Data.Connection.Interval

Documentation

data Interval a

An interval in a poset P.

An interval in a poset P is a subset I of P with the following property:

\( \forall x, y \in I, z \in P: x \leq z \leq y \Rightarrow z \in I \)

Instances

Eq a => Eq (Interval a)

Defined in Data.Connection.Interval

Methods

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

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

Show a => Show (Interval a)

Defined in Data.Connection.Interval

Methods

showsPrec :: Int -> Interval a -> ShowS

show :: Interval a -> String

showList :: [Interval a] -> ShowS

Preorder a => Preorder (Interval a)

Containment order

See https://en.wikipedia.org/wiki/Containment_order.

Defined in Data.Connection.Interval

Methods

(<~) :: Interval a -> Interval a -> Bool

(>~) :: Interval a -> Interval a -> Bool

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

(>) :: Interval a -> Interval a -> Bool

(?~) :: Interval a -> Interval a -> Bool

(~~) :: Interval a -> Interval a -> Bool

(/~) :: Interval a -> Interval a -> Bool

pmax :: Interval a -> Interval a -> Maybe (Interval a)

pmin :: Interval a -> Interval a -> Maybe (Interval a)

similar :: Interval a -> Interval a -> Bool

pcompare :: Interval a -> Interval a -> Maybe Ordering

Bounded a => Triple (Interval a) a

Defined in Data.Connection

Methods

triple :: Trip (Interval a) a

imap :: Preorder b => (a -> b) -> Interval a -> Interval b

Map over an interval.

Note this is not a functor, as a non-monotonic map may cause the interval to collapse to the empty interval.

(...) :: Preorder a => a -> a -> Interval a infix 3

Construct an interval from a pair of points.

empty :: Interval a

The empty interval.

>>> empty
Empty

singleton :: a -> Interval a

Construct an interval containing a single point.

>>> singleton 1
1 ... 1

contains :: Preorder a => Interval a -> a -> Bool

endpts :: Interval a -> Maybe (a, a)

Obtain the endpoints of an interval.

upper :: UpperBounded a => a -> Interval a

\( X_\geq(x) = \{ y \in X | y \geq x \} \)

Construct the upper set of an element x.

This function is monotone:

x <~ y <=> upper x <~ upper y

by the Yoneda lemma for preorders.

lower :: LowerBounded a => a -> Interval a

\( X_\leq(x) = \{ y \in X | y \leq x \} \)

Construct the lower set of an element x.

This function is antitone:

x <~ y <=> lower x >~ lower y

interval :: Bounded a => Trip (Interval a) a

TODO: Document