Algebra.Boolean

Documentation

data BooleanAlgebra a

Boolean is the left adjoint functor from the category of Heyting algebras to the category of Boolean algebras; its right adjoint is the inclusion.

Instances

Bounded a => Bounded (BooleanAlgebra a)

Defined in Algebra.Boolean

Methods

minBound :: BooleanAlgebra a

maxBound :: BooleanAlgebra a

Eq a => Eq (BooleanAlgebra a)

Defined in Algebra.Boolean

Methods

(==) :: BooleanAlgebra a -> BooleanAlgebra a -> Bool

(/=) :: BooleanAlgebra a -> BooleanAlgebra a -> Bool

Data a => Data (BooleanAlgebra a)

Defined in Algebra.Boolean

Methods

gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> BooleanAlgebra a -> c (BooleanAlgebra a)

gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (BooleanAlgebra a)

toConstr :: BooleanAlgebra a -> Constr

dataTypeOf :: BooleanAlgebra a -> DataType

dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c (BooleanAlgebra a))

dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (BooleanAlgebra a))

gmapT :: (forall b. Data b => b -> b) -> BooleanAlgebra a -> BooleanAlgebra a

gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> BooleanAlgebra a -> r

gmapQr :: forall r r'. (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> BooleanAlgebra a -> r

gmapQ :: (forall d. Data d => d -> u) -> BooleanAlgebra a -> [u]

gmapQi :: Int -> (forall d. Data d => d -> u) -> BooleanAlgebra a -> u

gmapM :: Monad m => (forall d. Data d => d -> m d) -> BooleanAlgebra a -> m (BooleanAlgebra a)

gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> BooleanAlgebra a -> m (BooleanAlgebra a)

gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> BooleanAlgebra a -> m (BooleanAlgebra a)

Ord a => Ord (BooleanAlgebra a)

Defined in Algebra.Boolean

Methods

compare :: BooleanAlgebra a -> BooleanAlgebra a -> Ordering

(<) :: BooleanAlgebra a -> BooleanAlgebra a -> Bool

(<=) :: BooleanAlgebra a -> BooleanAlgebra a -> Bool

(>) :: BooleanAlgebra a -> BooleanAlgebra a -> Bool

(>=) :: BooleanAlgebra a -> BooleanAlgebra a -> Bool

max :: BooleanAlgebra a -> BooleanAlgebra a -> BooleanAlgebra a

min :: BooleanAlgebra a -> BooleanAlgebra a -> BooleanAlgebra a

Read a => Read (BooleanAlgebra a)

Defined in Algebra.Boolean

Methods

readsPrec :: Int -> ReadS (BooleanAlgebra a)

readList :: ReadS [BooleanAlgebra a]

readPrec :: ReadPrec (BooleanAlgebra a)

readListPrec :: ReadPrec [BooleanAlgebra a]

Show a => Show (BooleanAlgebra a)

Defined in Algebra.Boolean

Methods

showsPrec :: Int -> BooleanAlgebra a -> ShowS

show :: BooleanAlgebra a -> String

showList :: [BooleanAlgebra a] -> ShowS

Generic (BooleanAlgebra a)

Defined in Algebra.Boolean

Associated Types

type Rep (BooleanAlgebra a) :: Type -> Type

Methods

from :: BooleanAlgebra a -> Rep (BooleanAlgebra a) x

to :: Rep (BooleanAlgebra a) x -> BooleanAlgebra a

Heyting a => Heyting (BooleanAlgebra a)

Defined in Algebra.Boolean

Methods

(==>) :: BooleanAlgebra a -> BooleanAlgebra a -> BooleanAlgebra a

neg :: BooleanAlgebra a -> BooleanAlgebra a

(<=>) :: BooleanAlgebra a -> BooleanAlgebra a -> BooleanAlgebra a

Lattice a => Lattice (BooleanAlgebra a)

Defined in Algebra.Boolean

Methods

(\/) :: BooleanAlgebra a -> BooleanAlgebra a -> BooleanAlgebra a

(/\) :: BooleanAlgebra a -> BooleanAlgebra a -> BooleanAlgebra a

BoundedJoinSemiLattice a => BoundedJoinSemiLattice (BooleanAlgebra a)

Defined in Algebra.Boolean

Methods

bottom :: BooleanAlgebra a

BoundedMeetSemiLattice a => BoundedMeetSemiLattice (BooleanAlgebra a)

Defined in Algebra.Boolean

Methods

top :: BooleanAlgebra a

Heyting a => Boolean (BooleanAlgebra a)

Defined in Algebra.Boolean

type Rep (BooleanAlgebra a)

Defined in Algebra.Boolean

type Rep (BooleanAlgebra a) = D1 ('MetaData "BooleanAlgebra" "Algebra.Boolean" "heyting-algebras-0.2.0.1-ESmFQmKgxne8K6ZOTNu8qI" 'True) (C1 ('MetaCons "BooleanAlgebra" 'PrefixI 'True) (S1 ('MetaSel ('Just "runBooleanAlgebra") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 a)))

(==>) :: Heyting a => a -> a -> a infixr 5

Implication.

neg :: Heyting a => a -> a

Negation.

neg x = x ==> bottom

Adjunction between Boolean and Heyting algebras

class Heyting a => Boolean a

Instances

Boolean Bool

Defined in Algebra.Boolean

Boolean ()

Defined in Algebra.Boolean

Boolean All

Defined in Algebra.Boolean

Boolean Any

Defined in Algebra.Boolean

Boolean a => Boolean (Identity a)

Defined in Algebra.Boolean

Boolean a => Boolean (Endo a)

Defined in Algebra.Boolean

Heyting a => Boolean (BooleanAlgebra a)

Defined in Algebra.Boolean

Boolean (FreeBoolean a)

Defined in Algebra.Boolean.Free

Boolean b => Boolean (a -> b)

Defined in Algebra.Boolean

Boolean (Proxy a)

Defined in Algebra.Boolean

Boolean a => Boolean (Const a b)

Defined in Algebra.Boolean

Boolean a => Boolean (Tagged t a)

Defined in Algebra.Boolean

runBooleanAlgebra :: BooleanAlgebra a -> a

extract value from Boolean

boolean :: Heyting a => a -> BooleanAlgebra a

Smart constructro of the Boolean type.