Algebra.Lattice.Ordered

Documentation

newtype Ordered a

A total order gives rise to a lattice. Join is max, meet is min.

Constructors

Ordered

Fields

• getOrdered :: a

Instances

Monad Ordered

Defined in Algebra.Lattice.Ordered

Methods

(>>=) :: Ordered a -> (a -> Ordered b) -> Ordered b

(>>) :: Ordered a -> Ordered b -> Ordered b

return :: a -> Ordered a

fail :: String -> Ordered a

Functor Ordered

Defined in Algebra.Lattice.Ordered

Methods

fmap :: (a -> b) -> Ordered a -> Ordered b

(<$) :: a -> Ordered b -> Ordered a

Applicative Ordered

Defined in Algebra.Lattice.Ordered

Methods

pure :: a -> Ordered a

(<*>) :: Ordered (a -> b) -> Ordered a -> Ordered b

liftA2 :: (a -> b -> c) -> Ordered a -> Ordered b -> Ordered c

(*>) :: Ordered a -> Ordered b -> Ordered b

(<*) :: Ordered a -> Ordered b -> Ordered a

Foldable Ordered

Defined in Algebra.Lattice.Ordered

Methods

fold :: Monoid m => Ordered m -> m

foldMap :: Monoid m => (a -> m) -> Ordered a -> m

foldr :: (a -> b -> b) -> b -> Ordered a -> b

foldr' :: (a -> b -> b) -> b -> Ordered a -> b

foldl :: (b -> a -> b) -> b -> Ordered a -> b

foldl' :: (b -> a -> b) -> b -> Ordered a -> b

foldr1 :: (a -> a -> a) -> Ordered a -> a

foldl1 :: (a -> a -> a) -> Ordered a -> a

toList :: Ordered a -> [a]

null :: Ordered a -> Bool

length :: Ordered a -> Int

elem :: Eq a => a -> Ordered a -> Bool

maximum :: Ord a => Ordered a -> a

minimum :: Ord a => Ordered a -> a

sum :: Num a => Ordered a -> a

product :: Num a => Ordered a -> a

Traversable Ordered

Defined in Algebra.Lattice.Ordered

Methods

traverse :: Applicative f => (a -> f b) -> Ordered a -> f (Ordered b)

sequenceA :: Applicative f => Ordered (f a) -> f (Ordered a)

mapM :: Monad m => (a -> m b) -> Ordered a -> m (Ordered b)

sequence :: Monad m => Ordered (m a) -> m (Ordered a)

Eq a => Eq (Ordered a)

Defined in Algebra.Lattice.Ordered

Methods

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

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

Data a => Data (Ordered a)

Defined in Algebra.Lattice.Ordered

Methods

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

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

toConstr :: Ordered a -> Constr

dataTypeOf :: Ordered a -> DataType

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

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

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

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

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

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

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

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

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

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

Ord a => Ord (Ordered a)

Defined in Algebra.Lattice.Ordered

Methods

compare :: Ordered a -> Ordered a -> Ordering

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

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

(>) :: Ordered a -> Ordered a -> Bool

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

max :: Ordered a -> Ordered a -> Ordered a

min :: Ordered a -> Ordered a -> Ordered a

Read a => Read (Ordered a)

Defined in Algebra.Lattice.Ordered

Methods

readsPrec :: Int -> ReadS (Ordered a)

readList :: ReadS [Ordered a]

readPrec :: ReadPrec (Ordered a)

readListPrec :: ReadPrec [Ordered a]

Show a => Show (Ordered a)

Defined in Algebra.Lattice.Ordered

Methods

showsPrec :: Int -> Ordered a -> ShowS

show :: Ordered a -> String

showList :: [Ordered a] -> ShowS

Generic (Ordered a)

Defined in Algebra.Lattice.Ordered

Associated Types

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

Methods

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

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

Function a => Function (Ordered a)

Defined in Algebra.Lattice.Ordered

Methods

function :: (Ordered a -> b) -> Ordered a :-> b

Arbitrary a => Arbitrary (Ordered a)

Defined in Algebra.Lattice.Ordered

Methods

arbitrary :: Gen (Ordered a)

shrink :: Ordered a -> [Ordered a]

CoArbitrary a => CoArbitrary (Ordered a)

Defined in Algebra.Lattice.Ordered

Methods

coarbitrary :: Ordered a -> Gen b -> Gen b

NFData a => NFData (Ordered a)

Defined in Algebra.Lattice.Ordered

Methods

rnf :: Ordered a -> ()

Hashable a => Hashable (Ordered a)

Defined in Algebra.Lattice.Ordered

Methods

hashWithSalt :: Int -> Ordered a -> Int

hash :: Ordered a -> Int

Universe a => Universe (Ordered a)

Defined in Algebra.Lattice.Ordered

Methods

universe :: [Ordered a]

Finite a => Finite (Ordered a)

Defined in Algebra.Lattice.Ordered

Methods

universeF :: [Ordered a]

cardinality :: Tagged (Ordered a) Natural

Ord a => PartialOrd (Ordered a)

Defined in Algebra.Lattice.Ordered

Methods

leq :: Ordered a -> Ordered a -> Bool

comparable :: Ordered a -> Ordered a -> Bool

(Ord a, Bounded a) => BoundedMeetSemiLattice (Ordered a)

Defined in Algebra.Lattice.Ordered

Methods

top :: Ordered a

(Ord a, Bounded a) => BoundedJoinSemiLattice (Ordered a)

Defined in Algebra.Lattice.Ordered

Methods

bottom :: Ordered a

Ord a => Lattice (Ordered a)

Defined in Algebra.Lattice.Ordered

Methods

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

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

(Ord a, Bounded a) => Heyting (Ordered a)

This is interesting logic, as it satisfies both de Morgan laws; but isn't Boolean: i.e. law of exluded middle doesn't hold.

Negation "smashes" value into minBound or maxBound.

Defined in Algebra.Lattice.Ordered

Methods

(==>) :: Ordered a -> Ordered a -> Ordered a

neg :: Ordered a -> Ordered a

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

Generic1 Ordered

Defined in Algebra.Lattice.Ordered

Associated Types

type Rep1 Ordered :: k -> Type

Methods

from1 :: Ordered a -> Rep1 Ordered a

to1 :: Rep1 Ordered a -> Ordered a

type Rep (Ordered a)

Defined in Algebra.Lattice.Ordered

type Rep (Ordered a) = D1 (MetaData "Ordered" "Algebra.Lattice.Ordered" "lattices-2.0.1-DhjhXTEEVqrG2Ss9wckwFx" True) (C1 (MetaCons "Ordered" PrefixI True) (S1 (MetaSel (Just "getOrdered") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 a)))

type Rep1 Ordered

Defined in Algebra.Lattice.Ordered

type Rep1 Ordered = D1 (MetaData "Ordered" "Algebra.Lattice.Ordered" "lattices-2.0.1-DhjhXTEEVqrG2Ss9wckwFx" True) (C1 (MetaCons "Ordered" PrefixI True) (S1 (MetaSel (Just "getOrdered") NoSourceUnpackedness NoSourceStrictness DecidedLazy) Par1))