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


-- | Constructive abstract algebra
--   
--   Constructive abstract algebra
@package algebra
@version 0.7.0

module Numeric.Natural.Internal
newtype Natural
Natural :: Integer -> Natural
runNatural :: Natural -> Integer
class Integral n => Whole n
toNatural :: Whole n => n -> Natural
unsafePred :: Whole n => n -> n
instance Eq Natural
instance Ord Natural
instance Whole Natural
instance Whole Word64
instance Whole Word32
instance Whole Word16
instance Whole Word8
instance Whole Word
instance Integral Natural
instance Enum Natural
instance Real Natural
instance Num Natural
instance Read Natural
instance Show Natural

module Numeric.Natural
data Natural
class Integral n => Whole n
toNatural :: Whole n => n -> Natural

module Numeric.Band.Rectangular

-- | a rectangular band is a nowhere commutative semigroup. That is to say,
--   if ab = ba then a = b. From this it follows classically that aa = a
--   and that such a band is isomorphic to the following structure
data Rect i j
Rect :: i -> j -> Rect i j
instance (Eq i, Eq j) => Eq (Rect i j)
instance (Ord i, Ord j) => Ord (Rect i j)
instance (Show i, Show j) => Show (Rect i j)
instance (Read i, Read j) => Read (Rect i j)
instance Band (Rect i j)
instance Multiplicative (Rect i j)
instance Semigroupoid Rect

module Numeric.Covector

-- | Linear functionals from elements of an (infinite) free module to a
--   scalar
newtype Covector r a
Covector :: ((a -> r) -> r) -> Covector r a
counitM :: UnitalAlgebra r a => a -> Covector r ()
unitM :: CounitalCoalgebra r c => Covector r c
comultM :: Algebra r a => a -> Covector r (a, a)
multM :: Coalgebra r c => c -> c -> Covector r c
invM :: InvolutiveAlgebra r h => h -> Covector r h
coinvM :: InvolutiveCoalgebra r h => h -> Covector r h

-- | convolveM antipodeM return = convolveM return antipodeM = comultM
--   >=> uncurry joinM
antipodeM :: HopfAlgebra r h => h -> Covector r h
convolveM :: (Algebra r c, Coalgebra r a) => (c -> Covector r a) -> (c -> Covector r a) -> c -> Covector r a
memoM :: HasTrie a => a -> Covector s a
instance RightModule r s => RightModule r (Covector s m)
instance Coalgebra r m => RightModule (Covector r m) (Covector r m)
instance LeftModule r s => LeftModule r (Covector s m)
instance Coalgebra r m => LeftModule (Covector r m) (Covector r m)
instance Group s => Group (Covector s a)
instance Abelian s => Abelian (Covector s a)
instance Monoidal s => Monoidal (Covector s a)
instance (Idempotent r, IdempotentCoalgebra r a) => Band (Covector r a)
instance Idempotent r => Idempotent (Covector r a)
instance (Ring r, CounitalCoalgebra r m) => Ring (Covector r m)
instance (Rig r, CounitalCoalgebra r m) => Rig (Covector r m)
instance CounitalCoalgebra r m => Unital (Covector r m)
instance Coalgebra r m => Semiring (Covector r m)
instance (Commutative m, Coalgebra r m) => Commutative (Covector r m)
instance Coalgebra r m => Multiplicative (Covector r m)
instance Additive r => Additive (Covector r a)
instance Monoidal r => MonadPlus (Covector r)
instance Monoidal r => Alternative (Covector r)
instance Monoidal r => Plus (Covector r)
instance Additive r => Alt (Covector r)
instance Monad (Covector r)
instance Bind (Covector r)
instance Applicative (Covector r)
instance Apply (Covector r)
instance Functor (Covector r)

module Numeric.Algebra.Distinguished.Class
class Distinguished t
e :: Distinguished t => t
instance Distinguished a => Distinguished (Covector r a)

module Numeric.Algebra.Complex.Class
class Distinguished r => Complicated r
i :: Complicated r => r
instance Complicated a => Complicated (Covector r a)

module Numeric.Algebra.Quaternion.Class
class Complicated t => Hamiltonian t
j :: Hamiltonian t => t
k :: Hamiltonian t => t
instance Hamiltonian a => Hamiltonian (Covector r a)

module Numeric.Algebra.Dual.Class
class Distinguished t => Infinitesimal t
d :: Infinitesimal t => t
instance Infinitesimal a => Infinitesimal (Covector r a)

module Numeric.Coalgebra.Hyperbolic.Class
class Hyperbolic r
cosh :: Hyperbolic r => r
sinh :: Hyperbolic r => r
instance Hyperbolic a => Hyperbolic (Covector r a)

module Numeric.Coalgebra.Trigonometric.Class
class Trigonometric r
cos :: Trigonometric r => r
sin :: Trigonometric r => r
instance Trigonometric a => Trigonometric (Covector r a)

module Numeric.Algebra

-- | <pre>
--   (a + b) + c = a + (b + c)
--   replicate 1 a = a
--   replicate (2 * n) a = replicate n a + replicate n a
--   replicate (2 * n + 1) a = replicate n a + replicate n a + a
--   </pre>
class Additive r
(+) :: Additive r => r -> r -> r
replicate1p :: (Additive r, Whole n) => n -> r -> r
sumWith1 :: (Additive r, Foldable1 f) => (a -> r) -> f a -> r
sum1 :: (Foldable1 f, Additive r) => f r -> r

-- | an additive abelian semigroup
--   
--   a + b = b + a
class Additive r => Abelian r

-- | An additive semigroup with idempotent addition.
--   
--   <pre>
--   a + a = a
--   </pre>
class Additive r => Idempotent r
replicate1pIdempotent :: Natural -> r -> r
replicateIdempotent :: (Integral n, Idempotent r, Monoidal r) => n -> r -> r
class Additive m => Partitionable m
partitionWith :: Partitionable m => (m -> m -> r) -> m -> NonEmpty r

-- | An additive monoid
--   
--   <pre>
--   zero + a = a = a + zero
--   </pre>
class (LeftModule Natural m, RightModule Natural m) => Monoidal m
zero :: Monoidal m => m
replicate :: (Monoidal m, Whole n) => n -> m -> m
sumWith :: (Monoidal m, Foldable f) => (a -> m) -> f a -> m
sum :: (Foldable f, Monoidal m) => f m -> m
class (LeftModule Integer r, RightModule Integer r, Monoidal r) => Group r
(-) :: Group r => r -> r -> r
negate :: Group r => r -> r
subtract :: Group r => r -> r -> r
times :: (Group r, Integral n) => n -> r -> r

-- | A multiplicative semigroup
class Multiplicative r
(*) :: Multiplicative r => r -> r -> r
pow1p :: (Multiplicative r, Whole n) => r -> n -> r
productWith1 :: (Multiplicative r, Foldable1 f) => (a -> r) -> f a -> r
product1 :: (Foldable1 f, Multiplicative r) => f r -> r

-- | A commutative multiplicative semigroup
class Multiplicative r => Commutative r
class Multiplicative r => Unital r
one :: Unital r => r
pow :: (Unital r, Whole n) => r -> n -> r
productWith :: (Unital r, Foldable f) => (a -> r) -> f a -> r
product :: (Foldable f, Unital r) => f r -> r

-- | An multiplicative semigroup with idempotent multiplication.
--   
--   <pre>
--   a * a = a
--   </pre>
class Multiplicative r => Band r
pow1pBand :: Whole n => r -> n -> r
powBand :: (Unital r, Whole n) => r -> n -> r
class Unital r => Division r
recip :: Division r => r -> r
(/) :: Division r => r -> r -> r
(\\) :: Division r => r -> r -> r
(^) :: (Division r, Integral n) => r -> n -> r

-- | `factorWith f c` returns a non-empty list containing `f a b` for all
--   `a, b` such that `a * b = c`.
--   
--   Results of factorWith f 0 are undefined and may result in either an
--   error or an infinite list.
class Multiplicative m => Factorable m
factorWith :: Factorable m => (m -> m -> r) -> m -> NonEmpty r

-- | An semigroup with involution
--   
--   <pre>
--   adjoint a * adjoint b = adjoint (b * a)
--   </pre>
class Multiplicative r => InvolutiveMultiplication r
adjoint :: InvolutiveMultiplication r => r -> r

-- | <pre>
--   adjoint = id
--   </pre>
class (Commutative r, InvolutiveMultiplication r) => TriviallyInvolutive r

-- | A pair of an additive abelian semigroup, and a multiplicative
--   semigroup, with the distributive laws:
--   
--   <pre>
--   a(b + c) = ab + ac -- left distribution (we are a LeftNearSemiring)
--   (a + b)c = ac + bc -- right distribution (we are a [Right]NearSemiring)
--   </pre>
--   
--   Common notation includes the laws for additive and multiplicative
--   identity in semiring.
--   
--   If you want that, look at <tt>Rig</tt> instead.
--   
--   Ideally we'd use the cyclic definition:
--   
--   <pre>
--   class (LeftModule r r, RightModule r r, Additive r, Abelian r, Multiplicative r) =&gt; Semiring r
--   </pre>
--   
--   to enforce that every semiring r is an r-module over itself, but
--   Haskell doesn't like that.
class (Additive r, Abelian r, Multiplicative r) => Semiring r

-- | adjoint (x + y) = adjoint x + adjoint y
class (Semiring r, InvolutiveMultiplication r) => InvolutiveSemiring r
class (Semiring r, Idempotent r) => Dioid r

-- | A Ring without an <i>i</i>dentity.
class (Group r, Semiring r) => Rng r

-- | A Ring without (n)egation
class (Semiring r, Unital r, Monoidal r) => Rig r
fromNatural :: Rig r => Natural -> r
class (Rig r, Rng r) => Ring r
fromInteger :: Ring r => Integer -> r
class (Division r, Ring r) => DivisionRing r
class (Commutative r, DivisionRing r) => Field r
class (Semiring r, Additive m) => LeftModule r m
(.*) :: LeftModule r m => r -> m -> m
class (Semiring r, Additive m) => RightModule r m
(*.) :: RightModule r m => m -> r -> m
class (LeftModule r m, RightModule r m) => Module r m

-- | An associative algebra built with a free module over a semiring
class Semiring r => Algebra r a
mult :: Algebra r a => (a -> a -> r) -> a -> r
class Semiring r => Coalgebra r c
comult :: Coalgebra r c => (c -> r) -> c -> c -> r

-- | An associative unital algebra over a semiring, built using a free
--   module
class Algebra r a => UnitalAlgebra r a
unit :: UnitalAlgebra r a => r -> a -> r
class Coalgebra r c => CounitalCoalgebra r c
counit :: CounitalCoalgebra r c => (c -> r) -> r

-- | A bialgebra is both a unital algebra and counital coalgebra where the
--   <a>mult</a> and <a>unit</a> are compatible in some sense with the
--   <a>comult</a> and <a>counit</a>. That is to say that <a>mult</a> and
--   <a>unit</a> are a coalgebra homomorphisms or (equivalently) that
--   <a>comult</a> and <a>counit</a> are an algebra homomorphisms.
class (UnitalAlgebra r a, CounitalCoalgebra r a) => Bialgebra r a
class (InvolutiveSemiring r, Algebra r a) => InvolutiveAlgebra r a
inv :: InvolutiveAlgebra r a => (a -> r) -> a -> r
class (InvolutiveSemiring r, Coalgebra r c) => InvolutiveCoalgebra r c
coinv :: InvolutiveCoalgebra r c => (c -> r) -> c -> r
class (Bialgebra r h, InvolutiveAlgebra r h, InvolutiveCoalgebra r h) => InvolutiveBialgebra r h
class (CommutativeAlgebra r a, TriviallyInvolutive r, InvolutiveAlgebra r a) => TriviallyInvolutiveAlgebra r a
class (CocommutativeCoalgebra r a, TriviallyInvolutive r, InvolutiveCoalgebra r a) => TriviallyInvolutiveCoalgebra r a
class (InvolutiveBialgebra r h, TriviallyInvolutiveAlgebra r h, TriviallyInvolutiveCoalgebra r h) => TriviallyInvolutiveBialgebra r h
class Algebra r a => IdempotentAlgebra r a
class (Bialgebra r h, IdempotentAlgebra r h, IdempotentCoalgebra r h) => IdempotentBialgebra r h
class Algebra r a => CommutativeAlgebra r a
class (Bialgebra r h, CommutativeAlgebra r h, CocommutativeCoalgebra r h) => CommutativeBialgebra r h
class Coalgebra r c => CocommutativeCoalgebra r c
class UnitalAlgebra r a => DivisionAlgebra r a
recipriocal :: DivisionAlgebra r a => (a -> r) -> a -> r

-- | A HopfAlgebra algebra on a semiring, where the module is free.
--   
--   When <tt>antipode . antipode = id</tt> and antipode is an
--   antihomomorphism then we are an InvolutiveBialgebra with <tt>inv =
--   antipode</tt> as well
class Bialgebra r h => HopfAlgebra r h
antipode :: HopfAlgebra r h => (h -> r) -> h -> r
class Rig r => Characteristic r
char :: Characteristic r => proxy r -> Natural
charInt :: (Integral s, Bounded s) => proxy s -> Natural
charWord :: (Whole s, Bounded s) => proxy s -> Natural
class Order a
(<~) :: Order a => a -> a -> Bool
(<) :: Order a => a -> a -> Bool
(>~) :: Order a => a -> a -> Bool
(>) :: Order a => a -> a -> Bool
(~~) :: Order a => a -> a -> Bool
(/~) :: Order a => a -> a -> Bool
order :: Order a => a -> a -> Maybe Ordering
comparable :: Order a => a -> a -> Bool
class (AdditiveOrder r, Rig r) => OrderedRig r

-- | z + x &lt;= z + y = x &lt;= y = x + z &lt;= y + z
class (Additive r, Order r) => AdditiveOrder r
class Monoidal r => DecidableZero r
class Unital r => DecidableUnits r
class Unital r => DecidableAssociates r
data Natural
class Integral n => Whole n
toNatural :: Whole n => n -> Natural

-- | `Additive.(+)` default definition
addRep :: (Zip m, Additive r) => m r -> m r -> m r

-- | <tt>Additive.replicate1p</tt> default definition
replicate1pRep :: (Whole n, Functor m, Additive r) => n -> m r -> m r

-- | <tt>Monoidal.zero</tt> default definition
zeroRep :: (Applicative m, Monoidal r) => m r

-- | <tt>Monoidal.replicate</tt> default definition
replicateRep :: (Whole n, Functor m, Monoidal r) => n -> m r -> m r

-- | <tt>Group.negate</tt> default definition
negateRep :: (Functor m, Group r) => m r -> m r

-- | `Group.(-)` default definition
minusRep :: (Zip m, Group r) => m r -> m r -> m r

-- | <tt>Group.subtract</tt> default definition
subtractRep :: (Zip m, Group r) => m r -> m r -> m r

-- | <tt>Group.times</tt> default definition
timesRep :: (Integral n, Functor m, Group r) => n -> m r -> m r

-- | `Multiplicative.(*)` default definition
mulRep :: (Representable m, Algebra r (Key m)) => m r -> m r -> m r

-- | <tt>Unital.one</tt> default definition
oneRep :: (Representable m, Unital r, UnitalAlgebra r (Key m)) => m r

-- | <tt>Rig.fromNatural</tt> default definition
fromNaturalRep :: (UnitalAlgebra r (Key m), Representable m, Rig r) => Natural -> m r

-- | <tt>Ring.fromInteger</tt> default definition
fromIntegerRep :: (UnitalAlgebra r (Key m), Representable m, Ring r) => Integer -> m r
class Additive r => Quadrance r m
quadrance :: Quadrance r m => m -> r

-- | Linear functionals from elements of an (infinite) free module to a
--   scalar
newtype Covector r a
Covector :: ((a -> r) -> r) -> Covector r a
counitM :: UnitalAlgebra r a => a -> Covector r ()
unitM :: CounitalCoalgebra r c => Covector r c
comultM :: Algebra r a => a -> Covector r (a, a)
multM :: Coalgebra r c => c -> c -> Covector r c
invM :: InvolutiveAlgebra r h => h -> Covector r h
coinvM :: InvolutiveCoalgebra r h => h -> Covector r h

-- | convolveM antipodeM return = convolveM return antipodeM = comultM
--   &gt;=&gt; uncurry joinM
antipodeM :: HopfAlgebra r h => h -> Covector r h
convolveM :: (Algebra r c, Coalgebra r a) => (c -> Covector r a) -> (c -> Covector r a) -> c -> Covector r a
memoM :: HasTrie a => a -> Covector s a

module Numeric.Algebra.Complex
class Distinguished t
e :: Distinguished t => t
class Distinguished r => Complicated r
i :: Complicated r => r
data ComplexBasis
E :: ComplexBasis
I :: ComplexBasis
data Complex a
Complex :: a -> a -> Complex a
realPart :: (Representable f, (Key f) ~ ComplexBasis) => f a -> a
imagPart :: (Representable f, (Key f) ~ ComplexBasis) => f a -> a

-- | half of the Cayley-Dickson quaternion isomorphism
uncomplicate :: Hamiltonian q => ComplexBasis -> ComplexBasis -> q
instance Typeable ComplexBasis
instance Typeable1 Complex
instance Eq ComplexBasis
instance Ord ComplexBasis
instance Show ComplexBasis
instance Read ComplexBasis
instance Enum ComplexBasis
instance Ix ComplexBasis
instance Bounded ComplexBasis
instance Data ComplexBasis
instance Eq a => Eq (Complex a)
instance Show a => Show (Complex a)
instance Read a => Read (Complex a)
instance Data a => Data (Complex a)
instance (Commutative r, InvolutiveSemiring r, DivisionRing r) => Division (Complex r)
instance (Commutative r, Rng r, InvolutiveSemiring r) => Quadrance r (Complex r)
instance (Commutative r, Rng r, InvolutiveSemiring r) => InvolutiveSemiring (Complex r)
instance (Commutative r, Rng r, InvolutiveMultiplication r) => InvolutiveMultiplication (Complex r)
instance (Commutative r, Rng r) => RightModule (Complex r) (Complex r)
instance (Commutative r, Rng r) => LeftModule (Complex r) (Complex r)
instance (Commutative r, Ring r) => Ring (Complex r)
instance (Commutative r, Ring r) => Rig (Complex r)
instance (Commutative r, Ring r) => Unital (Complex r)
instance (Commutative r, Rng r) => Semiring (Complex r)
instance (TriviallyInvolutive r, Rng r) => Commutative (Complex r)
instance (Commutative r, Rng r) => Multiplicative (Complex r)
instance (InvolutiveSemiring k, Rng k) => HopfAlgebra k ComplexBasis
instance (InvolutiveSemiring k, Rng k) => InvolutiveCoalgebra k ComplexBasis
instance (InvolutiveSemiring k, Rng k) => InvolutiveAlgebra k ComplexBasis
instance Rng k => Bialgebra k ComplexBasis
instance Rng k => CounitalCoalgebra k ComplexBasis
instance Rng k => Coalgebra k ComplexBasis
instance Rng k => UnitalAlgebra k ComplexBasis
instance Rng k => Algebra k ComplexBasis
instance Partitionable r => Partitionable (Complex r)
instance Idempotent r => Idempotent (Complex r)
instance Abelian r => Abelian (Complex r)
instance Group r => Group (Complex r)
instance Monoidal r => Monoidal (Complex r)
instance RightModule r s => RightModule r (Complex s)
instance LeftModule r s => LeftModule r (Complex s)
instance Additive r => Additive (Complex r)
instance HasTrie ComplexBasis
instance TraversableWithKey1 Complex
instance Traversable1 Complex
instance FoldableWithKey1 Complex
instance Foldable1 Complex
instance TraversableWithKey Complex
instance Traversable Complex
instance FoldableWithKey Complex
instance Foldable Complex
instance MonadReader ComplexBasis Complex
instance Monad Complex
instance Bind Complex
instance Applicative Complex
instance Apply Complex
instance Keyed Complex
instance ZipWithKey Complex
instance Zip Complex
instance Functor Complex
instance Distributive Complex
instance Adjustable Complex
instance Lookup Complex
instance Indexable Complex
instance Representable Complex
instance Rig r => Complicated (ComplexBasis :->: r)
instance Rig r => Distinguished (ComplexBasis :->: r)
instance Rig r => Complicated (ComplexBasis -> r)
instance Rig r => Distinguished (ComplexBasis -> r)
instance Rig r => Complicated (Complex r)
instance Rig r => Distinguished (Complex r)
instance Complicated ComplexBasis
instance Distinguished ComplexBasis

module Numeric.Algebra.Quaternion
class Distinguished t
e :: Distinguished t => t
class Distinguished r => Complicated r
i :: Complicated r => r
class Complicated t => Hamiltonian t
j :: Hamiltonian t => t
k :: Hamiltonian t => t
data QuaternionBasis
E :: QuaternionBasis
I :: QuaternionBasis
J :: QuaternionBasis
K :: QuaternionBasis
data Quaternion a
Quaternion :: a -> a -> a -> a -> Quaternion a

-- | Cayley-Dickson quaternion isomorphism (one way)
complicate :: Complicated c => QuaternionBasis -> (c, c)
vectorPart :: (Representable f, (Key f) ~ QuaternionBasis) => f r -> (r, r, r)
scalarPart :: (Representable f, (Key f) ~ QuaternionBasis) => f r -> r
instance Typeable QuaternionBasis
instance Typeable1 Quaternion
instance Eq QuaternionBasis
instance Ord QuaternionBasis
instance Enum QuaternionBasis
instance Read QuaternionBasis
instance Show QuaternionBasis
instance Bounded QuaternionBasis
instance Ix QuaternionBasis
instance Data QuaternionBasis
instance Eq a => Eq (Quaternion a)
instance Show a => Show (Quaternion a)
instance Read a => Read (Quaternion a)
instance Data a => Data (Quaternion a)
instance (TriviallyInvolutive r, Ring r, Division r) => Division (Quaternion r)
instance (TriviallyInvolutive r, Rng r) => Quadrance r (Quaternion r)
instance (TriviallyInvolutive r, Rng r) => InvolutiveMultiplication (Quaternion r)
instance (TriviallyInvolutive r, Rng r) => RightModule (Quaternion r) (Quaternion r)
instance (TriviallyInvolutive r, Rng r) => LeftModule (Quaternion r) (Quaternion r)
instance (TriviallyInvolutive r, Ring r) => Ring (Quaternion r)
instance (TriviallyInvolutive r, Ring r) => Rig (Quaternion r)
instance (TriviallyInvolutive r, Ring r) => Unital (Quaternion r)
instance (TriviallyInvolutive r, Rng r) => Semiring (Quaternion r)
instance (TriviallyInvolutive r, Rng r) => Multiplicative (Quaternion r)
instance (TriviallyInvolutive r, InvolutiveSemiring r, Rng r) => HopfAlgebra r QuaternionBasis
instance (TriviallyInvolutive r, InvolutiveSemiring r, Rng r) => InvolutiveCoalgebra r QuaternionBasis
instance (TriviallyInvolutive r, InvolutiveSemiring r, Rng r) => InvolutiveAlgebra r QuaternionBasis
instance (TriviallyInvolutive r, Rng r) => Bialgebra r QuaternionBasis
instance (TriviallyInvolutive r, Rng r) => CounitalCoalgebra r QuaternionBasis
instance (TriviallyInvolutive r, Rng r) => Coalgebra r QuaternionBasis
instance (TriviallyInvolutive r, Rng r) => UnitalAlgebra r QuaternionBasis
instance (TriviallyInvolutive r, Rng r) => Algebra r QuaternionBasis
instance Partitionable r => Partitionable (Quaternion r)
instance Idempotent r => Idempotent (Quaternion r)
instance Abelian r => Abelian (Quaternion r)
instance Group r => Group (Quaternion r)
instance Monoidal r => Monoidal (Quaternion r)
instance RightModule r s => RightModule r (Quaternion s)
instance LeftModule r s => LeftModule r (Quaternion s)
instance Additive r => Additive (Quaternion r)
instance HasTrie QuaternionBasis
instance TraversableWithKey1 Quaternion
instance Traversable1 Quaternion
instance FoldableWithKey1 Quaternion
instance Foldable1 Quaternion
instance TraversableWithKey Quaternion
instance Traversable Quaternion
instance FoldableWithKey Quaternion
instance Foldable Quaternion
instance MonadReader QuaternionBasis Quaternion
instance Monad Quaternion
instance Bind Quaternion
instance Applicative Quaternion
instance Apply Quaternion
instance Keyed Quaternion
instance ZipWithKey Quaternion
instance Zip Quaternion
instance Functor Quaternion
instance Distributive Quaternion
instance Adjustable Quaternion
instance Lookup Quaternion
instance Indexable Quaternion
instance Representable Quaternion
instance Rig r => Hamiltonian (QuaternionBasis -> r)
instance Rig r => Complicated (QuaternionBasis -> r)
instance Rig r => Distinguished (QuaternionBasis -> r)
instance Rig r => Hamiltonian (QuaternionBasis :->: r)
instance Rig r => Complicated (QuaternionBasis :->: r)
instance Rig r => Distinguished (QuaternionBasis :->: r)
instance Rig r => Hamiltonian (Quaternion r)
instance Rig r => Complicated (Quaternion r)
instance Rig r => Distinguished (Quaternion r)
instance Hamiltonian QuaternionBasis
instance Complicated QuaternionBasis
instance Distinguished QuaternionBasis

module Numeric.Algebra.Dual
class Distinguished t
e :: Distinguished t => t
class Distinguished t => Infinitesimal t
d :: Infinitesimal t => t

-- | dual number basis, D^2 = 0. D /= 0.
data DualBasis
E :: DualBasis
D :: DualBasis
data Dual a
Dual :: a -> a -> Dual a
instance Typeable DualBasis
instance Typeable1 Dual
instance Eq DualBasis
instance Ord DualBasis
instance Show DualBasis
instance Read DualBasis
instance Enum DualBasis
instance Ix DualBasis
instance Bounded DualBasis
instance Data DualBasis
instance Eq a => Eq (Dual a)
instance Show a => Show (Dual a)
instance Read a => Read (Dual a)
instance Data a => Data (Dual a)
instance (Commutative r, InvolutiveSemiring r, DivisionRing r) => Division (Dual r)
instance (Commutative r, Rng r, InvolutiveSemiring r) => Quadrance r (Dual r)
instance (Commutative r, Rng r, InvolutiveSemiring r) => InvolutiveSemiring (Dual r)
instance (Commutative r, Rng r, InvolutiveSemiring r) => InvolutiveMultiplication (Dual r)
instance (Commutative r, Rng r) => RightModule (Dual r) (Dual r)
instance (Commutative r, Rng r) => LeftModule (Dual r) (Dual r)
instance (Commutative r, Ring r) => Ring (Dual r)
instance (Commutative r, Ring r) => Rig (Dual r)
instance (Commutative r, Ring r) => Unital (Dual r)
instance (Commutative r, Rng r) => Semiring (Dual r)
instance (TriviallyInvolutive r, Rng r) => Commutative (Dual r)
instance (Commutative r, Rng r) => Multiplicative (Dual r)
instance (InvolutiveSemiring k, Rng k) => HopfAlgebra k DualBasis
instance (InvolutiveSemiring k, Rng k) => InvolutiveCoalgebra k DualBasis
instance (InvolutiveSemiring k, Rng k) => InvolutiveAlgebra k DualBasis
instance Rng k => Bialgebra k DualBasis
instance Rng k => CounitalCoalgebra k DualBasis
instance Rng k => Coalgebra k DualBasis
instance Rng k => UnitalAlgebra k DualBasis
instance Rng k => Algebra k DualBasis
instance Partitionable r => Partitionable (Dual r)
instance Idempotent r => Idempotent (Dual r)
instance Abelian r => Abelian (Dual r)
instance Group r => Group (Dual r)
instance Monoidal r => Monoidal (Dual r)
instance RightModule r s => RightModule r (Dual s)
instance LeftModule r s => LeftModule r (Dual s)
instance Additive r => Additive (Dual r)
instance HasTrie DualBasis
instance TraversableWithKey1 Dual
instance Traversable1 Dual
instance FoldableWithKey1 Dual
instance Foldable1 Dual
instance TraversableWithKey Dual
instance Traversable Dual
instance FoldableWithKey Dual
instance Foldable Dual
instance MonadReader DualBasis Dual
instance Monad Dual
instance Bind Dual
instance Applicative Dual
instance Apply Dual
instance Keyed Dual
instance ZipWithKey Dual
instance Zip Dual
instance Functor Dual
instance Distributive Dual
instance Adjustable Dual
instance Lookup Dual
instance Indexable Dual
instance Representable Dual
instance Rig r => Infinitesimal (DualBasis -> r)
instance Rig r => Distinguished (DualBasis -> r)
instance Rig r => Infinitesimal (Dual r)
instance Rig r => Distinguished (Dual r)
instance Infinitesimal DualBasis
instance Distinguished DualBasis

module Numeric.Algebra.Hyperbolic
class Hyperbolic r
cosh :: Hyperbolic r => r
sinh :: Hyperbolic r => r
data HyperBasis'
Cosh' :: HyperBasis'
Sinh' :: HyperBasis'
data Hyper' a
Hyper' :: a -> a -> Hyper' a
instance Typeable HyperBasis'
instance Typeable1 Hyper'
instance Eq HyperBasis'
instance Ord HyperBasis'
instance Show HyperBasis'
instance Read HyperBasis'
instance Enum HyperBasis'
instance Ix HyperBasis'
instance Bounded HyperBasis'
instance Data HyperBasis'
instance Eq a => Eq (Hyper' a)
instance Show a => Show (Hyper' a)
instance Read a => Read (Hyper' a)
instance Data a => Data (Hyper' a)
instance (Commutative r, InvolutiveSemiring r, DivisionRing r) => Division (Hyper' r)
instance (Commutative r, InvolutiveSemiring r, Rng r) => Quadrance r (Hyper' r)
instance (Commutative r, InvolutiveSemiring r, Rng r) => InvolutiveSemiring (Hyper' r)
instance (Commutative r, InvolutiveSemiring r, Rng r) => InvolutiveMultiplication (Hyper' r)
instance (Commutative r, Semiring r) => RightModule (Hyper' r) (Hyper' r)
instance (Commutative r, Semiring r) => LeftModule (Hyper' r) (Hyper' r)
instance (Commutative r, Ring r) => Ring (Hyper' r)
instance (Commutative r, Rig r) => Rig (Hyper' r)
instance (Commutative k, Rig k) => Unital (Hyper' k)
instance (Commutative k, Semiring k) => Semiring (Hyper' k)
instance (Commutative k, Semiring k) => Commutative (Hyper' k)
instance (Commutative k, Semiring k) => Multiplicative (Hyper' k)
instance (Commutative k, Group k, InvolutiveSemiring k) => HopfAlgebra k HyperBasis'
instance (Commutative k, Group k, InvolutiveSemiring k) => InvolutiveCoalgebra k HyperBasis'
instance (Commutative k, Group k, InvolutiveSemiring k) => InvolutiveAlgebra k HyperBasis'
instance (Commutative k, Monoidal k, Semiring k) => Bialgebra k HyperBasis'
instance (Commutative k, Monoidal k, Semiring k) => CounitalCoalgebra k HyperBasis'
instance (Commutative k, Monoidal k, Semiring k) => Coalgebra k HyperBasis'
instance (Commutative k, Monoidal k, Semiring k) => UnitalAlgebra k HyperBasis'
instance (Commutative k, Semiring k) => Algebra k HyperBasis'
instance Partitionable r => Partitionable (Hyper' r)
instance Idempotent r => Idempotent (Hyper' r)
instance Abelian r => Abelian (Hyper' r)
instance Group r => Group (Hyper' r)
instance Monoidal r => Monoidal (Hyper' r)
instance RightModule r s => RightModule r (Hyper' s)
instance LeftModule r s => LeftModule r (Hyper' s)
instance Additive r => Additive (Hyper' r)
instance HasTrie HyperBasis'
instance TraversableWithKey1 Hyper'
instance Traversable1 Hyper'
instance FoldableWithKey1 Hyper'
instance Foldable1 Hyper'
instance TraversableWithKey Hyper'
instance Traversable Hyper'
instance FoldableWithKey Hyper'
instance Foldable Hyper'
instance MonadReader HyperBasis' Hyper'
instance Monad Hyper'
instance Bind Hyper'
instance Applicative Hyper'
instance Apply Hyper'
instance Keyed Hyper'
instance ZipWithKey Hyper'
instance Zip Hyper'
instance Functor Hyper'
instance Distributive Hyper'
instance Adjustable Hyper'
instance Lookup Hyper'
instance Indexable Hyper'
instance Representable Hyper'
instance Rig r => Hyperbolic (HyperBasis' -> r)
instance Rig r => Hyperbolic (Hyper' r)
instance Hyperbolic HyperBasis'

module Numeric.Coalgebra.Hyperbolic
class Hyperbolic r
cosh :: Hyperbolic r => r
sinh :: Hyperbolic r => r
data HyperBasis
Cosh :: HyperBasis
Sinh :: HyperBasis
data Hyper a
Hyper :: a -> a -> Hyper a
instance Typeable HyperBasis
instance Typeable1 Hyper
instance Eq HyperBasis
instance Ord HyperBasis
instance Show HyperBasis
instance Read HyperBasis
instance Enum HyperBasis
instance Ix HyperBasis
instance Bounded HyperBasis
instance Data HyperBasis
instance Eq a => Eq (Hyper a)
instance Show a => Show (Hyper a)
instance Read a => Read (Hyper a)
instance Data a => Data (Hyper a)
instance (Commutative r, Group r, InvolutiveSemiring r) => InvolutiveSemiring (Hyper r)
instance (Commutative r, Group r, InvolutiveSemiring r) => InvolutiveMultiplication (Hyper r)
instance (Commutative r, Semiring r) => RightModule (Hyper r) (Hyper r)
instance (Commutative r, Semiring r) => LeftModule (Hyper r) (Hyper r)
instance (Commutative r, Ring r) => Ring (Hyper r)
instance (Commutative r, Rig r) => Rig (Hyper r)
instance (Commutative k, Rig k) => Unital (Hyper k)
instance (Commutative k, Semiring k) => Semiring (Hyper k)
instance (Commutative k, Semiring k) => Commutative (Hyper k)
instance (Commutative k, Semiring k) => Multiplicative (Hyper k)
instance (Commutative k, Group k, InvolutiveSemiring k) => HopfAlgebra k HyperBasis
instance (Commutative k, Group k, InvolutiveSemiring k) => InvolutiveCoalgebra k HyperBasis
instance (Commutative k, Group k, InvolutiveSemiring k) => InvolutiveAlgebra k HyperBasis
instance (Commutative k, Semiring k) => Bialgebra k HyperBasis
instance (Commutative k, Semiring k) => CounitalCoalgebra k HyperBasis
instance (Commutative k, Semiring k) => Coalgebra k HyperBasis
instance Semiring k => UnitalAlgebra k HyperBasis
instance Semiring k => Algebra k HyperBasis
instance Partitionable r => Partitionable (Hyper r)
instance Idempotent r => Idempotent (Hyper r)
instance Abelian r => Abelian (Hyper r)
instance Group r => Group (Hyper r)
instance Monoidal r => Monoidal (Hyper r)
instance RightModule r s => RightModule r (Hyper s)
instance LeftModule r s => LeftModule r (Hyper s)
instance Additive r => Additive (Hyper r)
instance HasTrie HyperBasis
instance TraversableWithKey1 Hyper
instance Traversable1 Hyper
instance FoldableWithKey1 Hyper
instance Foldable1 Hyper
instance TraversableWithKey Hyper
instance Traversable Hyper
instance FoldableWithKey Hyper
instance Foldable Hyper
instance MonadReader HyperBasis Hyper
instance Monad Hyper
instance Bind Hyper
instance Applicative Hyper
instance Apply Hyper
instance Keyed Hyper
instance ZipWithKey Hyper
instance Zip Hyper
instance Functor Hyper
instance Distributive Hyper
instance Adjustable Hyper
instance Lookup Hyper
instance Indexable Hyper
instance Representable Hyper
instance Rig r => Hyperbolic (HyperBasis -> r)
instance Rig r => Hyperbolic (Hyper r)
instance Hyperbolic HyperBasis

module Numeric.Coalgebra.Dual
class Distinguished t
e :: Distinguished t => t
class Distinguished t => Infinitesimal t
d :: Infinitesimal t => t

-- | dual number basis, D^2 = 0. D /= 0.
data DualBasis'
E :: DualBasis'
D :: DualBasis'
data Dual' a
Dual' :: a -> a -> Dual' a
instance Typeable DualBasis'
instance Typeable1 Dual'
instance Eq DualBasis'
instance Ord DualBasis'
instance Show DualBasis'
instance Read DualBasis'
instance Enum DualBasis'
instance Ix DualBasis'
instance Bounded DualBasis'
instance Data DualBasis'
instance Eq a => Eq (Dual' a)
instance Show a => Show (Dual' a)
instance Read a => Read (Dual' a)
instance Data a => Data (Dual' a)
instance (Commutative r, InvolutiveSemiring r, DivisionRing r) => Division (Dual' r)
instance (Commutative r, Rng r, InvolutiveSemiring r) => Quadrance r (Dual' r)
instance (Commutative r, Rng r, InvolutiveSemiring r) => InvolutiveSemiring (Dual' r)
instance (Commutative r, Rng r, InvolutiveSemiring r) => InvolutiveMultiplication (Dual' r)
instance (Commutative r, Rng r) => RightModule (Dual' r) (Dual' r)
instance (Commutative r, Rng r) => LeftModule (Dual' r) (Dual' r)
instance (Commutative r, Ring r) => Ring (Dual' r)
instance (Commutative r, Ring r) => Rig (Dual' r)
instance (Commutative r, Ring r) => Unital (Dual' r)
instance (Commutative r, Rng r) => Semiring (Dual' r)
instance (TriviallyInvolutive r, Rng r) => Commutative (Dual' r)
instance (Commutative r, Rng r) => Multiplicative (Dual' r)
instance (InvolutiveSemiring k, Rng k) => HopfAlgebra k DualBasis'
instance (InvolutiveSemiring k, Rng k) => InvolutiveCoalgebra k DualBasis'
instance (InvolutiveSemiring k, Rng k) => InvolutiveAlgebra k DualBasis'
instance Rng k => Bialgebra k DualBasis'
instance Rng k => CounitalCoalgebra k DualBasis'
instance Rng k => Coalgebra k DualBasis'
instance Semiring k => UnitalAlgebra k DualBasis'
instance Semiring k => Algebra k DualBasis'
instance Partitionable r => Partitionable (Dual' r)
instance Idempotent r => Idempotent (Dual' r)
instance Abelian r => Abelian (Dual' r)
instance Group r => Group (Dual' r)
instance Monoidal r => Monoidal (Dual' r)
instance RightModule r s => RightModule r (Dual' s)
instance LeftModule r s => LeftModule r (Dual' s)
instance Additive r => Additive (Dual' r)
instance HasTrie DualBasis'
instance TraversableWithKey1 Dual'
instance Traversable1 Dual'
instance FoldableWithKey1 Dual'
instance Foldable1 Dual'
instance TraversableWithKey Dual'
instance Traversable Dual'
instance FoldableWithKey Dual'
instance Foldable Dual'
instance MonadReader DualBasis' Dual'
instance Monad Dual'
instance Bind Dual'
instance Applicative Dual'
instance Apply Dual'
instance Keyed Dual'
instance ZipWithKey Dual'
instance Zip Dual'
instance Functor Dual'
instance Distributive Dual'
instance Adjustable Dual'
instance Lookup Dual'
instance Indexable Dual'
instance Representable Dual'
instance Rig r => Infinitesimal (DualBasis' -> r)
instance Rig r => Distinguished (DualBasis' -> r)
instance Rig r => Infinitesimal (Dual' r)
instance Rig r => Distinguished (Dual' r)
instance Infinitesimal DualBasis'
instance Distinguished DualBasis'

module Numeric.Coalgebra.Trigonometric
class Trigonometric r
cos :: Trigonometric r => r
sin :: Trigonometric r => r
data TrigBasis
Cos :: TrigBasis
Sin :: TrigBasis
data Trig a
Trig :: a -> a -> Trig a
instance Typeable TrigBasis
instance Typeable1 Trig
instance Eq TrigBasis
instance Ord TrigBasis
instance Show TrigBasis
instance Read TrigBasis
instance Enum TrigBasis
instance Ix TrigBasis
instance Bounded TrigBasis
instance Data TrigBasis
instance Eq a => Eq (Trig a)
instance Show a => Show (Trig a)
instance Read a => Read (Trig a)
instance Data a => Data (Trig a)
instance (Commutative r, Rng r, InvolutiveSemiring r) => InvolutiveSemiring (Trig r)
instance (Commutative r, Rng r, InvolutiveMultiplication r) => InvolutiveMultiplication (Trig r)
instance (Commutative r, Rng r) => RightModule (Trig r) (Trig r)
instance (Commutative r, Rng r) => LeftModule (Trig r) (Trig r)
instance (Commutative r, Ring r) => Ring (Trig r)
instance (Commutative r, Ring r) => Rig (Trig r)
instance (Commutative k, Ring k) => Unital (Trig k)
instance (Commutative k, Rng k) => Semiring (Trig k)
instance (Commutative k, Rng k) => Commutative (Trig k)
instance (Commutative k, Rng k) => Multiplicative (Trig k)
instance (Commutative k, Rng k) => CounitalCoalgebra k TrigBasis
instance (Commutative k, Group k, InvolutiveSemiring k) => HopfAlgebra k TrigBasis
instance (Commutative k, Group k, InvolutiveSemiring k) => InvolutiveCoalgebra k TrigBasis
instance (Commutative k, Group k, InvolutiveSemiring k) => InvolutiveAlgebra k TrigBasis
instance (Commutative k, Rng k) => Bialgebra k TrigBasis
instance (Commutative k, Rng k) => Coalgebra k TrigBasis
instance (Commutative k, Rng k) => UnitalAlgebra k TrigBasis
instance (Commutative k, Rng k) => Algebra k TrigBasis
instance Partitionable r => Partitionable (Trig r)
instance Idempotent r => Idempotent (Trig r)
instance Abelian r => Abelian (Trig r)
instance Group r => Group (Trig r)
instance Monoidal r => Monoidal (Trig r)
instance RightModule r s => RightModule r (Trig s)
instance LeftModule r s => LeftModule r (Trig s)
instance Additive r => Additive (Trig r)
instance HasTrie TrigBasis
instance TraversableWithKey1 Trig
instance Traversable1 Trig
instance FoldableWithKey1 Trig
instance Foldable1 Trig
instance TraversableWithKey Trig
instance Traversable Trig
instance FoldableWithKey Trig
instance Foldable Trig
instance MonadReader TrigBasis Trig
instance Monad Trig
instance Bind Trig
instance Applicative Trig
instance Apply Trig
instance Keyed Trig
instance ZipWithKey Trig
instance Zip Trig
instance Functor Trig
instance Distributive Trig
instance Adjustable Trig
instance Lookup Trig
instance Indexable Trig
instance Representable Trig
instance Rig r => Complicated (TrigBasis :->: r)
instance Rig r => Distinguished (TrigBasis :->: r)
instance Rig r => Trigonometric (TrigBasis :->: r)
instance Rig r => Trigonometric (TrigBasis -> r)
instance Rig r => Complicated (TrigBasis -> r)
instance Rig r => Distinguished (TrigBasis -> r)
instance Rig r => Trigonometric (Trig r)
instance Rig r => Complicated (Trig r)
instance Rig r => Distinguished (Trig r)
instance Trigonometric TrigBasis
instance Complicated TrigBasis
instance Distinguished TrigBasis

module Numeric.Coalgebra.Geometric
newtype BasisCoblade m
BasisCoblade :: Word64 -> BasisCoblade m
runBasisCoblade :: BasisCoblade m -> Word64
type Comultivector r m = Covector r (BasisCoblade m)
class Eigenbasis m
euclidean :: Eigenbasis m => proxy m -> Bool
antiEuclidean :: Eigenbasis m => proxy m -> Bool
v :: Eigenbasis m => m -> BasisCoblade m
e :: Eigenbasis m => Int -> m
class (Ring r, Eigenbasis m) => Eigenmetric r m
metric :: Eigenmetric r m => m -> r
grade :: BasisCoblade m -> Int
filterGrade :: Monoidal r => BasisCoblade m -> Int -> Comultivector r m
reverse :: Group r => BasisCoblade m -> Comultivector r m
gradeInversion :: Group r => BasisCoblade m -> Comultivector r m
cliffordConjugate :: Group r => BasisCoblade m -> Comultivector r m
geometric :: Eigenmetric r m => BasisCoblade m -> BasisCoblade m -> Comultivector r m
outer :: Eigenmetric r m => BasisCoblade m -> BasisCoblade m -> Comultivector r m
contractL :: Eigenmetric r m => BasisCoblade m -> BasisCoblade m -> Comultivector r m
contractR :: Eigenmetric r m => BasisCoblade m -> BasisCoblade m -> Comultivector r m
hestenes :: Eigenmetric r m => BasisCoblade m -> BasisCoblade m -> Comultivector r m
dot :: Eigenmetric r m => BasisCoblade m -> BasisCoblade m -> Comultivector r m
liftProduct :: (BasisCoblade m -> BasisCoblade m -> Comultivector r m) -> Comultivector r m -> Comultivector r m -> Comultivector r m
instance Typeable Euclidean
instance Eq (BasisCoblade m)
instance Ord (BasisCoblade m)
instance Num (BasisCoblade m)
instance Bits (BasisCoblade m)
instance Enum (BasisCoblade m)
instance Ix (BasisCoblade m)
instance Bounded (BasisCoblade m)
instance Show (BasisCoblade m)
instance Read (BasisCoblade m)
instance Real (BasisCoblade m)
instance Integral (BasisCoblade m)
instance Additive (BasisCoblade m)
instance Abelian (BasisCoblade m)
instance LeftModule Natural (BasisCoblade m)
instance RightModule Natural (BasisCoblade m)
instance Monoidal (BasisCoblade m)
instance Multiplicative (BasisCoblade m)
instance Unital (BasisCoblade m)
instance Commutative (BasisCoblade m)
instance Semiring (BasisCoblade m)
instance Rig (BasisCoblade m)
instance DecidableZero (BasisCoblade m)
instance DecidableAssociates (BasisCoblade m)
instance DecidableUnits (BasisCoblade m)
instance Eq Euclidean
instance Ord Euclidean
instance Show Euclidean
instance Read Euclidean
instance Num Euclidean
instance Ix Euclidean
instance Enum Euclidean
instance Real Euclidean
instance Integral Euclidean
instance Data Euclidean
instance Additive Euclidean
instance LeftModule Natural Euclidean
instance RightModule Natural Euclidean
instance Monoidal Euclidean
instance Abelian Euclidean
instance LeftModule Integer Euclidean
instance RightModule Integer Euclidean
instance Group Euclidean
instance Multiplicative Euclidean
instance TriviallyInvolutive Euclidean
instance InvolutiveMultiplication Euclidean
instance InvolutiveSemiring Euclidean
instance Unital Euclidean
instance Commutative Euclidean
instance Semiring Euclidean
instance Rig Euclidean
instance Ring Euclidean
instance Eigenmetric r m => CounitalCoalgebra r (BasisCoblade m)
instance Eigenmetric r m => Coalgebra r (BasisCoblade m)
instance Ring r => Eigenmetric r Euclidean
instance Eigenbasis Euclidean
instance HasTrie Euclidean
instance HasTrie (BasisCoblade m)

module Numeric.Coalgebra.Quaternion
class Distinguished t
e :: Distinguished t => t
class Distinguished r => Complicated r
i :: Complicated r => r
class Complicated t => Hamiltonian t
j :: Hamiltonian t => t
k :: Hamiltonian t => t
data QuaternionBasis'
E' :: QuaternionBasis'
I' :: QuaternionBasis'
J' :: QuaternionBasis'
K' :: QuaternionBasis'
data Quaternion' a
Quaternion' :: a -> a -> a -> a -> Quaternion' a

-- | Cayley-Dickson quaternion isomorphism (one way)
complicate' :: Complicated c => QuaternionBasis' -> (c, c)
vectorPart' :: (Representable f, (Key f) ~ QuaternionBasis') => f r -> (r, r, r)
scalarPart' :: (Representable f, (Key f) ~ QuaternionBasis') => f r -> r
instance Typeable QuaternionBasis'
instance Typeable1 Quaternion'
instance Eq QuaternionBasis'
instance Ord QuaternionBasis'
instance Enum QuaternionBasis'
instance Read QuaternionBasis'
instance Show QuaternionBasis'
instance Bounded QuaternionBasis'
instance Ix QuaternionBasis'
instance Data QuaternionBasis'
instance Eq a => Eq (Quaternion' a)
instance Show a => Show (Quaternion' a)
instance Read a => Read (Quaternion' a)
instance Data a => Data (Quaternion' a)
instance (TriviallyInvolutive r, Ring r, Division r) => Division (Quaternion' r)
instance (TriviallyInvolutive r, Rng r) => Quadrance r (Quaternion' r)
instance (TriviallyInvolutive r, Rng r) => InvolutiveMultiplication (Quaternion' r)
instance (TriviallyInvolutive r, Rng r) => RightModule (Quaternion' r) (Quaternion' r)
instance (TriviallyInvolutive r, Rng r) => LeftModule (Quaternion' r) (Quaternion' r)
instance (TriviallyInvolutive r, Ring r) => Ring (Quaternion' r)
instance (TriviallyInvolutive r, Ring r) => Rig (Quaternion' r)
instance (TriviallyInvolutive r, Ring r) => Unital (Quaternion' r)
instance (TriviallyInvolutive r, Semiring r) => Semiring (Quaternion' r)
instance (TriviallyInvolutive r, Semiring r) => Multiplicative (Quaternion' r)
instance (TriviallyInvolutive r, InvolutiveSemiring r, Rng r) => HopfAlgebra r QuaternionBasis'
instance (TriviallyInvolutive r, InvolutiveSemiring r, Rng r) => InvolutiveCoalgebra r QuaternionBasis'
instance (TriviallyInvolutive r, InvolutiveSemiring r, Rng r) => InvolutiveAlgebra r QuaternionBasis'
instance (TriviallyInvolutive r, Rng r) => Bialgebra r QuaternionBasis'
instance (TriviallyInvolutive r, Rng r) => CounitalCoalgebra r QuaternionBasis'
instance (TriviallyInvolutive r, Rng r) => Coalgebra r QuaternionBasis'
instance (TriviallyInvolutive r, Semiring r) => UnitalAlgebra r QuaternionBasis'
instance (TriviallyInvolutive r, Semiring r) => Algebra r QuaternionBasis'
instance Partitionable r => Partitionable (Quaternion' r)
instance Idempotent r => Idempotent (Quaternion' r)
instance Abelian r => Abelian (Quaternion' r)
instance Group r => Group (Quaternion' r)
instance Monoidal r => Monoidal (Quaternion' r)
instance RightModule r s => RightModule r (Quaternion' s)
instance LeftModule r s => LeftModule r (Quaternion' s)
instance Additive r => Additive (Quaternion' r)
instance HasTrie QuaternionBasis'
instance TraversableWithKey1 Quaternion'
instance Traversable1 Quaternion'
instance FoldableWithKey1 Quaternion'
instance Foldable1 Quaternion'
instance TraversableWithKey Quaternion'
instance Traversable Quaternion'
instance FoldableWithKey Quaternion'
instance Foldable Quaternion'
instance MonadReader QuaternionBasis' Quaternion'
instance Monad Quaternion'
instance Bind Quaternion'
instance Applicative Quaternion'
instance Apply Quaternion'
instance Keyed Quaternion'
instance ZipWithKey Quaternion'
instance Zip Quaternion'
instance Functor Quaternion'
instance Distributive Quaternion'
instance Adjustable Quaternion'
instance Lookup Quaternion'
instance Indexable Quaternion'
instance Representable Quaternion'
instance Rig r => Hamiltonian (QuaternionBasis' -> r)
instance Rig r => Complicated (QuaternionBasis' -> r)
instance Rig r => Distinguished (QuaternionBasis' -> r)
instance Rig r => Hamiltonian (QuaternionBasis' :->: r)
instance Rig r => Complicated (QuaternionBasis' :->: r)
instance Rig r => Distinguished (QuaternionBasis' :->: r)
instance Rig r => Hamiltonian (Quaternion' r)
instance Rig r => Complicated (Quaternion' r)
instance Rig r => Distinguished (Quaternion' r)
instance Hamiltonian QuaternionBasis'
instance Complicated QuaternionBasis'
instance Distinguished QuaternionBasis'

module Numeric.Exp
newtype Exp r
Exp :: r -> Exp r
runExp :: Exp r -> r
instance Partitionable r => Factorable (Exp r)
instance Idempotent r => Band (Exp r)
instance Abelian r => Commutative (Exp r)
instance Group r => Division (Exp r)
instance Monoidal r => Unital (Exp r)
instance Additive r => Multiplicative (Exp r)

module Numeric.Log
newtype Log r
Log :: r -> Log r
runLog :: Log r -> r
instance Factorable r => Partitionable (Log r)
instance Band r => Idempotent (Log r)
instance Commutative r => Abelian (Log r)
instance Division r => Group (Log r)
instance Division r => RightModule Integer (Log r)
instance Division r => LeftModule Integer (Log r)
instance Unital r => Monoidal (Log r)
instance Unital r => RightModule Natural (Log r)
instance Unital r => LeftModule Natural (Log r)
instance Multiplicative r => Additive (Log r)

module Numeric.Map

-- | linear maps from elements of a free module to another free module over
--   r
--   
--   <pre>
--   f $# x + y = (f $# x) + (f $# y)
--   f $# (r .* x) = r .* (f $# x)
--   </pre>
--   
--   <tt>Map r b a</tt> represents a linear mapping from a free module with
--   basis <tt>a</tt> over <tt>r</tt> to a free module with basis
--   <tt>b</tt> over <tt>r</tt>.
--   
--   Note well the reversed direction of the arrow, due to the
--   contravariance of change of basis!
--   
--   This way enables we can employ arbitrary pure functions as linear maps
--   by lifting them using <a>arr</a>, or build them by using the monad
--   instance for Map r b. As a consequence Map is an instance of, well,
--   almost everything.
newtype Map r b a
Map :: ((a -> r) -> b -> r) -> Map r b a

-- | extract a linear functional from a linear map
($@) :: Map r b a -> b -> Covector r a
multMap :: Coalgebra r c => Map r (c, c) c
unitMap :: CounitalCoalgebra r c => Map r () c

-- | (inefficiently) combine a linear combination of basis vectors to make
--   a map. arrMap :: (Monoidal r, Semiring r) =&gt; (b -&gt; [(r, a)])
--   -&gt; Map r b a arrMap f = Map $ k b -&gt; sum [ r * k a | (r, a)
--   &lt;- f b ]
--   
--   Memoize the results of this linear map
memoMap :: HasTrie a => Map r a a
comultMap :: Algebra r a => Map r a (a, a)
counitMap :: UnitalAlgebra r a => Map r a ()
invMap :: InvolutiveCoalgebra r c => Map r c c
coinvMap :: InvolutiveAlgebra r a => Map r a a
antipodeMap :: HopfAlgebra r h => Map r h h

-- | convolution given an associative algebra and coassociative coalgebra
convolveMap :: (Algebra r a, Coalgebra r c) => Map r a c -> Map r a c -> Map r a c
instance (Ring r, CounitalCoalgebra r m) => Ring (Map r a m)
instance (Rig r, CounitalCoalgebra r m) => Rig (Map r b m)
instance (Commutative m, Coalgebra r m) => Commutative (Map r b m)
instance Group s => Group (Map s b a)
instance Abelian s => Abelian (Map s b a)
instance Monoidal s => Monoidal (Map s b a)
instance Monoidal r => MonadPlus (Map r b)
instance Monoidal r => Alternative (Map r b)
instance Monoidal r => Plus (Map r b)
instance Additive r => Alt (Map r b)
instance RightModule r s => RightModule r (Map s b m)
instance Coalgebra r m => RightModule (Map r b m) (Map r b m)
instance LeftModule r s => LeftModule r (Map s b m)
instance Coalgebra r m => LeftModule (Map r b m) (Map r b m)
instance Coalgebra r m => Semiring (Map r b m)
instance CounitalCoalgebra r m => Unital (Map r b m)
instance Coalgebra r m => Multiplicative (Map r b m)
instance Additive r => Additive (Map r b a)
instance ArrowChoice (Map r)
instance Monoidal r => ArrowPlus (Map r)
instance Monoidal r => ArrowZero (Map r)
instance MonadReader b (Map r b)
instance ArrowApply (Map r)
instance Arrow (Map r)
instance Monoidal (Map r) Either
instance Comonoidal (Map r) Either
instance PreCoCartesian (Map r)
instance Symmetric (Map r) Either
instance Braided (Map r) Either
instance Disassociative (Map r) Either
instance Associative (Map r) Either
instance Bifunctor Either (Map r) (Map r) (Map r)
instance QFunctor Either (Map r) (Map r)
instance PFunctor Either (Map r) (Map r)
instance Distributive (Map r)
instance CCC (Map r)
instance PreCartesian (Map r)
instance Comonoidal (Map r) (,)
instance Monoidal (Map r) (,)
instance Symmetric (Map r) (,)
instance Braided (Map r) (,)
instance Disassociative (Map r) (,)
instance Associative (Map r) (,)
instance Bifunctor (,) (Map r) (Map r) (Map r)
instance QFunctor (,) (Map r) (Map r)
instance PFunctor (,) (Map r) (Map r)
instance Monad (Map r b)
instance Bind (Map r b)
instance Applicative (Map r b)
instance Apply (Map r b)
instance Functor (Map r b)
instance Semigroupoid (Map r)
instance Category (Map r)

module Numeric.Rng.Zero
newtype ZeroRng r
ZeroRng :: r -> ZeroRng r
runZeroRng :: ZeroRng r -> r
instance Eq r => Eq (ZeroRng r)
instance Ord r => Ord (ZeroRng r)
instance Show r => Show (ZeroRng r)
instance Read r => Read (ZeroRng r)
instance Group r => RightModule Integer (ZeroRng r)
instance Group r => LeftModule Integer (ZeroRng r)
instance Monoidal r => RightModule Natural (ZeroRng r)
instance Monoidal r => LeftModule Natural (ZeroRng r)
instance (Group r, Abelian r) => Rng (ZeroRng r)
instance Monoidal r => Commutative (ZeroRng r)
instance (Monoidal r, Abelian r) => Semiring (ZeroRng r)
instance Monoidal r => Multiplicative (ZeroRng r)
instance Group r => Group (ZeroRng r)
instance Monoidal r => Monoidal (ZeroRng r)
instance Abelian r => Abelian (ZeroRng r)
instance Idempotent r => Idempotent (ZeroRng r)
instance Additive r => Additive (ZeroRng r)

module Numeric.Ring.Rng

-- | The free Ring given a Rng obtained by adjoining Z, such that
--   
--   <pre>
--   RngRing r = n*1 + r
--   </pre>
--   
--   This ring is commonly denoted r^.
data RngRing r
RngRing :: !Integer -> r -> RngRing r

-- | The rng homomorphism from r to RngRing r
rngRingHom :: r -> RngRing r

-- | given a rng homomorphism from a rng r into a ring s, liftRngHom yields
--   a ring homomorphism from the ring `r^` into <tt>s</tt>.
liftRngHom :: Ring s => (r -> s) -> RngRing r -> s
instance Show r => Show (RngRing r)
instance Read r => Read (RngRing r)
instance Rng r => Ring (RngRing r)
instance Rng r => Rig (RngRing r)
instance Rng r => Semiring (RngRing r)
instance (Rng r, Division r) => Division (RngRing r)
instance Rng r => Unital (RngRing r)
instance Rng s => RightModule (RngRing s) (RngRing s)
instance Rng s => LeftModule (RngRing s) (RngRing s)
instance (Commutative r, Rng r) => Commutative (RngRing r)
instance Rng r => Multiplicative (RngRing r)
instance (Abelian r, Group r) => Group (RngRing r)
instance (Abelian r, Group r) => RightModule Integer (RngRing r)
instance (Abelian r, Group r) => LeftModule Integer (RngRing r)
instance (Abelian r, Monoidal r) => Monoidal (RngRing r)
instance (Abelian r, Monoidal r) => RightModule Natural (RngRing r)
instance (Abelian r, Monoidal r) => LeftModule Natural (RngRing r)
instance Abelian r => Abelian (RngRing r)
instance Abelian r => Additive (RngRing r)

module Numeric.Ring.Opposite

-- | http:<i></i>en.wikipedia.org<i>wiki</i>Opposite_ring
newtype Opposite r
Opposite :: r -> Opposite r
runOpposite :: Opposite r -> r
instance Show r => Show (Opposite r)
instance Read r => Read (Opposite r)
instance Ring r => Ring (Opposite r)
instance Rig r => Rig (Opposite r)
instance Semiring r => Semiring (Opposite r)
instance Division r => Division (Opposite r)
instance Unital r => Unital (Opposite r)
instance Band r => Band (Opposite r)
instance Idempotent r => Idempotent (Opposite r)
instance Commutative r => Commutative (Opposite r)
instance Multiplicative r => Multiplicative (Opposite r)
instance DecidableAssociates r => DecidableAssociates (Opposite r)
instance DecidableUnits r => DecidableUnits (Opposite r)
instance DecidableZero r => DecidableZero (Opposite r)
instance Abelian r => Abelian (Opposite r)
instance Group r => Group (Opposite r)
instance Semiring r => RightModule (Opposite r) (Opposite r)
instance LeftModule r s => RightModule r (Opposite s)
instance RightModule r s => LeftModule r (Opposite s)
instance Semiring r => LeftModule (Opposite r) (Opposite r)
instance Monoidal r => Monoidal (Opposite r)
instance Additive r => Additive (Opposite r)
instance Traversable1 Opposite
instance Foldable1 Opposite
instance Traversable Opposite
instance Foldable Opposite
instance Functor Opposite
instance Ord r => Ord (Opposite r)
instance Eq r => Eq (Opposite r)

module Numeric.Ring.Endomorphism

-- | The endomorphism ring of an abelian group or the endomorphism semiring
--   of an abelian monoid
--   
--   http:<i></i>en.wikipedia.org<i>wiki</i>Endomorphism_ring
newtype End a
End :: (a -> a) -> End a
appEnd :: End a -> a -> a
toEnd :: Multiplicative r => r -> End r
fromEnd :: Unital r => End r -> r
frobenius :: Characteristic r => End r
instance RightModule r m => RightModule r (End m)
instance LeftModule r m => LeftModule r (End m)
instance (Monoidal m, Abelian m) => RightModule (End m) (End m)
instance (Monoidal m, Abelian m) => LeftModule (End m) (End m)
instance (Abelian r, Group r) => Ring (End r)
instance (Abelian r, Monoidal r) => Rig (End r)
instance (Abelian r, Monoidal r) => Semiring (End r)
instance (Abelian r, Commutative r) => Commutative (End r)
instance Unital (End r)
instance Multiplicative (End r)
instance Group r => Group (End r)
instance Monoidal r => Monoidal (End r)
instance Abelian r => Abelian (End r)
instance Additive r => Additive (End r)
instance Monoid (End r)