Copyright	(c) Masahiro Sakai 2011-2014
License	BSD-style
Maintainer	masahiro.sakai@gmail.com
Stability	provisional
Portability	non-portable (ScopedTypeVariables, DeriveDataTypeable)
Safe Haskell	Safe-Inferred
Language	Haskell2010

Data.IntegerInterval

Contents

Interval type
Construction
Query
Universal comparison operators
Existential comparison operators
Existential comparison operators that produce witnesses (experimental)
Combine
Operations
Conversion

Description

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).

Synopsis

Interval type

data IntegerInterval Source

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

module Data.ExtendedReal

Construction

interval Source

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 Source

Arguments

:: Extended Integer	lower bound `l`
-> Extended Integer	upper bound `u`
-> IntegerInterval

closed interval [l,u]

(<..<=) infix 5 Source

Arguments

:: Extended Integer	lower bound `l`
-> Extended Integer	upper bound `u`
-> IntegerInterval

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

(<=..<) infix 5 Source

Arguments

:: Extended Integer	lower bound `l`
-> Extended Integer	upper bound `u`
-> IntegerInterval

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

(<..<) infix 5 Source

Arguments

:: Extended Integer	lower bound `l`
-> Extended Integer	upper bound `u`
-> IntegerInterval

open interval (l, u)

whole :: IntegerInterval Source

whole real number line (-∞, ∞)

empty :: IntegerInterval Source

empty (contradicting) interval

singleton :: Integer -> IntegerInterval Source

singleton set [x,x]

Query

null :: IntegerInterval -> Bool Source

Is the interval empty?

member :: Integer -> IntegerInterval -> Bool Source

Is the element in the interval?

notMember :: Integer -> IntegerInterval -> Bool Source

Is the element not in the interval?

isSubsetOf :: IntegerInterval -> IntegerInterval -> Bool Source

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

isProperSubsetOf :: IntegerInterval -> IntegerInterval -> Bool Source

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

lowerBound :: IntegerInterval -> Extended Integer Source

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 Source

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) Source

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) Source

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 Source

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

Universal comparison operators

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

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

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

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

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

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

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

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

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

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

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

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

Existential comparison operators

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

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

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

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

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

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

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

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

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

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

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

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 Source

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

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

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

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

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

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

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

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

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

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

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

Combine

intersection :: IntegerInterval -> IntegerInterval -> IntegerInterval Source

intersection of two intervals

intersections :: [IntegerInterval] -> IntegerInterval Source

intersection of a list of intervals.

hull :: IntegerInterval -> IntegerInterval -> IntegerInterval Source

convex hull of two intervals

hulls :: [IntegerInterval] -> IntegerInterval Source

convex hull of a list of intervals.

Operations

pickup :: IntegerInterval -> Maybe Integer Source

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

simplestIntegerWithin :: IntegerInterval -> Maybe Integer Source

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 Source

Convert the interval to Interval data type.

fromInterval :: Interval Integer -> IntegerInterval Source

Conversion from Interval data type.

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

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

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

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

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