certlin.intervals¶

Intervals¶

Classes

`Interval`(lower, upper[, lower_closed, ...])	An interval.
`Intervals`(intervals)	A Cartesian product of intervals.

class certlin.intervals.Interval(lower, upper, lower_closed: bool = True, upper_closed: bool = True)¶

An interval.

Also supports variables.

EXAMPLES:

sage: from certlin import *
sage: Interval(0, 1)
[0, 1]
sage: Interval(0, 1, False, True)
(0, 1]
sage: Interval(0, 1, False, False)
(0, 1)
sage: Interval(0, 1, True, False)
[0, 1)
sage: Interval(0, 0)
{0}
sage: Interval(0, 0, False, False)
{}
sage: Interval(0, 0, True, True)
{0}
sage: Interval(0, 0, True, False)
{}
sage: Interval(0, 0, False, True)
{}
sage: Interval(-oo, 4)
(-oo, 4]
sage: Interval(-oo, 4, False, False)
(-oo, 4)
sage: I = Interval(-3, +oo, False, False)
sage: I
(-3, +oo)
sage: 0 in I
True
sage: -3 in I
False
sage: Interval.random() # random
(-1, 1)

Variables are supported, too:

sage: var("a")
a
sage: Interval(a, 1)
[a, 1]
sage: I = Interval(a, 1, False, False)
sage: I
(a, 1)
sage: I.an_element()
1/2*a + 1/2

an_element()¶

Return an element of the interval.

EXAMPLES:

sage: from certlin import *
sage: Interval(0, 1).an_element()
0
sage: Interval(0, 1, False, True).an_element()
1
sage: Interval(0, 1, False, False).an_element()
1/2
sage: Interval(-oo, 0).an_element()
0
sage: Interval(-oo, +oo).an_element()
0
sage: Interval(5, +oo, False, False).an_element()
6
sage: Interval(0, 0, False, False).an_element()
Traceback (most recent call last):
...
ValueError: The interval is empty.

classmethod closed(lower, upper) → Interval¶

Return a closed interval.

EXAMPLES:

sage: from certlin import *
sage: Interval.closed(0, 1)
[0, 1]

classmethod empty() → Interval¶

Return the empty interval.

EXAMPLES:

sage: from certlin import *
sage: Interval.empty()
{}

infimum()¶

Return the infimum of the interval.

EXAMPLES:

sage: from certlin import *
sage: Interval(0, 1).infimum()
0
sage: Interval(-oo, 0).infimum()
-Infinity
sage: Interval(0, 0, False, False).infimum()
+Infinity

is_bounded() → bool¶: Return whether the interval is bounded.

is_closed() → bool¶: Return whether the interval is closed.

is_empty() → bool¶: Return whether the interval is empty.

is_open() → bool¶: Return whether the interval is open.

is_pointed() → bool¶

Return whether the interval is a point.

An interval is pointed if it has the same lower and upper bound, and both are closed.

EXAMPLES:

sage: from certlin import *
sage: Interval(0, 0).is_pointed()
True
sage: Interval(0, 1).is_pointed()
False

is_unbounded() → bool¶: Return whether the interval is unbounded.

classmethod open(lower, upper) → Interval¶

Return an open interval.

EXAMPLES:

sage: from certlin import *
sage: Interval.open(0, 1)
(0, 1)

classmethod random(ring=Rational Field, allow_infinity: bool = True, allow_empty: bool = False) → Interval¶

Generate a random interval.

EXAMPLES:

sage: from certlin import *
sage: Interval.random() # random
(-1, 1)
sage: Interval.random(ring=ZZ) # random
[-5, 3]

simplest_element()¶

Return the simplest rational in this interval.

OUTPUT: If possible, an integer with smallest possible absolute value will be returned. Otherwise, a rational number with smallest possible denominator is constructed.

EXAMPLES:

sage: from certlin import *
sage: Interval(1/2, +oo, False, False).simplest_element()
1
sage: Interval(-oo, 1/2, False, False).simplest_element()
0
sage: Interval(-19, 0, False, False).simplest_element()
-1
sage: Interval(0, 1, False, False).simplest_element()
1/2
sage: Interval(-2/3, 0, False, False).simplest_element()
-1/2
sage: Interval(4/3, 3/2, False, False).simplest_element()
7/5
sage: Interval(0, 0).simplest_element()
0
sage: Interval(5, 5).simplest_element()
5
sage: Interval(sqrt(2), pi/2).simplest_element()
3/2
sage: Interval(1/2, 1/2).simplest_element()
1/2

supremum()¶

Return the supremum of the interval.

EXAMPLES:

sage: from certlin import *
sage: Interval(0, 1).supremum()
1
sage: Interval(0, +oo).supremum()
+Infinity
sage: Interval(0, 0, False, False).supremum()
-Infinity

class certlin.intervals.Intervals(intervals: list[Interval])¶

A Cartesian product of intervals.

EXAMPLES:

sage: from certlin import *
sage: Intervals([Interval(0, 1), Interval(-5, 2, False, False), Interval(0, 1)])
[0, 1] x (-5, 2) x [0, 1]
sage: vector([0, 1]) in Intervals([Interval(0, 1), Interval(-5, 2)])
True
sage: Intervals.random(3) # random
[0, +oo) x (-5, 2) x (0, 1]

an_element()¶: Return an element from each interval.

classmethod from_bounds(lower: list, upper: list, lower_closed: bool = True, upper_closed: bool = True) → Intervals¶

Return intervals that are determined by bounds.

EXAMPLES:

sage: from certlin import *
sage: Intervals.from_bounds([0, -5, 0], [1, 2, +oo])
[0, 1] x [-5, 2] x [0, +oo)
sage: Intervals.from_bounds([0, -5, 0], [1, 2, +oo], False, False)
(0, 1) x (-5, 2) x (0, +oo)
sage: Intervals.from_bounds([0, -5, 0], [1, 2, +oo], True, False)
[0, 1) x [-5, 2) x [0, +oo)
sage: Intervals.from_bounds([0, -5, 0], [1, 2, +oo], [True, False, True], [True, True, False])
[0, 1] x (-5, 2] x [0, +oo)

is_bounded() → bool¶: Return whether all intervals are bounded.

is_closed() → bool¶: Return whether all intervals are closed.

is_empty() → bool¶: Return whether the intervals are empty.

is_open() → bool¶: Return whether all intervals are open.

is_pointed() → bool¶: Return whether all intervals are pointed.

is_unbounded() → bool¶: Return whether any interval is unbounded.

classmethod random(length: int, ring=Rational Field) → Intervals¶

Generate a random list of intervals.

EXAMPLES:

sage: from certlin import *
sage: Intervals.random(3) # random
[0, +oo) x (-5, 2) x (0, 1]

simplest_element()¶: Return the simplest element from each interval.

certlin.intervals.getrandbits(k) → x. Generates an int with k random bits.¶