Data.IntegerInterval

Interval datatype and interval arithmetic over integers.

Since 1.2.0

For the purpose of abstract interpretation, it might be convenient to use Lattice instance. See also lattices package (http://hackage.haskell.org/package/lattices).

Interval type

data IntegerInterval

The intervals (i.e. connected and convex subsets) over integers (Z).

Instances

Eq IntegerInterval

Data IntegerInterval

Num IntegerInterval

Read IntegerInterval

Show IntegerInterval

NFData IntegerInterval

Hashable IntegerInterval

JoinSemiLattice IntegerInterval

MeetSemiLattice IntegerInterval

Lattice IntegerInterval

BoundedJoinSemiLattice IntegerInterval

BoundedMeetSemiLattice IntegerInterval

BoundedLattice IntegerInterval

Typeable * IntegerInterval

module Data.ExtendedReal

Construction

interval

Arguments

:: (Extended Integer, Bool)

lower bound and whether it is included

-> (Extended Integer, Bool)

upper bound and whether it is included

-> IntegerInterval

smart constructor for Interval

(<=..<=) infix 5

Arguments

:: Extended Integer

lower bound l

-> Extended Integer

upper bound u

-> IntegerInterval

closed interval [l,u]

(<..<=) infix 5

Arguments

:: Extended Integer

lower bound l

-> Extended Integer

upper bound u

-> IntegerInterval

left-open right-closed interval (l,u]

(<=..<) infix 5

Arguments

:: Extended Integer

lower bound l

-> Extended Integer

upper bound u

-> IntegerInterval

left-closed right-open interval [l, u)

(<..<) infix 5

Arguments

:: Extended Integer

lower bound l

-> Extended Integer

upper bound u

-> IntegerInterval

open interval (l, u)

whole :: IntegerInterval

whole real number line (-∞, ∞)

empty :: IntegerInterval

empty (contradicting) interval

singleton :: Integer -> IntegerInterval

singleton set [x,x]

Query

null :: IntegerInterval -> Bool

Is the interval empty?

member :: Integer -> IntegerInterval -> Bool

Is the element in the interval?

notMember :: Integer -> IntegerInterval -> Bool

Is the element not in the interval?

isSubsetOf :: IntegerInterval -> IntegerInterval -> Bool

Is this a subset? (i1 `isSubsetOf` i2) tells whether i1 is a subset of i2.

isProperSubsetOf :: IntegerInterval -> IntegerInterval -> Bool

Is this a proper subset? (i.e. a subset but not equal).

lowerBound :: IntegerInterval -> Extended Integer

Lower endpoint (i.e. greatest lower bound) of the interval.

• lowerBound of the empty interval is PosInf.
• lowerBound of a left unbounded interval is NegInf.
• lowerBound of an interval may or may not be a member of the interval.

upperBound :: IntegerInterval -> Extended Integer

Upper endpoint (i.e. least upper bound) of the interval.

• upperBound of the empty interval is NegInf.
• upperBound of a right unbounded interval is PosInf.
• upperBound of an interval is a member of the interval.

lowerBound' :: IntegerInterval -> (Extended Integer, Bool)

lowerBound of the interval and whether it is included in the interval. The result is convenient to use as an argument for interval.

upperBound' :: IntegerInterval -> (Extended Integer, Bool)

upperBound of the interval and whether it is included in the interval. The result is convenient to use as an argument for interval.

width :: IntegerInterval -> Integer

Width of a interval. Width of an unbounded interval is undefined.

Universal comparison operators

(<!) :: IntegerInterval -> IntegerInterval -> Bool infix 4

For all x in X, y in Y. x < y?

(<=!) :: IntegerInterval -> IntegerInterval -> Bool infix 4

For all x in X, y in Y. x <= y?

(==!) :: IntegerInterval -> IntegerInterval -> Bool infix 4

For all x in X, y in Y. x == y?

(>=!) :: IntegerInterval -> IntegerInterval -> Bool infix 4

For all x in X, y in Y. x >= y?

(>!) :: IntegerInterval -> IntegerInterval -> Bool infix 4

For all x in X, y in Y. x > y?

(/=!) :: IntegerInterval -> IntegerInterval -> Bool infix 4

For all x in X, y in Y. x /= y?

Existential comparison operators

(<?) :: IntegerInterval -> IntegerInterval -> Bool infix 4

Does there exist an x in X, y in Y such that x < y?

(<=?) :: IntegerInterval -> IntegerInterval -> Bool infix 4

Does there exist an x in X, y in Y such that x <= y?

(==?) :: IntegerInterval -> IntegerInterval -> Bool infix 4

Does there exist an x in X, y in Y such that x == y?

(>=?) :: IntegerInterval -> IntegerInterval -> Bool infix 4

Does there exist an x in X, y in Y such that x >= y?

(>?) :: IntegerInterval -> IntegerInterval -> Bool infix 4

Does there exist an x in X, y in Y such that x > y?

(/=?) :: IntegerInterval -> IntegerInterval -> Bool infix 4

Does there exist an x in X, y in Y such that x /= y?

Existential comparison operators that produce witnesses (experimental)

(<??) :: IntegerInterval -> IntegerInterval -> Maybe (Integer, Integer) infix 4

Does there exist an x in X, y in Y such that x < y?

(<=??) :: IntegerInterval -> IntegerInterval -> Maybe (Integer, Integer) infix 4

Does there exist an x in X, y in Y such that x <= y?

(==??) :: IntegerInterval -> IntegerInterval -> Maybe (Integer, Integer) infix 4

Does there exist an x in X, y in Y such that x == y?

(>=??) :: IntegerInterval -> IntegerInterval -> Maybe (Integer, Integer) infix 4

Does there exist an x in X, y in Y such that x >= y?

(>??) :: IntegerInterval -> IntegerInterval -> Maybe (Integer, Integer) infix 4

Does there exist an x in X, y in Y such that x > y?

(/=??) :: IntegerInterval -> IntegerInterval -> Maybe (Integer, Integer) infix 4

Does there exist an x in X, y in Y such that x /= y?

Combine

intersection :: IntegerInterval -> IntegerInterval -> IntegerInterval

intersection of two intervals

intersections :: [IntegerInterval] -> IntegerInterval

intersection of a list of intervals.

hull :: IntegerInterval -> IntegerInterval -> IntegerInterval

convex hull of two intervals

hulls :: [IntegerInterval] -> IntegerInterval

convex hull of a list of intervals.

Operations

pickup :: IntegerInterval -> Maybe Integer

pick up an element from the interval if the interval is not empty.

simplestIntegerWithin :: IntegerInterval -> Maybe Integer

simplestIntegerWithin returns the simplest rational number within the interval.

An integer y is said to be simpler than another y' if

• abs y <= abs y, and

(see also approxRational)

Conversion

toInterval :: Real r => IntegerInterval -> Interval r

Convert the interval to Interval data type.

fromInterval :: Interval Integer -> IntegerInterval

Conversion from Interval data type.

fromIntervalOver :: RealFrac r => Interval r -> IntegerInterval

Given a Interval I over R, compute the smallest IntegerInterval J such that I ⊆ J.

fromIntervalUnder :: RealFrac r => Interval r -> IntegerInterval

Given a Interval I over R, compute the largest IntegerInterval J such that J ⊆ I.