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


-- | A library of constructive algebra.
--   
--   A library of constructive algebra.
@package constructive-algebra
@version 0.1


-- | The representation of the ring structure.
module Algebra.Structures.Ring

-- | Definition of rings
class Ring a
(<+>) :: (Ring a) => a -> a -> a
(<*>) :: (Ring a) => a -> a -> a
neg :: (Ring a) => a -> a
zero :: (Ring a) => a
one :: (Ring a) => a

-- | Specification of rings. Test that the arguments satisfy the ring
--   axioms.
propRing :: (Ring a, Eq a) => a -> a -> a -> Property

-- | Subtraction
(<->) :: (Ring a) => a -> a -> a

-- | Exponentiation
(<^>) :: (Ring a) => a -> Integer -> a

-- | Summation
sumRing :: (Ring a) => [a] -> a

-- | Product
productRing :: (Ring a) => [a] -> a

module Algebra.Structures.CommutativeRing

-- | Definition of commutative rings
class (Ring a) => CommutativeRing a

-- | Specification of commutative rings. Test that multiplication is
--   commutative and that it satisfies the ring axioms.
propCommutativeRing :: (CommutativeRing a, Eq a) => a -> a -> a -> Property


-- | Finitely generated ideals in commutative rings.
module Algebra.Ideal

-- | Ideals characterized by their list of generators.
data (CommutativeRing a) => Ideal a
Id :: [a] -> Ideal a

-- | The zero ideal.
zeroIdeal :: (CommutativeRing a) => Ideal a

-- | Test if an ideal is principal.
isPrincipal :: (CommutativeRing a) => Ideal a -> Bool

-- | Evaluate the ideal at a certain point.
eval :: (CommutativeRing a) => a -> Ideal a -> a

-- | Addition of ideals.
addId :: (CommutativeRing a, Eq a) => Ideal a -> Ideal a -> Ideal a

-- | Multiplication of ideals.
mulId :: (CommutativeRing a, Eq a) => Ideal a -> Ideal a -> Ideal a
instance (CommutativeRing a, Arbitrary a, Eq a) => Arbitrary (Ideal a)
instance (CommutativeRing a, Show a) => Show (Ideal a)

module Algebra.Structures.StronglyDiscrete

-- | Strongly discrete rings
--   
--   A ring is called strongly discrete if ideal membership is decidable.
--   Nothing correspond to that x is not in the ideal and Just is the
--   witness. Examples include all euclidean domains and the polynomial
--   ring.
class (Ring a) => StronglyDiscrete a
member :: (StronglyDiscrete a) => a -> Ideal a -> Maybe [a]

-- | Test that the witness is actually a witness that the element is in the
--   ideal.
propStronglyDiscrete :: (CommutativeRing a, StronglyDiscrete a, Eq a) => a -> Ideal a -> Bool

module Algebra.Structures.IntegralDomain

-- | Definition of integral domains
class (CommutativeRing a) => IntegralDomain a

-- | Specification of commutative rings. Test that there are no
--   zero-divisors commutative and that it satisfies the axioms of
--   commutative rings.
propIntegralDomain :: (IntegralDomain a, Eq a) => a -> a -> a -> Property

module Algebra.Z

-- | Type synonym for integers.
type Z = Integer
instance IntegralDomain Z
instance CommutativeRing Z
instance Ring Z

module Algebra.Structures.Field

-- | Definition of fields
class (IntegralDomain a) => Field a
inv :: (Field a) => a -> a

-- | Specification of fields. Test that the multiplicative inverses behave
--   as expected and that it satisfies the axioms of integral domains.
propField :: (Field a, Eq a) => a -> a -> a -> Property

-- | Division
(</>) :: (Field a) => a -> a -> a