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


-- | Ring-like objects.
--   
--   Semirings, rings, division rings, modules, and algebras.
@package rings
@version 0.1.2

module Data.Semigroup.Additive

-- | Hyphenation operator.
type ( g - f ) a = f (g a)
infixr 1 -

-- | A commutative <a>Semigroup</a> under <a>+</a>.
newtype Additive a
Additive :: a -> Additive a
[unAdditive] :: Additive a -> a

-- | Additive unit of a semiring.
zero :: (Additive - Monoid) a => a

-- | Additive semigroup operation on a semiring.
--   
--   <pre>
--   &gt;&gt;&gt; Dual [2] + Dual [3] :: Dual [Int]
--   Dual {getDual = [3,2]}
--   </pre>
(+) :: (Additive - Semigroup) a => a -> a -> a
infixl 6 +

-- | Subtract two elements.
subtract :: (Additive - Group) a => a -> a -> a

-- | A (potentially non-commutative) <a>Semigroup</a> under <a>*</a>.
newtype Multiplicative a
Multiplicative :: a -> Multiplicative a
[unMultiplicative] :: Multiplicative a -> a

-- | Multiplicative unit of a semiring.
one :: (Multiplicative - Monoid) a => a

-- | Multiplicative semigroup operation on a semiring.
--   
--   <pre>
--   &gt;&gt;&gt; Dual [2] * Dual [3] :: Dual [Int]
--   Dual {getDual = [5]}
--   </pre>
(*) :: (Multiplicative - Semigroup) a => a -> a -> a
infixl 7 *

-- | Reciprocal of a multiplicative group element.
--   
--   <pre>
--   x <a>/</a> y = x <a>*</a> <a>recip</a> y
--   x <a>\\</a> y = <a>recip</a> x <a>*</a> y
--   </pre>
--   
--   <pre>
--   &gt;&gt;&gt; recip (3 :+ 4) :: Complex Rational
--   3 % 25 :+ (-4) % 25
--   
--   &gt;&gt;&gt; recip (3 :+ 4) :: Complex Double
--   0.12 :+ (-0.16)
--   
--   &gt;&gt;&gt; recip (3 :+ 4) :: Complex Pico
--   0.120000000000 :+ -0.160000000000
--   </pre>
recip :: (Multiplicative - Group) a => a -> a

-- | Right division by a multiplicative group element.
(/) :: (Multiplicative - Group) a => a -> a -> a
infixl 7 /

-- | Left division by a multiplicative group element.
--   
--   When <a>*</a> is commutative we must have:
--   
--   <pre>
--   x <a>\\</a> y = y <a>/</a> x
--   </pre>
(\\) :: (Multiplicative - Group) a => a -> a -> a
infixl 7 \\
instance GHC.Base.Functor Data.Semigroup.Additive.Multiplicative
instance GHC.Show.Show a => GHC.Show.Show (Data.Semigroup.Additive.Multiplicative a)
instance GHC.Classes.Ord a => GHC.Classes.Ord (Data.Semigroup.Additive.Multiplicative a)
instance GHC.Generics.Generic (Data.Semigroup.Additive.Multiplicative a)
instance GHC.Classes.Eq a => GHC.Classes.Eq (Data.Semigroup.Additive.Multiplicative a)
instance GHC.Base.Functor Data.Semigroup.Additive.Additive
instance GHC.Show.Show a => GHC.Show.Show (Data.Semigroup.Additive.Additive a)
instance GHC.Classes.Ord a => GHC.Classes.Ord (Data.Semigroup.Additive.Additive a)
instance GHC.Generics.Generic (Data.Semigroup.Additive.Additive a)
instance GHC.Classes.Eq a => GHC.Classes.Eq (Data.Semigroup.Additive.Additive a)
instance GHC.Base.Applicative Data.Semigroup.Additive.Multiplicative
instance Data.Distributive.Distributive Data.Semigroup.Additive.Multiplicative
instance Data.Functor.Rep.Representable Data.Semigroup.Additive.Multiplicative
instance ((Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a, (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Multiplicative GHC.Base.Semigroup a) => GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative (Data.Complex.Complex a))
instance ((Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a, (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Multiplicative GHC.Base.Monoid a) => GHC.Base.Monoid (Data.Semigroup.Additive.Multiplicative (Data.Complex.Complex a))
instance ((Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a, (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Multiplicative Data.Group.Group a) => Data.Magma.Magma (Data.Semigroup.Additive.Multiplicative (Data.Complex.Complex a))
instance ((Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a, (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Multiplicative Data.Group.Group a) => Data.Group.Quasigroup (Data.Semigroup.Additive.Multiplicative (Data.Complex.Complex a))
instance ((Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a, (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Multiplicative Data.Group.Group a) => Data.Group.Loop (Data.Semigroup.Additive.Multiplicative (Data.Complex.Complex a))
instance ((Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a, (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Multiplicative Data.Group.Group a) => Data.Group.Group (Data.Semigroup.Additive.Multiplicative (Data.Complex.Complex a))
instance ((Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup a, (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Multiplicative GHC.Base.Semigroup a) => GHC.Base.Semigroup (Data.Semigroup.Additive.Additive (GHC.Real.Ratio a))
instance ((Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid a, (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Multiplicative GHC.Base.Monoid a) => GHC.Base.Monoid (Data.Semigroup.Additive.Additive (GHC.Real.Ratio a))
instance ((Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a, (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Multiplicative GHC.Base.Monoid a) => Data.Magma.Magma (Data.Semigroup.Additive.Additive (GHC.Real.Ratio a))
instance ((Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a, (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Multiplicative GHC.Base.Monoid a) => Data.Group.Quasigroup (Data.Semigroup.Additive.Additive (GHC.Real.Ratio a))
instance ((Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a, (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Multiplicative GHC.Base.Monoid a) => Data.Group.Loop (Data.Semigroup.Additive.Additive (GHC.Real.Ratio a))
instance ((Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a, (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Multiplicative GHC.Base.Monoid a) => Data.Group.Group (Data.Semigroup.Additive.Additive (GHC.Real.Ratio a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative [a])
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semigroup.Additive.Multiplicative [a])
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative (GHC.Base.NonEmpty a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative (Data.Semigroup.Min a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semigroup.Additive.Multiplicative (Data.Semigroup.Min a))
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative GHC.Types.Int)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative GHC.Int.Int8)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative GHC.Int.Int16)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative GHC.Int.Int32)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative GHC.Int.Int64)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative GHC.Integer.Type.Integer)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative GHC.Types.Word)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative GHC.Word.Word8)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative GHC.Word.Word16)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative GHC.Word.Word32)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative GHC.Word.Word64)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative GHC.Natural.Natural)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative Data.Fixed.Uni)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative Data.Fixed.Deci)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative Data.Fixed.Centi)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative Data.Fixed.Milli)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative Data.Fixed.Micro)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative Data.Fixed.Nano)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative Data.Fixed.Pico)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative GHC.Types.Float)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative Foreign.C.Types.CFloat)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative GHC.Types.Double)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative Foreign.C.Types.CDouble)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Multiplicative GHC.Types.Int)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Multiplicative GHC.Int.Int8)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Multiplicative GHC.Int.Int16)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Multiplicative GHC.Int.Int32)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Multiplicative GHC.Int.Int64)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Multiplicative GHC.Integer.Type.Integer)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Multiplicative GHC.Types.Word)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Multiplicative GHC.Word.Word8)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Multiplicative GHC.Word.Word16)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Multiplicative GHC.Word.Word32)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Multiplicative GHC.Word.Word64)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Multiplicative GHC.Natural.Natural)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Multiplicative Data.Fixed.Uni)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Multiplicative Data.Fixed.Deci)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Multiplicative Data.Fixed.Centi)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Multiplicative Data.Fixed.Milli)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Multiplicative Data.Fixed.Micro)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Multiplicative Data.Fixed.Nano)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Multiplicative Data.Fixed.Pico)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Multiplicative GHC.Types.Float)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Multiplicative Foreign.C.Types.CFloat)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Multiplicative GHC.Types.Double)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Multiplicative Foreign.C.Types.CDouble)
instance Data.Magma.Magma (Data.Semigroup.Additive.Multiplicative Data.Fixed.Uni)
instance Data.Magma.Magma (Data.Semigroup.Additive.Multiplicative Data.Fixed.Deci)
instance Data.Magma.Magma (Data.Semigroup.Additive.Multiplicative Data.Fixed.Centi)
instance Data.Magma.Magma (Data.Semigroup.Additive.Multiplicative Data.Fixed.Milli)
instance Data.Magma.Magma (Data.Semigroup.Additive.Multiplicative Data.Fixed.Micro)
instance Data.Magma.Magma (Data.Semigroup.Additive.Multiplicative Data.Fixed.Nano)
instance Data.Magma.Magma (Data.Semigroup.Additive.Multiplicative Data.Fixed.Pico)
instance Data.Magma.Magma (Data.Semigroup.Additive.Multiplicative GHC.Types.Float)
instance Data.Magma.Magma (Data.Semigroup.Additive.Multiplicative Foreign.C.Types.CFloat)
instance Data.Magma.Magma (Data.Semigroup.Additive.Multiplicative GHC.Types.Double)
instance Data.Magma.Magma (Data.Semigroup.Additive.Multiplicative Foreign.C.Types.CDouble)
instance Data.Group.Quasigroup (Data.Semigroup.Additive.Multiplicative Data.Fixed.Uni)
instance Data.Group.Quasigroup (Data.Semigroup.Additive.Multiplicative Data.Fixed.Deci)
instance Data.Group.Quasigroup (Data.Semigroup.Additive.Multiplicative Data.Fixed.Centi)
instance Data.Group.Quasigroup (Data.Semigroup.Additive.Multiplicative Data.Fixed.Milli)
instance Data.Group.Quasigroup (Data.Semigroup.Additive.Multiplicative Data.Fixed.Micro)
instance Data.Group.Quasigroup (Data.Semigroup.Additive.Multiplicative Data.Fixed.Nano)
instance Data.Group.Quasigroup (Data.Semigroup.Additive.Multiplicative Data.Fixed.Pico)
instance Data.Group.Quasigroup (Data.Semigroup.Additive.Multiplicative GHC.Types.Float)
instance Data.Group.Quasigroup (Data.Semigroup.Additive.Multiplicative Foreign.C.Types.CFloat)
instance Data.Group.Quasigroup (Data.Semigroup.Additive.Multiplicative GHC.Types.Double)
instance Data.Group.Quasigroup (Data.Semigroup.Additive.Multiplicative Foreign.C.Types.CDouble)
instance Data.Group.Loop (Data.Semigroup.Additive.Multiplicative Data.Fixed.Uni)
instance Data.Group.Loop (Data.Semigroup.Additive.Multiplicative Data.Fixed.Deci)
instance Data.Group.Loop (Data.Semigroup.Additive.Multiplicative Data.Fixed.Centi)
instance Data.Group.Loop (Data.Semigroup.Additive.Multiplicative Data.Fixed.Milli)
instance Data.Group.Loop (Data.Semigroup.Additive.Multiplicative Data.Fixed.Micro)
instance Data.Group.Loop (Data.Semigroup.Additive.Multiplicative Data.Fixed.Nano)
instance Data.Group.Loop (Data.Semigroup.Additive.Multiplicative Data.Fixed.Pico)
instance Data.Group.Loop (Data.Semigroup.Additive.Multiplicative GHC.Types.Float)
instance Data.Group.Loop (Data.Semigroup.Additive.Multiplicative Foreign.C.Types.CFloat)
instance Data.Group.Loop (Data.Semigroup.Additive.Multiplicative GHC.Types.Double)
instance Data.Group.Loop (Data.Semigroup.Additive.Multiplicative Foreign.C.Types.CDouble)
instance Data.Group.Group (Data.Semigroup.Additive.Multiplicative Data.Fixed.Uni)
instance Data.Group.Group (Data.Semigroup.Additive.Multiplicative Data.Fixed.Deci)
instance Data.Group.Group (Data.Semigroup.Additive.Multiplicative Data.Fixed.Centi)
instance Data.Group.Group (Data.Semigroup.Additive.Multiplicative Data.Fixed.Milli)
instance Data.Group.Group (Data.Semigroup.Additive.Multiplicative Data.Fixed.Micro)
instance Data.Group.Group (Data.Semigroup.Additive.Multiplicative Data.Fixed.Nano)
instance Data.Group.Group (Data.Semigroup.Additive.Multiplicative Data.Fixed.Pico)
instance Data.Group.Group (Data.Semigroup.Additive.Multiplicative GHC.Types.Float)
instance Data.Group.Group (Data.Semigroup.Additive.Multiplicative Foreign.C.Types.CFloat)
instance Data.Group.Group (Data.Semigroup.Additive.Multiplicative GHC.Types.Double)
instance Data.Group.Group (Data.Semigroup.Additive.Multiplicative Foreign.C.Types.CDouble)
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Multiplicative GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative (GHC.Real.Ratio a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Multiplicative GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semigroup.Additive.Multiplicative (GHC.Real.Ratio a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Multiplicative GHC.Base.Monoid a => Data.Magma.Magma (Data.Semigroup.Additive.Multiplicative (GHC.Real.Ratio a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Multiplicative GHC.Base.Monoid a => Data.Group.Quasigroup (Data.Semigroup.Additive.Multiplicative (GHC.Real.Ratio a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Multiplicative GHC.Base.Monoid a => Data.Group.Loop (Data.Semigroup.Additive.Multiplicative (GHC.Real.Ratio a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Multiplicative GHC.Base.Monoid a => Data.Group.Group (Data.Semigroup.Additive.Multiplicative (GHC.Real.Ratio a))
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative ())
instance GHC.Base.Monoid (Data.Semigroup.Additive.Multiplicative ())
instance Data.Magma.Magma (Data.Semigroup.Additive.Multiplicative ())
instance Data.Group.Quasigroup (Data.Semigroup.Additive.Multiplicative ())
instance Data.Group.Loop (Data.Semigroup.Additive.Multiplicative ())
instance Data.Group.Group (Data.Semigroup.Additive.Multiplicative ())
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative GHC.Types.Bool)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Multiplicative GHC.Types.Bool)
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Multiplicative GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative (Data.Semigroup.Internal.Dual a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Multiplicative GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semigroup.Additive.Multiplicative (Data.Semigroup.Internal.Dual a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Multiplicative GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative (Data.Ord.Down a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Multiplicative GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semigroup.Additive.Multiplicative (Data.Ord.Down a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Multiplicative GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative (Data.Semigroup.Max a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Multiplicative GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semigroup.Additive.Multiplicative (Data.Semigroup.Max a))
instance ((Data.Semigroup.Additive.-) Data.Semigroup.Additive.Multiplicative GHC.Base.Semigroup a, (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Multiplicative GHC.Base.Semigroup b) => GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative (a, b))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Multiplicative GHC.Base.Semigroup b => GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative (a -> b))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Multiplicative GHC.Base.Monoid b => GHC.Base.Monoid (Data.Semigroup.Additive.Multiplicative (a -> b))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Multiplicative GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative (GHC.Maybe.Maybe a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Multiplicative GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semigroup.Additive.Multiplicative (GHC.Maybe.Maybe a))
instance ((Data.Semigroup.Additive.-) Data.Semigroup.Additive.Multiplicative GHC.Base.Semigroup a, (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Multiplicative GHC.Base.Semigroup b) => GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative (Data.Either.Either a b))
instance GHC.Classes.Ord a => GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative (Data.Set.Internal.Set a))
instance (GHC.Classes.Ord k, (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Multiplicative GHC.Base.Semigroup a) => GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative (Data.Map.Internal.Map k a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Multiplicative GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative (Data.IntMap.Internal.IntMap a))
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative Data.IntSet.Internal.IntSet)
instance (GHC.Classes.Ord k, (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Multiplicative GHC.Base.Monoid k, (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Multiplicative GHC.Base.Monoid a) => GHC.Base.Monoid (Data.Semigroup.Additive.Multiplicative (Data.Map.Internal.Map k a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Multiplicative GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semigroup.Additive.Multiplicative (Data.IntMap.Internal.IntMap a))
instance GHC.Base.Applicative Data.Semigroup.Additive.Additive
instance Data.Distributive.Distributive Data.Semigroup.Additive.Additive
instance Data.Functor.Rep.Representable Data.Semigroup.Additive.Additive
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Additive GHC.Types.Int)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Additive GHC.Int.Int8)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Additive GHC.Int.Int16)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Additive GHC.Int.Int32)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Additive GHC.Int.Int64)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Additive GHC.Integer.Type.Integer)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Additive GHC.Types.Word)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Additive GHC.Word.Word8)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Additive GHC.Word.Word16)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Additive GHC.Word.Word32)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Additive GHC.Word.Word64)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Additive GHC.Natural.Natural)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Additive Data.Fixed.Uni)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Additive Data.Fixed.Deci)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Additive Data.Fixed.Centi)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Additive Data.Fixed.Milli)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Additive Data.Fixed.Micro)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Additive Data.Fixed.Nano)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Additive Data.Fixed.Pico)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Additive GHC.Types.Float)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Additive Foreign.C.Types.CFloat)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Additive GHC.Types.Double)
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Additive Foreign.C.Types.CDouble)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Additive GHC.Types.Int)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Additive GHC.Int.Int8)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Additive GHC.Int.Int16)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Additive GHC.Int.Int32)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Additive GHC.Int.Int64)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Additive GHC.Integer.Type.Integer)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Additive GHC.Types.Word)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Additive GHC.Word.Word8)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Additive GHC.Word.Word16)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Additive GHC.Word.Word32)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Additive GHC.Word.Word64)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Additive GHC.Natural.Natural)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Additive Data.Fixed.Uni)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Additive Data.Fixed.Deci)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Additive Data.Fixed.Centi)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Additive Data.Fixed.Milli)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Additive Data.Fixed.Micro)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Additive Data.Fixed.Nano)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Additive Data.Fixed.Pico)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Additive GHC.Types.Float)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Additive Foreign.C.Types.CFloat)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Additive GHC.Types.Double)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Additive Foreign.C.Types.CDouble)
instance Data.Magma.Magma (Data.Semigroup.Additive.Additive GHC.Types.Int)
instance Data.Magma.Magma (Data.Semigroup.Additive.Additive GHC.Int.Int8)
instance Data.Magma.Magma (Data.Semigroup.Additive.Additive GHC.Int.Int16)
instance Data.Magma.Magma (Data.Semigroup.Additive.Additive GHC.Int.Int32)
instance Data.Magma.Magma (Data.Semigroup.Additive.Additive GHC.Int.Int64)
instance Data.Magma.Magma (Data.Semigroup.Additive.Additive GHC.Integer.Type.Integer)
instance Data.Magma.Magma (Data.Semigroup.Additive.Additive Data.Fixed.Uni)
instance Data.Magma.Magma (Data.Semigroup.Additive.Additive Data.Fixed.Deci)
instance Data.Magma.Magma (Data.Semigroup.Additive.Additive Data.Fixed.Centi)
instance Data.Magma.Magma (Data.Semigroup.Additive.Additive Data.Fixed.Milli)
instance Data.Magma.Magma (Data.Semigroup.Additive.Additive Data.Fixed.Micro)
instance Data.Magma.Magma (Data.Semigroup.Additive.Additive Data.Fixed.Nano)
instance Data.Magma.Magma (Data.Semigroup.Additive.Additive Data.Fixed.Pico)
instance Data.Magma.Magma (Data.Semigroup.Additive.Additive GHC.Types.Float)
instance Data.Magma.Magma (Data.Semigroup.Additive.Additive Foreign.C.Types.CFloat)
instance Data.Magma.Magma (Data.Semigroup.Additive.Additive GHC.Types.Double)
instance Data.Magma.Magma (Data.Semigroup.Additive.Additive Foreign.C.Types.CDouble)
instance Data.Group.Quasigroup (Data.Semigroup.Additive.Additive GHC.Types.Int)
instance Data.Group.Quasigroup (Data.Semigroup.Additive.Additive GHC.Int.Int8)
instance Data.Group.Quasigroup (Data.Semigroup.Additive.Additive GHC.Int.Int16)
instance Data.Group.Quasigroup (Data.Semigroup.Additive.Additive GHC.Int.Int32)
instance Data.Group.Quasigroup (Data.Semigroup.Additive.Additive GHC.Int.Int64)
instance Data.Group.Quasigroup (Data.Semigroup.Additive.Additive GHC.Integer.Type.Integer)
instance Data.Group.Quasigroup (Data.Semigroup.Additive.Additive Data.Fixed.Uni)
instance Data.Group.Quasigroup (Data.Semigroup.Additive.Additive Data.Fixed.Deci)
instance Data.Group.Quasigroup (Data.Semigroup.Additive.Additive Data.Fixed.Centi)
instance Data.Group.Quasigroup (Data.Semigroup.Additive.Additive Data.Fixed.Milli)
instance Data.Group.Quasigroup (Data.Semigroup.Additive.Additive Data.Fixed.Micro)
instance Data.Group.Quasigroup (Data.Semigroup.Additive.Additive Data.Fixed.Nano)
instance Data.Group.Quasigroup (Data.Semigroup.Additive.Additive Data.Fixed.Pico)
instance Data.Group.Quasigroup (Data.Semigroup.Additive.Additive GHC.Types.Float)
instance Data.Group.Quasigroup (Data.Semigroup.Additive.Additive Foreign.C.Types.CFloat)
instance Data.Group.Quasigroup (Data.Semigroup.Additive.Additive GHC.Types.Double)
instance Data.Group.Quasigroup (Data.Semigroup.Additive.Additive Foreign.C.Types.CDouble)
instance Data.Group.Loop (Data.Semigroup.Additive.Additive GHC.Types.Int)
instance Data.Group.Loop (Data.Semigroup.Additive.Additive GHC.Int.Int8)
instance Data.Group.Loop (Data.Semigroup.Additive.Additive GHC.Int.Int16)
instance Data.Group.Loop (Data.Semigroup.Additive.Additive GHC.Int.Int32)
instance Data.Group.Loop (Data.Semigroup.Additive.Additive GHC.Int.Int64)
instance Data.Group.Loop (Data.Semigroup.Additive.Additive GHC.Integer.Type.Integer)
instance Data.Group.Loop (Data.Semigroup.Additive.Additive Data.Fixed.Uni)
instance Data.Group.Loop (Data.Semigroup.Additive.Additive Data.Fixed.Deci)
instance Data.Group.Loop (Data.Semigroup.Additive.Additive Data.Fixed.Centi)
instance Data.Group.Loop (Data.Semigroup.Additive.Additive Data.Fixed.Milli)
instance Data.Group.Loop (Data.Semigroup.Additive.Additive Data.Fixed.Micro)
instance Data.Group.Loop (Data.Semigroup.Additive.Additive Data.Fixed.Nano)
instance Data.Group.Loop (Data.Semigroup.Additive.Additive Data.Fixed.Pico)
instance Data.Group.Loop (Data.Semigroup.Additive.Additive GHC.Types.Float)
instance Data.Group.Loop (Data.Semigroup.Additive.Additive Foreign.C.Types.CFloat)
instance Data.Group.Loop (Data.Semigroup.Additive.Additive GHC.Types.Double)
instance Data.Group.Loop (Data.Semigroup.Additive.Additive Foreign.C.Types.CDouble)
instance Data.Group.Group (Data.Semigroup.Additive.Additive GHC.Types.Int)
instance Data.Group.Group (Data.Semigroup.Additive.Additive GHC.Int.Int8)
instance Data.Group.Group (Data.Semigroup.Additive.Additive GHC.Int.Int16)
instance Data.Group.Group (Data.Semigroup.Additive.Additive GHC.Int.Int32)
instance Data.Group.Group (Data.Semigroup.Additive.Additive GHC.Int.Int64)
instance Data.Group.Group (Data.Semigroup.Additive.Additive GHC.Integer.Type.Integer)
instance Data.Group.Group (Data.Semigroup.Additive.Additive Data.Fixed.Uni)
instance Data.Group.Group (Data.Semigroup.Additive.Additive Data.Fixed.Deci)
instance Data.Group.Group (Data.Semigroup.Additive.Additive Data.Fixed.Centi)
instance Data.Group.Group (Data.Semigroup.Additive.Additive Data.Fixed.Milli)
instance Data.Group.Group (Data.Semigroup.Additive.Additive Data.Fixed.Micro)
instance Data.Group.Group (Data.Semigroup.Additive.Additive Data.Fixed.Nano)
instance Data.Group.Group (Data.Semigroup.Additive.Additive Data.Fixed.Pico)
instance Data.Group.Group (Data.Semigroup.Additive.Additive GHC.Types.Float)
instance Data.Group.Group (Data.Semigroup.Additive.Additive Foreign.C.Types.CFloat)
instance Data.Group.Group (Data.Semigroup.Additive.Additive GHC.Types.Double)
instance Data.Group.Group (Data.Semigroup.Additive.Additive Foreign.C.Types.CDouble)
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semigroup.Additive.Additive (Data.Complex.Complex a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semigroup.Additive.Additive (Data.Complex.Complex a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Magma.Magma (Data.Semigroup.Additive.Additive (Data.Complex.Complex a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Quasigroup (Data.Semigroup.Additive.Additive (Data.Complex.Complex a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Loop (Data.Semigroup.Additive.Additive (Data.Complex.Complex a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Group (Data.Semigroup.Additive.Additive (Data.Complex.Complex a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup b => GHC.Base.Semigroup (Data.Semigroup.Additive.Additive (a -> b))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group b => Data.Magma.Magma (Data.Semigroup.Additive.Additive (a -> b))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group b => Data.Group.Quasigroup (Data.Semigroup.Additive.Additive (a -> b))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group b => Data.Group.Loop (Data.Semigroup.Additive.Additive (a -> b))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group b => Data.Group.Group (Data.Semigroup.Additive.Additive (a -> b))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid b => GHC.Base.Monoid (Data.Semigroup.Additive.Additive (a -> b))
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Additive [a])
instance GHC.Base.Monoid (Data.Semigroup.Additive.Additive [a])
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Additive (GHC.Base.NonEmpty a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semigroup.Additive.Additive (Data.Ord.Down a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semigroup.Additive.Additive (Data.Ord.Down a))
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Additive ())
instance GHC.Base.Monoid (Data.Semigroup.Additive.Additive ())
instance Data.Magma.Magma (Data.Semigroup.Additive.Additive ())
instance Data.Group.Quasigroup (Data.Semigroup.Additive.Additive ())
instance Data.Group.Loop (Data.Semigroup.Additive.Additive ())
instance Data.Group.Group (Data.Semigroup.Additive.Additive ())
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Additive GHC.Types.Bool)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Additive GHC.Types.Bool)
instance ((Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup a, (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup b) => GHC.Base.Semigroup (Data.Semigroup.Additive.Additive (a, b))
instance ((Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid a, (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid b) => GHC.Base.Monoid (Data.Semigroup.Additive.Additive (a, b))
instance ((Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup a, (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup b, (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup c) => GHC.Base.Semigroup (Data.Semigroup.Additive.Additive (a, b, c))
instance ((Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid a, (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid b, (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid c) => GHC.Base.Monoid (Data.Semigroup.Additive.Additive (a, b, c))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semigroup.Additive.Additive (GHC.Maybe.Maybe a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup a => GHC.Base.Monoid (Data.Semigroup.Additive.Additive (GHC.Maybe.Maybe a))
instance ((Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup a, (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup b) => GHC.Base.Semigroup (Data.Semigroup.Additive.Additive (Data.Either.Either a b))
instance GHC.Classes.Ord a => GHC.Base.Semigroup (Data.Semigroup.Additive.Additive (Data.Set.Internal.Set a))
instance (GHC.Classes.Ord k, (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup a) => GHC.Base.Semigroup (Data.Semigroup.Additive.Additive (Data.Map.Internal.Map k a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semigroup.Additive.Additive (Data.IntMap.Internal.IntMap a))
instance GHC.Base.Semigroup (Data.Semigroup.Additive.Additive Data.IntSet.Internal.IntSet)
instance GHC.Base.Monoid (Data.Semigroup.Additive.Additive Data.IntSet.Internal.IntSet)
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup a => GHC.Base.Monoid (Data.Semigroup.Additive.Additive (Data.IntMap.Internal.IntMap a))
instance GHC.Classes.Ord a => GHC.Base.Monoid (Data.Semigroup.Additive.Additive (Data.Set.Internal.Set a))
instance (GHC.Classes.Ord k, (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup a) => GHC.Base.Monoid (Data.Semigroup.Additive.Additive (Data.Map.Internal.Map k a))

module Data.Semigroup.Property

-- | &lt;math&gt;
--   
--   A semigroup must right-associate addition.
--   
--   This is a required property for semigroups.
associative_addition_on :: (Additive - Semigroup) r => Rel r b -> r -> r -> r -> b

-- | &lt;math&gt;
--   
--   A semigroup must right-associate multiplication.
--   
--   This is a required property for semigroups.
associative_multiplication_on :: (Multiplicative - Semigroup) r => Rel r b -> r -> r -> r -> b

-- | &lt;math&gt;
--   
--   A semigroup with a right-neutral additive identity must satisfy:
--   
--   <pre>
--   <a>neutral_addition_on</a> (<a>==</a>) <a>zero</a> r = <a>True</a>
--   </pre>
--   
--   Or, equivalently:
--   
--   <pre>
--   <a>zero</a> <a>+</a> r = r
--   </pre>
--   
--   This is a required property for additive monoids.
neutral_addition_on :: (Additive - Monoid) r => Rel r b -> r -> b

-- | &lt;math&gt;
--   
--   A semigroup with a right-neutral multiplicative identity must satisfy:
--   
--   <pre>
--   <a>neutral_multiplication_on</a> (<a>==</a>) <a>one</a> r = <a>True</a>
--   </pre>
--   
--   Or, equivalently:
--   
--   <pre>
--   <a>one</a> <a>*</a> r = r
--   </pre>
--   
--   This is a required property for multiplicative monoids.
neutral_multiplication_on :: (Multiplicative - Monoid) r => Rel r b -> r -> b

-- | &lt;math&gt;
--   
--   This is a an optional property for semigroups, and a required property
--   for semirings.
commutative_addition_on :: (Additive - Semigroup) r => Rel r b -> r -> r -> b

-- | &lt;math&gt;
--   
--   This is a an optional property for semigroups, and a optional property
--   for semirings and rings.
commutative_multiplication_on :: (Multiplicative - Semigroup) r => Rel r b -> r -> r -> b

-- | &lt;math&gt;
--   
--   If <i>R</i> is right-cancellative wrt addition then for all <i>a</i>
--   the section <i>(a +)</i> is injective.
--   
--   See <a>https://en.wikipedia.org/wiki/Cancellation_property</a>
cancellative_addition_on :: (Additive - Semigroup) r => Rel r Bool -> r -> r -> r -> Bool

-- | &lt;math&gt;
--   
--   If <i>R</i> is right-cancellative wrt multiplication then for all
--   <i>a</i> the section <i>(a *)</i> is injective.
cancellative_multiplication_on :: (Multiplicative - Semigroup) r => Rel r Bool -> r -> r -> r -> Bool

-- | Idempotency property for additive semigroups.
--   
--   See <a>https://en.wikipedia.org/wiki/Band_(mathematics)</a>.
--   
--   This is a an optional property for semigroups and semirings.
--   
--   This is a required property for lattices.
idempotent_addition_on :: (Additive - Semigroup) r => Rel r b -> r -> b

-- | Idempotency property for multplicative semigroups.
--   
--   See <a>https://en.wikipedia.org/wiki/Band_(mathematics)</a>.
--   
--   This is a an optional property for semigroups and semirings.
--   
--   This is a required property for lattices.
idempotent_multiplication_on :: (Multiplicative - Semigroup) r => Rel r b -> r -> b

-- | This is a required property for additive semigroup morphisms.
morphism_additive_on :: (Additive - Semigroup) r => (Additive - Semigroup) s => Rel s b -> (r -> s) -> r -> r -> b

-- | This is a required property for multiplicative semigroup morphisms.
morphism_multiplicative_on :: (Multiplicative - Semigroup) r => (Multiplicative - Semigroup) s => Rel s b -> (r -> s) -> r -> r -> b

-- | This is a required property for additive monoid morphisms.
morphism_additive_on' :: (Additive - Monoid) r => (Additive - Monoid) s => Rel s b -> (r -> s) -> b

-- | This is a required property for multiplicative monoid morphisms.
morphism_multiplicative_on' :: (Multiplicative - Monoid) r => (Multiplicative - Monoid) s => Rel s b -> (r -> s) -> b

module Data.Semiring

-- | Hyphenation operator.
type ( g - f ) a = f (g a)
infixr 1 -
type PresemiringLaw a = ((Additive - Semigroup) a, (Multiplicative - Semigroup) a)

-- | Right pre-semirings. and (non-unital and unital) right semirings.
--   
--   A right pre-semiring (sometimes referred to as a bisemigroup) is a
--   type <i>R</i> endowed with two associative binary (i.e. semigroup)
--   operations: <a>+</a> and <a>*</a>, along with a right-distributivity
--   property connecting them:
--   
--   <i>Distributivity</i>
--   
--   <pre>
--   (a <a>+</a> b) <a>*</a> c = (a <a>*</a> c) <a>+</a> (b <a>*</a> c)
--   </pre>
--   
--   Note that addition and multiplication needn't be commutative.
--   
--   See the properties module for a detailed specification of the laws.
class PresemiringLaw a => Presemiring a

-- | Additive semigroup operation on a semiring.
--   
--   <pre>
--   &gt;&gt;&gt; Dual [2] + Dual [3] :: Dual [Int]
--   Dual {getDual = [3,2]}
--   </pre>
(+) :: (Additive - Semigroup) a => a -> a -> a
infixl 6 +

-- | Multiplicative semigroup operation on a semiring.
--   
--   <pre>
--   &gt;&gt;&gt; Dual [2] * Dual [3] :: Dual [Int]
--   Dual {getDual = [5]}
--   </pre>
(*) :: (Multiplicative - Semigroup) a => a -> a -> a
infixl 7 *

-- | Evaluate a non-empty presemiring sum.
sum1 :: (Additive - Semigroup) a => Foldable1 f => f a -> a

-- | Evaluate a non-empty presemiring sum using a given presemiring.
--   
--   <pre>
--   &gt;&gt;&gt; evalWith1 Max $ (1 :| [2..5 :: Int]) :| [1 :| [2..5 :: Int]]
--   | Fold over a non-empty collection using the additive operation of an arbitrary semiring.
--   </pre>
--   
--   <pre>
--   &gt;&gt;&gt; sumWith1 First $ (1 :| [2..5 :: Int]) * (1 :| [2..5 :: Int])
--   First {getFirst = 1}
--   
--   &gt;&gt;&gt; sumWith1 First $ Nothing :| [Just (5 :: Int), Just 6,  Nothing]
--   First {getFirst = Nothing}
--   
--   &gt;&gt;&gt; sumWith1 Just $ 1 :| [2..5 :: Int]
--   Just 15
--   </pre>
sumWith1 :: (Additive - Semigroup) a => Foldable1 t => (b -> a) -> t b -> a

-- | Evaluate a non-empty presemiring product.
product1 :: (Multiplicative - Semigroup) a => Foldable1 f => f a -> a

-- | Evaluate a non-empty presemiring product using a given presemiring.
--   
--   As the collection is non-empty this does not require a distinct
--   multiplicative unit:
--   
--   <pre>
--   &gt;&gt;&gt; productWith1 Just $ 1 :| [2..5 :: Int]
--   Just 120
--   
--   &gt;&gt;&gt; productWith1 First $ 1 :| [2..(5 :: Int)]
--   First {getFirst = 15}
--   
--   &gt;&gt;&gt; productWith1 First $ Nothing :| [Just (5 :: Int), Just 6,  Nothing]
--   First {getFirst = Just 11}
--   </pre>
productWith1 :: (Multiplicative - Semigroup) a => Foldable1 t => (b -> a) -> t b -> a

-- | Cross-multiply two non-empty collections.
--   
--   <pre>
--   &gt;&gt;&gt; xmult1 (Right 2 :| [Left "oops"]) (Right 2 :| [Right 3]) :: Either [Char] Int
--   Right 4
--   </pre>
xmult1 :: Presemiring a => Foldable1 f => Apply f => f a -> f a -> a

-- | Evaluate a presemiring expression.
eval1 :: Presemiring a => Functor f => Foldable1 f => Foldable1 g => f (g a) -> a

-- | Evaluate a presemiring expression using a given presemiring.
--   
--   <pre>
--   &gt;&gt;&gt; evalWith1 (Max . Down) $ (1 :| [2..5 :: Int]) :| [-5 :| [2..5 :: Int]]
--   Max {getMax = Down 9}
--   
--   &gt;&gt;&gt; evalWith1 Max $ (1 :| [2..5 :: Int]) :| [-5 :| [2..5 :: Int]]
--   Max {getMax = 15}
--   </pre>
evalWith1 :: Presemiring r => Functor f => Functor g => Foldable1 f => Foldable1 g => (a -> r) -> f (g a) -> r
type SemiringLaw a = ((Additive - Monoid) a, (Multiplicative - Monoid) a)

-- | Right semirings.
--   
--   A right semiring is a pre-semiring with two distinct neutral elements,
--   <a>zero</a> and <a>one</a>, such that <a>zero</a> is right-neutral wrt
--   addition, <a>one</a> is right-neutral wrt multiplication, and
--   <a>zero</a> is right-annihilative wrt multiplication.
--   
--   <i>Neutrality</i>
--   
--   <pre>
--   <a>zero</a> <a>+</a> r = r
--   <a>one</a> <a>*</a> r = r
--   </pre>
--   
--   <i>Absorbtion</i>
--   
--   <pre>
--   <a>zero</a> <a>*</a> a = <a>zero</a>
--   </pre>
class (Presemiring a, SemiringLaw a) => Semiring a

-- | Additive unit of a semiring.
zero :: (Additive - Monoid) a => a

-- | Multiplicative unit of a semiring.
one :: (Multiplicative - Monoid) a => a

-- | <pre>
--   <a>two</a> = <a>one</a> <a>+</a> <a>one</a>
--   </pre>
two :: Semiring a => a

-- | Evaluate a natural-numbered power of a semiring element.
--   
--   <pre>
--   <a>one</a> = a <a>^</a> 0
--   </pre>
--   
--   <pre>
--   &gt;&gt;&gt; 8 ^ 0 :: Int
--   1
--   </pre>
(^) :: Semiring a => a -> Natural -> a
infixr 8 ^

-- | Evaluate a semiring sum.
--   
--   <pre>
--   &gt;&gt;&gt; sum [1..5 :: Int]
--   15
--   </pre>
sum :: (Additive - Monoid) a => Foldable f => f a -> a

-- | Evaluate a semiring sum using a given semiring.
sumWith :: (Additive - Monoid) a => Foldable f => (b -> a) -> f b -> a

-- | Evaluate a semiring product.
--   
--   <pre>
--   &gt;&gt;&gt; product [1..5 :: Int]
--   120
--   </pre>
product :: (Multiplicative - Monoid) a => Foldable f => f a -> a

-- | Evaluate a semiring product using a given semiring.
--   
--   <pre>
--   <a>product</a> f = <a>foldr'</a> ((<a>*</a>) . f) <a>one</a>
--   </pre>
--   
--   <pre>
--   &gt;&gt;&gt; productWith Just [1..5 :: Int]
--   Just 120
--   </pre>
productWith :: (Multiplicative - Monoid) a => Foldable f => (b -> a) -> f b -> a

-- | Cross-multiply two collections.
--   
--   <pre>
--   &gt;&gt;&gt; xmult (V3 1 2 3) (V3 1 2 3)
--   14
--   
--   &gt;&gt;&gt; xmult [1,2,3 :: Int] [1,2,3]
--   36
--   
--   &gt;&gt;&gt; xmult [1,2,3 :: Int] []
--   0
--   </pre>
xmult :: Semiring a => Foldable f => Applicative f => f a -> f a -> a

-- | Evaluate a semiring expression.
--   
--   <pre>
--   (a11 * .. * a1m) + (a21 * .. * a2n) + ...
--   </pre>
--   
--   <pre>
--   &gt;&gt;&gt; eval [[1, 2], [3, 4 :: Int]] -- 1 * 2 + 3 * 4
--   14
--   
--   &gt;&gt;&gt; eval $ sequence [[1, 2], [3, 4 :: Int]] -- 1 + 2 * 3 + 4
--   21
--   </pre>
eval :: Semiring a => Functor f => Foldable f => Foldable g => f (g a) -> a

-- | Evaluate a semiring expression using a given semiring.
--   
--   <pre>
--   &gt;&gt;&gt; evalWith Max [[1..4 :: Int], [0..2 :: Int]]
--   Max {getMax = 24}
--   </pre>
evalWith :: Semiring r => Functor f => Functor g => Foldable f => Foldable g => (a -> r) -> f (g a) -> r
type RingLaw a = ((Additive - Group) a, (Multiplicative - Monoid) a)

-- | Rings.
--   
--   A ring <i>R</i> is a commutative group with a second monoidal
--   operation <a>*</a> that distributes over <a>+</a>.
--   
--   The basic properties of a ring follow immediately from the axioms:
--   
--   <pre>
--   r <a>*</a> <a>zero</a> = <a>zero</a> = <a>zero</a> <a>*</a> r
--   </pre>
--   
--   <pre>
--   <a>negate</a> <a>one</a> <a>*</a> r = <a>negate</a> r
--   </pre>
--   
--   Furthermore, the binomial formula holds for any commuting pair of
--   elements (that is, any <i>a</i> and <i>b</i> such that <i>a * b = b *
--   a</i>).
--   
--   If <i>zero = one</i> in a ring <i>R</i>, then <i>R</i> has only one
--   element, and is called the zero ring. Otherwise the additive identity,
--   the additive inverse of each element, and the multiplicative identity
--   are unique.
--   
--   See <a>https://en.wikipedia.org/wiki/Ring_(mathematics)</a>.
--   
--   If the ring is <a>ordered</a> (i.e. has an <a>Ord</a> instance), then
--   the following additional properties must hold:
--   
--   <pre>
--   a <a>&lt;=</a> b ⇒ a <a>+</a> c <a>&lt;=</a> b <a>+</a> c
--   </pre>
--   
--   <pre>
--   <a>zero</a> <a>&lt;=</a> a <a>&amp;&amp;</a> <a>zero</a> <a>&lt;=</a> b ⇒ <a>zero</a> <a>&lt;=</a> a <a>*</a> b
--   </pre>
--   
--   See the properties module for a detailed specification of the laws.
class (Semiring a, RingLaw a) => Ring a

-- | Subtract two elements.
--   
--   <pre>
--   a <a>-</a> b = <a>subtract</a> b a
--   </pre>
(-) :: (Additive - Group) a => a -> a -> a
infixl 6 -

-- | Subtract two elements.
subtract :: (Additive - Group) a => a -> a -> a

-- | Reverse the sign of an element.
negate :: (Additive - Group) a => a -> a

-- | Absolute value of an element.
--   
--   <pre>
--   <a>abs</a> r = r <a>*</a> (<a>signum</a> r)
--   </pre>
abs :: (Additive - Group) a => Ord a => a -> a

-- | Extract the sign of an element.
--   
--   <a>signum</a> satisfies a trichotomy law:
--   
--   <pre>
--   <a>signum</a> r = <a>negate</a> r || <a>zero</a> || r
--   </pre>
--   
--   This follows from the fact that ordered rings are abelian, linearly
--   ordered groups with respect to addition.
--   
--   See <a>https://en.wikipedia.org/wiki/Linearly_ordered_group</a>.
signum :: Ring a => Ord a => a -> a

-- | A generalization of <a>replicate</a> to an arbitrary <a>Monoid</a>.
--   
--   Adapted from
--   <a>http://augustss.blogspot.com/2008/07/lost-and-found-if-i-write-108-in.html</a>.
mreplicate :: Monoid a => Natural -> a -> a

-- | A commutative <a>Semigroup</a> under <a>+</a>.
newtype Additive a
Additive :: a -> Additive a
[unAdditive] :: Additive a -> a

-- | A (potentially non-commutative) <a>Semigroup</a> under <a>*</a>.
newtype Multiplicative a
Multiplicative :: a -> Multiplicative a
[unMultiplicative] :: Multiplicative a -> a
class Magma a
(<<) :: Magma a => a -> a -> a
infixl 6 <<
class Magma a => Quasigroup a
class Quasigroup a => Loop a
class (Loop a, Monoid a) => Group a
inv :: Group a => a -> a
greplicate :: Group a => Integer -> a -> a
instance Data.Semiring.Ring a => Data.Semiring.Presemiring (Data.Complex.Complex a)
instance Data.Semiring.Ring a => Data.Semiring.Semiring (Data.Complex.Complex a)
instance Data.Semiring.Ring ()
instance Data.Semiring.Ring GHC.Types.Int
instance Data.Semiring.Ring GHC.Int.Int8
instance Data.Semiring.Ring GHC.Int.Int16
instance Data.Semiring.Ring GHC.Int.Int32
instance Data.Semiring.Ring GHC.Int.Int64
instance Data.Semiring.Ring GHC.Integer.Type.Integer
instance Data.Semiring.Ring (GHC.Real.Ratio GHC.Integer.Type.Integer)
instance Data.Semiring.Ring Data.Fixed.Uni
instance Data.Semiring.Ring Data.Fixed.Deci
instance Data.Semiring.Ring Data.Fixed.Centi
instance Data.Semiring.Ring Data.Fixed.Milli
instance Data.Semiring.Ring Data.Fixed.Micro
instance Data.Semiring.Ring Data.Fixed.Nano
instance Data.Semiring.Ring Data.Fixed.Pico
instance Data.Semiring.Ring GHC.Types.Float
instance Data.Semiring.Ring GHC.Types.Double
instance Data.Semiring.Ring Foreign.C.Types.CFloat
instance Data.Semiring.Ring Foreign.C.Types.CDouble
instance Data.Semiring.Ring a => Data.Semiring.Ring (Data.Complex.Complex a)
instance Data.Semiring.Semiring ()
instance Data.Semiring.Semiring GHC.Types.Bool
instance Data.Semiring.Semiring GHC.Types.Word
instance Data.Semiring.Semiring GHC.Word.Word8
instance Data.Semiring.Semiring GHC.Word.Word16
instance Data.Semiring.Semiring GHC.Word.Word32
instance Data.Semiring.Semiring GHC.Word.Word64
instance Data.Semiring.Semiring GHC.Natural.Natural
instance Data.Semiring.Semiring (GHC.Real.Ratio GHC.Natural.Natural)
instance Data.Semiring.Semiring GHC.Types.Int
instance Data.Semiring.Semiring GHC.Int.Int8
instance Data.Semiring.Semiring GHC.Int.Int16
instance Data.Semiring.Semiring GHC.Int.Int32
instance Data.Semiring.Semiring GHC.Int.Int64
instance Data.Semiring.Semiring GHC.Integer.Type.Integer
instance Data.Semiring.Semiring (GHC.Real.Ratio GHC.Integer.Type.Integer)
instance Data.Semiring.Semiring Data.Fixed.Uni
instance Data.Semiring.Semiring Data.Fixed.Deci
instance Data.Semiring.Semiring Data.Fixed.Centi
instance Data.Semiring.Semiring Data.Fixed.Milli
instance Data.Semiring.Semiring Data.Fixed.Micro
instance Data.Semiring.Semiring Data.Fixed.Nano
instance Data.Semiring.Semiring Data.Fixed.Pico
instance Data.Semiring.Semiring GHC.Types.Float
instance Data.Semiring.Semiring GHC.Types.Double
instance Data.Semiring.Semiring Foreign.C.Types.CFloat
instance Data.Semiring.Semiring Foreign.C.Types.CDouble
instance Data.Semiring.Semiring a => Data.Semiring.Semiring (r -> a)
instance Data.Semiring.Semiring a => Data.Semiring.Semiring (GHC.Maybe.Maybe a)
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid a => Data.Semiring.Semiring [a]
instance Data.Semiring.Semiring a => Data.Semiring.Semiring (Data.IntMap.Internal.IntMap a)
instance (GHC.Classes.Ord k, (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Multiplicative GHC.Base.Monoid k, Data.Semiring.Semiring a) => Data.Semiring.Semiring (Data.Map.Internal.Map k a)
instance Data.Semiring.Presemiring ()
instance Data.Semiring.Presemiring GHC.Types.Bool
instance Data.Semiring.Presemiring GHC.Types.Word
instance Data.Semiring.Presemiring GHC.Word.Word8
instance Data.Semiring.Presemiring GHC.Word.Word16
instance Data.Semiring.Presemiring GHC.Word.Word32
instance Data.Semiring.Presemiring GHC.Word.Word64
instance Data.Semiring.Presemiring GHC.Natural.Natural
instance Data.Semiring.Presemiring (GHC.Real.Ratio GHC.Natural.Natural)
instance Data.Semiring.Presemiring GHC.Types.Int
instance Data.Semiring.Presemiring GHC.Int.Int8
instance Data.Semiring.Presemiring GHC.Int.Int16
instance Data.Semiring.Presemiring GHC.Int.Int32
instance Data.Semiring.Presemiring GHC.Int.Int64
instance Data.Semiring.Presemiring GHC.Integer.Type.Integer
instance Data.Semiring.Presemiring (GHC.Real.Ratio GHC.Integer.Type.Integer)
instance Data.Semiring.Presemiring Data.Fixed.Uni
instance Data.Semiring.Presemiring Data.Fixed.Deci
instance Data.Semiring.Presemiring Data.Fixed.Centi
instance Data.Semiring.Presemiring Data.Fixed.Milli
instance Data.Semiring.Presemiring Data.Fixed.Micro
instance Data.Semiring.Presemiring Data.Fixed.Nano
instance Data.Semiring.Presemiring Data.Fixed.Pico
instance Data.Semiring.Presemiring GHC.Types.Float
instance Data.Semiring.Presemiring GHC.Types.Double
instance Data.Semiring.Presemiring Foreign.C.Types.CFloat
instance Data.Semiring.Presemiring Foreign.C.Types.CDouble
instance Data.Semiring.Presemiring a => Data.Semiring.Presemiring (r -> a)
instance (Data.Semiring.Presemiring a, Data.Semiring.Presemiring b) => Data.Semiring.Presemiring (Data.Either.Either a b)
instance Data.Semiring.Presemiring a => Data.Semiring.Presemiring (GHC.Maybe.Maybe a)
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup a => Data.Semiring.Presemiring [a]
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup a => Data.Semiring.Presemiring (GHC.Base.NonEmpty a)
instance Data.Semiring.Presemiring Data.IntSet.Internal.IntSet
instance GHC.Classes.Ord a => Data.Semiring.Presemiring (Data.Set.Internal.Set a)
instance Data.Semiring.Presemiring a => Data.Semiring.Presemiring (Data.IntMap.Internal.IntMap a)
instance (GHC.Classes.Ord k, Data.Semiring.Presemiring a) => Data.Semiring.Presemiring (Data.Map.Internal.Map k a)

module Data.Semifield
type SemifieldLaw a = ((Additive - Monoid) a, (Multiplicative - Group) a)

-- | A semifield, near-field, or division ring.
--   
--   Instances needn't have commutative multiplication or additive
--   inverses, however addition must be commutative, and addition and
--   multiplication must be associative as usual.
--   
--   See also the wikipedia definitions of:
--   
--   <ul>
--   <li><a>semifield</a></li>
--   <li><a>near-field</a></li>
--   <li><a>division ring</a></li>
--   </ul>
class (Semiring a, SemifieldLaw a) => Semifield a

-- | The <i>NaN</i> value of the semifield.
--   
--   <pre>
--   <a>anan</a> = <a>zero</a> <a>/</a> <a>zero</a>
--   </pre>
anan :: Semifield a => a

-- | The positive infinity of the semifield.
--   
--   <pre>
--   <a>pinf</a> = <a>one</a> <a>/</a> <a>zero</a>
--   </pre>
pinf :: Semifield a => a

-- | Right division by a multiplicative group element.
(/) :: (Multiplicative - Group) a => a -> a -> a
infixl 7 /

-- | Left division by a multiplicative group element.
--   
--   When <a>*</a> is commutative we must have:
--   
--   <pre>
--   x <a>\\</a> y = y <a>/</a> x
--   </pre>
(\\) :: (Multiplicative - Group) a => a -> a -> a
infixl 7 \\

-- | Reciprocal of a multiplicative group element.
--   
--   <pre>
--   x <a>/</a> y = x <a>*</a> <a>recip</a> y
--   x <a>\\</a> y = <a>recip</a> x <a>*</a> y
--   </pre>
--   
--   <pre>
--   &gt;&gt;&gt; recip (3 :+ 4) :: Complex Rational
--   3 % 25 :+ (-4) % 25
--   
--   &gt;&gt;&gt; recip (3 :+ 4) :: Complex Double
--   0.12 :+ (-0.16)
--   
--   &gt;&gt;&gt; recip (3 :+ 4) :: Complex Pico
--   0.120000000000 :+ -0.160000000000
--   </pre>
recip :: (Multiplicative - Group) a => a -> a
type FieldLaw a = ((Additive - Group) a, (Multiplicative - Group) a)

-- | A <a>field</a>.
class (Ring a, Semifield a, FieldLaw a) => Field a

-- | A type modeling the real numbers.
class Field a => Real a

-- | The <a>one</a> <a>/</a> <a>zero</a> value of the field.
--   
--   <pre>
--   <a>ninf</a> = <a>negate</a> <a>one</a> <a>/</a> <a>zero</a>
--   </pre>
ninf :: Field a => a

-- | Integral power of a multiplicative group element.
--   
--   <pre>
--   <a>one</a> <tt>==</tt> a <a>^^</a> 0
--   </pre>
--   
--   <pre>
--   &gt;&gt;&gt; 8 ^^ 0 :: Double
--   1.0
--   
--   &gt;&gt;&gt; 8 ^^ 0 :: Pico
--   1.000000000000
--   </pre>
(^^) :: (Multiplicative - Group) a => a -> Integer -> a
infixr 8 ^^
instance Data.Semifield.Real GHC.Real.Rational
instance Data.Semifield.Real Data.Fixed.Uni
instance Data.Semifield.Real Data.Fixed.Deci
instance Data.Semifield.Real Data.Fixed.Centi
instance Data.Semifield.Real Data.Fixed.Milli
instance Data.Semifield.Real Data.Fixed.Micro
instance Data.Semifield.Real Data.Fixed.Nano
instance Data.Semifield.Real Data.Fixed.Pico
instance Data.Semifield.Real GHC.Types.Float
instance Data.Semifield.Real GHC.Types.Double
instance Data.Semifield.Real Foreign.C.Types.CFloat
instance Data.Semifield.Real Foreign.C.Types.CDouble
instance Data.Semifield.Field a => Data.Semifield.Semifield (Data.Complex.Complex a)
instance Data.Semifield.Field ()
instance Data.Semifield.Field GHC.Real.Rational
instance Data.Semifield.Field Data.Fixed.Uni
instance Data.Semifield.Field Data.Fixed.Deci
instance Data.Semifield.Field Data.Fixed.Centi
instance Data.Semifield.Field Data.Fixed.Milli
instance Data.Semifield.Field Data.Fixed.Micro
instance Data.Semifield.Field Data.Fixed.Nano
instance Data.Semifield.Field Data.Fixed.Pico
instance Data.Semifield.Field GHC.Types.Float
instance Data.Semifield.Field GHC.Types.Double
instance Data.Semifield.Field Foreign.C.Types.CFloat
instance Data.Semifield.Field Foreign.C.Types.CDouble
instance Data.Semifield.Field a => Data.Semifield.Field (Data.Complex.Complex a)
instance Data.Semifield.Semifield ()
instance Data.Semifield.Semifield (GHC.Real.Ratio GHC.Natural.Natural)
instance Data.Semifield.Semifield GHC.Real.Rational
instance Data.Semifield.Semifield Data.Fixed.Uni
instance Data.Semifield.Semifield Data.Fixed.Deci
instance Data.Semifield.Semifield Data.Fixed.Centi
instance Data.Semifield.Semifield Data.Fixed.Milli
instance Data.Semifield.Semifield Data.Fixed.Micro
instance Data.Semifield.Semifield Data.Fixed.Nano
instance Data.Semifield.Semifield Data.Fixed.Pico
instance Data.Semifield.Semifield GHC.Types.Float
instance Data.Semifield.Semifield GHC.Types.Double
instance Data.Semifield.Semifield Foreign.C.Types.CFloat
instance Data.Semifield.Semifield Foreign.C.Types.CDouble

module Data.Semimodule

-- | A tensor product of semimodule morphisms.
type f ** g = Compose f g
infixr 2 **

-- | A direct sum of free semimodule elements.
type f ++ g = Product f g
infixr 1 ++
type Free f = (Representable f)
type Basis b f = (Free f, Rep f ~ b, Eq b)
type Basis2 b c f g = (Basis b f, Basis c g)
type Basis3 b c d f g h = (Basis b f, Basis c g, Basis d h)
type FreeModule a f = (Free f, (Additive - Group) (f a), Bimodule a a (f a))
type FreeSemimodule a f = (Free f, Bisemimodule a a (f a))
type LeftModule l a = (Ring l, (Additive - Group) a, LeftSemimodule l a)

-- | <a>Left semimodule</a> over a commutative semiring.
--   
--   All instances must satisfy the following identities:
--   
--   <pre>
--   <a>lscale</a> s (x <a>+</a> y) = <a>lscale</a> s x <a>+</a> <a>lscale</a> s y
--   <a>lscale</a> (s1 <a>+</a> s2) x = <a>lscale</a> s1 x <a>+</a> <a>lscale</a> s2 x
--   <a>lscale</a> (s1 <a>*</a> s2) = <a>lscale</a> s1 . <a>lscale</a> s2
--   <a>lscale</a> <a>zero</a> = <a>zero</a>
--   </pre>
--   
--   When the ring of coefficients <i>s</i> is unital we must additionally
--   have: <tt> <a>lscale</a> <a>one</a> = <a>id</a> </tt>
--   
--   See the properties module for a detailed specification of the laws.
class (Semiring l, (Additive - Monoid) a) => LeftSemimodule l a

-- | Left-multiply by a scalar.
lscale :: LeftSemimodule l a => l -> a -> a

-- | Left-multiply a module element by a scalar.
(*.) :: LeftSemimodule l a => l -> a -> a
infixr 7 *.

-- | Right-divide a vector by a scalar (on the left).
(/.) :: Semifield a => Functor f => a -> f a -> f a
infixr 7 /.

-- | Left-divide a vector by a scalar.
(\.) :: Semifield a => Functor f => a -> f a -> f a
infixr 7 \.

-- | Linearly interpolate between two vectors.
--   
--   <pre>
--   &gt;&gt;&gt; u = V3 (1 :% 1) (2 :% 1) (3 :% 1) :: V3 Rational
--   
--   &gt;&gt;&gt; v = V3 (2 :% 1) (4 :% 1) (6 :% 1) :: V3 Rational
--   
--   &gt;&gt;&gt; r = 1 :% 2 :: Rational
--   
--   &gt;&gt;&gt; lerp r u v
--   V3 (6 % 4) (12 % 4) (18 % 4)
--   </pre>
lerp :: LeftModule r a => r -> a -> a -> a

-- | Default definition of <a>lscale</a> for a free module.
lscaleDef :: Semiring a => Functor f => a -> f a -> f a

-- | Default definition of <a>&lt;&lt;</a> for a commutative group.
negateDef :: LeftModule Integer a => a -> a
type RightModule r a = (Ring r, (Additive - Group) a, RightSemimodule r a)

-- | <a>Right semimodule</a> over a commutative semiring.
--   
--   The laws for right semimodules are analagous to those of left
--   semimodules.
--   
--   See the properties module for a detailed specification.
class (Semiring r, (Additive - Monoid) a) => RightSemimodule r a

-- | Right-multiply by a scalar.
rscale :: RightSemimodule r a => r -> a -> a

-- | Right-multiply a module element by a scalar.
(.*) :: RightSemimodule r a => a -> r -> a
infixl 7 .*

-- | Right-divide a vector by a scalar.
(./) :: Semifield a => Functor f => f a -> a -> f a
infixl 7 ./

-- | Left-divide a vector by a scalar (on the right).
(.\) :: Semifield a => Functor f => f a -> a -> f a
infixl 7 .\

-- | Default definition of <a>rscale</a> for a free module.
rscaleDef :: Semiring a => Functor f => a -> f a -> f a
type Bimodule l r a = (LeftModule l a, RightModule r a, Bisemimodule l r a)

-- | <a>Bisemimodule</a> over a commutative semiring.
--   
--   <pre>
--   <a>lscale</a> l . <a>rscale</a> r = <a>rscale</a> r . <a>lscale</a> l
--   </pre>
class (LeftSemimodule l a, RightSemimodule r a) => Bisemimodule l r a

-- | Left and right-multiply by two scalars.
discale :: Bisemimodule l r a => l -> r -> a -> a
instance Data.Semiring.Semiring r => Data.Semimodule.Bisemimodule r r ()
instance Data.Semimodule.Bisemimodule r r a => Data.Semimodule.Bisemimodule r r (e -> a)
instance (Data.Semimodule.Bisemimodule r r a, Data.Semimodule.Bisemimodule r r b) => Data.Semimodule.Bisemimodule r r (a, b)
instance (Data.Semimodule.Bisemimodule r r a, Data.Semimodule.Bisemimodule r r b, Data.Semimodule.Bisemimodule r r c) => Data.Semimodule.Bisemimodule r r (a, b, c)
instance Data.Semiring.Semiring a => Data.Semimodule.Bisemimodule a a (GHC.Real.Ratio a)
instance Data.Semiring.Ring a => Data.Semimodule.Bisemimodule a a (Data.Complex.Complex a)
instance Data.Semiring.Ring a => Data.Semimodule.Bisemimodule (Data.Complex.Complex a) (Data.Complex.Complex a) (Data.Complex.Complex a)
instance Data.Semimodule.Bisemimodule GHC.Types.Bool GHC.Types.Bool GHC.Types.Bool
instance Data.Semimodule.Bisemimodule GHC.Types.Int GHC.Types.Int GHC.Types.Int
instance Data.Semimodule.Bisemimodule GHC.Int.Int8 GHC.Int.Int8 GHC.Int.Int8
instance Data.Semimodule.Bisemimodule GHC.Int.Int16 GHC.Int.Int16 GHC.Int.Int16
instance Data.Semimodule.Bisemimodule GHC.Int.Int32 GHC.Int.Int32 GHC.Int.Int32
instance Data.Semimodule.Bisemimodule GHC.Int.Int64 GHC.Int.Int64 GHC.Int.Int64
instance Data.Semimodule.Bisemimodule GHC.Types.Word GHC.Types.Word GHC.Types.Word
instance Data.Semimodule.Bisemimodule GHC.Word.Word8 GHC.Word.Word8 GHC.Word.Word8
instance Data.Semimodule.Bisemimodule GHC.Word.Word16 GHC.Word.Word16 GHC.Word.Word16
instance Data.Semimodule.Bisemimodule GHC.Word.Word32 GHC.Word.Word32 GHC.Word.Word32
instance Data.Semimodule.Bisemimodule GHC.Word.Word64 GHC.Word.Word64 GHC.Word.Word64
instance Data.Semimodule.Bisemimodule Data.Fixed.Uni Data.Fixed.Uni Data.Fixed.Uni
instance Data.Semimodule.Bisemimodule Data.Fixed.Deci Data.Fixed.Deci Data.Fixed.Deci
instance Data.Semimodule.Bisemimodule Data.Fixed.Centi Data.Fixed.Centi Data.Fixed.Centi
instance Data.Semimodule.Bisemimodule Data.Fixed.Milli Data.Fixed.Milli Data.Fixed.Milli
instance Data.Semimodule.Bisemimodule Data.Fixed.Micro Data.Fixed.Micro Data.Fixed.Micro
instance Data.Semimodule.Bisemimodule Data.Fixed.Nano Data.Fixed.Nano Data.Fixed.Nano
instance Data.Semimodule.Bisemimodule Data.Fixed.Pico Data.Fixed.Pico Data.Fixed.Pico
instance Data.Semimodule.Bisemimodule GHC.Types.Float GHC.Types.Float GHC.Types.Float
instance Data.Semimodule.Bisemimodule GHC.Types.Double GHC.Types.Double GHC.Types.Double
instance Data.Semimodule.Bisemimodule Foreign.C.Types.CFloat Foreign.C.Types.CFloat Foreign.C.Types.CFloat
instance Data.Semimodule.Bisemimodule Foreign.C.Types.CDouble Foreign.C.Types.CDouble Foreign.C.Types.CDouble
instance Data.Semimodule.Bisemimodule (GHC.Real.Ratio GHC.Integer.Type.Integer) (GHC.Real.Ratio GHC.Integer.Type.Integer) (GHC.Real.Ratio GHC.Integer.Type.Integer)
instance Data.Semimodule.Bisemimodule (GHC.Real.Ratio GHC.Natural.Natural) (GHC.Real.Ratio GHC.Natural.Natural) (GHC.Real.Ratio GHC.Natural.Natural)
instance Data.Semiring.Semiring r => Data.Semimodule.RightSemimodule r ()
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid a => Data.Semimodule.RightSemimodule () a
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid a => Data.Semimodule.RightSemimodule GHC.Natural.Natural a
instance ((Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid a, (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a) => Data.Semimodule.RightSemimodule GHC.Integer.Type.Integer a
instance Data.Semimodule.RightSemimodule r a => Data.Semimodule.RightSemimodule r (e -> a)
instance (Data.Semimodule.RightSemimodule r a, Data.Semimodule.RightSemimodule r b) => Data.Semimodule.RightSemimodule r (a, b)
instance (Data.Semimodule.RightSemimodule r a, Data.Semimodule.RightSemimodule r b, Data.Semimodule.RightSemimodule r c) => Data.Semimodule.RightSemimodule r (a, b, c)
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule a (GHC.Real.Ratio a)
instance Data.Semiring.Ring a => Data.Semimodule.RightSemimodule a (Data.Complex.Complex a)
instance Data.Semiring.Ring a => Data.Semimodule.RightSemimodule (Data.Complex.Complex a) (Data.Complex.Complex a)
instance Data.Semimodule.RightSemimodule GHC.Types.Bool GHC.Types.Bool
instance Data.Semimodule.RightSemimodule GHC.Types.Int GHC.Types.Int
instance Data.Semimodule.RightSemimodule GHC.Int.Int8 GHC.Int.Int8
instance Data.Semimodule.RightSemimodule GHC.Int.Int16 GHC.Int.Int16
instance Data.Semimodule.RightSemimodule GHC.Int.Int32 GHC.Int.Int32
instance Data.Semimodule.RightSemimodule GHC.Int.Int64 GHC.Int.Int64
instance Data.Semimodule.RightSemimodule GHC.Types.Word GHC.Types.Word
instance Data.Semimodule.RightSemimodule GHC.Word.Word8 GHC.Word.Word8
instance Data.Semimodule.RightSemimodule GHC.Word.Word16 GHC.Word.Word16
instance Data.Semimodule.RightSemimodule GHC.Word.Word32 GHC.Word.Word32
instance Data.Semimodule.RightSemimodule GHC.Word.Word64 GHC.Word.Word64
instance Data.Semimodule.RightSemimodule Data.Fixed.Uni Data.Fixed.Uni
instance Data.Semimodule.RightSemimodule Data.Fixed.Deci Data.Fixed.Deci
instance Data.Semimodule.RightSemimodule Data.Fixed.Centi Data.Fixed.Centi
instance Data.Semimodule.RightSemimodule Data.Fixed.Milli Data.Fixed.Milli
instance Data.Semimodule.RightSemimodule Data.Fixed.Micro Data.Fixed.Micro
instance Data.Semimodule.RightSemimodule Data.Fixed.Nano Data.Fixed.Nano
instance Data.Semimodule.RightSemimodule Data.Fixed.Pico Data.Fixed.Pico
instance Data.Semimodule.RightSemimodule GHC.Types.Float GHC.Types.Float
instance Data.Semimodule.RightSemimodule GHC.Types.Double GHC.Types.Double
instance Data.Semimodule.RightSemimodule Foreign.C.Types.CFloat Foreign.C.Types.CFloat
instance Data.Semimodule.RightSemimodule Foreign.C.Types.CDouble Foreign.C.Types.CDouble
instance Data.Semimodule.RightSemimodule (GHC.Real.Ratio GHC.Integer.Type.Integer) (GHC.Real.Ratio GHC.Integer.Type.Integer)
instance Data.Semimodule.RightSemimodule (GHC.Real.Ratio GHC.Natural.Natural) (GHC.Real.Ratio GHC.Natural.Natural)
instance Data.Semiring.Semiring l => Data.Semimodule.LeftSemimodule l ()
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid a => Data.Semimodule.LeftSemimodule () a
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid a => Data.Semimodule.LeftSemimodule GHC.Natural.Natural a
instance ((Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid a, (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a) => Data.Semimodule.LeftSemimodule GHC.Integer.Type.Integer a
instance Data.Semimodule.LeftSemimodule l a => Data.Semimodule.LeftSemimodule l (e -> a)
instance (Data.Semimodule.LeftSemimodule l a, Data.Semimodule.LeftSemimodule l b) => Data.Semimodule.LeftSemimodule l (a, b)
instance (Data.Semimodule.LeftSemimodule l a, Data.Semimodule.LeftSemimodule l b, Data.Semimodule.LeftSemimodule l c) => Data.Semimodule.LeftSemimodule l (a, b, c)
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule a (GHC.Real.Ratio a)
instance Data.Semiring.Ring a => Data.Semimodule.LeftSemimodule a (Data.Complex.Complex a)
instance Data.Semiring.Ring a => Data.Semimodule.LeftSemimodule (Data.Complex.Complex a) (Data.Complex.Complex a)
instance Data.Semimodule.LeftSemimodule GHC.Types.Bool GHC.Types.Bool
instance Data.Semimodule.LeftSemimodule GHC.Types.Int GHC.Types.Int
instance Data.Semimodule.LeftSemimodule GHC.Int.Int8 GHC.Int.Int8
instance Data.Semimodule.LeftSemimodule GHC.Int.Int16 GHC.Int.Int16
instance Data.Semimodule.LeftSemimodule GHC.Int.Int32 GHC.Int.Int32
instance Data.Semimodule.LeftSemimodule GHC.Int.Int64 GHC.Int.Int64
instance Data.Semimodule.LeftSemimodule GHC.Types.Word GHC.Types.Word
instance Data.Semimodule.LeftSemimodule GHC.Word.Word8 GHC.Word.Word8
instance Data.Semimodule.LeftSemimodule GHC.Word.Word16 GHC.Word.Word16
instance Data.Semimodule.LeftSemimodule GHC.Word.Word32 GHC.Word.Word32
instance Data.Semimodule.LeftSemimodule GHC.Word.Word64 GHC.Word.Word64
instance Data.Semimodule.LeftSemimodule Data.Fixed.Uni Data.Fixed.Uni
instance Data.Semimodule.LeftSemimodule Data.Fixed.Deci Data.Fixed.Deci
instance Data.Semimodule.LeftSemimodule Data.Fixed.Centi Data.Fixed.Centi
instance Data.Semimodule.LeftSemimodule Data.Fixed.Milli Data.Fixed.Milli
instance Data.Semimodule.LeftSemimodule Data.Fixed.Micro Data.Fixed.Micro
instance Data.Semimodule.LeftSemimodule Data.Fixed.Nano Data.Fixed.Nano
instance Data.Semimodule.LeftSemimodule Data.Fixed.Pico Data.Fixed.Pico
instance Data.Semimodule.LeftSemimodule GHC.Types.Float GHC.Types.Float
instance Data.Semimodule.LeftSemimodule GHC.Types.Double GHC.Types.Double
instance Data.Semimodule.LeftSemimodule Foreign.C.Types.CFloat Foreign.C.Types.CFloat
instance Data.Semimodule.LeftSemimodule Foreign.C.Types.CDouble Foreign.C.Types.CDouble
instance Data.Semimodule.LeftSemimodule (GHC.Real.Ratio GHC.Integer.Type.Integer) (GHC.Real.Ratio GHC.Integer.Type.Integer)
instance Data.Semimodule.LeftSemimodule (GHC.Real.Ratio GHC.Natural.Natural) (GHC.Real.Ratio GHC.Natural.Natural)

module Data.Semimodule.Algebra

-- | An algebra over a free module <i>f</i>.
--   
--   Note that this is distinct from a <a>free algebra</a>.
type FreeAlgebra a f = (FreeSemimodule a f, Algebra a (Rep f))

-- | An algebra <a>algebra</a> over a semiring.
--   
--   Note that the algebra <a>needn't be associative</a>.
class Semiring a => Algebra a b

-- | <pre>
--   <a>joined</a> = <a>runTran</a> <a>diagonal</a> <a>.</a> <a>uncurry</a>
--   </pre>
joined :: Algebra a b => (b -> b -> a) -> b -> a

-- | <pre>
--   <a>rmap'</a> (((c1,()),(c2,())) -&gt; (c1,c2)) <a>$</a> (<a>id</a> <a>***</a> <a>initial</a>) <a>.</a> <a>diagonal</a> = <a>id</a>
--   <a>rmap'</a> ((((),c1),((),c2)) -&gt; (c1,c2)) <a>$</a> (<a>initial</a> <a>***</a> <a>id</a>) <a>.</a> <a>diagonal</a> = <a>id</a>
--   </pre>
diagonal :: Algebra a b => Tran a b (b, b)

-- | Obtain the diagonal of a tensor product as a vector.
--   
--   When the coalgebra is trivial we have:
--   
--   <pre>
--   <a>diag</a> f = <a>tabulate</a> $ <a>joined</a> (<a>index</a> . <a>index</a> (<tt>getCompose</tt> f))
--   </pre>
--   
--   <pre>
--   &gt;&gt;&gt; diag $ m22 1.0 2.0 3.0 4.0
--   V2 1.0 4.0
--   </pre>
diag :: FreeAlgebra a f => (f ** f) a -> f a

-- | Multiplication operator on an algebra over a free semimodule.
--   
--   <i>Caution</i> in general (.*.) needn't be commutative, nor
--   associative.
(.*.) :: FreeAlgebra a f => f a -> f a -> f a
infixl 7 .*.

-- | A unital algebra over a free semimodule <i>f</i>.
type FreeUnital a f = (FreeAlgebra a f, Unital a (Rep f))

-- | A <a>unital algebra</a> over a semiring.
class Algebra a b => Unital a b

-- | <pre>
--   <a>unital</a> = <a>runTran</a> <a>initial</a> <a>.</a> <a>const</a>
--   </pre>
unital :: Unital a b => a -> b -> a
initial :: Unital a b => Tran a b ()

-- | Insert an element into an algebra.
--   
--   <pre>
--   &gt;&gt;&gt; V4 1 2 3 4 .*. unit two :: V4 Int
--   V4 2 4 6 8
--   </pre>
unit :: FreeUnital a f => a -> f a

-- | Unital element of a unital algebra over a free semimodule.
--   
--   <pre>
--   &gt;&gt;&gt; unit one :: Complex Int
--   1 :+ 0
--   </pre>
unit' :: FreeUnital a f => f a

-- | A coalgebra over a free semimodule <i>f</i>.
type FreeCoalgebra a f = (FreeSemimodule a f, Coalgebra a (Rep f))

-- | A coalgebra over a semiring.
class Semiring a => Coalgebra a c

-- | <pre>
--   <a>cojoined</a> = <a>curry</a> <a>.</a> <a>runTran</a> <a>codiagonal</a>
--   </pre>
cojoined :: Coalgebra a c => (c -> a) -> c -> c -> a

-- | <pre>
--   <a>lmap'</a> ((c1,c2) -&gt; ((c1,()),(c2,()))) <a>$</a> (<a>id</a> <a>***</a> <a>coinitial</a>) <a>.</a> <a>codiagonal</a> = <a>id</a>
--   <a>lmap'</a> ((c1,c2) -&gt; (((),c1),((),c2))) <a>$</a> (<a>coinitial</a> <a>***</a> <a>id</a>) <a>.</a> <a>codiagonal</a> = <a>id</a>
--   </pre>
codiagonal :: Coalgebra a c => Tran a (c, c) c

-- | Obtain a tensor from a vector.
--   
--   When the coalgebra is trivial we have:
--   
--   <pre>
--   <a>codiag</a> = <a>flip</a> <a>bindRep</a> <a>id</a> <a>.</a> <tt>getCompose</tt>
--   </pre>
codiag :: FreeCoalgebra a f => f a -> (f ** f) a

-- | Convolution with an associative algebra and coassociative coalgebra
convolve :: Algebra a b => Coalgebra a c => Tran a b c -> Tran a b c -> Tran a b c

-- | A counital coalgebra over a free semimodule <i>f</i>.
type FreeCounital a f = (FreeCoalgebra a f, Counital a (Rep f))

-- | A counital coalgebra over a semiring.
class Coalgebra a c => Counital a c
counital :: Counital a c => (c -> a) -> a
coinitial :: Counital a c => Tran a () c

-- | Obtain an element from a coalgebra over a free semimodule.
counit :: FreeCounital a f => f a -> a

-- | A bialgebra over a free semimodule <i>f</i>.
type FreeBialgebra a f = (FreeAlgebra a f, FreeCoalgebra a f, Bialgebra a (Rep f))

-- | A <a>bialgebra</a> over a semiring.
class (Unital a b, Counital a b) => Bialgebra a b

-- | A linear transformation between free semimodules indexed with bases
--   <i>b</i> and <i>c</i>.
--   
--   <pre>
--   f <a>!#</a> x <a>+</a> y = (f <a>!#</a> x) + (f <a>!#</a> y)
--   f <a>!#</a> (r <a>.*</a> x) = r <a>.*</a> (f <a>!#</a> x)
--   </pre>
--   
--   <i>Caution</i>: You must ensure these laws hold when using the default
--   constructor.
--   
--   Prefer <a>image</a> or <a>tran</a> where appropriate.
newtype Tran a b c
Tran :: ((c -> a) -> b -> a) -> Tran a b c
[runTran] :: Tran a b c -> (c -> a) -> b -> a

-- | An endomorphism over a free semimodule.
--   
--   <pre>
--   &gt;&gt;&gt; one + two !# V2 1 2 :: V2 Double
--   V2 3.0 6.0
--   </pre>
type Endo a b = Tran a b b

-- | Create a <a>Tran</a> from a linear combination of basis vectors.
--   
--   <pre>
--   &gt;&gt;&gt; image (e2 [(2, E31),(3, E32)] [(1, E33)]) !# V3 1 1 1 :: V2 Int
--   V2 5 1
--   </pre>
image :: Semiring a => (b -> [(a, c)]) -> Tran a b c

-- | Apply a transformation to a vector.
(!#) :: Free f => Free g => Tran a (Rep f) (Rep g) -> g a -> f a
infixr 2 !#

-- | Apply a transformation to a vector.
(#!) :: Free f => Free g => g a -> Tran a (Rep f) (Rep g) -> f a
infixl 2 #!

-- | Compose two transformations.
(!#!) :: Tran a c d -> Tran a b c -> Tran a b d
infix 2 !#!

-- | <a>Tran</a> is a profunctor in the category of semimodules.
--   
--   <i>Caution</i>: Arbitrary mapping functions may violate linearity.
--   
--   <pre>
--   &gt;&gt;&gt; dimap' id (e3 True True False) (arr id) !# 4 :+ 5 :: V3 Int
--   V3 5 5 4
--   </pre>
dimap' :: (b1 -> b2) -> (c1 -> c2) -> Tran a b2 c1 -> Tran a b1 c2
lmap' :: (b1 -> b2) -> Tran a b2 c -> Tran a b1 c
rmap' :: (c1 -> c2) -> Tran a b c1 -> Tran a b c2

-- | <a>Tran</a> is an invariant functor.
--   
--   See also <a>http://comonad.com/reader/2008/rotten-bananas/</a>.
invmap :: (a1 -> a2) -> (a2 -> a1) -> Tran a1 b c -> Tran a2 b c

-- | Swap components of a tensor product.
braid :: Tran a (b, c) (c, b)

-- | Swap components of a direct sum.
cobraid :: Tran a (b + c) (c + b)

-- | TODO: Document
split :: (b -> (b1, b2)) -> Tran a b1 c -> Tran a b2 c -> Tran a b c

-- | TODO: Document
cosplit :: ((c1 + c2) -> c) -> Tran a b c1 -> Tran a b c2 -> Tran a b c

-- | Project onto the left-hand component of a direct sum.
projl :: Free f => Free g => (f ++ g) a -> f a

-- | Project onto the right-hand component of a direct sum.
projr :: Free f => Free g => (f ++ g) a -> g a

-- | Left (post) composition with a linear transformation.
compl :: Free f1 => Free f2 => Free g => Tran a (Rep f1) (Rep f2) -> (f2 ** g) a -> (f1 ** g) a

-- | Right (pre) composition with a linear transformation.
compr :: Free f => Free g1 => Free g2 => Tran a (Rep g1) (Rep g2) -> (f ** g2) a -> (f ** g1) a

-- | Left and right composition with a linear transformation.
--   
--   <pre>
--   <a>complr</a> f g = <a>compl</a> f <a>&gt;&gt;&gt;</a> <a>compr</a> g
--   </pre>
complr :: Free f1 => Free f2 => Free g1 => Free g2 => Tran a (Rep f1) (Rep f2) -> Tran a (Rep g1) (Rep g2) -> (f2 ** g2) a -> (f1 ** g1) a
instance Data.Semiring.Semiring a => Data.Semimodule.Algebra.Bialgebra a ()
instance (Data.Semimodule.Algebra.Bialgebra a b1, Data.Semimodule.Algebra.Bialgebra a b2) => Data.Semimodule.Algebra.Bialgebra a (b1, b2)
instance (Data.Semimodule.Algebra.Bialgebra a b1, Data.Semimodule.Algebra.Bialgebra a b2, Data.Semimodule.Algebra.Bialgebra a b3) => Data.Semimodule.Algebra.Bialgebra a (b1, b2, b3)
instance Data.Semiring.Semiring a => Data.Semimodule.Algebra.Bialgebra a [b]
instance Data.Semiring.Semiring a => Data.Semimodule.Algebra.Bialgebra a (Data.Sequence.Internal.Seq b)
instance Data.Semiring.Semiring a => Data.Semimodule.Algebra.Unital a ()
instance (Data.Semimodule.Algebra.Unital a b1, Data.Semimodule.Algebra.Unital a b2) => Data.Semimodule.Algebra.Unital a (b1, b2)
instance (Data.Semimodule.Algebra.Unital a b1, Data.Semimodule.Algebra.Unital a b2, Data.Semimodule.Algebra.Unital a b3) => Data.Semimodule.Algebra.Unital a (b1, b2, b3)
instance Data.Semiring.Semiring a => Data.Semimodule.Algebra.Unital a [b]
instance Data.Semiring.Semiring a => Data.Semimodule.Algebra.Unital a (Data.Sequence.Internal.Seq b)
instance (Data.Semiring.Semiring a, GHC.Classes.Ord b) => Data.Semimodule.Algebra.Unital a (Data.Set.Internal.Set b)
instance Data.Semiring.Semiring a => Data.Semimodule.Algebra.Unital a Data.IntSet.Internal.IntSet
instance Data.Semimodule.Algebra.Unital a b => Data.Semimodule.Algebra.Counital a (b -> a)
instance Data.Semiring.Semiring a => Data.Semimodule.Algebra.Algebra a ()
instance (Data.Semimodule.Algebra.Algebra a b1, Data.Semimodule.Algebra.Algebra a b2) => Data.Semimodule.Algebra.Algebra a (b1, b2)
instance (Data.Semimodule.Algebra.Algebra a b1, Data.Semimodule.Algebra.Algebra a b2, Data.Semimodule.Algebra.Algebra a b3) => Data.Semimodule.Algebra.Algebra a (b1, b2, b3)
instance Data.Semiring.Semiring a => Data.Semimodule.Algebra.Algebra a [b]
instance Data.Semiring.Semiring a => Data.Semimodule.Algebra.Algebra a (Data.Sequence.Internal.Seq b)
instance (Data.Semiring.Semiring a, GHC.Classes.Ord b) => Data.Semimodule.Algebra.Algebra a (Data.Set.Internal.Set b)
instance Data.Semiring.Semiring a => Data.Semimodule.Algebra.Algebra a Data.IntSet.Internal.IntSet
instance Data.Semimodule.Algebra.Algebra a b => Data.Semimodule.Algebra.Coalgebra a (b -> a)
instance Data.Semiring.Semiring a => Data.Semimodule.Algebra.Counital a ()
instance (Data.Semimodule.Algebra.Counital a c1, Data.Semimodule.Algebra.Counital a c2) => Data.Semimodule.Algebra.Counital a (c1, c2)
instance (Data.Semimodule.Algebra.Counital a c1, Data.Semimodule.Algebra.Counital a c2, Data.Semimodule.Algebra.Counital a c3) => Data.Semimodule.Algebra.Counital a (c1, c2, c3)
instance Data.Semiring.Semiring a => Data.Semimodule.Algebra.Counital a [c]
instance Data.Semiring.Semiring a => Data.Semimodule.Algebra.Counital a (Data.Sequence.Internal.Seq c)
instance (Data.Semiring.Semiring a, GHC.Classes.Ord c) => Data.Semimodule.Algebra.Counital a (Data.Set.Internal.Set c)
instance Data.Semiring.Semiring a => Data.Semimodule.Algebra.Counital a Data.IntSet.Internal.IntSet
instance Data.Semimodule.Algebra.Counital a c => GHC.Base.Monoid (Data.Semigroup.Additive.Multiplicative (Data.Semimodule.Algebra.Tran a b c))
instance Data.Semimodule.Algebra.Counital a c => Data.Semiring.Semiring (Data.Semimodule.Algebra.Tran a b c)
instance Data.Semimodule.Algebra.Counital a m => Data.Semimodule.LeftSemimodule (Data.Semimodule.Algebra.Tran a b m) (Data.Semimodule.Algebra.Tran a b m)
instance Data.Semimodule.Algebra.Counital a m => Data.Semimodule.RightSemimodule (Data.Semimodule.Algebra.Tran a b m) (Data.Semimodule.Algebra.Tran a b m)
instance (Data.Semiring.Ring a, Data.Semimodule.Algebra.Counital a c) => Data.Semiring.Ring (Data.Semimodule.Algebra.Tran a b c)
instance Data.Semiring.Semiring a => Data.Semimodule.Algebra.Coalgebra a ()
instance (Data.Semimodule.Algebra.Coalgebra a c1, Data.Semimodule.Algebra.Coalgebra a c2) => Data.Semimodule.Algebra.Coalgebra a (c1, c2)
instance (Data.Semimodule.Algebra.Coalgebra a c1, Data.Semimodule.Algebra.Coalgebra a c2, Data.Semimodule.Algebra.Coalgebra a c3) => Data.Semimodule.Algebra.Coalgebra a (c1, c2, c3)
instance Data.Semiring.Semiring a => Data.Semimodule.Algebra.Coalgebra a [c]
instance Data.Semiring.Semiring a => Data.Semimodule.Algebra.Coalgebra a (Data.Sequence.Internal.Seq c)
instance (Data.Semiring.Semiring a, GHC.Classes.Ord c) => Data.Semimodule.Algebra.Coalgebra a (Data.Set.Internal.Set c)
instance Data.Semiring.Semiring a => Data.Semimodule.Algebra.Coalgebra a Data.IntSet.Internal.IntSet
instance Data.Semimodule.Algebra.Coalgebra a c => GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative (Data.Semimodule.Algebra.Tran a b c))
instance Data.Semimodule.Algebra.Coalgebra a c => Data.Semiring.Presemiring (Data.Semimodule.Algebra.Tran a b c)
instance GHC.Base.Functor (Data.Semimodule.Algebra.Tran a b)
instance GHC.Base.Applicative (Data.Semimodule.Algebra.Tran a b)
instance GHC.Base.Monad (Data.Semimodule.Algebra.Tran a b)
instance Control.Category.Category (Data.Semimodule.Algebra.Tran a)
instance Control.Arrow.Arrow (Data.Semimodule.Algebra.Tran a)
instance Control.Arrow.ArrowChoice (Data.Semimodule.Algebra.Tran a)
instance Control.Arrow.ArrowApply (Data.Semimodule.Algebra.Tran a)
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid a => Control.Arrow.ArrowZero (Data.Semimodule.Algebra.Tran a)
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid a => Control.Arrow.ArrowPlus (Data.Semimodule.Algebra.Tran a)
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Algebra.Tran a b c))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semigroup.Additive.Additive (Data.Semimodule.Algebra.Tran a b c))
instance Data.Semimodule.LeftSemimodule r s => Data.Semimodule.LeftSemimodule r (Data.Semimodule.Algebra.Tran s b m)
instance Data.Semimodule.RightSemimodule r s => Data.Semimodule.RightSemimodule r (Data.Semimodule.Algebra.Tran s b m)
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Magma.Magma (Data.Semigroup.Additive.Additive (Data.Semimodule.Algebra.Tran a b c))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Quasigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Algebra.Tran a b c))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Loop (Data.Semigroup.Additive.Additive (Data.Semimodule.Algebra.Tran a b c))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Group (Data.Semigroup.Additive.Additive (Data.Semimodule.Algebra.Tran a b c))

module Data.Semimodule.Dual

-- | Linear functionals from elements of a free semimodule to a scalar.
--   
--   <pre>
--   f <a>!*</a> (x <a>+</a> y) = (f <a>!*</a> x) <a>+</a> (f <a>!*</a> y)
--   f <a>!*</a> (x <a>.*</a> a) = a <a>*</a> (f <a>!*</a> x)
--   </pre>
--   
--   <i>Caution</i>: You must ensure these laws hold when using the default
--   constructor.
newtype Dual a c
Dual :: ((c -> a) -> a) -> Dual a c
[runDual] :: Dual a c -> (c -> a) -> a
infixr 3 `runDual`

-- | Create a <a>Dual</a> from a linear combination of basis vectors.
--   
--   <pre>
--   &gt;&gt;&gt; image' [(2, E31),(3, E32)] !* V3 1 1 1 :: Int
--   5
--   </pre>
image' :: Semiring a => Foldable f => f (a, c) -> Dual a c

-- | Apply a linear functional to a vector.
(!*) :: Free f => Dual a (Rep f) -> f a -> a
infixr 3 !*

-- | Apply a linear functional to a vector.
(*!) :: Free f => f a -> Dual a (Rep f) -> a
infixl 3 *!

-- | Obtain a linear transfrom from a linear functional.
toTran :: (b -> Dual a c) -> Tran a b c

-- | Obtain a linear functional from a linear transform.
fromTran :: Tran a b c -> b -> Dual a c

-- | TODO: Document
init :: Unital a b => b -> Dual a ()

-- | TODO: Document
coinit :: Counital a c => Dual a c

-- | TODO: Document
joined' :: Algebra a b => b -> Dual a (b, b)

-- | TODO: Document
--   
--   <pre>
--   <a>cojoined'</a> = <a>curry</a> <a>$</a> <a>fromTran</a> <a>codiagonal</a>
--   </pre>
cojoined' :: Coalgebra a c => c -> c -> Dual a c

-- | TODO: Document
convolve' :: Algebra a b => Coalgebra a c => (b -> Dual a c) -> (b -> Dual a c) -> b -> Dual a c
instance GHC.Base.Functor (Data.Semimodule.Dual.Dual a)
instance GHC.Base.Applicative (Data.Semimodule.Dual.Dual a)
instance GHC.Base.Monad (Data.Semimodule.Dual.Dual a)
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid a => GHC.Base.Alternative (Data.Semimodule.Dual.Dual a)
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid a => GHC.Base.MonadPlus (Data.Semimodule.Dual.Dual a)
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Dual.Dual a b))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semigroup.Additive.Additive (Data.Semimodule.Dual.Dual a b))
instance Data.Semimodule.Algebra.Coalgebra a b => GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative (Data.Semimodule.Dual.Dual a b))
instance Data.Semimodule.Algebra.Counital a b => GHC.Base.Monoid (Data.Semigroup.Additive.Multiplicative (Data.Semimodule.Dual.Dual a b))
instance Data.Semimodule.Algebra.Coalgebra a b => Data.Semiring.Presemiring (Data.Semimodule.Dual.Dual a b)
instance Data.Semimodule.Algebra.Counital a b => Data.Semiring.Semiring (Data.Semimodule.Dual.Dual a b)
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Magma.Magma (Data.Semigroup.Additive.Additive (Data.Semimodule.Dual.Dual a b))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Quasigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Dual.Dual a b))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Loop (Data.Semigroup.Additive.Additive (Data.Semimodule.Dual.Dual a b))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Group (Data.Semigroup.Additive.Additive (Data.Semimodule.Dual.Dual a b))
instance (Data.Semiring.Ring a, Data.Semimodule.Algebra.Counital a b) => Data.Semiring.Ring (Data.Semimodule.Dual.Dual a b)
instance Data.Semimodule.Algebra.Counital r m => Data.Semimodule.LeftSemimodule (Data.Semimodule.Dual.Dual r m) (Data.Semimodule.Dual.Dual r m)
instance Data.Semimodule.LeftSemimodule r s => Data.Semimodule.LeftSemimodule r (Data.Semimodule.Dual.Dual s m)
instance Data.Semimodule.Algebra.Counital r m => Data.Semimodule.RightSemimodule (Data.Semimodule.Dual.Dual r m) (Data.Semimodule.Dual.Dual r m)
instance Data.Semimodule.RightSemimodule r s => Data.Semimodule.RightSemimodule r (Data.Semimodule.Dual.Dual s m)

module Data.Semimodule.Operator
type Free f = (Representable f)
type Basis b f = (Free f, Rep f ~ b, Eq b)
type Basis2 b c f g = (Basis b f, Basis c g)
type Basis3 b c d f g h = (Basis b f, Basis c g, Basis d h)

-- | Linear functionals from elements of a free semimodule to a scalar.
--   
--   <pre>
--   f <a>!*</a> (x <a>+</a> y) = (f <a>!*</a> x) <a>+</a> (f <a>!*</a> y)
--   f <a>!*</a> (x <a>.*</a> a) = a <a>*</a> (f <a>!*</a> x)
--   </pre>
--   
--   <i>Caution</i>: You must ensure these laws hold when using the default
--   constructor.
newtype Dual a c
Dual :: ((c -> a) -> a) -> Dual a c
[runDual] :: Dual a c -> (c -> a) -> a
infixr 3 `runDual`

-- | Take the dual of a vector.
--   
--   <pre>
--   &gt;&gt;&gt; dual (V2 3 4) !% V2 1 2 :: Int
--   11
--   </pre>
dual :: FreeCounital a f => f a -> Dual a (Rep f)

-- | Create a <a>Dual</a> from a linear combination of basis vectors.
--   
--   <pre>
--   &gt;&gt;&gt; image' [(2, E31),(3, E32)] !* V3 1 1 1 :: Int
--   5
--   </pre>
image' :: Semiring a => Foldable f => f (a, c) -> Dual a c

-- | Dirac delta function.
dirac :: Eq i => Semiring a => i -> i -> a

-- | Create a unit vector at an index.
--   
--   <pre>
--   &gt;&gt;&gt; idx E21 :: V2 Int
--   V2 1 0
--   </pre>
--   
--   <pre>
--   &gt;&gt;&gt; idx E42 :: V4 Int
--   V4 0 1 0 0
--   </pre>
idx :: Semiring a => Basis b f => b -> f a

-- | Retrieve an element of a vector.
--   
--   <pre>
--   &gt;&gt;&gt; elt E21 (V2 1 2)
--   1
--   </pre>
elt :: Basis b f => b -> f a -> a

-- | Create a lens from a representable functor.
lensRep :: Basis b f => b -> forall g. Functor g => (a -> g a) -> f a -> g (f a)

-- | Create an indexed grate from a representable functor.
grateRep :: Basis b f => forall g. Functor g => (b -> g a1 -> a2) -> g (f a1) -> f a2

-- | Right-multiply a module element by a scalar.
(.*) :: RightSemimodule r a => a -> r -> a
infixl 7 .*

-- | Apply a linear functional to a vector.
(!*) :: Free f => Dual a (Rep f) -> f a -> a
infixr 3 !*

-- | Multiply a matrix on the right by a column vector.
--   
--   <pre>
--   (<a>.#</a>) = (<a>!#</a>) . <a>tran</a>
--   </pre>
--   
--   <pre>
--   &gt;&gt;&gt; tran (m23 1 2 3 4 5 6) !# V3 7 8 9 :: V2 Int
--   V2 50 122
--   
--   &gt;&gt;&gt; m23 1 2 3 4 5 6 .# V3 7 8 9 :: V2 Int
--   V2 50 122
--   
--   &gt;&gt;&gt; m22 1 0 0 0 .# m23 1 2 3 4 5 6 .# V3 7 8 9 :: V2 Int
--   V2 50 0
--   </pre>
(.#) :: Free f => FreeCounital a g => (f ** g) a -> g a -> f a
infixr 7 .#

-- | Apply a transformation to a vector.
(!#) :: Free f => Free g => Tran a (Rep f) (Rep g) -> g a -> f a
infixr 2 !#

-- | Left-multiply a module element by a scalar.
(*.) :: LeftSemimodule l a => l -> a -> a
infixr 7 *.

-- | Apply a linear functional to a vector.
(*!) :: Free f => f a -> Dual a (Rep f) -> a
infixl 3 *!

-- | Multiply a matrix on the left by a row vector.
--   
--   <pre>
--   &gt;&gt;&gt; V2 1 2 #. m23 3 4 5 6 7 8
--   V3 15 18 21
--   </pre>
--   
--   <pre>
--   &gt;&gt;&gt; V2 1 2 #. m23 3 4 5 6 7 8 #. m32 1 0 0 0 0 0 :: V2 Int
--   V2 15 0
--   </pre>
(#.) :: FreeCounital a f => Free g => f a -> (f ** g) a -> g a
infixl 7 #.

-- | Apply a transformation to a vector.
(#!) :: Free f => Free g => g a -> Tran a (Rep f) (Rep g) -> f a
infixl 2 #!

-- | Inner product.
--   
--   This is a variant of <a>xmult</a> restricted to free functors.
--   
--   <pre>
--   &gt;&gt;&gt; V3 1 2 3 `inner` V3 1 2 3
--   14
--   </pre>
inner :: FreeCounital a f => f a -> f a -> a
infix 6 `inner`

-- | Outer product.
--   
--   <pre>
--   &gt;&gt;&gt; V2 1 1 `outer` V2 1 1
--   Compose (V2 (V2 1 1) (V2 1 1))
--   </pre>
outer :: Semiring a => Free f => Free g => f a -> g a -> (f ** g) a

-- | Linearly interpolate between two vectors.
--   
--   <pre>
--   &gt;&gt;&gt; u = V3 (1 :% 1) (2 :% 1) (3 :% 1) :: V3 Rational
--   
--   &gt;&gt;&gt; v = V3 (2 :% 1) (4 :% 1) (6 :% 1) :: V3 Rational
--   
--   &gt;&gt;&gt; r = 1 :% 2 :: Rational
--   
--   &gt;&gt;&gt; lerp r u v
--   V3 (6 % 4) (12 % 4) (18 % 4)
--   </pre>
lerp :: LeftModule r a => r -> a -> a -> a

-- | Squared <i>l2</i> norm of a vector.
quadrance :: FreeCounital a f => f a -> a

-- | A linear transformation between free semimodules indexed with bases
--   <i>b</i> and <i>c</i>.
--   
--   <pre>
--   f <a>!#</a> x <a>+</a> y = (f <a>!#</a> x) + (f <a>!#</a> y)
--   f <a>!#</a> (r <a>.*</a> x) = r <a>.*</a> (f <a>!#</a> x)
--   </pre>
--   
--   <i>Caution</i>: You must ensure these laws hold when using the default
--   constructor.
--   
--   Prefer <a>image</a> or <a>tran</a> where appropriate.
newtype Tran a b c
Tran :: ((c -> a) -> b -> a) -> Tran a b c
[runTran] :: Tran a b c -> (c -> a) -> b -> a

-- | Lift a matrix into a linear transformation
--   
--   <pre>
--   (<a>.#</a>) = (<a>!#</a>) . <a>tran</a>
--   </pre>
tran :: Free f => FreeCounital a g => (f ** g) a -> Tran a (Rep f) (Rep g)

-- | Create a <a>Tran</a> from a linear combination of basis vectors.
--   
--   <pre>
--   &gt;&gt;&gt; image (e2 [(2, E31),(3, E32)] [(1, E33)]) !# V3 1 1 1 :: V2 Int
--   V2 5 1
--   </pre>
image :: Semiring a => (b -> [(a, c)]) -> Tran a b c

-- | Retrieve an element of a matrix.
--   
--   <pre>
--   &gt;&gt;&gt; elt2 E21 E21 $ m22 1 2 3 4
--   1
--   </pre>
elt2 :: Basis2 b c f g => b -> c -> (f ** g) a -> a

-- | Retrieve a row of a matrix.
--   
--   <pre>
--   &gt;&gt;&gt; row E22 $ m23 1 2 3 4 5 6
--   V3 4 5 6
--   </pre>
row :: Free f => Rep f -> (f ** g) a -> g a

-- | Obtain a matrix by stacking rows.
--   
--   <pre>
--   &gt;&gt;&gt; rows (V2 1 2) :: M22 Int
--   V2 (V2 1 2) (V2 1 2)
--   </pre>
rows :: Free f => Free g => g a -> (f ** g) a

-- | Retrieve a column of a matrix.
--   
--   <pre>
--   &gt;&gt;&gt; elt E22 . col E31 $ m23 1 2 3 4 5 6
--   4
--   </pre>
col :: Free f => Free g => Rep g -> (f ** g) a -> f a

-- | Obtain a matrix by stacking columns.
--   
--   <pre>
--   &gt;&gt;&gt; cols (V2 1 2) :: M22 Int
--   V2 (V2 1 1) (V2 2 2)
--   </pre>
cols :: Free f => Free g => f a -> (f ** g) a

-- | Obtain the diagonal of a tensor product as a vector.
--   
--   When the coalgebra is trivial we have:
--   
--   <pre>
--   <a>diag</a> f = <a>tabulate</a> $ <a>joined</a> (<a>index</a> . <a>index</a> (<tt>getCompose</tt> f))
--   </pre>
--   
--   <pre>
--   &gt;&gt;&gt; diag $ m22 1.0 2.0 3.0 4.0
--   V2 1.0 4.0
--   </pre>
diag :: FreeAlgebra a f => (f ** f) a -> f a

-- | Obtain a tensor from a vector.
--   
--   When the coalgebra is trivial we have:
--   
--   <pre>
--   <a>codiag</a> = <a>flip</a> <a>bindRep</a> <a>id</a> <a>.</a> <tt>getCompose</tt>
--   </pre>
codiag :: FreeCoalgebra a f => f a -> (f ** f) a

-- | Obtain a <a>scalar matrix</a> from a scalar.
--   
--   <pre>
--   &gt;&gt;&gt; scalar 4.0 :: M22 Double
--   Compose (V2 (V2 4.0 0.0) (V2 0.0 4.0))
--   </pre>
scalar :: FreeCoalgebra a f => a -> (f ** f) a

-- | Obtain an identity matrix.
--   
--   <pre>
--   &gt;&gt;&gt; identity :: M33 Int
--   Compose (V3 (V3 1 0 0) (V3 0 1 0) (V3 0 0 1))
--   </pre>
identity :: FreeCoalgebra a f => (f ** f) a

-- | Multiply two matrices.
--   
--   <pre>
--   &gt;&gt;&gt; m22 1 2 3 4 .#. m22 1 2 3 4 :: M22 Int
--   Compose (V2 (V2 7 10) (V2 15 22))
--   </pre>
--   
--   <pre>
--   &gt;&gt;&gt; m23 1 2 3 4 5 6 .#. m32 1 2 3 4 4 5 :: M22 Int
--   Compose (V2 (V2 19 25) (V2 43 58))
--   </pre>
(.#.) :: Free f => FreeCounital a g => Free h => (f ** g) a -> (g ** h) a -> (f ** h) a
infixr 7 .#.

-- | Compose two transformations.
(!#!) :: Tran a c d -> Tran a b c -> Tran a b d
infix 2 !#!

-- | Trace of an endomorphism.
--   
--   <pre>
--   &gt;&gt;&gt; trace $ m22 1.0 2.0 3.0 4.0
--   5.0
--   </pre>
trace :: FreeBialgebra a f => (f ** f) a -> a

-- | Transpose a matrix.
--   
--   <pre>
--   &gt;&gt;&gt; transpose $ m23 1 2 3 4 5 6 :: M32 Int
--   V3 (V2 1 4) (V2 2 5) (V2 3 6)
--   </pre>
transpose :: Free f => Free g => (f ** g) a -> (g ** f) a

module Data.Semimodule.Basis
data E1
E11 :: E1
e1 :: a -> E1 -> a
fillE1 :: Basis E1 f => a -> f a
data E2
E21 :: E2
E22 :: E2
e2 :: a -> a -> E2 -> a
fillE2 :: Basis E2 f => a -> a -> f a
data E3
E31 :: E3
E32 :: E3
E33 :: E3
e3 :: a -> a -> a -> E3 -> a
fillE3 :: Basis E3 f => a -> a -> a -> f a
data E4
E41 :: E4
E42 :: E4
E43 :: E4
E44 :: E4
e4 :: a -> a -> a -> a -> E4 -> a
fillE4 :: Basis E4 f => a -> a -> a -> a -> f a
instance GHC.Show.Show Data.Semimodule.Basis.E4
instance GHC.Classes.Ord Data.Semimodule.Basis.E4
instance GHC.Classes.Eq Data.Semimodule.Basis.E4
instance GHC.Show.Show Data.Semimodule.Basis.E3
instance GHC.Classes.Ord Data.Semimodule.Basis.E3
instance GHC.Classes.Eq Data.Semimodule.Basis.E3
instance GHC.Show.Show Data.Semimodule.Basis.E2
instance GHC.Classes.Ord Data.Semimodule.Basis.E2
instance GHC.Classes.Eq Data.Semimodule.Basis.E2
instance GHC.Show.Show Data.Semimodule.Basis.E1
instance GHC.Classes.Ord Data.Semimodule.Basis.E1
instance GHC.Classes.Eq Data.Semimodule.Basis.E1
instance Data.Semiring.Semiring r => Data.Semimodule.Algebra.Algebra r Data.Semimodule.Basis.E4
instance Data.Semiring.Semiring r => Data.Semimodule.Algebra.Unital r Data.Semimodule.Basis.E4
instance Data.Semiring.Semiring r => Data.Semimodule.Algebra.Coalgebra r Data.Semimodule.Basis.E4
instance Data.Semiring.Semiring r => Data.Semimodule.Algebra.Counital r Data.Semimodule.Basis.E4
instance Data.Semiring.Semiring r => Data.Semimodule.Algebra.Bialgebra r Data.Semimodule.Basis.E4
instance Data.Semiring.Semiring r => Data.Semimodule.Algebra.Algebra r Data.Semimodule.Basis.E3
instance Data.Semiring.Semiring r => Data.Semimodule.Algebra.Unital r Data.Semimodule.Basis.E3
instance Data.Semiring.Semiring r => Data.Semimodule.Algebra.Coalgebra r Data.Semimodule.Basis.E3
instance Data.Semiring.Semiring r => Data.Semimodule.Algebra.Counital r Data.Semimodule.Basis.E3
instance Data.Semiring.Semiring r => Data.Semimodule.Algebra.Bialgebra r Data.Semimodule.Basis.E3
instance Data.Semiring.Semiring r => Data.Semimodule.Algebra.Algebra r Data.Semimodule.Basis.E2
instance Data.Semiring.Semiring r => Data.Semimodule.Algebra.Unital r Data.Semimodule.Basis.E2
instance Data.Semiring.Semiring r => Data.Semimodule.Algebra.Coalgebra r Data.Semimodule.Basis.E2
instance Data.Semiring.Semiring r => Data.Semimodule.Algebra.Counital r Data.Semimodule.Basis.E2
instance Data.Semiring.Semiring r => Data.Semimodule.Algebra.Bialgebra r Data.Semimodule.Basis.E2
instance Data.Semiring.Semiring r => Data.Semimodule.Algebra.Algebra r Data.Semimodule.Basis.E1
instance Data.Semiring.Semiring r => Data.Semimodule.Algebra.Unital r Data.Semimodule.Basis.E1
instance Data.Semiring.Semiring r => Data.Semimodule.Algebra.Coalgebra r Data.Semimodule.Basis.E1
instance Data.Semiring.Semiring r => Data.Semimodule.Algebra.Counital r Data.Semimodule.Basis.E1
instance Data.Semiring.Semiring r => Data.Semimodule.Algebra.Bialgebra r Data.Semimodule.Basis.E1

module Data.Semimodule.Free
newtype V1 a
V1 :: a -> V1 a
unV1 :: V1 a -> a
data V2 a
V2 :: !a -> !a -> V2 a
data V3 a
V3 :: !a -> !a -> !a -> V3 a

-- | Cross product.
--   
--   <pre>
--   a `<tt>cross'</tt> a = <a>zero</a>
--   a `<tt>cross'</tt> b = <a>negate</a> ( b `<tt>cross'</tt> a ) , 
--   a `<tt>cross'</tt> ( b <a>+</a> c ) = ( a `<tt>cross'</tt> b ) <a>+</a> ( a `<tt>cross'</tt> c ) , 
--   ( r a ) `<tt>cross'</tt> b = a `<tt>cross'</tt> ( r b ) = r ( a `<tt>cross'</tt> b ) . 
--   a `<tt>cross'</tt> ( b `<tt>cross'</tt> c ) <a>+</a> b `<tt>cross'</tt> ( c `<tt>cross'</tt> a ) <a>+</a> c `<tt>cross'</tt> ( a `<tt>cross'</tt> b ) = <a>zero</a> . 
--   </pre>
--   
--   See <a>Jacobi identity</a>.
cross :: Ring a => V3 a -> V3 a -> V3 a

-- | Scalar triple product.
--   
--   <pre>
--   <a>triple</a> x y z = <a>triple</a> z x y = <a>triple</a> y z x
--   <a>triple</a> x y z = <a>negate</a> <a>$</a> <a>triple</a> x z y = <a>negate</a> <a>$</a> <a>triple</a> y x z
--   <a>triple</a> x x y = <a>triple</a> x y y = <a>triple</a> x y x = <a>zero</a>
--   (<a>triple</a> x y z) <a>*.</a> x = (x `<tt>cross'</tt> y) `<tt>cross'</tt> (x `<tt>cross'</tt> z)
--   </pre>
--   
--   <pre>
--   &gt;&gt;&gt; triple (V3 0 0 1) (V3 1 0 0) (V3 0 1 0) :: Double
--   1.0
--   </pre>
triple :: Ring a => V3 a -> V3 a -> V3 a -> a
data V4 a
V4 :: !a -> !a -> !a -> !a -> V4 a

-- | A 1x1 matrix.
type M11 = Compose V1 V1

-- | A 1x2 matrix.
type M12 = Compose V1 V2

-- | A 1x3 matrix.
type M13 = Compose V1 V3

-- | A 1x4 matrix.
type M14 = Compose V1 V4

-- | A 2x1 matrix.
type M21 = Compose V2 V1

-- | A 3x1 matrix.
type M31 = Compose V3 V1

-- | A 4x1 matrix.
type M41 = Compose V4 V1

-- | A 2x2 matrix.
type M22 = Compose V2 V2

-- | A 2x3 matrix.
type M23 = Compose V2 V3

-- | A 2x4 matrix.
type M24 = Compose V2 V4

-- | A 3x2 matrix.
type M32 = Compose V3 V2

-- | A 3x3 matrix.
type M33 = Compose V3 V3

-- | A 3x4 matrix.
type M34 = Compose V3 V4

-- | A 4x2 matrix.
type M42 = Compose V4 V2

-- | A 4x3 matrix.
type M43 = Compose V4 V3

-- | A 4x4 matrix.
type M44 = Compose V4 V4

-- | Construct a 1x1 matrix.
--   
--   <pre>
--   &gt;&gt;&gt; m11 1 :: M11 Int
--   Compose (V1 (V1 1))
--   </pre>
m11 :: a -> M11 a

-- | Construct a 1x2 matrix.
--   
--   <pre>
--   &gt;&gt;&gt; m12 1 2 :: M12 Int
--   Compose (V1 (V2 1 2))
--   </pre>
m12 :: a -> a -> M12 a

-- | Construct a 1x3 matrix.
--   
--   <pre>
--   &gt;&gt;&gt; m13 1 2 3 :: M13 Int
--   Compose (V1 (V3 1 2 3))
--   </pre>
m13 :: a -> a -> a -> M13 a

-- | Construct a 1x4 matrix.
--   
--   <pre>
--   &gt;&gt;&gt; m14 1 2 3 4 :: M14 Int
--   Compose (V1 (V4 1 2 3 4))
--   </pre>
m14 :: a -> a -> a -> a -> M14 a

-- | Construct a 2x1 matrix.
--   
--   <pre>
--   &gt;&gt;&gt; m21 1 2 :: M21 Int
--   Compose (V2 (V1 1) (V1 2))
--   </pre>
m21 :: a -> a -> M21 a

-- | Construct a 3x1 matrix.
--   
--   <pre>
--   &gt;&gt;&gt; m31 1 2 3 :: M31 Int
--   Compose (V3 (V1 1) (V1 2) (V1 3))
--   </pre>
m31 :: a -> a -> a -> M31 a

-- | Construct a 4x1 matrix.
--   
--   <pre>
--   &gt;&gt;&gt; m41 1 2 3 4 :: M41 Int
--   Compose (V4 (V1 1) (V1 2) (V1 3) (V1 4))
--   </pre>
m41 :: a -> a -> a -> a -> M41 a

-- | Construct a 2x2 matrix.
--   
--   Arguments are in row-major order.
--   
--   <pre>
--   &gt;&gt;&gt; m22 1 2 3 4 :: M22 Int
--   Compose (V2 (V2 1 2) (V2 3 4))
--   </pre>
m22 :: a -> a -> a -> a -> M22 a

-- | Construct a 2x3 matrix.
--   
--   Arguments are in row-major order.
m23 :: a -> a -> a -> a -> a -> a -> M23 a

-- | Construct a 2x4 matrix.
--   
--   Arguments are in row-major order.
m24 :: a -> a -> a -> a -> a -> a -> a -> a -> M24 a

-- | Construct a 3x2 matrix.
--   
--   Arguments are in row-major order.
m32 :: a -> a -> a -> a -> a -> a -> M32 a

-- | Construct a 3x3 matrix.
--   
--   Arguments are in row-major order.
m33 :: a -> a -> a -> a -> a -> a -> a -> a -> a -> M33 a

-- | Construct a 3x4 matrix.
--   
--   Arguments are in row-major order.
m34 :: a -> a -> a -> a -> a -> a -> a -> a -> a -> a -> a -> a -> M34 a

-- | Construct a 4x2 matrix.
--   
--   Arguments are in row-major order.
m42 :: a -> a -> a -> a -> a -> a -> a -> a -> M42 a

-- | Construct a 4x3 matrix.
--   
--   Arguments are in row-major order.
m43 :: a -> a -> a -> a -> a -> a -> a -> a -> a -> a -> a -> a -> M43 a

-- | Construct a 4x4 matrix.
--   
--   Arguments are in row-major order.
m44 :: a -> a -> a -> a -> a -> a -> a -> a -> a -> a -> a -> a -> a -> a -> a -> a -> M44 a

-- | 1x1 matrix inverse over a field.
--   
--   <pre>
--   &gt;&gt;&gt; inv1 $ m11 4.0 :: M11 Double
--   Compose (V1 (V1 0.25))
--   </pre>
inv1 :: Field a => M11 a -> M11 a

-- | 2x2 matrix inverse over a field.
--   
--   <pre>
--   &gt;&gt;&gt; inv2 $ m22 1 2 3 4 :: M22 Double
--   Compose (V2 (V2 (-2.0) 1.0) (V2 1.5 (-0.5)))
--   </pre>
inv2 :: Field a => M22 a -> M22 a

-- | 2x2 matrix bdeterminant over a commutative semiring.
--   
--   <pre>
--   &gt;&gt;&gt; bdet2 $ m22 1 2 3 4
--   (4,6)
--   </pre>
bdet2 :: Semiring a => Basis2 E2 E2 f g => (f ** g) a -> (a, a)

-- | 2x2 matrix determinant over a commutative ring.
--   
--   <pre>
--   <a>det2</a> = <a>uncurry</a> (<a>-</a>) . <a>bdet2</a>
--   </pre>
--   
--   <pre>
--   &gt;&gt;&gt; det2 $ m22 1 2 3 4 :: Double
--   -2.0
--   </pre>
det2 :: Ring a => Basis2 E2 E2 f g => (f ** g) a -> a

-- | 3x3 matrix bdeterminant over a commutative semiring.
--   
--   <pre>
--   &gt;&gt;&gt; bdet3 (V3 (V3 1 2 3) (V3 4 5 6) (V3 7 8 9))
--   (225, 225)
--   </pre>
bdet3 :: Semiring a => Basis2 E3 E3 f g => (f ** g) a -> (a, a)

-- | 3x3 double-precision matrix determinant.
--   
--   <pre>
--   <a>det3</a> = <a>uncurry</a> (<a>-</a>) . <a>bdet3</a>
--   </pre>
--   
--   Implementation uses a cofactor expansion to avoid loss of precision.
--   
--   <pre>
--   &gt;&gt;&gt; det3 $ m33 1 2 3 4 5 6 7 8 9
--   0
--   </pre>
det3 :: Ring a => Basis2 E3 E3 f g => (f ** g) a -> a

-- | 3x3 matrix inverse.
--   
--   <pre>
--   &gt;&gt;&gt; inv3 $ m33 1 2 4 4 2 2 1 1 1 :: M33 Double
--   Compose (V3 (V3 0.0 0.5 (-1.0)) (V3 (-0.5) (-0.75) 3.5) (V3 0.5 0.25 (-1.5)))
--   </pre>
inv3 :: Field a => M33 a -> M33 a

-- | 4x4 matrix bdeterminant over a commutative semiring.
--   
--   <pre>
--   &gt;&gt;&gt; bdet4 $ m44 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
--   (27728,27728)
--   </pre>
bdet4 :: Semiring a => Basis2 E4 E4 f g => (f ** g) a -> (a, a)

-- | 4x4 matrix determinant over a commutative ring.
--   
--   <pre>
--   <a>det4</a> = <a>uncurry</a> (<a>-</a>) . <a>bdet4</a>
--   </pre>
--   
--   This implementation uses a cofactor expansion to avoid loss of
--   precision.
--   
--   <pre>
--   &gt;&gt;&gt; det4 $ m44 1 0 3 2 2 0 2 1 0 0 0 1 0 3 4 0 :: Rational
--   (-12) % 1
--   </pre>
det4 :: Ring a => Basis2 E4 E4 f g => (f ** g) a -> a

-- | 4x4 matrix inverse.
--   
--   <pre>
--   &gt;&gt;&gt; row E41 . inv4 $ m44 1 0 3 2 2 0 2 1 0 0 0 1 0 3 4 0 :: V4 Rational
--   V4 (6 % (-12)) ((-9) % (-12)) ((-3) % (-12)) (0 % (-12))
--   </pre>
inv4 :: Field a => M44 a -> M44 a
instance GHC.Show.Show a => GHC.Show.Show (Data.Semimodule.Free.V4 a)
instance GHC.Classes.Ord a => GHC.Classes.Ord (Data.Semimodule.Free.V4 a)
instance GHC.Classes.Eq a => GHC.Classes.Eq (Data.Semimodule.Free.V4 a)
instance GHC.Show.Show a => GHC.Show.Show (Data.Semimodule.Free.V3 a)
instance GHC.Classes.Ord a => GHC.Classes.Ord (Data.Semimodule.Free.V3 a)
instance GHC.Classes.Eq a => GHC.Classes.Eq (Data.Semimodule.Free.V3 a)
instance GHC.Show.Show a => GHC.Show.Show (Data.Semimodule.Free.V2 a)
instance GHC.Classes.Ord a => GHC.Classes.Ord (Data.Semimodule.Free.V2 a)
instance GHC.Classes.Eq a => GHC.Classes.Eq (Data.Semimodule.Free.V2 a)
instance GHC.Show.Show a => GHC.Show.Show (Data.Semimodule.Free.V1 a)
instance GHC.Classes.Ord a => GHC.Classes.Ord (Data.Semimodule.Free.V1 a)
instance GHC.Classes.Eq a => GHC.Classes.Eq (Data.Semimodule.Free.V1 a)
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule (Data.Semimodule.Free.M44 a) (Data.Semimodule.Free.M41 a)
instance Data.Semiring.Semiring a => Data.Semimodule.Bisemimodule (Data.Semimodule.Free.M44 a) (Data.Semimodule.Free.M11 a) (Data.Semimodule.Free.M41 a)
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule (Data.Semimodule.Free.M44 a) (Data.Semimodule.Free.M42 a)
instance Data.Semiring.Semiring a => Data.Semimodule.Bisemimodule (Data.Semimodule.Free.M44 a) (Data.Semimodule.Free.M22 a) (Data.Semimodule.Free.M42 a)
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule (Data.Semimodule.Free.M44 a) (Data.Semimodule.Free.M43 a)
instance Data.Semiring.Semiring a => Data.Semimodule.Bisemimodule (Data.Semimodule.Free.M44 a) (Data.Semimodule.Free.M33 a) (Data.Semimodule.Free.M43 a)
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule (Data.Semimodule.Free.M44 a) (Data.Semimodule.Free.M14 a)
instance Data.Semiring.Semiring a => Data.Semimodule.Bisemimodule (Data.Semimodule.Free.M11 a) (Data.Semimodule.Free.M44 a) (Data.Semimodule.Free.M14 a)
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule (Data.Semimodule.Free.M44 a) (Data.Semimodule.Free.M24 a)
instance Data.Semiring.Semiring a => Data.Semimodule.Bisemimodule (Data.Semimodule.Free.M22 a) (Data.Semimodule.Free.M44 a) (Data.Semimodule.Free.M24 a)
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule (Data.Semimodule.Free.M44 a) (Data.Semimodule.Free.M34 a)
instance Data.Semiring.Semiring a => Data.Semimodule.Bisemimodule (Data.Semimodule.Free.M33 a) (Data.Semimodule.Free.M44 a) (Data.Semimodule.Free.M34 a)
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M44 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M44 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Magma.Magma (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M44 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Quasigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M44 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Loop (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M44 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Group (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M44 a))
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule (Data.Semimodule.Free.M44 a) (Data.Semimodule.Free.M44 a)
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule (Data.Semimodule.Free.M44 a) (Data.Semimodule.Free.M44 a)
instance Data.Semiring.Semiring a => Data.Semimodule.Bisemimodule (Data.Semimodule.Free.M44 a) (Data.Semimodule.Free.M44 a) (Data.Semimodule.Free.M44 a)
instance Data.Semiring.Semiring a => GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative (Data.Semimodule.Free.M44 a))
instance Data.Semiring.Semiring a => GHC.Base.Monoid (Data.Semigroup.Additive.Multiplicative (Data.Semimodule.Free.M44 a))
instance Data.Semiring.Semiring a => Data.Semiring.Presemiring (Data.Semimodule.Free.M44 a)
instance Data.Semiring.Semiring a => Data.Semiring.Semiring (Data.Semimodule.Free.M44 a)
instance Data.Semiring.Ring a => Data.Semiring.Ring (Data.Semimodule.Free.M44 a)
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M43 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M43 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Magma.Magma (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M43 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Quasigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M43 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Loop (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M43 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Group (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M43 a))
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule (Data.Semimodule.Free.M33 a) (Data.Semimodule.Free.M43 a)
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M42 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M42 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Magma.Magma (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M42 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Quasigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M42 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Loop (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M42 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Group (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M42 a))
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule (Data.Semimodule.Free.M22 a) (Data.Semimodule.Free.M42 a)
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M34 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M34 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Magma.Magma (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M34 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Quasigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M34 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Loop (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M34 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Group (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M34 a))
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule (Data.Semimodule.Free.M33 a) (Data.Semimodule.Free.M34 a)
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule (Data.Semimodule.Free.M33 a) (Data.Semimodule.Free.M31 a)
instance Data.Semiring.Semiring a => Data.Semimodule.Bisemimodule (Data.Semimodule.Free.M33 a) (Data.Semimodule.Free.M11 a) (Data.Semimodule.Free.M31 a)
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule (Data.Semimodule.Free.M33 a) (Data.Semimodule.Free.M32 a)
instance Data.Semiring.Semiring a => Data.Semimodule.Bisemimodule (Data.Semimodule.Free.M33 a) (Data.Semimodule.Free.M22 a) (Data.Semimodule.Free.M32 a)
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule (Data.Semimodule.Free.M33 a) (Data.Semimodule.Free.M13 a)
instance Data.Semiring.Semiring a => Data.Semimodule.Bisemimodule (Data.Semimodule.Free.M11 a) (Data.Semimodule.Free.M33 a) (Data.Semimodule.Free.M13 a)
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule (Data.Semimodule.Free.M33 a) (Data.Semimodule.Free.M23 a)
instance Data.Semiring.Semiring a => Data.Semimodule.Bisemimodule (Data.Semimodule.Free.M22 a) (Data.Semimodule.Free.M33 a) (Data.Semimodule.Free.M23 a)
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M33 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M33 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Magma.Magma (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M33 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Quasigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M33 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Loop (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M33 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Group (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M33 a))
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule (Data.Semimodule.Free.M33 a) (Data.Semimodule.Free.M33 a)
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule (Data.Semimodule.Free.M33 a) (Data.Semimodule.Free.M33 a)
instance Data.Semiring.Semiring a => Data.Semimodule.Bisemimodule (Data.Semimodule.Free.M33 a) (Data.Semimodule.Free.M33 a) (Data.Semimodule.Free.M33 a)
instance Data.Semiring.Semiring a => GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative (Data.Semimodule.Free.M33 a))
instance Data.Semiring.Semiring a => GHC.Base.Monoid (Data.Semigroup.Additive.Multiplicative (Data.Semimodule.Free.M33 a))
instance Data.Semiring.Semiring a => Data.Semiring.Presemiring (Data.Semimodule.Free.M33 a)
instance Data.Semiring.Semiring a => Data.Semiring.Semiring (Data.Semimodule.Free.M33 a)
instance Data.Semiring.Ring a => Data.Semiring.Ring (Data.Semimodule.Free.M33 a)
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M32 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M32 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Magma.Magma (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M32 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Quasigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M32 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Loop (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M32 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Group (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M32 a))
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule (Data.Semimodule.Free.M22 a) (Data.Semimodule.Free.M32 a)
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M24 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M24 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Magma.Magma (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M24 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Quasigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M24 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Loop (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M24 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Group (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M24 a))
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule (Data.Semimodule.Free.M22 a) (Data.Semimodule.Free.M24 a)
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M23 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M23 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Magma.Magma (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M23 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Quasigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M23 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Loop (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M23 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Group (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M23 a))
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule (Data.Semimodule.Free.M22 a) (Data.Semimodule.Free.M23 a)
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule (Data.Semimodule.Free.M22 a) (Data.Semimodule.Free.M21 a)
instance Data.Semiring.Semiring a => Data.Semimodule.Bisemimodule (Data.Semimodule.Free.M22 a) (Data.Semimodule.Free.M11 a) (Data.Semimodule.Free.M21 a)
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule (Data.Semimodule.Free.M22 a) (Data.Semimodule.Free.M12 a)
instance Data.Semiring.Semiring a => Data.Semimodule.Bisemimodule (Data.Semimodule.Free.M11 a) (Data.Semimodule.Free.M22 a) (Data.Semimodule.Free.M12 a)
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M22 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M22 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Magma.Magma (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M22 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Quasigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M22 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Loop (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M22 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Group (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M22 a))
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule (Data.Semimodule.Free.M22 a) (Data.Semimodule.Free.M22 a)
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule (Data.Semimodule.Free.M22 a) (Data.Semimodule.Free.M22 a)
instance Data.Semiring.Semiring a => Data.Semimodule.Bisemimodule (Data.Semimodule.Free.M22 a) (Data.Semimodule.Free.M22 a) (Data.Semimodule.Free.M22 a)
instance Data.Semiring.Semiring a => GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative (Data.Semimodule.Free.M22 a))
instance Data.Semiring.Semiring a => GHC.Base.Monoid (Data.Semigroup.Additive.Multiplicative (Data.Semimodule.Free.M22 a))
instance Data.Semiring.Semiring a => Data.Semiring.Presemiring (Data.Semimodule.Free.M22 a)
instance Data.Semiring.Semiring a => Data.Semiring.Semiring (Data.Semimodule.Free.M22 a)
instance Data.Semiring.Ring a => Data.Semiring.Ring (Data.Semimodule.Free.M22 a)
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M41 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M41 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Magma.Magma (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M41 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Quasigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M41 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Loop (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M41 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Group (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M41 a))
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule (Data.Semimodule.Free.M11 a) (Data.Semimodule.Free.M41 a)
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M31 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M31 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Magma.Magma (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M31 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Quasigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M31 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Loop (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M31 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Group (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M31 a))
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule (Data.Semimodule.Free.M11 a) (Data.Semimodule.Free.M31 a)
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M21 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M21 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Magma.Magma (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M21 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Quasigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M21 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Loop (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M21 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Group (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M21 a))
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule (Data.Semimodule.Free.M11 a) (Data.Semimodule.Free.M21 a)
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M14 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M14 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Magma.Magma (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M14 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Quasigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M14 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Loop (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M14 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Group (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M14 a))
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule (Data.Semimodule.Free.M11 a) (Data.Semimodule.Free.M14 a)
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M13 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M13 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Magma.Magma (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M13 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Quasigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M13 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Loop (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M13 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Group (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M13 a))
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule (Data.Semimodule.Free.M11 a) (Data.Semimodule.Free.M13 a)
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M12 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M12 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Magma.Magma (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M12 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Quasigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M12 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Loop (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M12 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Group (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M12 a))
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule (Data.Semimodule.Free.M11 a) (Data.Semimodule.Free.M12 a)
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M11 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M11 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Magma.Magma (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M11 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Quasigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M11 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Loop (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M11 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Group (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.M11 a))
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule (Data.Semimodule.Free.M11 a) (Data.Semimodule.Free.M11 a)
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule (Data.Semimodule.Free.M11 a) (Data.Semimodule.Free.M11 a)
instance Data.Semiring.Semiring a => Data.Semimodule.Bisemimodule (Data.Semimodule.Free.M11 a) (Data.Semimodule.Free.M11 a) (Data.Semimodule.Free.M11 a)
instance Data.Semiring.Semiring a => GHC.Base.Semigroup (Data.Semigroup.Additive.Multiplicative (Data.Semimodule.Free.M11 a))
instance Data.Semiring.Semiring a => GHC.Base.Monoid (Data.Semigroup.Additive.Multiplicative (Data.Semimodule.Free.M11 a))
instance Data.Semiring.Semiring a => Data.Semiring.Presemiring (Data.Semimodule.Free.M11 a)
instance Data.Semiring.Semiring a => Data.Semiring.Semiring (Data.Semimodule.Free.M11 a)
instance Data.Semiring.Ring a => Data.Semiring.Ring (Data.Semimodule.Free.M11 a)
instance Data.Functor.Classes.Show1 Data.Semimodule.Free.V4
instance GHC.Base.Functor Data.Semimodule.Free.V4
instance GHC.Base.Applicative Data.Semimodule.Free.V4
instance Data.Foldable.Foldable Data.Semimodule.Free.V4
instance Data.Semigroup.Foldable.Class.Foldable1 Data.Semimodule.Free.V4
instance Data.Distributive.Distributive Data.Semimodule.Free.V4
instance Data.Functor.Rep.Representable Data.Semimodule.Free.V4
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.V4 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.V4 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Magma.Magma (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.V4 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Quasigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.V4 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Loop (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.V4 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Group (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.V4 a))
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule a (Data.Semimodule.Free.V4 a)
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule a (Data.Semimodule.Free.V4 a)
instance Data.Semiring.Semiring a => Data.Semimodule.Bisemimodule a a (Data.Semimodule.Free.V4 a)
instance Data.Functor.Classes.Eq1 Data.Semimodule.Free.V3
instance Data.Functor.Classes.Show1 Data.Semimodule.Free.V3
instance GHC.Base.Functor Data.Semimodule.Free.V3
instance GHC.Base.Applicative Data.Semimodule.Free.V3
instance Data.Foldable.Foldable Data.Semimodule.Free.V3
instance Data.Semigroup.Foldable.Class.Foldable1 Data.Semimodule.Free.V3
instance Data.Distributive.Distributive Data.Semimodule.Free.V3
instance Data.Functor.Rep.Representable Data.Semimodule.Free.V3
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.V3 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.V3 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Magma.Magma (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.V3 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Quasigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.V3 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Loop (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.V3 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Group (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.V3 a))
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule a (Data.Semimodule.Free.V3 a)
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule a (Data.Semimodule.Free.V3 a)
instance Data.Semiring.Semiring a => Data.Semimodule.Bisemimodule a a (Data.Semimodule.Free.V3 a)
instance Data.Functor.Classes.Show1 Data.Semimodule.Free.V2
instance GHC.Base.Functor Data.Semimodule.Free.V2
instance GHC.Base.Applicative Data.Semimodule.Free.V2
instance Data.Foldable.Foldable Data.Semimodule.Free.V2
instance Data.Semigroup.Foldable.Class.Foldable1 Data.Semimodule.Free.V2
instance Data.Distributive.Distributive Data.Semimodule.Free.V2
instance Data.Functor.Rep.Representable Data.Semimodule.Free.V2
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.V2 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.V2 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Magma.Magma (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.V2 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Quasigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.V2 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Loop (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.V2 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Group (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.V2 a))
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule a (Data.Semimodule.Free.V2 a)
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule a (Data.Semimodule.Free.V2 a)
instance Data.Semiring.Semiring a => Data.Semimodule.Bisemimodule a a (Data.Semimodule.Free.V2 a)
instance Data.Functor.Classes.Show1 Data.Semimodule.Free.V1
instance GHC.Base.Functor Data.Semimodule.Free.V1
instance GHC.Base.Applicative Data.Semimodule.Free.V1
instance Data.Foldable.Foldable Data.Semimodule.Free.V1
instance Data.Semigroup.Foldable.Class.Foldable1 Data.Semimodule.Free.V1
instance Data.Distributive.Distributive Data.Semimodule.Free.V1
instance Data.Functor.Rep.Representable Data.Semimodule.Free.V1
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.V1 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.V1 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Magma.Magma (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.V1 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Quasigroup (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.V1 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Loop (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.V1 a))
instance (Data.Semigroup.Additive.-) Data.Semigroup.Additive.Additive Data.Group.Group a => Data.Group.Group (Data.Semigroup.Additive.Additive (Data.Semimodule.Free.V1 a))
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule a (Data.Semimodule.Free.V1 a)
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule a (Data.Semimodule.Free.V1 a)
instance Data.Semiring.Semiring a => Data.Semimodule.Bisemimodule a a (Data.Semimodule.Free.V1 a)


-- | See the <i>connections</i> package for idempotent &amp; selective
--   semirings, and lattices.
module Data.Semiring.Property

-- | &lt;math&gt;
--   
--   If <i>R</i> is non-unital (i.e. <i>one</i> is not distinct from
--   <i>zero</i>) then it will instead satisfy a right-absorbtion property.
--   
--   This follows from right-neutrality and right-distributivity.
--   
--   When <i>R</i> is also left-distributive we get: &lt;math&gt;
--   
--   See also <a>Warning</a> and
--   <a>https://blogs.ncl.ac.uk/andreymokhov/united-monoids/#whatif</a>.
nonunital_on :: Presemiring r => Rel r b -> r -> r -> b

-- | Presemiring morphisms are distributive semigroup morphisms.
--   
--   This is a required property for presemiring morphisms.
morphism_presemiring :: Eq s => Presemiring r => Presemiring s => (r -> s) -> r -> r -> r -> Bool

-- | &lt;math&gt;
--   
--   A semigroup must right-associate addition.
--   
--   This is a required property for semigroups.
associative_addition_on :: (Additive - Semigroup) r => Rel r b -> r -> r -> r -> b

-- | &lt;math&gt;
--   
--   This is a an optional property for semigroups, and a required property
--   for semirings.
commutative_addition_on :: (Additive - Semigroup) r => Rel r b -> r -> r -> b

-- | &lt;math&gt;
--   
--   A semigroup must right-associate multiplication.
--   
--   This is a required property for semigroups.
associative_multiplication_on :: (Multiplicative - Semigroup) r => Rel r b -> r -> r -> r -> b

-- | &lt;math&gt;
--   
--   <i>R</i> must right-distribute multiplication.
--   
--   When <i>R</i> is a functor and the semiring structure is derived from
--   <a>Alternative</a>, this translates to:
--   
--   <pre>
--   (a <a>&lt;|&gt;</a> b) <a>*&gt;</a> c = (a <a>*&gt;</a> c) <a>&lt;|&gt;</a> (b <a>*&gt;</a> c)
--   </pre>
--   
--   See
--   <a>https://en.wikibooks.org/wiki/Haskell/Alternative_and_MonadPlus</a>.
--   
--   This is a required property.
distributive_on :: Presemiring r => Rel r b -> r -> r -> r -> b

-- | &lt;math&gt;
--   
--   <i>R</i> must right-distribute multiplication over finite (non-empty)
--   sums.
--   
--   For types with exact arithmetic this follows from
--   <a>distributive_on</a> and the universality of folds.
distributive_finite1_on :: Presemiring r => Foldable1 f => Rel r b -> f r -> r -> b

-- | &lt;math&gt;
--   
--   Presemiring morphisms must be compatible with right-distribution.
morphism_distribitive_on :: Presemiring r => Presemiring s => Rel s b -> (r -> s) -> r -> r -> r -> b

-- | Semiring morphisms are monoidal presemiring morphisms.
--   
--   This is a required property for semiring morphisms.
morphism_semiring :: Eq s => Semiring r => Semiring s => (r -> s) -> r -> r -> r -> Bool

-- | &lt;math&gt;
--   
--   A semigroup with a right-neutral additive identity must satisfy:
--   
--   <pre>
--   <a>neutral_addition_on</a> (<a>==</a>) <a>zero</a> r = <a>True</a>
--   </pre>
--   
--   Or, equivalently:
--   
--   <pre>
--   <a>zero</a> <a>+</a> r = r
--   </pre>
--   
--   This is a required property for additive monoids.
neutral_addition_on :: (Additive - Monoid) r => Rel r b -> r -> b

-- | &lt;math&gt;
--   
--   A semigroup with a right-neutral multiplicative identity must satisfy:
--   
--   <pre>
--   <a>neutral_multiplication_on</a> (<a>==</a>) <a>one</a> r = <a>True</a>
--   </pre>
--   
--   Or, equivalently:
--   
--   <pre>
--   <a>one</a> <a>*</a> r = r
--   </pre>
--   
--   This is a required property for multiplicative monoids.
neutral_multiplication_on :: (Multiplicative - Monoid) r => Rel r b -> r -> b

-- | &lt;math&gt;
--   
--   A <i>R</i> is semiring then its addititive one must be
--   right-annihilative, i.e.:
--   
--   <pre>
--   <a>zero</a> <a>*</a> a = <a>zero</a>
--   </pre>
--   
--   For <a>Alternative</a> instances this property translates to:
--   
--   <pre>
--   <a>empty</a> <a>*&gt;</a> a = <a>empty</a>
--   </pre>
--   
--   This is a required property.
annihilative_multiplication_on :: Semiring r => Rel r b -> r -> b

-- | &lt;math&gt;
--   
--   <i>R</i> must right-distribute multiplication between finite sums.
--   
--   For types with exact arithmetic this follows from
--   <a>distributive_on</a> &amp; <a>neutral_multiplication_on</a>.
distributive_finite_on :: Semiring r => Foldable f => Rel r b -> f r -> r -> b

-- | &lt;math&gt;
--   
--   If <i>R</i> is also left-distributive then it supports
--   xmult-multiplication.
distributive_xmult_on :: Semiring r => Applicative f => Foldable f => Rel r b -> f r -> f r -> b

-- | &lt;math&gt;
--   
--   If <i>R</i> is also left-distributive then it supports (non-empty)
--   xmult-multiplication.
distributive_xmult1_on :: Presemiring r => Apply f => Foldable1 f => Rel r b -> f r -> f r -> b

-- | &lt;math&gt;
--   
--   This is a an optional property for semigroups, and a optional property
--   for semirings and rings.
commutative_multiplication_on :: (Multiplicative - Semigroup) r => Rel r b -> r -> r -> b

-- | &lt;math&gt;
--   
--   If <i>R</i> is right-cancellative wrt addition then for all <i>a</i>
--   the section <i>(a +)</i> is injective.
--   
--   See <a>https://en.wikipedia.org/wiki/Cancellation_property</a>
cancellative_addition_on :: (Additive - Semigroup) r => Rel r Bool -> r -> r -> r -> Bool

-- | &lt;math&gt;
--   
--   If <i>R</i> is right-cancellative wrt multiplication then for all
--   <i>a</i> the section <i>(a *)</i> is injective.
cancellative_multiplication_on :: (Multiplicative - Semigroup) r => Rel r Bool -> r -> r -> r -> Bool