from sympy import (Dummy, S, symbols, Lambda, pi, Basic, sympify, ask, Q, Min,
Max)
from sympy.functions.elementary.integers import floor, ceiling
from sympy.functions.elementary.complexes import sign
from sympy.core.compatibility import iterable, as_int
from sympy.core.sets import Set, Interval, FiniteSet, Intersection
from sympy.core.singleton import Singleton, S
from sympy.solvers import solve
oo = S.Infinity
[docs]class Naturals(Set):
"""
Represents the Natural Numbers. The Naturals are available as a singleton
as S.Naturals
Examples
========
>>> from sympy import S, Interval, pprint
>>> 5 in S.Naturals
True
>>> iterable = iter(S.Naturals)
>>> print iterable.next()
1
>>> print iterable.next()
2
>>> print iterable.next()
3
>>> pprint(S.Naturals.intersect(Interval(0, 10)))
{1, 2, ..., 10}
"""
__metaclass__ = Singleton
is_iterable = True
def _intersect(self, other):
if other.is_Interval:
return Intersection(S.Integers, other, Interval(1, oo))
return None
def _contains(self, other):
if ask(Q.positive(other)) and ask(Q.integer(other)):
return True
return False
def __iter__(self):
i = S(1)
while True:
yield i
i = i + 1
@property
def _inf(self):
return S.One
@property
def _sup(self):
return oo
[docs]class Integers(Set):
"""
Represents the Integers. The Integers are available as a singleton
as S.Integers
Examples
========
>>> from sympy import S, Interval, pprint
>>> 5 in S.Naturals
True
>>> iterable = iter(S.Integers)
>>> print iterable.next()
0
>>> print iterable.next()
1
>>> print iterable.next()
-1
>>> print iterable.next()
2
>>> pprint(S.Integers.intersect(Interval(-4, 4)))
{-4, -3, ..., 4}
"""
__metaclass__ = Singleton
is_iterable = True
def _intersect(self, other):
if other.is_Interval and other.measure < oo:
s = Range(ceiling(other.left), floor(other.right) + 1)
return s.intersect(other) # take out endpoints if open interval
return None
def _contains(self, other):
if ask(Q.integer(other)):
return True
return False
def __iter__(self):
yield S.Zero
i = S(1)
while True:
yield i
yield -i
i = i + 1
@property
def _inf(self):
return -oo
@property
def _sup(self):
return oo
class Reals(Interval):
__metaclass__ = Singleton
def __new__(cls):
return Interval.__new__(cls, -oo, oo)
class Range(Set):
"""
Represents a range of integers.
Examples
========
>>> from sympy import Range
>>> list(Range(5)) # 0 to 5
[0, 1, 2, 3, 4]
>>> list(Range(10, 15)) # 10 to 15
[10, 11, 12, 13, 14]
>>> list(Range(10, 20, 2)) # 10 to 20 in steps of 2
[10, 12, 14, 16, 18]
>>> list(Range(20, 10, -2)) # 20 to 10 backward in steps of 2
[12, 14, 16, 18, 20]
"""
is_iterable = True
def __new__(cls, *args):
# expand range
slc = slice(*args)
start, stop, step = slc.start or 0, slc.stop, slc.step or 1
try:
start, stop, step = [S(as_int(w)) for w in (start, stop, step)]
except ValueError:
raise ValueError("Inputs to Range must be Integer Valued\n" +
"Use TransformationSets of Ranges for other cases")
n = ceiling((stop - start)/step)
if n <= 0:
return S.EmptySet
# normalize args: regardless of how they are entered they will show
# canonically as Range(inf, sup, step) with step > 0
start, stop = sorted((start, start + (n - 1)*step))
step = abs(step)
return Basic.__new__(cls, start, stop + step, step)
start = property(lambda self: self.args[0])
stop = property(lambda self: self.args[1])
step = property(lambda self: self.args[2])
def _intersect(self, other):
if other.is_Interval:
osup = other.sup
oinf = other.inf
# if other is [0, 10) we can only go up to 9
if osup.is_integer and other.right_open:
osup -= 1
if oinf.is_integer and other.left_open:
oinf += 1
# Take the most restrictive of the bounds set by the two sets
# round inwards
inf = ceiling(Max(self.inf, oinf))
sup = floor(Min(self.sup, osup))
# if we are off the sequence, get back on
off = (inf - self.inf) % self.step
if off:
inf += self.step - off
return Range(inf, sup + 1, self.step)
if other == S.Naturals:
return self._intersect(Interval(1, oo))
if other == S.Integers:
return self
return None
def _contains(self, other):
return (other >= self.inf and other <= self.sup and
ask(Q.integer((self.start - other)/self.step)))
def __iter__(self):
i = self.start
while(i < self.stop):
yield i
i = i + self.step
def __len__(self):
return ((self.stop - self.start)//self.step)
def _ith_element(self, i):
return self.start + i*self.step
@property
def _last_element(self):
return self._ith_element(len(self) - 1)
@property
def _inf(self):
return self.start
@property
def _sup(self):
return self.stop - self.step