Copyright	(C) 2010-2015 Maximilian Bolingbroke
License	BSD-3-Clause (see the file LICENSE)
Maintainer	Oleg Grenrus <oleg.grenrus@iki.fi>
Safe Haskell	Trustworthy
Language	Haskell2010

Algebra.Lattice

Contents

Unbounded lattices
Bounded lattices
Monoid wrappers
Fixed points of chains in lattices

Description

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.

Synopsis

Unbounded lattices

class JoinSemiLattice a where Source

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 Source

join :: a -> a -> a Source

Deprecated: Use \/ infix operator

Instances

JoinSemiLattice Bool Source
Methods (\/) :: Bool -> Bool -> Bool Source join :: Bool -> Bool -> Bool Source
JoinSemiLattice () Source
Methods (\/) :: () -> () -> () Source join :: () -> () -> () Source
JoinSemiLattice Void Source
Methods (\/) :: Void -> Void -> Void Source join :: Void -> Void -> Void Source
JoinSemiLattice All Source
Methods (\/) :: All -> All -> All Source join :: All -> All -> All Source
JoinSemiLattice Any Source
Methods (\/) :: Any -> Any -> Any Source join :: Any -> Any -> Any Source
JoinSemiLattice IntSet Source
Methods (\/) :: IntSet -> IntSet -> IntSet Source join :: IntSet -> IntSet -> IntSet Source
JoinSemiLattice a => JoinSemiLattice (Identity a) Source
Methods (\/) :: Identity a -> Identity a -> Identity a Source join :: Identity a -> Identity a -> Identity a Source
JoinSemiLattice a => JoinSemiLattice (Endo a) Source
Methods (\/) :: Endo a -> Endo a -> Endo a Source join :: Endo a -> Endo a -> Endo a Source
JoinSemiLattice v => JoinSemiLattice (IntMap v) Source
Methods (\/) :: IntMap v -> IntMap v -> IntMap v Source join :: IntMap v -> IntMap v -> IntMap v Source
Ord a => JoinSemiLattice (Set a) Source
Methods (\/) :: Set a -> Set a -> Set a Source join :: Set a -> Set a -> Set a Source
(Eq a, Hashable a) => JoinSemiLattice (HashSet a) Source
Methods (\/) :: HashSet a -> HashSet a -> HashSet a Source join :: HashSet a -> HashSet a -> HashSet a Source
JoinSemiLattice a => JoinSemiLattice (Dropped a) Source
Methods (\/) :: Dropped a -> Dropped a -> Dropped a Source join :: Dropped a -> Dropped a -> Dropped a Source
JoinSemiLattice a => JoinSemiLattice (Levitated a) Source
Methods (\/) :: Levitated a -> Levitated a -> Levitated a Source join :: Levitated a -> Levitated a -> Levitated a Source
JoinSemiLattice a => JoinSemiLattice (Lifted a) Source
Methods (\/) :: Lifted a -> Lifted a -> Lifted a Source join :: Lifted a -> Lifted a -> Lifted a Source
MeetSemiLattice a => JoinSemiLattice (Op a) Source
Methods (\/) :: Op a -> Op a -> Op a Source join :: Op a -> Op a -> Op a Source
Ord a => JoinSemiLattice (Ordered a) Source
Methods (\/) :: Ordered a -> Ordered a -> Ordered a Source join :: Ordered a -> Ordered a -> Ordered a Source
JoinSemiLattice v => JoinSemiLattice (k -> v) Source
Methods (\/) :: (k -> v) -> (k -> v) -> k -> v Source join :: (k -> v) -> (k -> v) -> k -> v Source
(JoinSemiLattice a, JoinSemiLattice b) => JoinSemiLattice (a, b) Source
Methods (\/) :: (a, b) -> (a, b) -> (a, b) Source join :: (a, b) -> (a, b) -> (a, b) Source
JoinSemiLattice a => JoinSemiLattice (Const a b) Source
Methods (\/) :: Const a b -> Const a b -> Const a b Source join :: Const a b -> Const a b -> Const a b Source
JoinSemiLattice (Proxy * a) Source
Methods (\/) :: Proxy * a -> Proxy * a -> Proxy * a Source join :: Proxy * a -> Proxy * a -> Proxy * a Source
(Ord k, JoinSemiLattice v) => JoinSemiLattice (Map k v) Source
Methods (\/) :: Map k v -> Map k v -> Map k v Source join :: Map k v -> Map k v -> Map k v Source
(Eq k, Hashable k) => JoinSemiLattice (HashMap k v) Source
Methods (\/) :: HashMap k v -> HashMap k v -> HashMap k v Source join :: HashMap k v -> HashMap k v -> HashMap k v Source
(PartialOrd k, JoinSemiLattice k, JoinSemiLattice v) => JoinSemiLattice (Lexicographic k v) Source
Methods (\/) :: Lexicographic k v -> Lexicographic k v -> Lexicographic k v Source join :: Lexicographic k v -> Lexicographic k v -> Lexicographic k v Source
JoinSemiLattice a => JoinSemiLattice (Tagged * t a) Source
Methods (\/) :: Tagged * t a -> Tagged * t a -> Tagged * t a Source join :: Tagged * t a -> Tagged * t a -> Tagged * t a Source

class MeetSemiLattice a where Source

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 Source

meet :: a -> a -> a Source

Deprecated: Use /\ infix operator

Instances

MeetSemiLattice Bool Source
Methods (/\) :: Bool -> Bool -> Bool Source meet :: Bool -> Bool -> Bool Source
MeetSemiLattice () Source
Methods (/\) :: () -> () -> () Source meet :: () -> () -> () Source
MeetSemiLattice Void Source
Methods (/\) :: Void -> Void -> Void Source meet :: Void -> Void -> Void Source
MeetSemiLattice All Source
Methods (/\) :: All -> All -> All Source meet :: All -> All -> All Source
MeetSemiLattice Any Source
Methods (/\) :: Any -> Any -> Any Source meet :: Any -> Any -> Any Source
MeetSemiLattice a => MeetSemiLattice (Identity a) Source
Methods (/\) :: Identity a -> Identity a -> Identity a Source meet :: Identity a -> Identity a -> Identity a Source
MeetSemiLattice a => MeetSemiLattice (Endo a) Source
Methods (/\) :: Endo a -> Endo a -> Endo a Source meet :: Endo a -> Endo a -> Endo a Source
Ord a => MeetSemiLattice (Set a) Source
Methods (/\) :: Set a -> Set a -> Set a Source meet :: Set a -> Set a -> Set a Source
(Eq a, Hashable a) => MeetSemiLattice (HashSet a) Source
Methods (/\) :: HashSet a -> HashSet a -> HashSet a Source meet :: HashSet a -> HashSet a -> HashSet a Source
MeetSemiLattice a => MeetSemiLattice (Dropped a) Source
Methods (/\) :: Dropped a -> Dropped a -> Dropped a Source meet :: Dropped a -> Dropped a -> Dropped a Source
MeetSemiLattice a => MeetSemiLattice (Levitated a) Source
Methods (/\) :: Levitated a -> Levitated a -> Levitated a Source meet :: Levitated a -> Levitated a -> Levitated a Source
MeetSemiLattice a => MeetSemiLattice (Lifted a) Source
Methods (/\) :: Lifted a -> Lifted a -> Lifted a Source meet :: Lifted a -> Lifted a -> Lifted a Source
JoinSemiLattice a => MeetSemiLattice (Op a) Source
Methods (/\) :: Op a -> Op a -> Op a Source meet :: Op a -> Op a -> Op a Source
Ord a => MeetSemiLattice (Ordered a) Source
Methods (/\) :: Ordered a -> Ordered a -> Ordered a Source meet :: Ordered a -> Ordered a -> Ordered a Source
MeetSemiLattice v => MeetSemiLattice (k -> v) Source
Methods (/\) :: (k -> v) -> (k -> v) -> k -> v Source meet :: (k -> v) -> (k -> v) -> k -> v Source
(MeetSemiLattice a, MeetSemiLattice b) => MeetSemiLattice (a, b) Source
Methods (/\) :: (a, b) -> (a, b) -> (a, b) Source meet :: (a, b) -> (a, b) -> (a, b) Source
MeetSemiLattice a => MeetSemiLattice (Const a b) Source
Methods (/\) :: Const a b -> Const a b -> Const a b Source meet :: Const a b -> Const a b -> Const a b Source
MeetSemiLattice (Proxy * a) Source
Methods (/\) :: Proxy * a -> Proxy * a -> Proxy * a Source meet :: Proxy * a -> Proxy * a -> Proxy * a Source
(Ord k, MeetSemiLattice v) => MeetSemiLattice (Map k v) Source
Methods (/\) :: Map k v -> Map k v -> Map k v Source meet :: Map k v -> Map k v -> Map k v Source
(Eq k, Hashable k) => MeetSemiLattice (HashMap k v) Source
Methods (/\) :: HashMap k v -> HashMap k v -> HashMap k v Source meet :: HashMap k v -> HashMap k v -> HashMap k v Source
(PartialOrd k, MeetSemiLattice k, MeetSemiLattice v) => MeetSemiLattice (Lexicographic k v) Source
Methods (/\) :: Lexicographic k v -> Lexicographic k v -> Lexicographic k v Source meet :: Lexicographic k v -> Lexicographic k v -> Lexicographic k v Source
MeetSemiLattice a => MeetSemiLattice (Tagged * t a) Source
Methods (/\) :: Tagged * t a -> Tagged * t a -> Tagged * t a Source meet :: Tagged * t a -> Tagged * t a -> Tagged * t a Source

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

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 Source

Lattice () Source

Lattice Void Source

Lattice All Source

Lattice Any Source

Lattice a => Lattice (Identity a) Source

Lattice a => Lattice (Endo a) Source

Ord a => Lattice (Set a) Source

Lattice a => Lattice (Dropped a) Source

Lattice a => Lattice (Levitated a) Source

Lattice a => Lattice (Lifted a) Source

(Lattice a, Ord a) => Lattice (Op a) Source

(Lattice a, Ord a) => Lattice (Ordered a) Source

Lattice v => Lattice (k -> v) Source

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

Lattice a => Lattice (Const a b) Source

Lattice (Proxy * a) Source

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

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

Lattice a => Lattice (Tagged * t a) Source

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

The partial ordering induced by the join-semilattice structure

joins1 :: JoinSemiLattice a => [a] -> a Source

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

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

The partial ordering induced by the meet-semilattice structure

meets1 :: MeetSemiLattice a => [a] -> a Source

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

Bounded lattices

class JoinSemiLattice a => BoundedJoinSemiLattice a where Source

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

Identity: x \/ bottom == x

Methods

bottom :: a Source

Instances

BoundedJoinSemiLattice Bool Source
Methods bottom :: Bool Source
BoundedJoinSemiLattice () Source
Methods bottom :: () Source
BoundedJoinSemiLattice All Source
Methods bottom :: All Source
BoundedJoinSemiLattice Any Source
Methods bottom :: Any Source
BoundedJoinSemiLattice IntSet Source
Methods bottom :: IntSet Source
BoundedJoinSemiLattice a => BoundedJoinSemiLattice (Identity a) Source
Methods bottom :: Identity a Source
BoundedJoinSemiLattice a => BoundedJoinSemiLattice (Endo a) Source
Methods bottom :: Endo a Source
JoinSemiLattice v => BoundedJoinSemiLattice (IntMap v) Source
Methods bottom :: IntMap v Source
Ord a => BoundedJoinSemiLattice (Set a) Source
Methods bottom :: Set a Source
(Eq a, Hashable a) => BoundedJoinSemiLattice (HashSet a) Source
Methods bottom :: HashSet a Source
BoundedJoinSemiLattice a => BoundedJoinSemiLattice (Dropped a) Source
Methods bottom :: Dropped a Source
JoinSemiLattice a => BoundedJoinSemiLattice (Levitated a) Source
Methods bottom :: Levitated a Source
JoinSemiLattice a => BoundedJoinSemiLattice (Lifted a) Source
Methods bottom :: Lifted a Source
BoundedMeetSemiLattice a => BoundedJoinSemiLattice (Op a) Source
Methods bottom :: Op a Source
(Ord a, Bounded a) => BoundedJoinSemiLattice (Ordered a) Source
Methods bottom :: Ordered a Source
BoundedJoinSemiLattice v => BoundedJoinSemiLattice (k -> v) Source
Methods bottom :: k -> v Source
(BoundedJoinSemiLattice a, BoundedJoinSemiLattice b) => BoundedJoinSemiLattice (a, b) Source
Methods bottom :: (a, b) Source
BoundedJoinSemiLattice a => BoundedJoinSemiLattice (Const a b) Source
Methods bottom :: Const a b Source
BoundedJoinSemiLattice (Proxy * a) Source
Methods bottom :: Proxy * a Source
(Ord k, JoinSemiLattice v) => BoundedJoinSemiLattice (Map k v) Source
Methods bottom :: Map k v Source
(Eq k, Hashable k) => BoundedJoinSemiLattice (HashMap k v) Source
Methods bottom :: HashMap k v Source
(PartialOrd k, BoundedJoinSemiLattice k, BoundedJoinSemiLattice v) => BoundedJoinSemiLattice (Lexicographic k v) Source
Methods bottom :: Lexicographic k v Source
BoundedJoinSemiLattice a => BoundedJoinSemiLattice (Tagged * t a) Source
Methods bottom :: Tagged * t a Source

class MeetSemiLattice a => BoundedMeetSemiLattice a where Source

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

Identity: x /\ top == x

Methods

top :: a Source

Instances

BoundedMeetSemiLattice Bool Source
Methods top :: Bool Source
BoundedMeetSemiLattice () Source
Methods top :: () Source
BoundedMeetSemiLattice All Source
Methods top :: All Source
BoundedMeetSemiLattice Any Source
Methods top :: Any Source
BoundedMeetSemiLattice a => BoundedMeetSemiLattice (Identity a) Source
Methods top :: Identity a Source
BoundedMeetSemiLattice a => BoundedMeetSemiLattice (Endo a) Source
Methods top :: Endo a Source
(Ord a, Finite a) => BoundedMeetSemiLattice (Set a) Source
Methods top :: Set a Source
MeetSemiLattice a => BoundedMeetSemiLattice (Dropped a) Source
Methods top :: Dropped a Source
MeetSemiLattice a => BoundedMeetSemiLattice (Levitated a) Source
Methods top :: Levitated a Source
BoundedMeetSemiLattice a => BoundedMeetSemiLattice (Lifted a) Source
Methods top :: Lifted a Source
BoundedJoinSemiLattice a => BoundedMeetSemiLattice (Op a) Source
Methods top :: Op a Source
(Ord a, Bounded a) => BoundedMeetSemiLattice (Ordered a) Source
Methods top :: Ordered a Source
BoundedMeetSemiLattice v => BoundedMeetSemiLattice (k -> v) Source
Methods top :: k -> v Source
(BoundedMeetSemiLattice a, BoundedMeetSemiLattice b) => BoundedMeetSemiLattice (a, b) Source
Methods top :: (a, b) Source
BoundedMeetSemiLattice a => BoundedMeetSemiLattice (Const a b) Source
Methods top :: Const a b Source
BoundedMeetSemiLattice (Proxy * a) Source
Methods top :: Proxy * a Source
(Ord k, Finite k, BoundedMeetSemiLattice v) => BoundedMeetSemiLattice (Map k v) Source
Methods top :: Map k v Source
(PartialOrd k, BoundedMeetSemiLattice k, BoundedMeetSemiLattice v) => BoundedMeetSemiLattice (Lexicographic k v) Source
Methods top :: Lexicographic k v Source
BoundedMeetSemiLattice a => BoundedMeetSemiLattice (Tagged * t a) Source
Methods top :: Tagged * t a Source

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

Lattices with both bounds

Instances

BoundedLattice Bool Source

BoundedLattice () Source

BoundedLattice All Source

BoundedLattice Any Source

BoundedLattice a => BoundedLattice (Identity a) Source

BoundedLattice a => BoundedLattice (Endo a) Source

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

BoundedLattice a => BoundedLattice (Dropped a) Source

Lattice a => BoundedLattice (Levitated a) Source

BoundedLattice a => BoundedLattice (Lifted a) Source

(BoundedLattice a, Ord a, Bounded a) => BoundedLattice (Op a) Source

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

BoundedLattice v => BoundedLattice (k -> v) Source

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

BoundedLattice a => BoundedLattice (Const a b) Source

BoundedLattice (Proxy * a) Source

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

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

BoundedLattice a => BoundedLattice (Tagged * t a) Source

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

The join of a list of join-semilattice elements

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

The meet of a list of meet-semilattice elements

fromBool :: BoundedLattice a => Bool -> a Source

True to top and False to bottom

Monoid wrappers

newtype Meet a Source

Monoid wrapper for MeetSemiLattice

Constructors

Meet
Fields getMeet :: a

Instances

Monad Meet Source
Methods (>>=) :: Meet a -> (a -> Meet b) -> Meet b (>>) :: Meet a -> Meet b -> Meet b return :: a -> Meet a fail :: String -> Meet a
Functor Meet Source
Methods fmap :: (a -> b) -> Meet a -> Meet b (<$) :: a -> Meet b -> Meet a
Applicative Meet Source
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 Source
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) Source
Methods minBound :: Meet a maxBound :: Meet a
Eq a => Eq (Meet a) Source
Methods (==) :: Meet a -> Meet a -> Bool (/=) :: Meet a -> Meet a -> Bool
Data a => Data (Meet a) Source
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) Source
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) Source
Methods readsPrec :: Int -> ReadS (Meet a) readList :: ReadS [Meet a] readPrec :: ReadPrec (Meet a) readListPrec :: ReadPrec [Meet a]
Show a => Show (Meet a) Source
Methods showsPrec :: Int -> Meet a -> ShowS show :: Meet a -> String showList :: [Meet a] -> ShowS
Generic (Meet a) Source
Associated Types type Rep (Meet a) :: * -> * Methods from :: Meet a -> Rep (Meet a) x to :: Rep (Meet a) x -> Meet a
BoundedMeetSemiLattice a => Monoid (Meet a) Source
Methods mempty :: Meet a mappend :: Meet a -> Meet a -> Meet a mconcat :: [Meet a] -> Meet a
MeetSemiLattice a => Semigroup (Meet a) Source
Methods (<>) :: Meet a -> Meet a -> Meet a sconcat :: NonEmpty (Meet a) -> Meet a times1p :: Natural -> Meet a -> Meet a
Universe a => Universe (Meet a) Source
Methods universe :: [Meet a]
Finite a => Finite (Meet a) Source
Methods universeF :: [Meet a]
type Rep (Meet a) Source

newtype Join a Source

Monoid wrapper for JoinSemiLattice

Constructors

Join
Fields getJoin :: a

Instances

Monad Join Source
Methods (>>=) :: Join a -> (a -> Join b) -> Join b (>>) :: Join a -> Join b -> Join b return :: a -> Join a fail :: String -> Join a
Functor Join Source
Methods fmap :: (a -> b) -> Join a -> Join b (<$) :: a -> Join b -> Join a
Applicative Join Source
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 Source
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) Source
Methods minBound :: Join a maxBound :: Join a
Eq a => Eq (Join a) Source
Methods (==) :: Join a -> Join a -> Bool (/=) :: Join a -> Join a -> Bool
Data a => Data (Join a) Source
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) Source
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) Source
Methods readsPrec :: Int -> ReadS (Join a) readList :: ReadS [Join a] readPrec :: ReadPrec (Join a) readListPrec :: ReadPrec [Join a]
Show a => Show (Join a) Source
Methods showsPrec :: Int -> Join a -> ShowS show :: Join a -> String showList :: [Join a] -> ShowS
Generic (Join a) Source
Associated Types type Rep (Join a) :: * -> * Methods from :: Join a -> Rep (Join a) x to :: Rep (Join a) x -> Join a
BoundedJoinSemiLattice a => Monoid (Join a) Source
Methods mempty :: Join a mappend :: Join a -> Join a -> Join a mconcat :: [Join a] -> Join a
JoinSemiLattice a => Semigroup (Join a) Source
Methods (<>) :: Join a -> Join a -> Join a sconcat :: NonEmpty (Join a) -> Join a times1p :: Natural -> Join a -> Join a
Universe a => Universe (Join a) Source
Methods universe :: [Join a]
Finite a => Finite (Join a) Source
Methods universeF :: [Join a]
type Rep (Join a) Source

Fixed points of chains in lattices

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

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 Source

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 Source

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 Source

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 Source

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 Source

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.