Algebra.Lattice

In mathematics, a lattice is a partially ordered set in which every two elements have a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet).

In this module lattices are defined using meet and join operators, as it's constructive one.

Unbounded lattices

class JoinSemiLattice a where

A algebraic structure with element joins: http://en.wikipedia.org/wiki/Semilattice

Associativity: x \/ (y \/ z) == (x \/ y) \/ z
Commutativity: x \/ y == y \/ x
Idempotency:   x \/ x == x

Minimal complete definition

(\/) | join

Methods

(\/) :: a -> a -> a infixr 5

join :: a -> a -> a

Deprecated: Use \/ infix operator

Instances

JoinSemiLattice Bool

Methods

(\/) :: Bool -> Bool -> Bool

join :: Bool -> Bool -> Bool

JoinSemiLattice ()

Methods

(\/) :: () -> () -> ()

join :: () -> () -> ()

JoinSemiLattice Void

Methods

(\/) :: Void -> Void -> Void

join :: Void -> Void -> Void

JoinSemiLattice All

Methods

(\/) :: All -> All -> All

join :: All -> All -> All

JoinSemiLattice Any

Methods

(\/) :: Any -> Any -> Any

join :: Any -> Any -> Any

JoinSemiLattice IntSet

Methods

(\/) :: IntSet -> IntSet -> IntSet

join :: IntSet -> IntSet -> IntSet

JoinSemiLattice a => JoinSemiLattice (Identity a)

Methods

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

join :: Identity a -> Identity a -> Identity a

JoinSemiLattice a => JoinSemiLattice (Endo a)

Methods

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

join :: Endo a -> Endo a -> Endo a

JoinSemiLattice v => JoinSemiLattice (IntMap v)

Methods

(\/) :: IntMap v -> IntMap v -> IntMap v

join :: IntMap v -> IntMap v -> IntMap v

Ord a => JoinSemiLattice (Set a)

Methods

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

join :: Set a -> Set a -> Set a

(Eq a, Hashable a) => JoinSemiLattice (HashSet a)

Methods

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

join :: HashSet a -> HashSet a -> HashSet a

Integral a => JoinSemiLattice (Divisibility a)

Methods

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

join :: Divisibility a -> Divisibility a -> Divisibility a

JoinSemiLattice a => JoinSemiLattice (Dropped a)

Methods

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

join :: Dropped a -> Dropped a -> Dropped a

JoinSemiLattice (FreeLattice a)

Methods

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

join :: FreeLattice a -> FreeLattice a -> FreeLattice a

JoinSemiLattice (FreeJoinSemiLattice a)

Methods

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

join :: FreeJoinSemiLattice a -> FreeJoinSemiLattice a -> FreeJoinSemiLattice a

JoinSemiLattice a => JoinSemiLattice (Levitated a)

Methods

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

join :: Levitated a -> Levitated a -> Levitated a

JoinSemiLattice a => JoinSemiLattice (Lifted a)

Methods

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

join :: Lifted a -> Lifted a -> Lifted a

MeetSemiLattice a => JoinSemiLattice (Op a)

Methods

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

join :: Op a -> Op a -> Op a

Ord a => JoinSemiLattice (Ordered a)

Methods

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

join :: Ordered a -> Ordered a -> Ordered a

JoinSemiLattice v => JoinSemiLattice (k -> v)

Methods

(\/) :: (k -> v) -> (k -> v) -> k -> v

join :: (k -> v) -> (k -> v) -> k -> v

(JoinSemiLattice a, JoinSemiLattice b) => JoinSemiLattice (a, b)

Methods

(\/) :: (a, b) -> (a, b) -> (a, b)

join :: (a, b) -> (a, b) -> (a, b)

JoinSemiLattice (Proxy * a)

Methods

(\/) :: Proxy * a -> Proxy * a -> Proxy * a

join :: Proxy * a -> Proxy * a -> Proxy * a

(Ord k, JoinSemiLattice v) => JoinSemiLattice (Map k v)

Methods

(\/) :: Map k v -> Map k v -> Map k v

join :: Map k v -> Map k v -> Map k v

(Eq k, Hashable k, JoinSemiLattice v) => JoinSemiLattice (HashMap k v)

Methods

(\/) :: HashMap k v -> HashMap k v -> HashMap k v

join :: HashMap k v -> HashMap k v -> HashMap k v

(PartialOrd k, JoinSemiLattice k, BoundedJoinSemiLattice v) => JoinSemiLattice (Lexicographic k v)

Methods

(\/) :: Lexicographic k v -> Lexicographic k v -> Lexicographic k v

join :: Lexicographic k v -> Lexicographic k v -> Lexicographic k v

JoinSemiLattice a => JoinSemiLattice (Const * a b)

Methods

(\/) :: Const * a b -> Const * a b -> Const * a b

join :: Const * a b -> Const * a b -> Const * a b

JoinSemiLattice a => JoinSemiLattice (Tagged * t a)

Methods

(\/) :: Tagged * t a -> Tagged * t a -> Tagged * t a

join :: Tagged * t a -> Tagged * t a -> Tagged * t a

class MeetSemiLattice a where

A algebraic structure with element meets: http://en.wikipedia.org/wiki/Semilattice

Associativity: x /\ (y /\ z) == (x /\ y) /\ z
Commutativity: x /\ y == y /\ x
Idempotency:   x /\ x == x

Minimal complete definition

(/\) | meet

Methods

(/\) :: a -> a -> a infixr 6

meet :: a -> a -> a

Deprecated: Use /\ infix operator

Instances

MeetSemiLattice Bool

Methods

(/\) :: Bool -> Bool -> Bool

meet :: Bool -> Bool -> Bool

MeetSemiLattice ()

Methods

(/\) :: () -> () -> ()

meet :: () -> () -> ()

MeetSemiLattice Void

Methods

(/\) :: Void -> Void -> Void

meet :: Void -> Void -> Void

MeetSemiLattice All

Methods

(/\) :: All -> All -> All

meet :: All -> All -> All

MeetSemiLattice Any

Methods

(/\) :: Any -> Any -> Any

meet :: Any -> Any -> Any

MeetSemiLattice IntSet

Methods

(/\) :: IntSet -> IntSet -> IntSet

meet :: IntSet -> IntSet -> IntSet

MeetSemiLattice a => MeetSemiLattice (Identity a)

Methods

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

meet :: Identity a -> Identity a -> Identity a

MeetSemiLattice a => MeetSemiLattice (Endo a)

Methods

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

meet :: Endo a -> Endo a -> Endo a

MeetSemiLattice v => MeetSemiLattice (IntMap v)

Methods

(/\) :: IntMap v -> IntMap v -> IntMap v

meet :: IntMap v -> IntMap v -> IntMap v

Ord a => MeetSemiLattice (Set a)

Methods

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

meet :: Set a -> Set a -> Set a

(Eq a, Hashable a) => MeetSemiLattice (HashSet a)

Methods

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

meet :: HashSet a -> HashSet a -> HashSet a

Integral a => MeetSemiLattice (Divisibility a)

Methods

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

meet :: Divisibility a -> Divisibility a -> Divisibility a

MeetSemiLattice a => MeetSemiLattice (Dropped a)

Methods

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

meet :: Dropped a -> Dropped a -> Dropped a

MeetSemiLattice (FreeLattice a)

Methods

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

meet :: FreeLattice a -> FreeLattice a -> FreeLattice a

MeetSemiLattice (FreeMeetSemiLattice a)

Methods

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

meet :: FreeMeetSemiLattice a -> FreeMeetSemiLattice a -> FreeMeetSemiLattice a

MeetSemiLattice a => MeetSemiLattice (Levitated a)

Methods

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

meet :: Levitated a -> Levitated a -> Levitated a

MeetSemiLattice a => MeetSemiLattice (Lifted a)

Methods

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

meet :: Lifted a -> Lifted a -> Lifted a

JoinSemiLattice a => MeetSemiLattice (Op a)

Methods

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

meet :: Op a -> Op a -> Op a

Ord a => MeetSemiLattice (Ordered a)

Methods

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

meet :: Ordered a -> Ordered a -> Ordered a

MeetSemiLattice v => MeetSemiLattice (k -> v)

Methods

(/\) :: (k -> v) -> (k -> v) -> k -> v

meet :: (k -> v) -> (k -> v) -> k -> v

(MeetSemiLattice a, MeetSemiLattice b) => MeetSemiLattice (a, b)

Methods

(/\) :: (a, b) -> (a, b) -> (a, b)

meet :: (a, b) -> (a, b) -> (a, b)

MeetSemiLattice (Proxy * a)

Methods

(/\) :: Proxy * a -> Proxy * a -> Proxy * a

meet :: Proxy * a -> Proxy * a -> Proxy * a

(Ord k, MeetSemiLattice v) => MeetSemiLattice (Map k v)

Methods

(/\) :: Map k v -> Map k v -> Map k v

meet :: Map k v -> Map k v -> Map k v

(Eq k, Hashable k, MeetSemiLattice v) => MeetSemiLattice (HashMap k v)

Methods

(/\) :: HashMap k v -> HashMap k v -> HashMap k v

meet :: HashMap k v -> HashMap k v -> HashMap k v

(PartialOrd k, MeetSemiLattice k, BoundedMeetSemiLattice v) => MeetSemiLattice (Lexicographic k v)

Methods

(/\) :: Lexicographic k v -> Lexicographic k v -> Lexicographic k v

meet :: Lexicographic k v -> Lexicographic k v -> Lexicographic k v

MeetSemiLattice a => MeetSemiLattice (Const * a b)

Methods

(/\) :: Const * a b -> Const * a b -> Const * a b

meet :: Const * a b -> Const * a b -> Const * a b

MeetSemiLattice a => MeetSemiLattice (Tagged * t a)

Methods

(/\) :: Tagged * t a -> Tagged * t a -> Tagged * t a

meet :: Tagged * t a -> Tagged * t a -> Tagged * t a

class (JoinSemiLattice a, MeetSemiLattice a) => Lattice a

The combination of two semi lattices makes a lattice if the absorption law holds: see http://en.wikipedia.org/wiki/Absorption_law and http://en.wikipedia.org/wiki/Lattice_(order)

Absorption: a \/ (a /\ b) == a /\ (a \/ b) == a

Instances

Lattice Bool

Lattice ()

Lattice Void

Lattice All

Lattice Any

Lattice IntSet

Lattice a => Lattice (Identity a)

Lattice a => Lattice (Endo a)

Lattice v => Lattice (IntMap v)

Ord a => Lattice (Set a)

(Eq a, Hashable a) => Lattice (HashSet a)

Integral a => Lattice (Divisibility a)

Lattice a => Lattice (Dropped a)

Lattice (FreeLattice a)

Lattice a => Lattice (Levitated a)

Lattice a => Lattice (Lifted a)

Lattice a => Lattice (Op a)

Ord a => Lattice (Ordered a)

Lattice v => Lattice (k -> v)

(Lattice a, Lattice b) => Lattice (a, b)

Lattice (Proxy * a)

(Ord k, Lattice v) => Lattice (Map k v)

(Eq k, Hashable k, Lattice v) => Lattice (HashMap k v)

(PartialOrd k, Lattice k, BoundedLattice v) => Lattice (Lexicographic k v)

Lattice a => Lattice (Const * a b)

Lattice a => Lattice (Tagged * t a)

joinLeq :: (Eq a, JoinSemiLattice a) => a -> a -> Bool

The partial ordering induced by the join-semilattice structure

joins1 :: (JoinSemiLattice a, Foldable1 f) => f a -> a

The join of at a list of join-semilattice elements (of length at least one)

meetLeq :: (Eq a, MeetSemiLattice a) => a -> a -> Bool

The partial ordering induced by the meet-semilattice structure

meets1 :: (MeetSemiLattice a, Foldable1 f) => f a -> a

The meet of at a list of meet-semilattice elements (of length at least one)

Bounded lattices

class JoinSemiLattice a => BoundedJoinSemiLattice a where

A join-semilattice with some element |bottom| that / approaches.

Identity: x \/ bottom == x

Minimal complete definition

bottom

Methods

bottom :: a

Instances

BoundedJoinSemiLattice Bool

Methods

bottom :: Bool

BoundedJoinSemiLattice ()

Methods

bottom :: ()

BoundedJoinSemiLattice All

Methods

bottom :: All

BoundedJoinSemiLattice Any

Methods

bottom :: Any

BoundedJoinSemiLattice IntSet

Methods

bottom :: IntSet

BoundedJoinSemiLattice a => BoundedJoinSemiLattice (Identity a)

Methods

bottom :: Identity a

BoundedJoinSemiLattice a => BoundedJoinSemiLattice (Endo a)

Methods

bottom :: Endo a

JoinSemiLattice v => BoundedJoinSemiLattice (IntMap v)

Methods

bottom :: IntMap v

Ord a => BoundedJoinSemiLattice (Set a)

Methods

bottom :: Set a

(Eq a, Hashable a) => BoundedJoinSemiLattice (HashSet a)

Methods

bottom :: HashSet a

Integral a => BoundedJoinSemiLattice (Divisibility a)

Methods

bottom :: Divisibility a

BoundedJoinSemiLattice a => BoundedJoinSemiLattice (Dropped a)

Methods

bottom :: Dropped a

BoundedJoinSemiLattice a => BoundedJoinSemiLattice (FreeLattice a)

Methods

bottom :: FreeLattice a

BoundedJoinSemiLattice a => BoundedJoinSemiLattice (FreeJoinSemiLattice a)

Methods

bottom :: FreeJoinSemiLattice a

JoinSemiLattice a => BoundedJoinSemiLattice (Levitated a)

Methods

bottom :: Levitated a

JoinSemiLattice a => BoundedJoinSemiLattice (Lifted a)

Methods

bottom :: Lifted a

BoundedMeetSemiLattice a => BoundedJoinSemiLattice (Op a)

Methods

bottom :: Op a

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

Methods

bottom :: Ordered a

BoundedJoinSemiLattice v => BoundedJoinSemiLattice (k -> v)

Methods

bottom :: k -> v

(BoundedJoinSemiLattice a, BoundedJoinSemiLattice b) => BoundedJoinSemiLattice (a, b)

Methods

bottom :: (a, b)

BoundedJoinSemiLattice (Proxy * a)

Methods

bottom :: Proxy * a

(Ord k, JoinSemiLattice v) => BoundedJoinSemiLattice (Map k v)

Methods

bottom :: Map k v

(Eq k, Hashable k, JoinSemiLattice v) => BoundedJoinSemiLattice (HashMap k v)

Methods

bottom :: HashMap k v

(PartialOrd k, BoundedJoinSemiLattice k, BoundedJoinSemiLattice v) => BoundedJoinSemiLattice (Lexicographic k v)

Methods

bottom :: Lexicographic k v

BoundedJoinSemiLattice a => BoundedJoinSemiLattice (Const * a b)

Methods

bottom :: Const * a b

BoundedJoinSemiLattice a => BoundedJoinSemiLattice (Tagged * t a)

Methods

bottom :: Tagged * t a

class MeetSemiLattice a => BoundedMeetSemiLattice a where

A meet-semilattice with some element |top| that / approaches.

Identity: x /\ top == x

Minimal complete definition

top

Methods

top :: a

Instances

BoundedMeetSemiLattice Bool

Methods

top :: Bool

BoundedMeetSemiLattice ()

Methods

top :: ()

BoundedMeetSemiLattice All

Methods

top :: All

BoundedMeetSemiLattice Any

Methods

top :: Any

BoundedMeetSemiLattice a => BoundedMeetSemiLattice (Identity a)

Methods

top :: Identity a

BoundedMeetSemiLattice a => BoundedMeetSemiLattice (Endo a)

Methods

top :: Endo a

(Ord a, Finite a) => BoundedMeetSemiLattice (Set a)

Methods

top :: Set a

(Eq a, Hashable a, Finite a) => BoundedMeetSemiLattice (HashSet a)

Methods

top :: HashSet a

MeetSemiLattice a => BoundedMeetSemiLattice (Dropped a)

Methods

top :: Dropped a

BoundedMeetSemiLattice a => BoundedMeetSemiLattice (FreeLattice a)

Methods

top :: FreeLattice a

BoundedMeetSemiLattice a => BoundedMeetSemiLattice (FreeMeetSemiLattice a)

Methods

top :: FreeMeetSemiLattice a

MeetSemiLattice a => BoundedMeetSemiLattice (Levitated a)

Methods

top :: Levitated a

BoundedMeetSemiLattice a => BoundedMeetSemiLattice (Lifted a)

Methods

top :: Lifted a

BoundedJoinSemiLattice a => BoundedMeetSemiLattice (Op a)

Methods

top :: Op a

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

Methods

top :: Ordered a

BoundedMeetSemiLattice v => BoundedMeetSemiLattice (k -> v)

Methods

top :: k -> v

(BoundedMeetSemiLattice a, BoundedMeetSemiLattice b) => BoundedMeetSemiLattice (a, b)

Methods

top :: (a, b)

BoundedMeetSemiLattice (Proxy * a)

Methods

top :: Proxy * a

(Ord k, Finite k, BoundedMeetSemiLattice v) => BoundedMeetSemiLattice (Map k v)

Methods

top :: Map k v

(Eq k, Hashable k, Finite k, BoundedMeetSemiLattice v) => BoundedMeetSemiLattice (HashMap k v)

Methods

top :: HashMap k v

(PartialOrd k, BoundedMeetSemiLattice k, BoundedMeetSemiLattice v) => BoundedMeetSemiLattice (Lexicographic k v)

Methods

top :: Lexicographic k v

BoundedMeetSemiLattice a => BoundedMeetSemiLattice (Const * a b)

Methods

top :: Const * a b

BoundedMeetSemiLattice a => BoundedMeetSemiLattice (Tagged * t a)

Methods

top :: Tagged * t a

class (Lattice a, BoundedJoinSemiLattice a, BoundedMeetSemiLattice a) => BoundedLattice a

Lattices with both bounds

Instances

BoundedLattice Bool

BoundedLattice ()

BoundedLattice All

BoundedLattice Any

BoundedLattice a => BoundedLattice (Identity a)

BoundedLattice a => BoundedLattice (Endo a)

(Ord a, Finite a) => BoundedLattice (Set a)

(Eq a, Hashable a, Finite a) => BoundedLattice (HashSet a)

BoundedLattice a => BoundedLattice (Dropped a)

BoundedLattice a => BoundedLattice (FreeLattice a)

Lattice a => BoundedLattice (Levitated a)

BoundedLattice a => BoundedLattice (Lifted a)

BoundedLattice a => BoundedLattice (Op a)

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

BoundedLattice v => BoundedLattice (k -> v)

(BoundedLattice a, BoundedLattice b) => BoundedLattice (a, b)

BoundedLattice (Proxy * a)

(Ord k, Finite k, BoundedLattice v) => BoundedLattice (Map k v)

(Eq k, Hashable k, Finite k, BoundedLattice v) => BoundedLattice (HashMap k v)

(PartialOrd k, BoundedLattice k, BoundedLattice v) => BoundedLattice (Lexicographic k v)

BoundedLattice a => BoundedLattice (Const * a b)

BoundedLattice a => BoundedLattice (Tagged * t a)

joins :: (BoundedJoinSemiLattice a, Foldable f) => f a -> a

The join of a list of join-semilattice elements

meets :: (BoundedMeetSemiLattice a, Foldable f) => f a -> a

The meet of a list of meet-semilattice elements

fromBool :: BoundedLattice a => Bool -> a

True to top and False to bottom

Monoid wrappers

newtype Meet a

Monoid wrapper for MeetSemiLattice

Constructors

Meet

Fields

• getMeet :: a

Instances

Monad Meet

Methods

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

(>>) :: Meet a -> Meet b -> Meet b

return :: a -> Meet a

fail :: String -> Meet a

Functor Meet

Methods

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

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

Applicative Meet

Methods

pure :: a -> Meet a

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

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

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

MonadZip Meet

Methods

mzip :: Meet a -> Meet b -> Meet (a, b)

mzipWith :: (a -> b -> c) -> Meet a -> Meet b -> Meet c

munzip :: Meet (a, b) -> (Meet a, Meet b)

Bounded a => Bounded (Meet a)

Methods

minBound :: Meet a

maxBound :: Meet a

Eq a => Eq (Meet a)

Methods

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

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

Data a => Data (Meet a)

Methods

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

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

toConstr :: Meet a -> Constr

dataTypeOf :: Meet a -> DataType

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

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

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

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

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

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

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

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

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

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

Ord a => Ord (Meet a)

Methods

compare :: Meet a -> Meet a -> Ordering

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

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

(>) :: Meet a -> Meet a -> Bool

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

max :: Meet a -> Meet a -> Meet a

min :: Meet a -> Meet a -> Meet a

Read a => Read (Meet a)

Methods

readsPrec :: Int -> ReadS (Meet a)

readList :: ReadS [Meet a]

readPrec :: ReadPrec (Meet a)

readListPrec :: ReadPrec [Meet a]

Show a => Show (Meet a)

Methods

showsPrec :: Int -> Meet a -> ShowS

show :: Meet a -> String

showList :: [Meet a] -> ShowS

Generic (Meet a)

Associated Types

type Rep (Meet a) :: * -> *

Methods

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

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

MeetSemiLattice a => Semigroup (Meet a)

Methods

(<>) :: Meet a -> Meet a -> Meet a

sconcat :: NonEmpty (Meet a) -> Meet a

stimes :: Integral b => b -> Meet a -> Meet a

BoundedMeetSemiLattice a => Monoid (Meet a)

Methods

mempty :: Meet a

mappend :: Meet a -> Meet a -> Meet a

mconcat :: [Meet a] -> Meet a

Universe a => Universe (Meet a)

Methods

universe :: [Meet a]

Finite a => Finite (Meet a)

Methods

universeF :: [Meet a]

(Eq a, MeetSemiLattice a) => PartialOrd (Meet a)

Methods

leq :: Meet a -> Meet a -> Bool

comparable :: Meet a -> Meet a -> Bool

type Rep (Meet a)

type Rep (Meet a) = D1 (MetaData "Meet" "Algebra.Lattice" "lattices-1.7-LVxdjRPvAj7FxWDUYmNHEJ" True) (C1 (MetaCons "Meet" PrefixI True) (S1 (MetaSel (Just Symbol "getMeet") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 a)))

newtype Join a

Monoid wrapper for JoinSemiLattice

Constructors

Join

Fields

• getJoin :: a

Instances

Monad Join

Methods

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

(>>) :: Join a -> Join b -> Join b

return :: a -> Join a

fail :: String -> Join a

Functor Join

Methods

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

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

Applicative Join

Methods

pure :: a -> Join a

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

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

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

MonadZip Join

Methods

mzip :: Join a -> Join b -> Join (a, b)

mzipWith :: (a -> b -> c) -> Join a -> Join b -> Join c

munzip :: Join (a, b) -> (Join a, Join b)

Bounded a => Bounded (Join a)

Methods

minBound :: Join a

maxBound :: Join a

Eq a => Eq (Join a)

Methods

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

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

Data a => Data (Join a)

Methods

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

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

toConstr :: Join a -> Constr

dataTypeOf :: Join a -> DataType

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

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

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

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

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

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

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

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

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

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

Ord a => Ord (Join a)

Methods

compare :: Join a -> Join a -> Ordering

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

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

(>) :: Join a -> Join a -> Bool

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

max :: Join a -> Join a -> Join a

min :: Join a -> Join a -> Join a

Read a => Read (Join a)

Methods

readsPrec :: Int -> ReadS (Join a)

readList :: ReadS [Join a]

readPrec :: ReadPrec (Join a)

readListPrec :: ReadPrec [Join a]

Show a => Show (Join a)

Methods

showsPrec :: Int -> Join a -> ShowS

show :: Join a -> String

showList :: [Join a] -> ShowS

Generic (Join a)

Associated Types

type Rep (Join a) :: * -> *

Methods

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

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

JoinSemiLattice a => Semigroup (Join a)

Methods

(<>) :: Join a -> Join a -> Join a

sconcat :: NonEmpty (Join a) -> Join a

stimes :: Integral b => b -> Join a -> Join a

BoundedJoinSemiLattice a => Monoid (Join a)

Methods

mempty :: Join a

mappend :: Join a -> Join a -> Join a

mconcat :: [Join a] -> Join a

Universe a => Universe (Join a)

Methods

universe :: [Join a]

Finite a => Finite (Join a)

Methods

universeF :: [Join a]

(Eq a, JoinSemiLattice a) => PartialOrd (Join a)

Methods

leq :: Join a -> Join a -> Bool

comparable :: Join a -> Join a -> Bool

type Rep (Join a)

type Rep (Join a) = D1 (MetaData "Join" "Algebra.Lattice" "lattices-1.7-LVxdjRPvAj7FxWDUYmNHEJ" True) (C1 (MetaCons "Join" PrefixI True) (S1 (MetaSel (Just Symbol "getJoin") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 a)))

Fixed points of chains in lattices

lfp :: (Eq a, BoundedJoinSemiLattice a) => (a -> a) -> a

Implementation of Kleene fixed-point theorem http://en.wikipedia.org/wiki/Kleene_fixed-point_theorem. Forces the function to be monotone.

lfpFrom :: (Eq a, BoundedJoinSemiLattice a) => a -> (a -> a) -> a

Implementation of Kleene fixed-point theorem http://en.wikipedia.org/wiki/Kleene_fixed-point_theorem. Forces the function to be monotone.

unsafeLfp :: (Eq a, BoundedJoinSemiLattice a) => (a -> a) -> a

Implementation of Kleene fixed-point theorem http://en.wikipedia.org/wiki/Kleene_fixed-point_theorem. Assumes that the function is monotone and does not check if that is correct.

gfp :: (Eq a, BoundedMeetSemiLattice a) => (a -> a) -> a

Implementation of Kleene fixed-point theorem http://en.wikipedia.org/wiki/Kleene_fixed-point_theorem. Forces the function to be antinone.

gfpFrom :: (Eq a, BoundedMeetSemiLattice a) => a -> (a -> a) -> a

Implementation of Kleene fixed-point theorem http://en.wikipedia.org/wiki/Kleene_fixed-point_theorem. Forces the function to be antinone.

unsafeGfp :: (Eq a, BoundedMeetSemiLattice a) => (a -> a) -> a

Implementation of Kleene fixed-point theorem http://en.wikipedia.org/wiki/Kleene_fixed-point_theorem. Assumes that the function is antinone and does not check if that is correct.