Source code for sympy.sets.fancysets

from __future__ import print_function, division

from sympy.core.basic import Basic
from sympy.core.compatibility import as_int, with_metaclass
from sympy.sets.sets import Set, Interval, Intersection, \
FiniteSet, Union, Complement, EmptySet
from sympy.core.singleton import Singleton, S
from sympy.core.symbol import symbols
from sympy.core.sympify import sympify, _sympify
from sympy.core.decorators import deprecated
from sympy.core.function import Lambda

[docs]class Naturals(with_metaclass(Singleton, Set)):
"""
Represents the natural numbers (or counting numbers) which are all
positive integers starting from 1. This set is also available as
the Singleton, S.Naturals.

Examples
========

>>> from sympy import S, Interval, pprint
>>> 5 in S.Naturals
True
>>> iterable = iter(S.Naturals)
>>> next(iterable)
1
>>> next(iterable)
2
>>> next(iterable)
3
>>> pprint(S.Naturals.intersect(Interval(0, 10)))
{1, 2, ..., 10}

========
Naturals0 : non-negative integers (i.e. includes 0, too)
Integers : also includes negative integers
"""

is_iterable = True
_inf = S.One
_sup = S.Infinity

def _intersect(self, other):
if other.is_Interval:
return Intersection(
S.Integers, other, Interval(self._inf, S.Infinity))
return None

def _contains(self, other):
if other.is_positive and other.is_integer:
return S.true
elif other.is_integer is False or other.is_positive is False:
return S.false

def __iter__(self):
i = self._inf
while True:
yield i
i = i + 1

@property
def _boundary(self):
return self

[docs]class Naturals0(Naturals):
"""Represents the whole numbers which are all the non-negative integers,
inclusive of zero.

========
Naturals : positive integers; does not include 0
Integers : also includes the negative integers
"""
_inf = S.Zero

def _contains(self, other):
if other.is_integer and other.is_nonnegative:
return S.true
elif other.is_integer is False or other.is_nonnegative is False:
return S.false

[docs]class Integers(with_metaclass(Singleton, Set)):
"""
Represents all integers: positive, negative and zero. This set is also
available as the Singleton, S.Integers.

Examples
========

>>> from sympy import S, Interval, pprint
>>> 5 in S.Naturals
True
>>> iterable = iter(S.Integers)
>>> next(iterable)
0
>>> next(iterable)
1
>>> next(iterable)
-1
>>> next(iterable)
2

>>> pprint(S.Integers.intersect(Interval(-4, 4)))
{-4, -3, ..., 4}

========
Naturals0 : non-negative integers
Integers : positive and negative integers and zero
"""

is_iterable = True

def _intersect(self, other):
from sympy.functions.elementary.integers import floor, ceiling
if other is Interval(S.NegativeInfinity, S.Infinity) or other is S.Reals:
return self
elif other.is_Interval:
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 other.is_integer:
return S.true
elif other.is_integer is False:
return S.false

def __iter__(self):
yield S.Zero
i = S(1)
while True:
yield i
yield -i
i = i + 1

@property
def _inf(self):
return -S.Infinity

@property
def _sup(self):
return S.Infinity

@property
def _boundary(self):
return self

def _eval_imageset(self, f):
from sympy import Wild
expr = f.expr
if len(f.variables) > 1:
return
n = f.variables[0]

a = Wild('a')
b = Wild('b')

match = expr.match(a*n + b)
if match[a].is_negative:
expr = -expr

match = expr.match(a*n + b)
if match[a] is S.One and match[b].is_integer:
expr = expr - match[b]

return ImageSet(Lambda(n, expr), S.Integers)

class Reals(with_metaclass(Singleton, Interval)):

def __new__(cls):
return Interval.__new__(cls, -S.Infinity, S.Infinity)

[docs]class ImageSet(Set):
"""
Image of a set under a mathematical function

Examples
========

>>> from sympy import Symbol, S, ImageSet, FiniteSet, Lambda

>>> x = Symbol('x')
>>> N = S.Naturals
>>> squares = ImageSet(Lambda(x, x**2), N) # {x**2 for x in N}
>>> 4 in squares
True
>>> 5 in squares
False

>>> FiniteSet(0, 1, 2, 3, 4, 5, 6, 7, 9, 10).intersect(squares)
{1, 4, 9}

>>> square_iterable = iter(squares)
>>> for i in range(4):
...     next(square_iterable)
1
4
9
16
"""
def __new__(cls, lamda, base_set):
return Basic.__new__(cls, lamda, base_set)

lamda = property(lambda self: self.args[0])
base_set = property(lambda self: self.args[1])

def __iter__(self):
for i in self.base_set:
val = self.lamda(i)
continue
else:
yield val

def _is_multivariate(self):
return len(self.lamda.variables) > 1

def _contains(self, other):
from sympy.solvers import solve
L = self.lamda
if self._is_multivariate():
solns = solve([expr - val for val, expr in zip(other, L.expr)],
L.variables)
else:
solns = solve(L.expr - other, L.variables[0])

for soln in solns:
try:
if soln in self.base_set:
return S.true
except TypeError:
if soln.evalf() in self.base_set:
return S.true
return S.false

@property
def is_iterable(self):
return self.base_set.is_iterable

def _intersect(self, other):
from sympy import Dummy
from sympy.solvers.diophantine import diophantine
from sympy.sets.sets import imageset
if self.base_set is S.Integers:
if isinstance(other, ImageSet) and other.base_set is S.Integers:
f, g = self.lamda.expr, other.lamda.expr
n, m = self.lamda.variables[0], other.lamda.variables[0]

# Diophantine sorts the solutions according to the alphabetic
# order of the variable names, since the result should not depend
# on the variable name, they are replaced by the dummy variables
# below
a, b = Dummy('a'), Dummy('b')
f, g = f.subs(n, a), g.subs(m, b)
solns_set = diophantine(f - g)
if solns_set == set():
return EmptySet()
solns = list(diophantine(f - g))
if len(solns) == 1:
t = list(solns[0][0].free_symbols)[0]
else:
return None

# since 'a' < 'b'
return imageset(Lambda(t, f.subs(a, solns[0][0])), S.Integers)

def TransformationSet(*args, **kwargs):
"""Deprecated alias for the ImageSet constructor."""
return ImageSet(*args, **kwargs)

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):
from sympy.functions.elementary.integers import ceiling
# expand range
slc = slice(*args)
start, stop, step = slc.start or 0, slc.stop, slc.step or 1
try:
start, stop, step = [w if w in [S.NegativeInfinity, S.Infinity] else S(as_int(w))
for w in (start, stop, step)]
except ValueError:
raise ValueError("Inputs to Range must be Integer Valued\n" +
"Use ImageSets of Ranges for other cases")

if not step.is_finite:
raise ValueError("Infinite step is not allowed")
if start == stop:
return S.EmptySet

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
if n.is_finite:
start, stop = sorted((start, start + (n - 1)*step))
else:
start, stop = sorted((start, stop - step))

step = abs(step)
if (start, stop) == (S.NegativeInfinity, S.Infinity):
raise ValueError("Both the start and end value of "
"Range cannot be unbounded")
else:
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):
from sympy.functions.elementary.integers import floor, ceiling
from sympy.functions.elementary.miscellaneous import Min, Max
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
if inf.is_finite and self.inf.is_finite:
off = (inf - self.inf) % self.step
else:
off = S.Zero
if off:
inf += self.step - off

return Range(inf, sup + 1, self.step)

if other == S.Naturals:
return self._intersect(Interval(1, S.Infinity))

if other == S.Integers:
return self

return None

def _contains(self, other):
if (((self.start - other)/self.step).is_integer or
((self.stop - other)/self.step).is_integer):
return _sympify(other >= self.inf and other <= self.sup)
elif (((self.start - other)/self.step).is_integer is False and
((self.stop - other)/self.step).is_integer is False):
return S.false

def __iter__(self):
if self.start is S.NegativeInfinity:
i = self.stop - self.step
step = -self.step
else:
i = self.start
step = self.step

while(i < self.stop and i >= self.start):
yield i
i += step

def __len__(self):
return (self.stop - self.start)//self.step

def __nonzero__(self):
return True

__bool__ = __nonzero__

def _ith_element(self, i):
return self.start + i*self.step

@property
def _last_element(self):
if self.stop is S.Infinity:
return S.Infinity
elif self.start is S.NegativeInfinity:
return self.stop - self.step
else:
return self._ith_element(len(self) - 1)

@property
def _inf(self):
return self.start

@property
def _sup(self):
return self.stop - self.step

@property
def _boundary(self):
return self