-- 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 0.1.3

module Data.Semiring

-- | Hyphenation operator.
type ( g - f ) a = f (g a)
infixl 6 -

-- | Tensor product.
type f ** g = Compose f g
infixr 2 **

-- | Direct sum.
type f ++ g = Product f g
infixr 1 ++
type Free = Representable
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 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

-- | 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
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 GHC.Base.Functor Data.Semiring.Multiplicative
instance GHC.Show.Show a => GHC.Show.Show (Data.Semiring.Multiplicative a)
instance GHC.Classes.Ord a => GHC.Classes.Ord (Data.Semiring.Multiplicative a)
instance GHC.Generics.Generic (Data.Semiring.Multiplicative a)
instance GHC.Classes.Eq a => GHC.Classes.Eq (Data.Semiring.Multiplicative a)
instance GHC.Base.Functor Data.Semiring.Additive
instance GHC.Show.Show a => GHC.Show.Show (Data.Semiring.Additive a)
instance GHC.Classes.Ord a => GHC.Classes.Ord (Data.Semiring.Additive a)
instance GHC.Generics.Generic (Data.Semiring.Additive a)
instance GHC.Classes.Eq a => GHC.Classes.Eq (Data.Semiring.Additive a)
instance Data.Semiring.Ring a => Data.Semiring.Presemiring (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 (e -> 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 (e -> a)
instance Data.Semiring.Semiring a => Data.Semiring.Semiring (GHC.Maybe.Maybe a)
instance (Data.Semiring.-) Data.Semiring.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.Semiring.-) Data.Semiring.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 (e -> 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.Semiring.-) Data.Semiring.Additive GHC.Base.Semigroup a => Data.Semiring.Presemiring [a]
instance (Data.Semiring.-) Data.Semiring.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)
instance GHC.Base.Applicative Data.Semiring.Multiplicative
instance Data.Distributive.Distributive Data.Semiring.Multiplicative
instance Data.Functor.Rep.Representable Data.Semiring.Multiplicative
instance ((Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a, (Data.Semiring.-) Data.Semiring.Multiplicative GHC.Base.Semigroup a) => GHC.Base.Semigroup (Data.Semiring.Multiplicative (Data.Complex.Complex a))
instance ((Data.Semiring.-) Data.Semiring.Additive GHC.Base.Semigroup a, (Data.Semiring.-) Data.Semiring.Multiplicative GHC.Base.Semigroup a) => GHC.Base.Semigroup (Data.Semiring.Additive (GHC.Real.Ratio a))
instance ((Data.Semiring.-) Data.Semiring.Additive GHC.Base.Monoid a, (Data.Semiring.-) Data.Semiring.Multiplicative GHC.Base.Monoid a) => GHC.Base.Monoid (Data.Semiring.Additive (GHC.Real.Ratio a))
instance ((Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a, (Data.Semiring.-) Data.Semiring.Multiplicative GHC.Base.Monoid a) => Data.Magma.Magma (Data.Semiring.Additive (GHC.Real.Ratio a))
instance ((Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a, (Data.Semiring.-) Data.Semiring.Multiplicative GHC.Base.Monoid a) => Data.Group.Quasigroup (Data.Semiring.Additive (GHC.Real.Ratio a))
instance ((Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a, (Data.Semiring.-) Data.Semiring.Multiplicative GHC.Base.Monoid a) => Data.Group.Loop (Data.Semiring.Additive (GHC.Real.Ratio a))
instance ((Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a, (Data.Semiring.-) Data.Semiring.Multiplicative GHC.Base.Monoid a) => Data.Group.Group (Data.Semiring.Additive (GHC.Real.Ratio a))
instance (Data.Semiring.-) Data.Semiring.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semiring.Multiplicative [a])
instance (Data.Semiring.-) Data.Semiring.Additive GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semiring.Multiplicative [a])
instance (Data.Semiring.-) Data.Semiring.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semiring.Multiplicative (GHC.Base.NonEmpty a))
instance (Data.Semiring.-) Data.Semiring.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semiring.Multiplicative (Data.Semigroup.Min a))
instance (Data.Semiring.-) Data.Semiring.Additive GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semiring.Multiplicative (Data.Semigroup.Min a))
instance GHC.Base.Semigroup (Data.Semiring.Multiplicative GHC.Types.Int)
instance GHC.Base.Semigroup (Data.Semiring.Multiplicative GHC.Int.Int8)
instance GHC.Base.Semigroup (Data.Semiring.Multiplicative GHC.Int.Int16)
instance GHC.Base.Semigroup (Data.Semiring.Multiplicative GHC.Int.Int32)
instance GHC.Base.Semigroup (Data.Semiring.Multiplicative GHC.Int.Int64)
instance GHC.Base.Semigroup (Data.Semiring.Multiplicative GHC.Integer.Type.Integer)
instance GHC.Base.Semigroup (Data.Semiring.Multiplicative GHC.Types.Word)
instance GHC.Base.Semigroup (Data.Semiring.Multiplicative GHC.Word.Word8)
instance GHC.Base.Semigroup (Data.Semiring.Multiplicative GHC.Word.Word16)
instance GHC.Base.Semigroup (Data.Semiring.Multiplicative GHC.Word.Word32)
instance GHC.Base.Semigroup (Data.Semiring.Multiplicative GHC.Word.Word64)
instance GHC.Base.Semigroup (Data.Semiring.Multiplicative GHC.Natural.Natural)
instance GHC.Base.Semigroup (Data.Semiring.Multiplicative Data.Fixed.Uni)
instance GHC.Base.Semigroup (Data.Semiring.Multiplicative Data.Fixed.Deci)
instance GHC.Base.Semigroup (Data.Semiring.Multiplicative Data.Fixed.Centi)
instance GHC.Base.Semigroup (Data.Semiring.Multiplicative Data.Fixed.Milli)
instance GHC.Base.Semigroup (Data.Semiring.Multiplicative Data.Fixed.Micro)
instance GHC.Base.Semigroup (Data.Semiring.Multiplicative Data.Fixed.Nano)
instance GHC.Base.Semigroup (Data.Semiring.Multiplicative Data.Fixed.Pico)
instance GHC.Base.Semigroup (Data.Semiring.Multiplicative GHC.Types.Float)
instance GHC.Base.Semigroup (Data.Semiring.Multiplicative Foreign.C.Types.CFloat)
instance GHC.Base.Semigroup (Data.Semiring.Multiplicative GHC.Types.Double)
instance GHC.Base.Semigroup (Data.Semiring.Multiplicative Foreign.C.Types.CDouble)
instance GHC.Base.Monoid (Data.Semiring.Multiplicative GHC.Types.Int)
instance GHC.Base.Monoid (Data.Semiring.Multiplicative GHC.Int.Int8)
instance GHC.Base.Monoid (Data.Semiring.Multiplicative GHC.Int.Int16)
instance GHC.Base.Monoid (Data.Semiring.Multiplicative GHC.Int.Int32)
instance GHC.Base.Monoid (Data.Semiring.Multiplicative GHC.Int.Int64)
instance GHC.Base.Monoid (Data.Semiring.Multiplicative GHC.Integer.Type.Integer)
instance GHC.Base.Monoid (Data.Semiring.Multiplicative GHC.Types.Word)
instance GHC.Base.Monoid (Data.Semiring.Multiplicative GHC.Word.Word8)
instance GHC.Base.Monoid (Data.Semiring.Multiplicative GHC.Word.Word16)
instance GHC.Base.Monoid (Data.Semiring.Multiplicative GHC.Word.Word32)
instance GHC.Base.Monoid (Data.Semiring.Multiplicative GHC.Word.Word64)
instance GHC.Base.Monoid (Data.Semiring.Multiplicative GHC.Natural.Natural)
instance GHC.Base.Monoid (Data.Semiring.Multiplicative Data.Fixed.Uni)
instance GHC.Base.Monoid (Data.Semiring.Multiplicative Data.Fixed.Deci)
instance GHC.Base.Monoid (Data.Semiring.Multiplicative Data.Fixed.Centi)
instance GHC.Base.Monoid (Data.Semiring.Multiplicative Data.Fixed.Milli)
instance GHC.Base.Monoid (Data.Semiring.Multiplicative Data.Fixed.Micro)
instance GHC.Base.Monoid (Data.Semiring.Multiplicative Data.Fixed.Nano)
instance GHC.Base.Monoid (Data.Semiring.Multiplicative Data.Fixed.Pico)
instance GHC.Base.Monoid (Data.Semiring.Multiplicative GHC.Types.Float)
instance GHC.Base.Monoid (Data.Semiring.Multiplicative Foreign.C.Types.CFloat)
instance GHC.Base.Monoid (Data.Semiring.Multiplicative GHC.Types.Double)
instance GHC.Base.Monoid (Data.Semiring.Multiplicative Foreign.C.Types.CDouble)
instance Data.Magma.Magma (Data.Semiring.Multiplicative Data.Fixed.Uni)
instance Data.Magma.Magma (Data.Semiring.Multiplicative Data.Fixed.Deci)
instance Data.Magma.Magma (Data.Semiring.Multiplicative Data.Fixed.Centi)
instance Data.Magma.Magma (Data.Semiring.Multiplicative Data.Fixed.Milli)
instance Data.Magma.Magma (Data.Semiring.Multiplicative Data.Fixed.Micro)
instance Data.Magma.Magma (Data.Semiring.Multiplicative Data.Fixed.Nano)
instance Data.Magma.Magma (Data.Semiring.Multiplicative Data.Fixed.Pico)
instance Data.Magma.Magma (Data.Semiring.Multiplicative GHC.Types.Float)
instance Data.Magma.Magma (Data.Semiring.Multiplicative Foreign.C.Types.CFloat)
instance Data.Magma.Magma (Data.Semiring.Multiplicative GHC.Types.Double)
instance Data.Magma.Magma (Data.Semiring.Multiplicative Foreign.C.Types.CDouble)
instance Data.Group.Quasigroup (Data.Semiring.Multiplicative Data.Fixed.Uni)
instance Data.Group.Quasigroup (Data.Semiring.Multiplicative Data.Fixed.Deci)
instance Data.Group.Quasigroup (Data.Semiring.Multiplicative Data.Fixed.Centi)
instance Data.Group.Quasigroup (Data.Semiring.Multiplicative Data.Fixed.Milli)
instance Data.Group.Quasigroup (Data.Semiring.Multiplicative Data.Fixed.Micro)
instance Data.Group.Quasigroup (Data.Semiring.Multiplicative Data.Fixed.Nano)
instance Data.Group.Quasigroup (Data.Semiring.Multiplicative Data.Fixed.Pico)
instance Data.Group.Quasigroup (Data.Semiring.Multiplicative GHC.Types.Float)
instance Data.Group.Quasigroup (Data.Semiring.Multiplicative Foreign.C.Types.CFloat)
instance Data.Group.Quasigroup (Data.Semiring.Multiplicative GHC.Types.Double)
instance Data.Group.Quasigroup (Data.Semiring.Multiplicative Foreign.C.Types.CDouble)
instance Data.Group.Loop (Data.Semiring.Multiplicative Data.Fixed.Uni)
instance Data.Group.Loop (Data.Semiring.Multiplicative Data.Fixed.Deci)
instance Data.Group.Loop (Data.Semiring.Multiplicative Data.Fixed.Centi)
instance Data.Group.Loop (Data.Semiring.Multiplicative Data.Fixed.Milli)
instance Data.Group.Loop (Data.Semiring.Multiplicative Data.Fixed.Micro)
instance Data.Group.Loop (Data.Semiring.Multiplicative Data.Fixed.Nano)
instance Data.Group.Loop (Data.Semiring.Multiplicative Data.Fixed.Pico)
instance Data.Group.Loop (Data.Semiring.Multiplicative GHC.Types.Float)
instance Data.Group.Loop (Data.Semiring.Multiplicative Foreign.C.Types.CFloat)
instance Data.Group.Loop (Data.Semiring.Multiplicative GHC.Types.Double)
instance Data.Group.Loop (Data.Semiring.Multiplicative Foreign.C.Types.CDouble)
instance Data.Group.Group (Data.Semiring.Multiplicative Data.Fixed.Uni)
instance Data.Group.Group (Data.Semiring.Multiplicative Data.Fixed.Deci)
instance Data.Group.Group (Data.Semiring.Multiplicative Data.Fixed.Centi)
instance Data.Group.Group (Data.Semiring.Multiplicative Data.Fixed.Milli)
instance Data.Group.Group (Data.Semiring.Multiplicative Data.Fixed.Micro)
instance Data.Group.Group (Data.Semiring.Multiplicative Data.Fixed.Nano)
instance Data.Group.Group (Data.Semiring.Multiplicative Data.Fixed.Pico)
instance Data.Group.Group (Data.Semiring.Multiplicative GHC.Types.Float)
instance Data.Group.Group (Data.Semiring.Multiplicative Foreign.C.Types.CFloat)
instance Data.Group.Group (Data.Semiring.Multiplicative GHC.Types.Double)
instance Data.Group.Group (Data.Semiring.Multiplicative Foreign.C.Types.CDouble)
instance (Data.Semiring.-) Data.Semiring.Multiplicative GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semiring.Multiplicative (GHC.Real.Ratio a))
instance (Data.Semiring.-) Data.Semiring.Multiplicative GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semiring.Multiplicative (GHC.Real.Ratio a))
instance (Data.Semiring.-) Data.Semiring.Multiplicative GHC.Base.Monoid a => Data.Magma.Magma (Data.Semiring.Multiplicative (GHC.Real.Ratio a))
instance (Data.Semiring.-) Data.Semiring.Multiplicative GHC.Base.Monoid a => Data.Group.Quasigroup (Data.Semiring.Multiplicative (GHC.Real.Ratio a))
instance (Data.Semiring.-) Data.Semiring.Multiplicative GHC.Base.Monoid a => Data.Group.Loop (Data.Semiring.Multiplicative (GHC.Real.Ratio a))
instance (Data.Semiring.-) Data.Semiring.Multiplicative GHC.Base.Monoid a => Data.Group.Group (Data.Semiring.Multiplicative (GHC.Real.Ratio a))
instance GHC.Base.Semigroup (Data.Semiring.Multiplicative ())
instance GHC.Base.Monoid (Data.Semiring.Multiplicative ())
instance Data.Magma.Magma (Data.Semiring.Multiplicative ())
instance Data.Group.Quasigroup (Data.Semiring.Multiplicative ())
instance Data.Group.Loop (Data.Semiring.Multiplicative ())
instance Data.Group.Group (Data.Semiring.Multiplicative ())
instance GHC.Base.Semigroup (Data.Semiring.Multiplicative GHC.Types.Bool)
instance GHC.Base.Monoid (Data.Semiring.Multiplicative GHC.Types.Bool)
instance (Data.Semiring.-) Data.Semiring.Multiplicative GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semiring.Multiplicative (Data.Semigroup.Internal.Dual a))
instance (Data.Semiring.-) Data.Semiring.Multiplicative GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semiring.Multiplicative (Data.Semigroup.Internal.Dual a))
instance (Data.Semiring.-) Data.Semiring.Multiplicative GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semiring.Multiplicative (Data.Ord.Down a))
instance (Data.Semiring.-) Data.Semiring.Multiplicative GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semiring.Multiplicative (Data.Ord.Down a))
instance (Data.Semiring.-) Data.Semiring.Multiplicative GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semiring.Multiplicative (Data.Semigroup.Max a))
instance (Data.Semiring.-) Data.Semiring.Multiplicative GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semiring.Multiplicative (Data.Semigroup.Max a))
instance ((Data.Semiring.-) Data.Semiring.Multiplicative GHC.Base.Semigroup a, (Data.Semiring.-) Data.Semiring.Multiplicative GHC.Base.Semigroup b) => GHC.Base.Semigroup (Data.Semiring.Multiplicative (a, b))
instance (Data.Semiring.-) Data.Semiring.Multiplicative GHC.Base.Semigroup b => GHC.Base.Semigroup (Data.Semiring.Multiplicative (a -> b))
instance (Data.Semiring.-) Data.Semiring.Multiplicative GHC.Base.Monoid b => GHC.Base.Monoid (Data.Semiring.Multiplicative (a -> b))
instance (Data.Semiring.-) Data.Semiring.Multiplicative GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semiring.Multiplicative (GHC.Maybe.Maybe a))
instance (Data.Semiring.-) Data.Semiring.Multiplicative GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semiring.Multiplicative (GHC.Maybe.Maybe a))
instance ((Data.Semiring.-) Data.Semiring.Multiplicative GHC.Base.Semigroup a, (Data.Semiring.-) Data.Semiring.Multiplicative GHC.Base.Semigroup b) => GHC.Base.Semigroup (Data.Semiring.Multiplicative (Data.Either.Either a b))
instance GHC.Classes.Ord a => GHC.Base.Semigroup (Data.Semiring.Multiplicative (Data.Set.Internal.Set a))
instance (GHC.Classes.Ord k, (Data.Semiring.-) Data.Semiring.Multiplicative GHC.Base.Semigroup a) => GHC.Base.Semigroup (Data.Semiring.Multiplicative (Data.Map.Internal.Map k a))
instance (Data.Semiring.-) Data.Semiring.Multiplicative GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semiring.Multiplicative (Data.IntMap.Internal.IntMap a))
instance GHC.Base.Semigroup (Data.Semiring.Multiplicative Data.IntSet.Internal.IntSet)
instance (GHC.Classes.Ord k, (Data.Semiring.-) Data.Semiring.Multiplicative GHC.Base.Monoid k, (Data.Semiring.-) Data.Semiring.Multiplicative GHC.Base.Monoid a) => GHC.Base.Monoid (Data.Semiring.Multiplicative (Data.Map.Internal.Map k a))
instance (Data.Semiring.-) Data.Semiring.Multiplicative GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semiring.Multiplicative (Data.IntMap.Internal.IntMap a))
instance GHC.Base.Applicative Data.Semiring.Additive
instance Data.Distributive.Distributive Data.Semiring.Additive
instance Data.Functor.Rep.Representable Data.Semiring.Additive
instance GHC.Base.Semigroup (Data.Semiring.Additive GHC.Types.Int)
instance GHC.Base.Semigroup (Data.Semiring.Additive GHC.Int.Int8)
instance GHC.Base.Semigroup (Data.Semiring.Additive GHC.Int.Int16)
instance GHC.Base.Semigroup (Data.Semiring.Additive GHC.Int.Int32)
instance GHC.Base.Semigroup (Data.Semiring.Additive GHC.Int.Int64)
instance GHC.Base.Semigroup (Data.Semiring.Additive GHC.Integer.Type.Integer)
instance GHC.Base.Semigroup (Data.Semiring.Additive GHC.Types.Word)
instance GHC.Base.Semigroup (Data.Semiring.Additive GHC.Word.Word8)
instance GHC.Base.Semigroup (Data.Semiring.Additive GHC.Word.Word16)
instance GHC.Base.Semigroup (Data.Semiring.Additive GHC.Word.Word32)
instance GHC.Base.Semigroup (Data.Semiring.Additive GHC.Word.Word64)
instance GHC.Base.Semigroup (Data.Semiring.Additive GHC.Natural.Natural)
instance GHC.Base.Semigroup (Data.Semiring.Additive Data.Fixed.Uni)
instance GHC.Base.Semigroup (Data.Semiring.Additive Data.Fixed.Deci)
instance GHC.Base.Semigroup (Data.Semiring.Additive Data.Fixed.Centi)
instance GHC.Base.Semigroup (Data.Semiring.Additive Data.Fixed.Milli)
instance GHC.Base.Semigroup (Data.Semiring.Additive Data.Fixed.Micro)
instance GHC.Base.Semigroup (Data.Semiring.Additive Data.Fixed.Nano)
instance GHC.Base.Semigroup (Data.Semiring.Additive Data.Fixed.Pico)
instance GHC.Base.Semigroup (Data.Semiring.Additive GHC.Types.Float)
instance GHC.Base.Semigroup (Data.Semiring.Additive Foreign.C.Types.CFloat)
instance GHC.Base.Semigroup (Data.Semiring.Additive GHC.Types.Double)
instance GHC.Base.Semigroup (Data.Semiring.Additive Foreign.C.Types.CDouble)
instance GHC.Base.Monoid (Data.Semiring.Additive GHC.Types.Int)
instance GHC.Base.Monoid (Data.Semiring.Additive GHC.Int.Int8)
instance GHC.Base.Monoid (Data.Semiring.Additive GHC.Int.Int16)
instance GHC.Base.Monoid (Data.Semiring.Additive GHC.Int.Int32)
instance GHC.Base.Monoid (Data.Semiring.Additive GHC.Int.Int64)
instance GHC.Base.Monoid (Data.Semiring.Additive GHC.Integer.Type.Integer)
instance GHC.Base.Monoid (Data.Semiring.Additive GHC.Types.Word)
instance GHC.Base.Monoid (Data.Semiring.Additive GHC.Word.Word8)
instance GHC.Base.Monoid (Data.Semiring.Additive GHC.Word.Word16)
instance GHC.Base.Monoid (Data.Semiring.Additive GHC.Word.Word32)
instance GHC.Base.Monoid (Data.Semiring.Additive GHC.Word.Word64)
instance GHC.Base.Monoid (Data.Semiring.Additive GHC.Natural.Natural)
instance GHC.Base.Monoid (Data.Semiring.Additive Data.Fixed.Uni)
instance GHC.Base.Monoid (Data.Semiring.Additive Data.Fixed.Deci)
instance GHC.Base.Monoid (Data.Semiring.Additive Data.Fixed.Centi)
instance GHC.Base.Monoid (Data.Semiring.Additive Data.Fixed.Milli)
instance GHC.Base.Monoid (Data.Semiring.Additive Data.Fixed.Micro)
instance GHC.Base.Monoid (Data.Semiring.Additive Data.Fixed.Nano)
instance GHC.Base.Monoid (Data.Semiring.Additive Data.Fixed.Pico)
instance GHC.Base.Monoid (Data.Semiring.Additive GHC.Types.Float)
instance GHC.Base.Monoid (Data.Semiring.Additive Foreign.C.Types.CFloat)
instance GHC.Base.Monoid (Data.Semiring.Additive GHC.Types.Double)
instance GHC.Base.Monoid (Data.Semiring.Additive Foreign.C.Types.CDouble)
instance Data.Magma.Magma (Data.Semiring.Additive GHC.Types.Int)
instance Data.Magma.Magma (Data.Semiring.Additive GHC.Int.Int8)
instance Data.Magma.Magma (Data.Semiring.Additive GHC.Int.Int16)
instance Data.Magma.Magma (Data.Semiring.Additive GHC.Int.Int32)
instance Data.Magma.Magma (Data.Semiring.Additive GHC.Int.Int64)
instance Data.Magma.Magma (Data.Semiring.Additive GHC.Integer.Type.Integer)
instance Data.Magma.Magma (Data.Semiring.Additive Data.Fixed.Uni)
instance Data.Magma.Magma (Data.Semiring.Additive Data.Fixed.Deci)
instance Data.Magma.Magma (Data.Semiring.Additive Data.Fixed.Centi)
instance Data.Magma.Magma (Data.Semiring.Additive Data.Fixed.Milli)
instance Data.Magma.Magma (Data.Semiring.Additive Data.Fixed.Micro)
instance Data.Magma.Magma (Data.Semiring.Additive Data.Fixed.Nano)
instance Data.Magma.Magma (Data.Semiring.Additive Data.Fixed.Pico)
instance Data.Magma.Magma (Data.Semiring.Additive GHC.Types.Float)
instance Data.Magma.Magma (Data.Semiring.Additive Foreign.C.Types.CFloat)
instance Data.Magma.Magma (Data.Semiring.Additive GHC.Types.Double)
instance Data.Magma.Magma (Data.Semiring.Additive Foreign.C.Types.CDouble)
instance Data.Group.Quasigroup (Data.Semiring.Additive GHC.Types.Int)
instance Data.Group.Quasigroup (Data.Semiring.Additive GHC.Int.Int8)
instance Data.Group.Quasigroup (Data.Semiring.Additive GHC.Int.Int16)
instance Data.Group.Quasigroup (Data.Semiring.Additive GHC.Int.Int32)
instance Data.Group.Quasigroup (Data.Semiring.Additive GHC.Int.Int64)
instance Data.Group.Quasigroup (Data.Semiring.Additive GHC.Integer.Type.Integer)
instance Data.Group.Quasigroup (Data.Semiring.Additive Data.Fixed.Uni)
instance Data.Group.Quasigroup (Data.Semiring.Additive Data.Fixed.Deci)
instance Data.Group.Quasigroup (Data.Semiring.Additive Data.Fixed.Centi)
instance Data.Group.Quasigroup (Data.Semiring.Additive Data.Fixed.Milli)
instance Data.Group.Quasigroup (Data.Semiring.Additive Data.Fixed.Micro)
instance Data.Group.Quasigroup (Data.Semiring.Additive Data.Fixed.Nano)
instance Data.Group.Quasigroup (Data.Semiring.Additive Data.Fixed.Pico)
instance Data.Group.Quasigroup (Data.Semiring.Additive GHC.Types.Float)
instance Data.Group.Quasigroup (Data.Semiring.Additive Foreign.C.Types.CFloat)
instance Data.Group.Quasigroup (Data.Semiring.Additive GHC.Types.Double)
instance Data.Group.Quasigroup (Data.Semiring.Additive Foreign.C.Types.CDouble)
instance Data.Group.Loop (Data.Semiring.Additive GHC.Types.Int)
instance Data.Group.Loop (Data.Semiring.Additive GHC.Int.Int8)
instance Data.Group.Loop (Data.Semiring.Additive GHC.Int.Int16)
instance Data.Group.Loop (Data.Semiring.Additive GHC.Int.Int32)
instance Data.Group.Loop (Data.Semiring.Additive GHC.Int.Int64)
instance Data.Group.Loop (Data.Semiring.Additive GHC.Integer.Type.Integer)
instance Data.Group.Loop (Data.Semiring.Additive Data.Fixed.Uni)
instance Data.Group.Loop (Data.Semiring.Additive Data.Fixed.Deci)
instance Data.Group.Loop (Data.Semiring.Additive Data.Fixed.Centi)
instance Data.Group.Loop (Data.Semiring.Additive Data.Fixed.Milli)
instance Data.Group.Loop (Data.Semiring.Additive Data.Fixed.Micro)
instance Data.Group.Loop (Data.Semiring.Additive Data.Fixed.Nano)
instance Data.Group.Loop (Data.Semiring.Additive Data.Fixed.Pico)
instance Data.Group.Loop (Data.Semiring.Additive GHC.Types.Float)
instance Data.Group.Loop (Data.Semiring.Additive Foreign.C.Types.CFloat)
instance Data.Group.Loop (Data.Semiring.Additive GHC.Types.Double)
instance Data.Group.Loop (Data.Semiring.Additive Foreign.C.Types.CDouble)
instance Data.Group.Group (Data.Semiring.Additive GHC.Types.Int)
instance Data.Group.Group (Data.Semiring.Additive GHC.Int.Int8)
instance Data.Group.Group (Data.Semiring.Additive GHC.Int.Int16)
instance Data.Group.Group (Data.Semiring.Additive GHC.Int.Int32)
instance Data.Group.Group (Data.Semiring.Additive GHC.Int.Int64)
instance Data.Group.Group (Data.Semiring.Additive GHC.Integer.Type.Integer)
instance Data.Group.Group (Data.Semiring.Additive Data.Fixed.Uni)
instance Data.Group.Group (Data.Semiring.Additive Data.Fixed.Deci)
instance Data.Group.Group (Data.Semiring.Additive Data.Fixed.Centi)
instance Data.Group.Group (Data.Semiring.Additive Data.Fixed.Milli)
instance Data.Group.Group (Data.Semiring.Additive Data.Fixed.Micro)
instance Data.Group.Group (Data.Semiring.Additive Data.Fixed.Nano)
instance Data.Group.Group (Data.Semiring.Additive Data.Fixed.Pico)
instance Data.Group.Group (Data.Semiring.Additive GHC.Types.Float)
instance Data.Group.Group (Data.Semiring.Additive Foreign.C.Types.CFloat)
instance Data.Group.Group (Data.Semiring.Additive GHC.Types.Double)
instance Data.Group.Group (Data.Semiring.Additive Foreign.C.Types.CDouble)
instance ((Data.Semiring.-) Data.Semiring.Additive GHC.Base.Semigroup a, Data.Semiring.Free f, Data.Semiring.Free g) => GHC.Base.Semigroup (Data.Semiring.Additive ((Data.Semiring.++) f g a))
instance ((Data.Semiring.-) Data.Semiring.Additive GHC.Base.Monoid a, Data.Semiring.Free f, Data.Semiring.Free g) => GHC.Base.Monoid (Data.Semiring.Additive ((Data.Semiring.++) f g a))
instance ((Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a, Data.Semiring.Free f, Data.Semiring.Free g) => Data.Magma.Magma (Data.Semiring.Additive ((Data.Semiring.++) f g a))
instance ((Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a, Data.Semiring.Free f, Data.Semiring.Free g) => Data.Group.Quasigroup (Data.Semiring.Additive ((Data.Semiring.++) f g a))
instance ((Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a, Data.Semiring.Free f, Data.Semiring.Free g) => Data.Group.Loop (Data.Semiring.Additive ((Data.Semiring.++) f g a))
instance ((Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a, Data.Semiring.Free f, Data.Semiring.Free g) => Data.Group.Group (Data.Semiring.Additive ((Data.Semiring.++) f g a))
instance ((Data.Semiring.-) Data.Semiring.Additive GHC.Base.Semigroup a, Data.Semiring.Free f, Data.Semiring.Free g) => GHC.Base.Semigroup (Data.Semiring.Additive ((Data.Semiring.**) f g a))
instance ((Data.Semiring.-) Data.Semiring.Additive GHC.Base.Monoid a, Data.Semiring.Free f, Data.Semiring.Free g) => GHC.Base.Monoid (Data.Semiring.Additive ((Data.Semiring.**) f g a))
instance ((Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a, Data.Semiring.Free f, Data.Semiring.Free g) => Data.Magma.Magma (Data.Semiring.Additive ((Data.Semiring.**) f g a))
instance ((Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a, Data.Semiring.Free f, Data.Semiring.Free g) => Data.Group.Quasigroup (Data.Semiring.Additive ((Data.Semiring.**) f g a))
instance ((Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a, Data.Semiring.Free f, Data.Semiring.Free g) => Data.Group.Loop (Data.Semiring.Additive ((Data.Semiring.**) f g a))
instance ((Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a, Data.Semiring.Free f, Data.Semiring.Free g) => Data.Group.Group (Data.Semiring.Additive ((Data.Semiring.**) f g a))
instance (Data.Semiring.-) Data.Semiring.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semiring.Additive (Data.Complex.Complex a))
instance (Data.Semiring.-) Data.Semiring.Additive GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semiring.Additive (Data.Complex.Complex a))
instance (Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a => Data.Magma.Magma (Data.Semiring.Additive (Data.Complex.Complex a))
instance (Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a => Data.Group.Quasigroup (Data.Semiring.Additive (Data.Complex.Complex a))
instance (Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a => Data.Group.Loop (Data.Semiring.Additive (Data.Complex.Complex a))
instance (Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a => Data.Group.Group (Data.Semiring.Additive (Data.Complex.Complex a))
instance (Data.Semiring.-) Data.Semiring.Additive GHC.Base.Semigroup b => GHC.Base.Semigroup (Data.Semiring.Additive (a -> b))
instance (Data.Semiring.-) Data.Semiring.Additive GHC.Base.Monoid b => GHC.Base.Monoid (Data.Semiring.Additive (a -> b))
instance (Data.Semiring.-) Data.Semiring.Additive Data.Group.Group b => Data.Magma.Magma (Data.Semiring.Additive (a -> b))
instance (Data.Semiring.-) Data.Semiring.Additive Data.Group.Group b => Data.Group.Quasigroup (Data.Semiring.Additive (a -> b))
instance (Data.Semiring.-) Data.Semiring.Additive Data.Group.Group b => Data.Group.Loop (Data.Semiring.Additive (a -> b))
instance (Data.Semiring.-) Data.Semiring.Additive Data.Group.Group b => Data.Group.Group (Data.Semiring.Additive (a -> b))
instance (Data.Semiring.-) Data.Semiring.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semiring.Additive (Data.Functor.Contravariant.Op a b))
instance (Data.Semiring.-) Data.Semiring.Additive GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semiring.Additive (Data.Functor.Contravariant.Op a b))
instance (Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a => Data.Magma.Magma (Data.Semiring.Additive (Data.Functor.Contravariant.Op a b))
instance (Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a => Data.Group.Quasigroup (Data.Semiring.Additive (Data.Functor.Contravariant.Op a b))
instance (Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a => Data.Group.Loop (Data.Semiring.Additive (Data.Functor.Contravariant.Op a b))
instance (Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a => Data.Group.Group (Data.Semiring.Additive (Data.Functor.Contravariant.Op a b))
instance GHC.Base.Semigroup (Data.Semiring.Additive [a])
instance GHC.Base.Monoid (Data.Semiring.Additive [a])
instance GHC.Base.Semigroup (Data.Semiring.Additive (GHC.Base.NonEmpty a))
instance (Data.Semiring.-) Data.Semiring.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semiring.Additive (Data.Ord.Down a))
instance (Data.Semiring.-) Data.Semiring.Additive GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semiring.Additive (Data.Ord.Down a))
instance GHC.Base.Semigroup (Data.Semiring.Additive ())
instance GHC.Base.Monoid (Data.Semiring.Additive ())
instance Data.Magma.Magma (Data.Semiring.Additive ())
instance Data.Group.Quasigroup (Data.Semiring.Additive ())
instance Data.Group.Loop (Data.Semiring.Additive ())
instance Data.Group.Group (Data.Semiring.Additive ())
instance GHC.Base.Semigroup (Data.Semiring.Additive GHC.Types.Bool)
instance GHC.Base.Monoid (Data.Semiring.Additive GHC.Types.Bool)
instance ((Data.Semiring.-) Data.Semiring.Additive GHC.Base.Semigroup a, (Data.Semiring.-) Data.Semiring.Additive GHC.Base.Semigroup b) => GHC.Base.Semigroup (Data.Semiring.Additive (a, b))
instance ((Data.Semiring.-) Data.Semiring.Additive GHC.Base.Monoid a, (Data.Semiring.-) Data.Semiring.Additive GHC.Base.Monoid b) => GHC.Base.Monoid (Data.Semiring.Additive (a, b))
instance ((Data.Semiring.-) Data.Semiring.Additive GHC.Base.Semigroup a, (Data.Semiring.-) Data.Semiring.Additive GHC.Base.Semigroup b, (Data.Semiring.-) Data.Semiring.Additive GHC.Base.Semigroup c) => GHC.Base.Semigroup (Data.Semiring.Additive (a, b, c))
instance ((Data.Semiring.-) Data.Semiring.Additive GHC.Base.Monoid a, (Data.Semiring.-) Data.Semiring.Additive GHC.Base.Monoid b, (Data.Semiring.-) Data.Semiring.Additive GHC.Base.Monoid c) => GHC.Base.Monoid (Data.Semiring.Additive (a, b, c))
instance (Data.Semiring.-) Data.Semiring.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semiring.Additive (GHC.Maybe.Maybe a))
instance (Data.Semiring.-) Data.Semiring.Additive GHC.Base.Semigroup a => GHC.Base.Monoid (Data.Semiring.Additive (GHC.Maybe.Maybe a))
instance ((Data.Semiring.-) Data.Semiring.Additive GHC.Base.Semigroup a, (Data.Semiring.-) Data.Semiring.Additive GHC.Base.Semigroup b) => GHC.Base.Semigroup (Data.Semiring.Additive (Data.Either.Either a b))
instance GHC.Classes.Ord a => GHC.Base.Semigroup (Data.Semiring.Additive (Data.Set.Internal.Set a))
instance (GHC.Classes.Ord k, (Data.Semiring.-) Data.Semiring.Additive GHC.Base.Semigroup a) => GHC.Base.Semigroup (Data.Semiring.Additive (Data.Map.Internal.Map k a))
instance (Data.Semiring.-) Data.Semiring.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semiring.Additive (Data.IntMap.Internal.IntMap a))
instance GHC.Base.Semigroup (Data.Semiring.Additive Data.IntSet.Internal.IntSet)
instance GHC.Base.Monoid (Data.Semiring.Additive Data.IntSet.Internal.IntSet)
instance (Data.Semiring.-) Data.Semiring.Additive GHC.Base.Semigroup a => GHC.Base.Monoid (Data.Semiring.Additive (Data.IntMap.Internal.IntMap a))
instance GHC.Classes.Ord a => GHC.Base.Monoid (Data.Semiring.Additive (Data.Set.Internal.Set a))
instance (GHC.Classes.Ord k, (Data.Semiring.-) Data.Semiring.Additive GHC.Base.Semigroup a) => GHC.Base.Monoid (Data.Semiring.Additive (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 ()
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.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
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
infixr 7 `lscaleDef`

-- | 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
infixl 7 `rscaleDef`
type Bimodule l r a = (LeftModule l a, RightModule r a, Bisemimodule l r a)
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))

-- | <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

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

-- | 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> = <tt>runLin</tt> <tt>initial</tt> <a>.</a> <a>const</a>
--   </pre>
unital :: Unital a b => a -> b -> 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> <tt>runLin</tt> <tt>codiagonal</tt>
--   </pre>
cojoined :: Coalgebra a c => (c -> a) -> c -> c -> a

-- | 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

-- | 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
instance Data.Semiring.Semiring a => Data.Semimodule.Bialgebra a ()
instance (Data.Semimodule.Bialgebra a b1, Data.Semimodule.Bialgebra a b2) => Data.Semimodule.Bialgebra a (b1, b2)
instance (Data.Semimodule.Bialgebra a b1, Data.Semimodule.Bialgebra a b2, Data.Semimodule.Bialgebra a b3) => Data.Semimodule.Bialgebra a (b1, b2, b3)
instance Data.Semiring.Semiring a => Data.Semimodule.Bialgebra a [b]
instance Data.Semiring.Semiring a => Data.Semimodule.Bialgebra a (Data.Sequence.Internal.Seq b)
instance Data.Semiring.Semiring a => Data.Semimodule.Counital a ()
instance (Data.Semimodule.Counital a c1, Data.Semimodule.Counital a c2) => Data.Semimodule.Counital a (c1, c2)
instance (Data.Semimodule.Counital a c1, Data.Semimodule.Counital a c2, Data.Semimodule.Counital a c3) => Data.Semimodule.Counital a (c1, c2, c3)
instance Data.Semimodule.Unital a b => Data.Semimodule.Counital a (b -> a)
instance Data.Semiring.Semiring a => Data.Semimodule.Counital a [c]
instance Data.Semiring.Semiring a => Data.Semimodule.Counital a (Data.Sequence.Internal.Seq c)
instance (Data.Semiring.Semiring a, GHC.Classes.Ord c) => Data.Semimodule.Counital a (Data.Set.Internal.Set c)
instance Data.Semiring.Semiring a => Data.Semimodule.Counital a Data.IntSet.Internal.IntSet
instance Data.Semiring.Semiring a => Data.Semimodule.Coalgebra a ()
instance (Data.Semimodule.Coalgebra a c1, Data.Semimodule.Coalgebra a c2) => Data.Semimodule.Coalgebra a (c1, c2)
instance (Data.Semimodule.Coalgebra a c1, Data.Semimodule.Coalgebra a c2, Data.Semimodule.Coalgebra a c3) => Data.Semimodule.Coalgebra a (c1, c2, c3)
instance Data.Semimodule.Algebra a b => Data.Semimodule.Coalgebra a (b -> a)
instance Data.Semiring.Semiring a => Data.Semimodule.Coalgebra a [c]
instance Data.Semiring.Semiring a => Data.Semimodule.Coalgebra a (Data.Sequence.Internal.Seq c)
instance (Data.Semiring.Semiring a, GHC.Classes.Ord c) => Data.Semimodule.Coalgebra a (Data.Set.Internal.Set c)
instance Data.Semiring.Semiring a => Data.Semimodule.Coalgebra a Data.IntSet.Internal.IntSet
instance Data.Semiring.Semiring a => Data.Semimodule.Unital a ()
instance (Data.Semimodule.Unital a b1, Data.Semimodule.Unital a b2) => Data.Semimodule.Unital a (b1, b2)
instance (Data.Semimodule.Unital a b1, Data.Semimodule.Unital a b2, Data.Semimodule.Unital a b3) => Data.Semimodule.Unital a (b1, b2, b3)
instance Data.Semiring.Semiring a => Data.Semimodule.Unital a [b]
instance Data.Semiring.Semiring a => Data.Semimodule.Unital a (Data.Sequence.Internal.Seq b)
instance (Data.Semiring.Semiring a, GHC.Classes.Ord b) => Data.Semimodule.Unital a (Data.Set.Internal.Set b)
instance Data.Semiring.Semiring a => Data.Semimodule.Unital a Data.IntSet.Internal.IntSet
instance Data.Semiring.Semiring a => Data.Semimodule.Algebra a ()
instance (Data.Semimodule.Algebra a b1, Data.Semimodule.Algebra a b2) => Data.Semimodule.Algebra a (b1, b2)
instance (Data.Semimodule.Algebra a b1, Data.Semimodule.Algebra a b2, Data.Semimodule.Algebra a b3) => Data.Semimodule.Algebra a (b1, b2, b3)
instance Data.Semiring.Semiring a => Data.Semimodule.Algebra a [b]
instance Data.Semiring.Semiring a => Data.Semimodule.Algebra a (Data.Sequence.Internal.Seq b)
instance (Data.Semiring.Semiring a, GHC.Classes.Ord b) => Data.Semimodule.Algebra a (Data.Set.Internal.Set b)
instance Data.Semiring.Semiring a => Data.Semimodule.Algebra a Data.IntSet.Internal.IntSet
instance Data.Semiring.Semiring a => Data.Semimodule.Bisemimodule a a a
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 (Data.Functor.Contravariant.Op a e)
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.Semiring a => Data.Semimodule.RightSemimodule a a
instance Data.Semimodule.RightSemimodule r a => Data.Semimodule.RightSemimodule r (e -> a)
instance Data.Semimodule.RightSemimodule r a => Data.Semimodule.RightSemimodule r (Data.Functor.Contravariant.Op a e)
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.Semiring a => Data.Semimodule.LeftSemimodule a a
instance Data.Semimodule.LeftSemimodule l a => Data.Semimodule.LeftSemimodule l (e -> a)
instance Data.Semimodule.LeftSemimodule l a => Data.Semimodule.LeftSemimodule l (Data.Functor.Contravariant.Op a e)
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)

module Data.Semimodule.Free
type Free = Representable

-- | A vector in a vector space or free semimodule.
--   
--   Equivalent to <a>Op</a>.
--   
--   Vectors transform contravariantly as a function of their bases.
--   
--   See
--   <a>https://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors#Definition</a>.
newtype Vec a b
Vec :: (b -> a) -> Vec a b
[runVec] :: Vec a b -> b -> a
infixr 3 `runVec`

-- | Use a linear transformation to map over a vector space.
--   
--   Note that the basis transforms
--   <a>https://en.wikipedia.org/wiki/Covariant_transformation#Contravariant_transformation</a>
--   contravariantly.
vmap :: Lin a b c -> Vec a c -> Vec a b

-- | Obtain a vector from a vector on the tensor product space.
join :: Algebra a b => Vec a (b, b) -> Vec a b

-- | Obtain a vector from the unit of a unital algebra.
--   
--   <pre>
--   <a>init</a> a = <a>vmap</a> <a>initial</a> (<a>Vec</a> $ _ -&gt; a)
--   </pre>
init :: Unital a b => a -> Vec a b

-- | Apply a covector to a vector on the right.
(!*) :: Vec a b -> Cov a b -> a
infixr 7 !*

-- | Apply a covector to a vector on the left.
(*!) :: Cov a b -> Vec a b -> a
infixl 7 *!

-- | Multiplication operator on an algebra over a free semimodule.
--   
--   <pre>
--   &gt;&gt;&gt; flip runVec E22 $ (vec $ V2 1 2) !*! (vec $ V2 7 4)
--   8
--   </pre>
--   
--   <i>Caution</i> in general <tt>mult</tt> needn't be commutative, nor
--   associative.
(!*!) :: Algebra a b => Vec a b -> Vec a b -> Vec a b
infixr 7 !*!

-- | 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.
--   
--   Co-vectors transform covariantly as a function of their bases.
--   
--   See
--   <a>https://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors#Definition</a>.
newtype Cov a c
Cov :: ((c -> a) -> a) -> Cov a c
[runCov] :: Cov a c -> (c -> a) -> a
infixr 3 `runCov`

-- | Obtain a covector from a linear combination of basis elements.
--   
--   <pre>
--   &gt;&gt;&gt; images [(2, E31),(3, E32)] !* vec (V3 1 1 1) :: Int
--   5
--   </pre>
images :: Semiring a => Foldable f => f (a, c) -> Cov a c

-- | Use a linear transformation to map over a dual space.
--   
--   Note that the basis transforms <a>covariantly</a>.
cmap :: Lin a b c -> Cov a b -> Cov a c

-- | Obtain a covector from a covector on the tensor product space.
cojoin :: Coalgebra a c => Cov a (c, c) -> Cov a c

-- | Obtain a covector from the counit of a counital coalgebra.
--   
--   <pre>
--   <a>coinit</a> = <a>cmap</a> <a>coinitial</a> (<a>Cov</a> $ f -&gt; f ())
--   </pre>
coinit :: Counital a c => Cov a c

-- | Multiplication operator on a coalgebra over a free semimodule.
--   
--   <pre>
--   &gt;&gt;&gt; flip runCov (e2 1 1) $ comult (cov $ V2 1 2) (cov $ V2 7 4)
--   11
--   </pre>
--   
--   <i>Caution</i> in general <a>comult</a> needn't be commutative, nor
--   coassociative.
comult :: Coalgebra a c => Cov a c -> Cov a c -> Cov a c
infixr 7 `comult`

-- | 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 Lin a b c
Lin :: ((c -> a) -> b -> a) -> Lin a b c
[runLin] :: Lin 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 End a b = Lin a b b

-- | Create a <a>Lin</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)]) -> Lin a b c

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

-- | The <a>augmentation</a> ring homomorphism.
augment :: Semiring a => Lin a b c -> b -> a

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

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

-- | Compose two transformations.
--   
--   <a>!#!</a> = <a>&lt;&lt;&lt;</a>
(!#!) :: Lin a c d -> Lin a b c -> Lin a b d
infixr 2 !#!

-- | Swap components of a tensor product.
--   
--   This is equivalent to a matrix transpose.
braid :: Lin a (b, c) (c, b)

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

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

-- | TODO: Document
cosplit :: ((c1 + c2) -> c) -> Lin a b c1 -> Lin a b c2 -> Lin 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 => Lin 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 => Lin 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 => Lin a (Rep f1) (Rep f2) -> Lin a (Rep g1) (Rep g2) -> (f2 ** g2) a -> (f1 ** g1) 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 => Lin a b (b, b)

-- | <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 => Lin a (c, c) c

-- | TODO: Document
initial :: Unital a b => Lin a b ()

-- | TODO: Document
coinitial :: Counital a c => Lin a () c

-- | Convolution with an associative algebra and coassociative coalgebra
convolve :: Algebra a b => Coalgebra a c => Lin a b c -> Lin a b c -> Lin a b c
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule (Data.Semimodule.Free.End a b) (Data.Semimodule.Free.Vec a b)
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule (Data.Semimodule.Free.End a b) (Data.Semimodule.Free.Vec a b)
instance Data.Semiring.Semiring a => Data.Semimodule.Bisemimodule (Data.Semimodule.Free.End a b) (Data.Semimodule.Free.End a b) (Data.Semimodule.Free.Vec a b)
instance Data.Semimodule.Counital a b => Data.Semimodule.LeftSemimodule (Data.Semimodule.Free.End a b) (Data.Semimodule.Free.Cov a b)
instance Data.Semimodule.Counital a b => Data.Semimodule.RightSemimodule (Data.Semimodule.Free.End a b) (Data.Semimodule.Free.Cov a b)
instance Data.Semimodule.Counital a b => Data.Semimodule.Bisemimodule (Data.Semimodule.Free.End a b) (Data.Semimodule.Free.End a b) (Data.Semimodule.Free.Cov a b)
instance Data.Semiring.Presemiring a => GHC.Base.Semigroup (Data.Semiring.Multiplicative (Data.Semimodule.Free.End a b))
instance Data.Semiring.Semiring a => GHC.Base.Monoid (Data.Semiring.Multiplicative (Data.Semimodule.Free.End a b))
instance Data.Semiring.Presemiring a => Data.Semiring.Presemiring (Data.Semimodule.Free.End a b)
instance Data.Semiring.Semiring a => Data.Semiring.Semiring (Data.Semimodule.Free.End a b)
instance Data.Semiring.Ring a => Data.Semiring.Ring (Data.Semimodule.Free.End a b)
instance GHC.Base.Functor (Data.Semimodule.Free.Lin a b)
instance Control.Category.Category (Data.Semimodule.Free.Lin a)
instance Data.Functor.Bind.Class.Apply (Data.Semimodule.Free.Lin a b)
instance GHC.Base.Applicative (Data.Semimodule.Free.Lin a b)
instance Data.Profunctor.Unsafe.Profunctor (Data.Semimodule.Free.Lin a)
instance Data.Profunctor.Strong.Strong (Data.Semimodule.Free.Lin a)
instance Data.Profunctor.Choice.Choice (Data.Semimodule.Free.Lin a)
instance Data.Profunctor.Sieve.Sieve (Data.Semimodule.Free.Lin a) (Data.Semimodule.Free.Cov a)
instance Data.Profunctor.Rep.Representable (Data.Semimodule.Free.Lin a)
instance GHC.Base.Monad (Data.Semimodule.Free.Lin a b)
instance Control.Arrow.Arrow (Data.Semimodule.Free.Lin a)
instance Control.Arrow.ArrowChoice (Data.Semimodule.Free.Lin a)
instance Control.Arrow.ArrowApply (Data.Semimodule.Free.Lin a)
instance (Data.Semiring.-) Data.Semiring.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semiring.Additive (Data.Semimodule.Free.Lin a b c))
instance (Data.Semiring.-) Data.Semiring.Additive GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semiring.Additive (Data.Semimodule.Free.Lin a b c))
instance Data.Semimodule.Counital a b => Data.Semimodule.LeftSemimodule (Data.Semimodule.Free.Lin a b b) (Data.Semimodule.Free.Lin a b c)
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule a (Data.Semimodule.Free.Lin a b c)
instance Data.Semimodule.Counital a c => Data.Semimodule.RightSemimodule (Data.Semimodule.Free.Lin a c c) (Data.Semimodule.Free.Lin a b c)
instance (Data.Semimodule.Counital a b, Data.Semimodule.Counital a c) => Data.Semimodule.Bisemimodule (Data.Semimodule.Free.Lin a b b) (Data.Semimodule.Free.Lin a c c) (Data.Semimodule.Free.Lin a b c)
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule a (Data.Semimodule.Free.Lin a b m)
instance Data.Semimodule.Bisemimodule a a a => Data.Semimodule.Bisemimodule a a (Data.Semimodule.Free.Lin a b c)
instance (Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a => Data.Magma.Magma (Data.Semiring.Additive (Data.Semimodule.Free.Lin a b c))
instance (Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a => Data.Group.Quasigroup (Data.Semiring.Additive (Data.Semimodule.Free.Lin a b c))
instance (Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a => Data.Group.Loop (Data.Semiring.Additive (Data.Semimodule.Free.Lin a b c))
instance (Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a => Data.Group.Group (Data.Semiring.Additive (Data.Semimodule.Free.Lin a b c))
instance GHC.Base.Functor (Data.Semimodule.Free.Cov a)
instance GHC.Base.Applicative (Data.Semimodule.Free.Cov a)
instance GHC.Base.Monad (Data.Semimodule.Free.Cov a)
instance (Data.Semiring.-) Data.Semiring.Additive GHC.Base.Monoid a => GHC.Base.Alternative (Data.Semimodule.Free.Cov a)
instance (Data.Semiring.-) Data.Semiring.Additive GHC.Base.Monoid a => GHC.Base.MonadPlus (Data.Semimodule.Free.Cov a)
instance (Data.Semiring.-) Data.Semiring.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semiring.Additive (Data.Semimodule.Free.Cov a b))
instance (Data.Semiring.-) Data.Semiring.Additive GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semiring.Additive (Data.Semimodule.Free.Cov a b))
instance (Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a => Data.Magma.Magma (Data.Semiring.Additive (Data.Semimodule.Free.Cov a b))
instance (Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a => Data.Group.Quasigroup (Data.Semiring.Additive (Data.Semimodule.Free.Cov a b))
instance (Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a => Data.Group.Loop (Data.Semiring.Additive (Data.Semimodule.Free.Cov a b))
instance (Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a => Data.Group.Group (Data.Semiring.Additive (Data.Semimodule.Free.Cov a b))
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule a (Data.Semimodule.Free.Cov a b)
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule a (Data.Semimodule.Free.Cov a b)
instance Data.Semimodule.Bisemimodule a a a => Data.Semimodule.Bisemimodule a a (Data.Semimodule.Free.Cov a b)
instance Data.Functor.Contravariant.Contravariant (Data.Semimodule.Free.Vec a)
instance Control.Category.Category Data.Semimodule.Free.Vec
instance (Data.Semiring.-) Data.Semiring.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semiring.Additive (Data.Semimodule.Free.Vec a b))
instance (Data.Semiring.-) Data.Semiring.Additive GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semiring.Additive (Data.Semimodule.Free.Vec a b))
instance (Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a => Data.Magma.Magma (Data.Semiring.Additive (Data.Semimodule.Free.Vec a b))
instance (Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a => Data.Group.Quasigroup (Data.Semiring.Additive (Data.Semimodule.Free.Vec a b))
instance (Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a => Data.Group.Loop (Data.Semiring.Additive (Data.Semimodule.Free.Vec a b))
instance (Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a => Data.Group.Group (Data.Semiring.Additive (Data.Semimodule.Free.Vec a b))
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule a (Data.Semimodule.Free.Vec a b)
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule a (Data.Semimodule.Free.Vec a b)
instance Data.Semimodule.Bisemimodule a a a => Data.Semimodule.Bisemimodule a a (Data.Semimodule.Free.Vec a b)

module Data.Semimodule.Combinator

-- | Retrieve the coefficient of a basis element
--   
--   <pre>
--   &gt;&gt;&gt; elt E21 (V2 1 2)
--   1
--   </pre>
elt :: Free f => Rep f -> f a -> a

-- | Obtain a vector from an array of coefficients and a basis.
vec :: Free f => f a -> Vec a (Rep f)

-- | Obtain a covector from an array of coefficients and a basis.
--   
--   <pre>
--   &gt;&gt;&gt; cov (V2 7 4) !* vec (V2 1 2) :: Int
--   11
--   </pre>
cov :: FreeCounital a f => f a -> Cov a (Rep f)

-- | Insert an element into an algebra.
--   
--   When the algebra is trivial this is equal to <a>pureRep</a>.
--   
--   <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' :: Complex Int
--   1 :+ 0
--   </pre>
unit' :: FreeUnital a f => f a

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

-- | Create a unit vector at an index.
--   
--   <pre>
--   &gt;&gt;&gt; dirac E21 :: V2 Int
--   V2 1 0
--   </pre>
--   
--   <pre>
--   &gt;&gt;&gt; dirac E42 :: V4 Int
--   V4 0 1 0 0
--   </pre>
dirac :: Semiring a => Free f => Eq (Rep f) => Rep f -> f a

-- | Create a lens from a representable functor.
lensRep :: Free f => Eq (Rep f) => Rep f -> forall g. Functor g => (a -> g a) -> f a -> g (f a)

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

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

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

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

-- | Apply a covector to a vector on the right.
(!*) :: Vec a b -> Cov a b -> a
infixr 7 !*

-- | Apply a covector to a vector on the left.
(*!) :: Cov a b -> Vec a b -> a
infixl 7 *!

-- | Multiplication operator on an algebra over a free semimodule.
--   
--   <pre>
--   &gt;&gt;&gt; flip runVec E22 $ (vec $ V2 1 2) !*! (vec $ V2 7 4)
--   8
--   </pre>
--   
--   <i>Caution</i> in general <tt>mult</tt> needn't be commutative, nor
--   associative.
(!*!) :: Algebra a b => Vec a b -> Vec a b -> Vec a b
infixr 7 !*!

-- | Use a linear transformation to map over a vector space.
--   
--   Note that the basis transforms
--   <a>https://en.wikipedia.org/wiki/Covariant_transformation#Contravariant_transformation</a>
--   contravariantly.
vmap :: Lin a b c -> Vec a c -> Vec a b

-- | Use a linear transformation to map over a dual space.
--   
--   Note that the basis transforms <a>covariantly</a>.
cmap :: Lin a b c -> Cov a b -> Cov a c

-- | Inner product.
--   
--   When the coalgebra is trivial 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

-- | Obtain a linear linsformation from a matrix.
--   
--   <pre>
--   (<a>.#</a>) = (<a>!#</a>) . <a>lin</a>
--   </pre>
lin :: Free f => FreeCounital a g => (f ** g) a -> Lin a (Rep f) (Rep g)

-- | Retrieve an element of a matrix.
--   
--   <pre>
--   &gt;&gt;&gt; elt2 E21 E21 $ m22 1 2 3 4
--   1
--   </pre>
elt2 :: Free f => Free g => Rep f -> Rep g -> (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 a vector from a tensor.
--   
--   When the algebra is trivial we have:
--   
--   <pre>
--   <a>diag</a> f = <a>tabulate</a> $ <a>joined</a> (<a>index</a> . <a>index</a> (<a>getCompose</a> 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> <a>getCompose</a>
--   </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 a matrix on the right by a column vector.
--   
--   <pre>
--   (<a>.#</a>) = (<a>!#</a>) . <a>lin</a>
--   </pre>
--   
--   <pre>
--   &gt;&gt;&gt; lin (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 .#

-- | 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 -> Lin a (Rep f) (Rep g) -> f a
infixl 2 #!

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

-- | 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 .#.

-- | 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
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)
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 r Data.Semimodule.Basis.E4
instance Data.Semiring.Semiring r => Data.Semimodule.Unital r Data.Semimodule.Basis.E4
instance Data.Semiring.Semiring r => Data.Semimodule.Coalgebra r Data.Semimodule.Basis.E4
instance Data.Semiring.Semiring r => Data.Semimodule.Counital r Data.Semimodule.Basis.E4
instance Data.Semiring.Semiring r => Data.Semimodule.Bialgebra r Data.Semimodule.Basis.E4
instance Data.Semiring.Semiring r => Data.Semimodule.Algebra r Data.Semimodule.Basis.E3
instance Data.Semiring.Semiring r => Data.Semimodule.Unital r Data.Semimodule.Basis.E3
instance Data.Semiring.Semiring r => Data.Semimodule.Coalgebra r Data.Semimodule.Basis.E3
instance Data.Semiring.Semiring r => Data.Semimodule.Counital r Data.Semimodule.Basis.E3
instance Data.Semiring.Semiring r => Data.Semimodule.Bialgebra r Data.Semimodule.Basis.E3
instance Data.Semiring.Semiring r => Data.Semimodule.Algebra r Data.Semimodule.Basis.E2
instance Data.Semiring.Semiring r => Data.Semimodule.Unital r Data.Semimodule.Basis.E2
instance Data.Semiring.Semiring r => Data.Semimodule.Coalgebra r Data.Semimodule.Basis.E2
instance Data.Semiring.Semiring r => Data.Semimodule.Counital r Data.Semimodule.Basis.E2
instance Data.Semiring.Semiring r => Data.Semimodule.Bialgebra r Data.Semimodule.Basis.E2
instance Data.Semiring.Semiring r => Data.Semimodule.Algebra r Data.Semimodule.Basis.E1
instance Data.Semiring.Semiring r => Data.Semimodule.Unital r Data.Semimodule.Basis.E1
instance Data.Semiring.Semiring r => Data.Semimodule.Coalgebra r Data.Semimodule.Basis.E1
instance Data.Semiring.Semiring r => Data.Semimodule.Counital r Data.Semimodule.Basis.E1
instance Data.Semiring.Semiring r => Data.Semimodule.Bialgebra r Data.Semimodule.Basis.E1

module Data.Semimodule.Finite
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.Finite.V4 a)
instance GHC.Classes.Ord a => GHC.Classes.Ord (Data.Semimodule.Finite.V4 a)
instance GHC.Classes.Eq a => GHC.Classes.Eq (Data.Semimodule.Finite.V4 a)
instance GHC.Show.Show a => GHC.Show.Show (Data.Semimodule.Finite.V3 a)
instance GHC.Classes.Ord a => GHC.Classes.Ord (Data.Semimodule.Finite.V3 a)
instance GHC.Classes.Eq a => GHC.Classes.Eq (Data.Semimodule.Finite.V3 a)
instance GHC.Show.Show a => GHC.Show.Show (Data.Semimodule.Finite.V2 a)
instance GHC.Classes.Ord a => GHC.Classes.Ord (Data.Semimodule.Finite.V2 a)
instance GHC.Classes.Eq a => GHC.Classes.Eq (Data.Semimodule.Finite.V2 a)
instance GHC.Show.Show a => GHC.Show.Show (Data.Semimodule.Finite.V1 a)
instance GHC.Classes.Ord a => GHC.Classes.Ord (Data.Semimodule.Finite.V1 a)
instance GHC.Classes.Eq a => GHC.Classes.Eq (Data.Semimodule.Finite.V1 a)
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule (Data.Semimodule.Finite.M44 a) (Data.Semimodule.Finite.M41 a)
instance Data.Semiring.Semiring a => Data.Semimodule.Bisemimodule (Data.Semimodule.Finite.M44 a) (Data.Semimodule.Finite.M11 a) (Data.Semimodule.Finite.M41 a)
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule (Data.Semimodule.Finite.M44 a) (Data.Semimodule.Finite.M42 a)
instance Data.Semiring.Semiring a => Data.Semimodule.Bisemimodule (Data.Semimodule.Finite.M44 a) (Data.Semimodule.Finite.M22 a) (Data.Semimodule.Finite.M42 a)
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule (Data.Semimodule.Finite.M44 a) (Data.Semimodule.Finite.M43 a)
instance Data.Semiring.Semiring a => Data.Semimodule.Bisemimodule (Data.Semimodule.Finite.M44 a) (Data.Semimodule.Finite.M33 a) (Data.Semimodule.Finite.M43 a)
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule (Data.Semimodule.Finite.M44 a) (Data.Semimodule.Finite.M14 a)
instance Data.Semiring.Semiring a => Data.Semimodule.Bisemimodule (Data.Semimodule.Finite.M11 a) (Data.Semimodule.Finite.M44 a) (Data.Semimodule.Finite.M14 a)
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule (Data.Semimodule.Finite.M44 a) (Data.Semimodule.Finite.M24 a)
instance Data.Semiring.Semiring a => Data.Semimodule.Bisemimodule (Data.Semimodule.Finite.M22 a) (Data.Semimodule.Finite.M44 a) (Data.Semimodule.Finite.M24 a)
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule (Data.Semimodule.Finite.M44 a) (Data.Semimodule.Finite.M34 a)
instance Data.Semiring.Semiring a => Data.Semimodule.Bisemimodule (Data.Semimodule.Finite.M33 a) (Data.Semimodule.Finite.M44 a) (Data.Semimodule.Finite.M34 a)
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule (Data.Semimodule.Finite.M44 a) (Data.Semimodule.Finite.M44 a)
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule (Data.Semimodule.Finite.M44 a) (Data.Semimodule.Finite.M44 a)
instance Data.Semiring.Semiring a => GHC.Base.Semigroup (Data.Semiring.Multiplicative (Data.Semimodule.Finite.M44 a))
instance Data.Semiring.Semiring a => GHC.Base.Monoid (Data.Semiring.Multiplicative (Data.Semimodule.Finite.M44 a))
instance Data.Semiring.Semiring a => Data.Semiring.Presemiring (Data.Semimodule.Finite.M44 a)
instance Data.Semiring.Semiring a => Data.Semiring.Semiring (Data.Semimodule.Finite.M44 a)
instance Data.Semiring.Ring a => Data.Semiring.Ring (Data.Semimodule.Finite.M44 a)
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule (Data.Semimodule.Finite.M33 a) (Data.Semimodule.Finite.M43 a)
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule (Data.Semimodule.Finite.M22 a) (Data.Semimodule.Finite.M42 a)
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule (Data.Semimodule.Finite.M33 a) (Data.Semimodule.Finite.M34 a)
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule (Data.Semimodule.Finite.M33 a) (Data.Semimodule.Finite.M31 a)
instance Data.Semiring.Semiring a => Data.Semimodule.Bisemimodule (Data.Semimodule.Finite.M33 a) (Data.Semimodule.Finite.M11 a) (Data.Semimodule.Finite.M31 a)
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule (Data.Semimodule.Finite.M33 a) (Data.Semimodule.Finite.M32 a)
instance Data.Semiring.Semiring a => Data.Semimodule.Bisemimodule (Data.Semimodule.Finite.M33 a) (Data.Semimodule.Finite.M22 a) (Data.Semimodule.Finite.M32 a)
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule (Data.Semimodule.Finite.M33 a) (Data.Semimodule.Finite.M13 a)
instance Data.Semiring.Semiring a => Data.Semimodule.Bisemimodule (Data.Semimodule.Finite.M11 a) (Data.Semimodule.Finite.M33 a) (Data.Semimodule.Finite.M13 a)
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule (Data.Semimodule.Finite.M33 a) (Data.Semimodule.Finite.M23 a)
instance Data.Semiring.Semiring a => Data.Semimodule.Bisemimodule (Data.Semimodule.Finite.M22 a) (Data.Semimodule.Finite.M33 a) (Data.Semimodule.Finite.M23 a)
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule (Data.Semimodule.Finite.M33 a) (Data.Semimodule.Finite.M33 a)
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule (Data.Semimodule.Finite.M33 a) (Data.Semimodule.Finite.M33 a)
instance Data.Semiring.Semiring a => GHC.Base.Semigroup (Data.Semiring.Multiplicative (Data.Semimodule.Finite.M33 a))
instance Data.Semiring.Semiring a => GHC.Base.Monoid (Data.Semiring.Multiplicative (Data.Semimodule.Finite.M33 a))
instance Data.Semiring.Semiring a => Data.Semiring.Presemiring (Data.Semimodule.Finite.M33 a)
instance Data.Semiring.Semiring a => Data.Semiring.Semiring (Data.Semimodule.Finite.M33 a)
instance Data.Semiring.Ring a => Data.Semiring.Ring (Data.Semimodule.Finite.M33 a)
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule (Data.Semimodule.Finite.M22 a) (Data.Semimodule.Finite.M32 a)
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule (Data.Semimodule.Finite.M22 a) (Data.Semimodule.Finite.M24 a)
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule (Data.Semimodule.Finite.M22 a) (Data.Semimodule.Finite.M23 a)
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule (Data.Semimodule.Finite.M22 a) (Data.Semimodule.Finite.M21 a)
instance Data.Semiring.Semiring a => Data.Semimodule.Bisemimodule (Data.Semimodule.Finite.M22 a) (Data.Semimodule.Finite.M11 a) (Data.Semimodule.Finite.M21 a)
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule (Data.Semimodule.Finite.M22 a) (Data.Semimodule.Finite.M12 a)
instance Data.Semiring.Semiring a => Data.Semimodule.Bisemimodule (Data.Semimodule.Finite.M11 a) (Data.Semimodule.Finite.M22 a) (Data.Semimodule.Finite.M12 a)
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule (Data.Semimodule.Finite.M22 a) (Data.Semimodule.Finite.M22 a)
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule (Data.Semimodule.Finite.M22 a) (Data.Semimodule.Finite.M22 a)
instance Data.Semiring.Semiring a => GHC.Base.Semigroup (Data.Semiring.Multiplicative (Data.Semimodule.Finite.M22 a))
instance Data.Semiring.Semiring a => GHC.Base.Monoid (Data.Semiring.Multiplicative (Data.Semimodule.Finite.M22 a))
instance Data.Semiring.Semiring a => Data.Semiring.Presemiring (Data.Semimodule.Finite.M22 a)
instance Data.Semiring.Semiring a => Data.Semiring.Semiring (Data.Semimodule.Finite.M22 a)
instance Data.Semiring.Ring a => Data.Semiring.Ring (Data.Semimodule.Finite.M22 a)
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule (Data.Semimodule.Finite.M11 a) (Data.Semimodule.Finite.M41 a)
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule (Data.Semimodule.Finite.M11 a) (Data.Semimodule.Finite.M31 a)
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule (Data.Semimodule.Finite.M11 a) (Data.Semimodule.Finite.M21 a)
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule (Data.Semimodule.Finite.M11 a) (Data.Semimodule.Finite.M14 a)
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule (Data.Semimodule.Finite.M11 a) (Data.Semimodule.Finite.M13 a)
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule (Data.Semimodule.Finite.M11 a) (Data.Semimodule.Finite.M12 a)
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule (Data.Semimodule.Finite.M11 a) (Data.Semimodule.Finite.M11 a)
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule (Data.Semimodule.Finite.M11 a) (Data.Semimodule.Finite.M11 a)
instance Data.Semiring.Semiring a => GHC.Base.Semigroup (Data.Semiring.Multiplicative (Data.Semimodule.Finite.M11 a))
instance Data.Semiring.Semiring a => GHC.Base.Monoid (Data.Semiring.Multiplicative (Data.Semimodule.Finite.M11 a))
instance Data.Semiring.Semiring a => Data.Semiring.Presemiring (Data.Semimodule.Finite.M11 a)
instance Data.Semiring.Semiring a => Data.Semiring.Semiring (Data.Semimodule.Finite.M11 a)
instance Data.Semiring.Ring a => Data.Semiring.Ring (Data.Semimodule.Finite.M11 a)
instance Data.Functor.Classes.Show1 Data.Semimodule.Finite.V4
instance GHC.Base.Functor Data.Semimodule.Finite.V4
instance GHC.Base.Applicative Data.Semimodule.Finite.V4
instance Data.Foldable.Foldable Data.Semimodule.Finite.V4
instance Data.Semigroup.Foldable.Class.Foldable1 Data.Semimodule.Finite.V4
instance Data.Distributive.Distributive Data.Semimodule.Finite.V4
instance Data.Functor.Rep.Representable Data.Semimodule.Finite.V4
instance (Data.Semiring.-) Data.Semiring.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semiring.Additive (Data.Semimodule.Finite.V4 a))
instance (Data.Semiring.-) Data.Semiring.Additive GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semiring.Additive (Data.Semimodule.Finite.V4 a))
instance (Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a => Data.Magma.Magma (Data.Semiring.Additive (Data.Semimodule.Finite.V4 a))
instance (Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a => Data.Group.Quasigroup (Data.Semiring.Additive (Data.Semimodule.Finite.V4 a))
instance (Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a => Data.Group.Loop (Data.Semiring.Additive (Data.Semimodule.Finite.V4 a))
instance (Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a => Data.Group.Group (Data.Semiring.Additive (Data.Semimodule.Finite.V4 a))
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule a (Data.Semimodule.Finite.V4 a)
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule a (Data.Semimodule.Finite.V4 a)
instance Data.Semiring.Semiring a => Data.Semimodule.Bisemimodule a a (Data.Semimodule.Finite.V4 a)
instance Data.Functor.Classes.Eq1 Data.Semimodule.Finite.V3
instance Data.Functor.Classes.Show1 Data.Semimodule.Finite.V3
instance GHC.Base.Functor Data.Semimodule.Finite.V3
instance GHC.Base.Applicative Data.Semimodule.Finite.V3
instance Data.Foldable.Foldable Data.Semimodule.Finite.V3
instance Data.Semigroup.Foldable.Class.Foldable1 Data.Semimodule.Finite.V3
instance Data.Distributive.Distributive Data.Semimodule.Finite.V3
instance Data.Functor.Rep.Representable Data.Semimodule.Finite.V3
instance (Data.Semiring.-) Data.Semiring.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semiring.Additive (Data.Semimodule.Finite.V3 a))
instance (Data.Semiring.-) Data.Semiring.Additive GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semiring.Additive (Data.Semimodule.Finite.V3 a))
instance (Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a => Data.Magma.Magma (Data.Semiring.Additive (Data.Semimodule.Finite.V3 a))
instance (Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a => Data.Group.Quasigroup (Data.Semiring.Additive (Data.Semimodule.Finite.V3 a))
instance (Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a => Data.Group.Loop (Data.Semiring.Additive (Data.Semimodule.Finite.V3 a))
instance (Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a => Data.Group.Group (Data.Semiring.Additive (Data.Semimodule.Finite.V3 a))
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule a (Data.Semimodule.Finite.V3 a)
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule a (Data.Semimodule.Finite.V3 a)
instance Data.Semiring.Semiring a => Data.Semimodule.Bisemimodule a a (Data.Semimodule.Finite.V3 a)
instance Data.Functor.Classes.Show1 Data.Semimodule.Finite.V2
instance GHC.Base.Functor Data.Semimodule.Finite.V2
instance GHC.Base.Applicative Data.Semimodule.Finite.V2
instance Data.Foldable.Foldable Data.Semimodule.Finite.V2
instance Data.Semigroup.Foldable.Class.Foldable1 Data.Semimodule.Finite.V2
instance Data.Distributive.Distributive Data.Semimodule.Finite.V2
instance Data.Functor.Rep.Representable Data.Semimodule.Finite.V2
instance (Data.Semiring.-) Data.Semiring.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semiring.Additive (Data.Semimodule.Finite.V2 a))
instance (Data.Semiring.-) Data.Semiring.Additive GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semiring.Additive (Data.Semimodule.Finite.V2 a))
instance (Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a => Data.Magma.Magma (Data.Semiring.Additive (Data.Semimodule.Finite.V2 a))
instance (Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a => Data.Group.Quasigroup (Data.Semiring.Additive (Data.Semimodule.Finite.V2 a))
instance (Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a => Data.Group.Loop (Data.Semiring.Additive (Data.Semimodule.Finite.V2 a))
instance (Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a => Data.Group.Group (Data.Semiring.Additive (Data.Semimodule.Finite.V2 a))
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule a (Data.Semimodule.Finite.V2 a)
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule a (Data.Semimodule.Finite.V2 a)
instance Data.Semiring.Semiring a => Data.Semimodule.Bisemimodule a a (Data.Semimodule.Finite.V2 a)
instance Data.Functor.Classes.Show1 Data.Semimodule.Finite.V1
instance GHC.Base.Functor Data.Semimodule.Finite.V1
instance GHC.Base.Applicative Data.Semimodule.Finite.V1
instance Data.Foldable.Foldable Data.Semimodule.Finite.V1
instance Data.Semigroup.Foldable.Class.Foldable1 Data.Semimodule.Finite.V1
instance Data.Distributive.Distributive Data.Semimodule.Finite.V1
instance Data.Functor.Rep.Representable Data.Semimodule.Finite.V1
instance (Data.Semiring.-) Data.Semiring.Additive GHC.Base.Semigroup a => GHC.Base.Semigroup (Data.Semiring.Additive (Data.Semimodule.Finite.V1 a))
instance (Data.Semiring.-) Data.Semiring.Additive GHC.Base.Monoid a => GHC.Base.Monoid (Data.Semiring.Additive (Data.Semimodule.Finite.V1 a))
instance (Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a => Data.Magma.Magma (Data.Semiring.Additive (Data.Semimodule.Finite.V1 a))
instance (Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a => Data.Group.Quasigroup (Data.Semiring.Additive (Data.Semimodule.Finite.V1 a))
instance (Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a => Data.Group.Loop (Data.Semiring.Additive (Data.Semimodule.Finite.V1 a))
instance (Data.Semiring.-) Data.Semiring.Additive Data.Group.Group a => Data.Group.Group (Data.Semiring.Additive (Data.Semimodule.Finite.V1 a))
instance Data.Semiring.Semiring a => Data.Semimodule.LeftSemimodule a (Data.Semimodule.Finite.V1 a)
instance Data.Semiring.Semiring a => Data.Semimodule.RightSemimodule a (Data.Semimodule.Finite.V1 a)
instance Data.Semiring.Semiring a => Data.Semimodule.Bisemimodule a a (Data.Semimodule.Finite.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

-- | 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

-- | &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

-- | &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

-- | 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

-- | 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;
--   
--   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

-- | 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