes.Ord a) => GHC.Base.Monoid (Coya.CoyaGroup a)
instance (GHC.Float.Floating a, GHC.Classes.Ord a) => Data.Group.Group (Coya.CoyaGroup a)
instance GHC.Float.Floating a => GHC.Base.Semigroup (Coya.Coya a)
instance GHC.Float.Floating a => GHC.Base.Monoid (Coya.Coya a)