| Safe Haskell | Safe-Inferred |
|---|
Algebra.Lattice
- class JoinSemiLattice a where
- join :: a -> a -> a
- class MeetSemiLattice a where
- meet :: a -> a -> a
- class (JoinSemiLattice a, MeetSemiLattice a) => Lattice a
- joinLeq :: (Eq a, JoinSemiLattice a) => a -> a -> Bool
- joins1 :: JoinSemiLattice a => [a] -> a
- meetLeq :: (Eq a, MeetSemiLattice a) => a -> a -> Bool
- meets1 :: MeetSemiLattice a => [a] -> a
- class JoinSemiLattice a => BoundedJoinSemiLattice a where
- bottom :: a
- class MeetSemiLattice a => BoundedMeetSemiLattice a where
- top :: a
- class (Lattice a, BoundedJoinSemiLattice a, BoundedMeetSemiLattice a) => BoundedLattice a
- joins :: BoundedJoinSemiLattice a => [a] -> a
- meets :: BoundedMeetSemiLattice a => [a] -> a
- lfp :: (Eq a, BoundedJoinSemiLattice a) => (a -> a) -> a
- lfpFrom :: (Eq a, BoundedJoinSemiLattice a) => a -> (a -> a) -> a
- unsafeLfp :: (Eq a, BoundedJoinSemiLattice a) => (a -> a) -> a
- gfp :: (Eq a, BoundedMeetSemiLattice a) => (a -> a) -> a
- gfpFrom :: (Eq a, BoundedMeetSemiLattice a) => a -> (a -> a) -> a
- unsafeGfp :: (Eq a, BoundedMeetSemiLattice a) => (a -> a) -> a
Unbounded lattices
class JoinSemiLattice a whereSource
A algebraic structure with element joins: http://en.wikipedia.org/wiki/Semilattice
Associativity: x join (y join z) == (x join y) join z
Commutativity: x join y == y join x
Idempotency: x join x == x
Instances
| JoinSemiLattice Bool | |
| JoinSemiLattice IntSet | |
| JoinSemiLattice v => JoinSemiLattice (IntMap v) | |
| Ord a => JoinSemiLattice (Set a) | |
| JoinSemiLattice a => JoinSemiLattice (Dropped a) | |
| JoinSemiLattice a => JoinSemiLattice (Levitated a) | |
| JoinSemiLattice a => JoinSemiLattice (Lifted a) | |
| JoinSemiLattice v => JoinSemiLattice (k -> v) | |
| (JoinSemiLattice a, JoinSemiLattice b) => JoinSemiLattice (a, b) | |
| (Ord k, JoinSemiLattice v) => JoinSemiLattice (Map k v) |
class MeetSemiLattice a whereSource
A algebraic structure with element meets: http://en.wikipedia.org/wiki/Semilattice
Associativity: x meet (y meet z) == (x meet y) meet z
Commutativity: x meet y == y meet x
Idempotency: x meet x == x
Instances
| MeetSemiLattice Bool | |
| (Ord a, Enumerable a) => MeetSemiLattice (Set (Enumerated a)) | |
| MeetSemiLattice a => MeetSemiLattice (Dropped a) | |
| MeetSemiLattice a => MeetSemiLattice (Levitated a) | |
| MeetSemiLattice a => MeetSemiLattice (Lifted a) | |
| MeetSemiLattice v => MeetSemiLattice (k -> v) | |
| (MeetSemiLattice a, MeetSemiLattice b) => MeetSemiLattice (a, b) | |
| (Ord k, Enumerable k, MeetSemiLattice v) => MeetSemiLattice (Map (Enumerated k) v) |
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)
Instances
| Lattice Bool | |
| (JoinSemiLattice (Set (Enumerated a)), MeetSemiLattice (Set (Enumerated a)), Ord a, Enumerable a) => Lattice (Set (Enumerated a)) | |
| (JoinSemiLattice (Dropped a), MeetSemiLattice (Dropped a), Lattice a) => Lattice (Dropped a) | |
| (JoinSemiLattice (Levitated a), MeetSemiLattice (Levitated a), Lattice a) => Lattice (Levitated a) | |
| (JoinSemiLattice (Lifted a), MeetSemiLattice (Lifted a), Lattice a) => Lattice (Lifted a) | |
| (JoinSemiLattice (k -> v), MeetSemiLattice (k -> v), Lattice v) => Lattice (k -> v) | |
| (JoinSemiLattice (a, b), MeetSemiLattice (a, b), Lattice a, Lattice b) => Lattice (a, b) | |
| (JoinSemiLattice (Map (Enumerated k) v), MeetSemiLattice (Map (Enumerated k) v), Ord k, Enumerable k, Lattice v) => Lattice (Map (Enumerated k) v) |
joinLeq :: (Eq a, JoinSemiLattice a) => a -> a -> BoolSource
The partial ordering induced by the join-semilattice structure
joins1 :: JoinSemiLattice a => [a] -> aSource
The join of at a list of join-semilattice elements (of length at least one)
meetLeq :: (Eq a, MeetSemiLattice a) => a -> a -> BoolSource
The partial ordering induced by the meet-semilattice structure
meets1 :: MeetSemiLattice a => [a] -> aSource
The meet of at a list of meet-semilattice elements (of length at least one)
Bounded lattices
class JoinSemiLattice a => BoundedJoinSemiLattice a whereSource
Instances
| BoundedJoinSemiLattice Bool | |
| BoundedJoinSemiLattice IntSet | |
| (JoinSemiLattice (IntMap v), JoinSemiLattice v) => BoundedJoinSemiLattice (IntMap v) | |
| (JoinSemiLattice (Set a), Ord a) => BoundedJoinSemiLattice (Set a) | |
| (JoinSemiLattice (Dropped a), BoundedJoinSemiLattice a) => BoundedJoinSemiLattice (Dropped a) | |
| (JoinSemiLattice (Levitated a), JoinSemiLattice a) => BoundedJoinSemiLattice (Levitated a) | |
| (JoinSemiLattice (Lifted a), JoinSemiLattice a) => BoundedJoinSemiLattice (Lifted a) | |
| (JoinSemiLattice (k -> v), BoundedJoinSemiLattice v) => BoundedJoinSemiLattice (k -> v) | |
| (JoinSemiLattice (a, b), BoundedJoinSemiLattice a, BoundedJoinSemiLattice b) => BoundedJoinSemiLattice (a, b) | |
| (JoinSemiLattice (Map k v), Ord k, JoinSemiLattice v) => BoundedJoinSemiLattice (Map k v) |
class MeetSemiLattice a => BoundedMeetSemiLattice a whereSource
Instances
| BoundedMeetSemiLattice Bool | |
| (MeetSemiLattice (Set (Enumerated a)), Ord a, Enumerable a) => BoundedMeetSemiLattice (Set (Enumerated a)) | |
| (MeetSemiLattice (Dropped a), MeetSemiLattice a) => BoundedMeetSemiLattice (Dropped a) | |
| (MeetSemiLattice (Levitated a), MeetSemiLattice a) => BoundedMeetSemiLattice (Levitated a) | |
| (MeetSemiLattice (Lifted a), BoundedMeetSemiLattice a) => BoundedMeetSemiLattice (Lifted a) | |
| (MeetSemiLattice (k -> v), BoundedMeetSemiLattice v) => BoundedMeetSemiLattice (k -> v) | |
| (MeetSemiLattice (a, b), BoundedMeetSemiLattice a, BoundedMeetSemiLattice b) => BoundedMeetSemiLattice (a, b) | |
| (MeetSemiLattice (Map (Enumerated k) v), Ord k, Enumerable k, BoundedMeetSemiLattice v) => BoundedMeetSemiLattice (Map (Enumerated k) v) |
class (Lattice a, BoundedJoinSemiLattice a, BoundedMeetSemiLattice a) => BoundedLattice a Source
Lattices with both bounds
Instances
joins :: BoundedJoinSemiLattice a => [a] -> aSource
The join of a list of join-semilattice elements
meets :: BoundedMeetSemiLattice a => [a] -> aSource
The meet of a list of meet-semilattice elements
Fixed points of chains in lattices
lfp :: (Eq a, BoundedJoinSemiLattice a) => (a -> a) -> aSource
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) -> aSource
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) -> aSource
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) -> aSource
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) -> aSource
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) -> aSource
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.