def __new__(cls, sym, condition, base_set=S.UniversalSet):
# nonlinsolve uses ConditionSet to return an unsolved system
# of equations (see _return_conditionset in solveset) so until
# that is changed we do minimal checking of the args
if isinstance(sym, (Tuple, tuple)): # unsolved eqns syntax
sym = Tuple(*sym)
condition = FiniteSet(*condition)
return Basic.__new__(cls, sym, condition, base_set)
condition = as_Boolean(condition)
if isinstance(base_set, set):
base_set = FiniteSet(*base_set)
elif not isinstance(base_set, Set):
raise TypeError('expecting set for base_set')
if condition is S.false:
return S.EmptySet
if condition is S.true:
return base_set
if isinstance(base_set, EmptySet):
return base_set
know = None
if isinstance(base_set, FiniteSet):
sifted = sift(
base_set, lambda _: fuzzy_bool(
condition.subs(sym, _)))
if sifted[None]:
know = FiniteSet(*sifted[True])
base_set = FiniteSet(*sifted[None])
else:
return FiniteSet(*sifted[True])
if isinstance(base_set, cls):
s, c, base_set = base_set.args
if sym == s:
condition = And(condition, c)
elif sym not in c.free_symbols:
condition = And(condition, c.xreplace({s: sym}))
elif s not in condition.free_symbols:
condition = And(condition.xreplace({sym: s}), c)
sym = s
else:
# user will have to use cls.sym to get symbol
dum = Symbol('lambda')
if dum in condition.free_symbols or \
dum in c.free_symbols:
dum = Dummy(str(dum))
condition = And(
condition.xreplace({sym: dum}),
c.xreplace({s: dum}))
sym = dum
if not isinstance(sym, Symbol):
s = Dummy('lambda')
if s not in condition.xreplace({sym: s}).free_symbols:
raise ValueError(
'non-symbol dummy not recognized in condition')
rv = Basic.__new__(cls, sym, condition, base_set)
return rv if know is None else Union(know, rv)
def __new__(cls, sets, polar=False):
from sympy import sin, cos
x, y, r, theta = symbols('x, y, r, theta', cls=Dummy)
I = S.ImaginaryUnit
polar = sympify(polar)
# Rectangular Form
if polar == False:
if all(_a.is_FiniteSet for _a in sets.args) and (len(sets.args) == 2):
# ** ProductSet of FiniteSets in the Complex Plane. **
# For Cases like ComplexRegion({2, 4}*{3}), It
# would return {2 + 3*I, 4 + 3*I}
complex_num = []
for x in sets.args[0]:
for y in sets.args[1]:
complex_num.append(x + I*y)
obj = FiniteSet(*complex_num)
else:
obj = ImageSet.__new__(cls, Lambda((x, y), x + I*y), sets)
obj._variables = (x, y)
obj._expr = x + I*y
# Polar Form
elif polar == True:
new_sets = []
# sets is Union of ProductSets
if not sets.is_ProductSet:
for k in sets.args:
new_sets.append(k)
# sets is ProductSets
else:
new_sets.append(sets)
# Normalize input theta
for k, v in enumerate(new_sets):
from sympy.sets import ProductSet
new_sets[k] = ProductSet(v.args[0],
normalize_theta_set(v.args[1]))
sets = Union(*new_sets)
obj = ImageSet.__new__(cls, Lambda((r, theta),
r*(cos(theta) + I*sin(theta))),
sets)
obj._variables = (r, theta)
obj._expr = r*(cos(theta) + I*sin(theta))
else:
raise ValueError("polar should be either True or False")
obj._sets = sets
obj._polar = polar
return obj
def __new__(cls, *partition):
"""
Generates a new partition object.
This method also verifies if the arguments passed are
valid and raises a ValueError if they are not.
Examples
========
>>> from sympy.combinatorics.partitions import Partition
>>> a = Partition([1, 2], [3])
>>> a
{{3}, {1, 2}}
>>> a.partition
[[1, 2], [3]]
>>> len(a)
2
>>> a.members
(1, 2, 3)
"""
args = partition
if not all(isinstance(part, (list, FiniteSet)) for part in args):
raise ValueError(
"Each argument to Partition should be a list or a FiniteSet")
# sort so we have a canonical reference for RGS
partition = sorted(sum(partition, []), key=default_sort_key)
if has_dups(partition):
raise ValueError("Partition contained duplicated elements.")
obj = FiniteSet.__new__(cls, *[FiniteSet(*x) for x in args])
obj.members = tuple(partition)
obj.size = len(partition)
return obj
def test_ImageSet():
raises(ValueError, lambda: ImageSet(x, S.Integers))
assert ImageSet(Lambda(x, 1), S.Integers) == FiniteSet(1)
assert ImageSet(Lambda(x, y), S.Integers) == {y}
assert ImageSet(Lambda(x, 1), S.EmptySet) == S.EmptySet
empty = Intersection(FiniteSet(log(2) / pi), S.Integers)
assert unchanged(ImageSet, Lambda(x, 1), empty) # issue #17471
squares = ImageSet(Lambda(x, x**2), S.Naturals)
assert 4 in squares
assert 5 not in squares
assert FiniteSet(*range(10)).intersect(squares) == FiniteSet(1, 4, 9)
assert 16 not in squares.intersect(Interval(0, 10))
si = iter(squares)
a, b, c, d = next(si), next(si), next(si), next(si)
assert (a, b, c, d) == (1, 4, 9, 16)
harmonics = ImageSet(Lambda(x, 1 / x), S.Naturals)
assert Rational(1, 5) in harmonics
assert Rational(.25) in harmonics
assert 0.25 not in harmonics
assert Rational(.3) not in harmonics
assert (1, 2) not in harmonics
assert harmonics.is_iterable
assert imageset(x, -x, Interval(0, 1)) == Interval(-1, 0)
assert ImageSet(Lambda(x, x**2), Interval(0, 2)).doit() == Interval(0, 4)
assert ImageSet(Lambda((x, y), 2 * x), {4}, {3}).doit() == FiniteSet(8)
assert (ImageSet(Lambda((x, y), x + y), {1, 2, 3},
{10, 20, 30}).doit() == FiniteSet(11, 12, 13, 21, 22, 23,
31, 32, 33))
c = Interval(1, 3) * Interval(1, 3)
assert Tuple(2, 6) in ImageSet(Lambda(((x, y), ), (x, 2 * y)), c)
assert Tuple(2, S.Half) in ImageSet(Lambda(((x, y), ), (x, 1 / y)), c)
assert Tuple(2, -2) not in ImageSet(Lambda(((x, y), ), (x, y**2)), c)
assert Tuple(2, -2) in ImageSet(Lambda(((x, y), ), (x, -2)), c)
c3 = ProductSet(Interval(3, 7), Interval(8, 11), Interval(5, 9))
assert Tuple(8, 3, 9) in ImageSet(Lambda(((t, y, x), ), (y, t, x)), c3)
assert Tuple(Rational(1, 8), 3,
9) in ImageSet(Lambda(((t, y, x), ), (1 / y, t, x)), c3)
assert 2 / pi not in ImageSet(Lambda(((x, y), ), 2 / x), c)
assert 2 / S(100) not in ImageSet(Lambda(((x, y), ), 2 / x), c)
assert Rational(2, 3) in ImageSet(Lambda(((x, y), ), 2 / x), c)
S1 = imageset(lambda x, y: x + y, S.Integers, S.Naturals)
assert S1.base_pset == ProductSet(S.Integers, S.Naturals)
assert S1.base_sets == (S.Integers, S.Naturals)
# Passing a set instead of a FiniteSet shouldn't raise
assert unchanged(ImageSet, Lambda(x, x**2), {1, 2, 3})
S2 = ImageSet(Lambda(((x, y), ), x + y), {(1, 2), (3, 4)})
assert 3 in S2.doit()
# FIXME: This doesn't yet work:
#assert 3 in S2
assert S2._contains(3) is None
raises(TypeError, lambda: ImageSet(Lambda(x, x**2), 1))
def elements(self):
return FiniteSet(*[frozenset(((self.symbol, elem), )) for elem in self.set])
def _density(self):
return dict((FiniteSet((self.symbol, val)), prob)
for val, prob in self.distribution.dict.items())
def spaces(self):
return FiniteSet(*self.args)
def __new__(cls, symbols, *args):
symbols = FiniteSet(*symbols)
return Basic.__new__(cls, symbols, *args)
def test_Lambda():
e = Lambda(x, x**2)
assert e(4) == 16
assert e(x) == x**2
assert e(y) == y**2
assert Lambda((), 42)() == 42
assert unchanged(Lambda, (), 42)
assert Lambda((), 42) != Lambda((), 43)
assert Lambda((), f(x))() == f(x)
assert Lambda((), 42).nargs == FiniteSet(0)
assert unchanged(Lambda, (x,), x**2)
assert Lambda(x, x**2) == Lambda((x,), x**2)
assert Lambda(x, x**2) != Lambda(x, x**2 + 1)
assert Lambda((x, y), x**y) != Lambda((y, x), y**x)
assert Lambda((x, y), x**y) != Lambda((x, y), y**x)
assert Lambda((x, y), x**y)(x, y) == x**y
assert Lambda((x, y), x**y)(3, 3) == 3**3
assert Lambda((x, y), x**y)(x, 3) == x**3
assert Lambda((x, y), x**y)(3, y) == 3**y
assert Lambda(x, f(x))(x) == f(x)
assert Lambda(x, x**2)(e(x)) == x**4
assert e(e(x)) == x**4
x1, x2 = (Indexed('x', i) for i in (1, 2))
assert Lambda((x1, x2), x1 + x2)(x, y) == x + y
assert Lambda((x, y), x + y).nargs == FiniteSet(2)
p = x, y, z, t
assert Lambda(p, t*(x + y + z))(*p) == t * (x + y + z)
eq = Lambda(x, 2*x) + Lambda(y, 2*y)
assert eq != 2*Lambda(x, 2*x)
assert eq.as_dummy() == 2*Lambda(x, 2*x).as_dummy()
assert Lambda(x, 2*x) not in [ Lambda(x, x) ]
raises(BadSignatureError, lambda: Lambda(1, x))
assert Lambda(x, 1)(1) is S.One
raises(BadSignatureError, lambda: Lambda((x, x), x + 2))
raises(BadSignatureError, lambda: Lambda(((x, x), y), x))
raises(BadSignatureError, lambda: Lambda(((y, x), x), x))
raises(BadSignatureError, lambda: Lambda(((y, 1), 2), x))
with warns_deprecated_sympy():
assert Lambda([x, y], x+y) == Lambda((x, y), x+y)
flam = Lambda(((x, y),), x + y)
assert flam((2, 3)) == 5
flam = Lambda(((x, y), z), x + y + z)
assert flam((2, 3), 1) == 6
flam = Lambda((((x, y), z),), x + y + z)
assert flam(((2, 3), 1)) == 6
raises(BadArgumentsError, lambda: flam(1, 2, 3))
flam = Lambda( (x,), (x, x))
assert flam(1,) == (1, 1)
assert flam((1,)) == ((1,), (1,))
flam = Lambda( ((x,),), (x, x))
raises(BadArgumentsError, lambda: flam(1))
assert flam((1,)) == (1, 1)
# Previously TypeError was raised so this is potentially needed for
# backwards compatibility.
assert issubclass(BadSignatureError, TypeError)
assert issubclass(BadArgumentsError, TypeError)
# These are tested to see they don't raise:
hash(Lambda(x, 2*x))
hash(Lambda(x, x)) # IdentityFunction subclass
def normalize_theta_set(theta):
"""
Normalize a Real Set `theta` in the Interval [0, 2*pi). It returns
a normalized value of theta in the Set. For Interval, a maximum of
one cycle [0, 2*pi], is returned i.e. for theta equal to [0, 10*pi],
returned normalized value would be [0, 2*pi). As of now intervals
with end points as non-multiples of `pi` is not supported.
Raises
======
NotImplementedError
The algorithms for Normalizing theta Set are not yet
implemented.
ValueError
The input is not valid, i.e. the input is not a real set.
RuntimeError
It is a bug, please report to the github issue tracker.
Examples
========
>>> from sympy.sets.fancysets import normalize_theta_set
>>> from sympy import Interval, FiniteSet, pi
>>> normalize_theta_set(Interval(9*pi/2, 5*pi))
Interval(pi/2, pi)
>>> normalize_theta_set(Interval(-3*pi/2, pi/2))
Interval.Ropen(0, 2*pi)
>>> normalize_theta_set(Interval(-pi/2, pi/2))
Union(Interval(0, pi/2), Interval.Ropen(3*pi/2, 2*pi))
>>> normalize_theta_set(Interval(-4*pi, 3*pi))
Interval.Ropen(0, 2*pi)
>>> normalize_theta_set(Interval(-3*pi/2, -pi/2))
Interval(pi/2, 3*pi/2)
>>> normalize_theta_set(FiniteSet(0, pi, 3*pi))
{0, pi}
"""
from sympy.functions.elementary.trigonometric import _pi_coeff as coeff
if theta.is_Interval:
interval_len = theta.measure
# one complete circle
if interval_len >= 2 * S.Pi:
if interval_len == 2 * S.Pi and theta.left_open and theta.right_open:
k = coeff(theta.start)
return Union(Interval(0, k * S.Pi, False, True),
Interval(k * S.Pi, 2 * S.Pi, True, True))
return Interval(0, 2 * S.Pi, False, True)
k_start, k_end = coeff(theta.start), coeff(theta.end)
if k_start is None or k_end is None:
raise NotImplementedError(
"Normalizing theta without pi as coefficient is "
"not yet implemented")
new_start = k_start * S.Pi
new_end = k_end * S.Pi
if new_start > new_end:
return Union(Interval(S.Zero, new_end, False, theta.right_open),
Interval(new_start, 2 * S.Pi, theta.left_open, True))
else:
return Interval(new_start, new_end, theta.left_open,
theta.right_open)
elif theta.is_FiniteSet:
new_theta = []
for element in theta:
k = coeff(element)
if k is None:
raise NotImplementedError('Normalizing theta without pi as '
'coefficient, is not Implemented.')
else:
new_theta.append(k * S.Pi)
return FiniteSet(*new_theta)
elif theta.is_Union:
return Union(
*[normalize_theta_set(interval) for interval in theta.args])
elif theta.is_subset(S.Reals):
raise NotImplementedError(
"Normalizing theta when, it is of type %s is not "
"implemented" % type(theta))
else:
raise ValueError(" %s is not a real set" % (theta))
def __new__(cls, *partition):
"""
Generates a new partition object.
This method also verifies if the arguments passed are
valid and raises a ValueError if they are not.
Examples
========
Creating Partition from Python lists:
>>> from sympy.combinatorics.partitions import Partition
>>> a = Partition([1, 2], [3])
>>> a
Partition(FiniteSet(1, 2), FiniteSet(3))
>>> a.partition
[[1, 2], [3]]
>>> len(a)
2
>>> a.members
(1, 2, 3)
Creating Partition from Python sets:
>>> Partition({1, 2, 3}, {4, 5})
Partition(FiniteSet(1, 2, 3), FiniteSet(4, 5))
Creating Partition from SymPy finite sets:
>>> from sympy.sets.sets import FiniteSet
>>> a = FiniteSet(1, 2, 3)
>>> b = FiniteSet(4, 5)
>>> Partition(a, b)
Partition(FiniteSet(1, 2, 3), FiniteSet(4, 5))
"""
args = []
dups = False
for arg in partition:
if isinstance(arg, list):
as_set = set(arg)
if len(as_set) < len(arg):
dups = True
break # error below
arg = as_set
args.append(_sympify(arg))
if not all(isinstance(part, FiniteSet) for part in args):
raise ValueError(
"Each argument to Partition should be " \
"a list, set, or a FiniteSet")
# sort so we have a canonical reference for RGS
U = Union(*args)
if dups or len(U) < sum(len(arg) for arg in args):
raise ValueError("Partition contained duplicate elements.")
obj = FiniteSet.__new__(cls, *args)
obj.members = tuple(U)
obj.size = len(U)
return obj
def test_normalize_theta_set():
# Interval
assert normalize_theta_set(Interval(pi, 2*pi)) == \
Union(FiniteSet(0), Interval.Ropen(pi, 2*pi))
assert normalize_theta_set(Interval(9 * pi / 2,
5 * pi)) == Interval(pi / 2, pi)
assert normalize_theta_set(Interval(-3 * pi / 2,
pi / 2)) == Interval.Ropen(0, 2 * pi)
assert normalize_theta_set(Interval.open(-3*pi/2, pi/2)) == \
Union(Interval.Ropen(0, pi/2), Interval.open(pi/2, 2*pi))
assert normalize_theta_set(Interval.open(-7*pi/2, -3*pi/2)) == \
Union(Interval.Ropen(0, pi/2), Interval.open(pi/2, 2*pi))
assert normalize_theta_set(Interval(-pi/2, pi/2)) == \
Union(Interval(0, pi/2), Interval.Ropen(3*pi/2, 2*pi))
assert normalize_theta_set(Interval.open(-pi/2, pi/2)) == \
Union(Interval.Ropen(0, pi/2), Interval.open(3*pi/2, 2*pi))
assert normalize_theta_set(Interval(-4 * pi,
3 * pi)) == Interval.Ropen(0, 2 * pi)
assert normalize_theta_set(Interval(-3 * pi / 2, -pi / 2)) == Interval(
pi / 2, 3 * pi / 2)
assert normalize_theta_set(Interval.open(0, 2 * pi)) == Interval.open(
0, 2 * pi)
assert normalize_theta_set(Interval.Ropen(-pi/2, pi/2)) == \
Union(Interval.Ropen(0, pi/2), Interval.Ropen(3*pi/2, 2*pi))
assert normalize_theta_set(Interval.Lopen(-pi/2, pi/2)) == \
Union(Interval(0, pi/2), Interval.open(3*pi/2, 2*pi))
assert normalize_theta_set(Interval(-pi/2, pi/2)) == \
Union(Interval(0, pi/2), Interval.Ropen(3*pi/2, 2*pi))
assert normalize_theta_set(Interval.open(4 * pi,
9 * pi / 2)) == Interval.open(
0, pi / 2)
assert normalize_theta_set(Interval.Lopen(4 * pi,
9 * pi / 2)) == Interval.Lopen(
0, pi / 2)
assert normalize_theta_set(Interval.Ropen(4 * pi,
9 * pi / 2)) == Interval.Ropen(
0, pi / 2)
assert normalize_theta_set(Interval.open(3*pi, 5*pi)) == \
Union(Interval.Ropen(0, pi), Interval.open(pi, 2*pi))
# FiniteSet
assert normalize_theta_set(FiniteSet(0, pi, 3 * pi)) == FiniteSet(0, pi)
assert normalize_theta_set(FiniteSet(0, pi / 2, pi,
2 * pi)) == FiniteSet(0, pi / 2, pi)
assert normalize_theta_set(FiniteSet(0, -pi / 2, -pi,
-2 * pi)) == FiniteSet(
0, pi, 3 * pi / 2)
assert normalize_theta_set(FiniteSet(-3*pi/2, pi/2)) == \
FiniteSet(pi/2)
assert normalize_theta_set(FiniteSet(2 * pi)) == FiniteSet(0)
# Unions
assert normalize_theta_set(Union(Interval(0, pi/3), Interval(pi/2, pi))) == \
Union(Interval(0, pi/3), Interval(pi/2, pi))
assert normalize_theta_set(Union(Interval(0, pi), Interval(2*pi, 7*pi/3))) == \
Interval(0, pi)
# ValueError for non-real sets
raises(ValueError, lambda: normalize_theta_set(S.Complexes))
def __pow__(self, other):
from sympy.functions.elementary.miscellaneous import real_root
if isinstance(other, Expr):
if other is S.Infinity:
if self.min.is_nonnegative:
if self.max < 1:
return S.Zero
if self.min > 1:
return S.Infinity
return AccumBounds(0, oo)
elif self.max.is_negative:
if self.min > -1:
return S.Zero
if self.max < -1:
return FiniteSet(-oo, oo)
return AccumBounds(-oo, oo)
else:
if self.min > -1:
if self.max < 1:
return S.Zero
return AccumBounds(0, oo)
return AccumBounds(-oo, oo)
if other is S.NegativeInfinity:
return (1 / self)**oo
if other.is_real and other.is_number:
if other.is_zero:
return S.One
if other.is_Integer:
if self.min.is_positive:
return AccumBounds(
Min(self.min**other, self.max**other),
Max(self.min**other, self.max**other))
elif self.max.is_negative:
return AccumBounds(
Min(self.max**other, self.min**other),
Max(self.max**other, self.min**other))
if other % 2 == 0:
if other.is_negative:
if self.min.is_zero:
return AccumBounds(self.max**other, oo)
if self.max.is_zero:
return AccumBounds(self.min**other, oo)
return AccumBounds(0, oo)
return AccumBounds(
S.Zero, Max(self.min**other, self.max**other))
else:
if other.is_negative:
if self.min.is_zero:
return AccumBounds(self.max**other, oo)
if self.max.is_zero:
return AccumBounds(-oo, self.min**other)
return AccumBounds(-oo, oo)
return AccumBounds(self.min**other, self.max**other)
num, den = other.as_numer_denom()
if num == S(1):
if den % 2 == 0:
if S.Zero in self:
if self.min.is_negative:
return AccumBounds(0, real_root(self.max, den))
return AccumBounds(real_root(self.min, den),
real_root(self.max, den))
num_pow = self**num
return num_pow**(1 / den)
return Pow(self, other, evaluate=False)
return NotImplemented
def function_range(f, symbol, domain):
"""
Finds the range of a function in a given domain.
This method is limited by the ability to determine the singularities and
determine limits.
Examples
========
>>> from sympy import Symbol, S, exp, log, pi, sqrt, sin, tan
>>> from sympy.sets import Interval
>>> from sympy.calculus.util import function_range
>>> x = Symbol('x')
>>> function_range(sin(x), x, Interval(0, 2*pi))
[-1, 1]
>>> function_range(tan(x), x, Interval(-pi/2, pi/2))
(-oo, oo)
>>> function_range(1/x, x, S.Reals)
(-oo, oo)
>>> function_range(exp(x), x, S.Reals)
(0, oo)
>>> function_range(log(x), x, S.Reals)
(-oo, oo)
>>> function_range(sqrt(x), x , Interval(-5, 9))
[0, 3]
"""
from sympy.solvers.solveset import solveset
vals = S.EmptySet
intervals = continuous_domain(f, symbol, domain)
range_int = S.EmptySet
if isinstance(intervals, Interval):
interval_iter = (intervals, )
else:
interval_iter = intervals.args
for interval in interval_iter:
critical_points = S.EmptySet
critical_values = S.EmptySet
bounds = ((interval.left_open, interval.inf, '+'),
(interval.right_open, interval.sup, '-'))
for is_open, limit_point, direction in bounds:
if is_open:
critical_values += FiniteSet(
limit(f, symbol, limit_point, direction))
vals += critical_values
else:
vals += FiniteSet(f.subs(symbol, limit_point))
critical_points += solveset(f.diff(symbol), symbol, domain)
for critical_point in critical_points:
vals += FiniteSet(f.subs(symbol, critical_point))
left_open, right_open = False, False
if critical_values is not S.EmptySet:
if critical_values.inf == vals.inf:
left_open = True
if critical_values.sup == vals.sup:
right_open = True
range_int += Interval(vals.inf, vals.sup, left_open, right_open)
return range_int
def __new__(cls, symbol, set):
if not isinstance(set, FiniteSet) and \
not isinstance(set, Intersection):
set = FiniteSet(*set)
return Basic.__new__(cls, symbol, set)
def dict(self):
return FiniteSet(*[Dict(dict(el)) for el in self.elements])
def symbols(self):
return FiniteSet(sym for sym, val in self.elements)
def test_ComplexRegion_from_real():
c1 = ComplexRegion(Interval(0, 1) * Interval(0, 2 * S.Pi), polar=True)
raises(ValueError, lambda: c1.from_real(c1))
assert c1.from_real(Interval(-1, 1)) == ComplexRegion(
Interval(-1, 1) * FiniteSet(0), False)
def test_not_empty_in():
from sympy.abc import x
a = Symbol('a', real=True)
assert not_empty_in(FiniteSet(x, 2*x).intersect(Interval(1, 2, True, False)), x) == \
Interval(S(1)/2, 2, True, False)
assert not_empty_in(FiniteSet(x, x**2).intersect(Interval(1, 2)), x) == \
Union(Interval(-sqrt(2), -1), Interval(1, 2))
assert not_empty_in(FiniteSet(x**2 + x, x).intersect(Interval(2, 4)), x) == \
Union(Interval(-sqrt(17)/2 - S(1)/2, -2), Interval(1, -S(1)/2 + sqrt(17)/2), Interval(2, 4))
assert not_empty_in(FiniteSet(x/(x - 1)).intersect(S.Reals), x) == Complement(S.Reals, FiniteSet(1))
assert not_empty_in(FiniteSet(a/(a - 1)).intersect(S.Reals), a) == Complement(S.Reals, FiniteSet(1))
assert not_empty_in(FiniteSet((x**2 - 3*x + 2)/(x - 1)).intersect(S.Reals), x) == \
Complement(S.Reals, FiniteSet(1))
assert not_empty_in(FiniteSet(3, 4, x/(x - 1)).intersect(Interval(2, 3)), x) == \
Union(Interval(S(3)/2, 2), FiniteSet(3))
assert not_empty_in(FiniteSet(x/(x**2 - 1)).intersect(S.Reals), x) == \
Complement(S.Reals, FiniteSet(-1, 1))
assert not_empty_in(FiniteSet(x, x**2).intersect(Union(Interval(1, 3, True, True), Interval(4, 5))), x) == \
Union(Interval(-sqrt(5), -2), Interval(-sqrt(3), -1, True, True), Interval(1, 3, True, True), Interval(4, 5))
assert not_empty_in(FiniteSet(1).intersect(Interval(3, 4)), x) == S.EmptySet
assert not_empty_in(FiniteSet(x**2/(x + 2)).intersect(Interval(1, oo)), x) == \
Union(Interval(-2, -1, True, False), Interval(2, oo))
def test_normalize_theta_set():
# Interval
assert normalize_theta_set(Interval(pi, 2*pi)) == \
Union(FiniteSet(0), Interval.Ropen(pi, 2*pi))
assert normalize_theta_set(Interval(pi * Rational(9, 2),
5 * pi)) == Interval(pi / 2, pi)
assert normalize_theta_set(Interval(pi * Rational(-3, 2),
pi / 2)) == Interval.Ropen(0, 2 * pi)
assert normalize_theta_set(Interval.open(pi*Rational(-3, 2), pi/2)) == \
Union(Interval.Ropen(0, pi/2), Interval.open(pi/2, 2*pi))
assert normalize_theta_set(Interval.open(pi*Rational(-7, 2), pi*Rational(-3, 2))) == \
Union(Interval.Ropen(0, pi/2), Interval.open(pi/2, 2*pi))
assert normalize_theta_set(Interval(-pi/2, pi/2)) == \
Union(Interval(0, pi/2), Interval.Ropen(pi*Rational(3, 2), 2*pi))
assert normalize_theta_set(Interval.open(-pi/2, pi/2)) == \
Union(Interval.Ropen(0, pi/2), Interval.open(pi*Rational(3, 2), 2*pi))
assert normalize_theta_set(Interval(-4 * pi,
3 * pi)) == Interval.Ropen(0, 2 * pi)
assert normalize_theta_set(Interval(pi * Rational(-3, 2),
-pi / 2)) == Interval(
pi / 2, pi * Rational(3, 2))
assert normalize_theta_set(Interval.open(0, 2 * pi)) == Interval.open(
0, 2 * pi)
assert normalize_theta_set(Interval.Ropen(-pi/2, pi/2)) == \
Union(Interval.Ropen(0, pi/2), Interval.Ropen(pi*Rational(3, 2), 2*pi))
assert normalize_theta_set(Interval.Lopen(-pi/2, pi/2)) == \
Union(Interval(0, pi/2), Interval.open(pi*Rational(3, 2), 2*pi))
assert normalize_theta_set(Interval(-pi/2, pi/2)) == \
Union(Interval(0, pi/2), Interval.Ropen(pi*Rational(3, 2), 2*pi))
assert normalize_theta_set(Interval.open(
4 * pi, pi * Rational(9, 2))) == Interval.open(0, pi / 2)
assert normalize_theta_set(Interval.Lopen(
4 * pi, pi * Rational(9, 2))) == Interval.Lopen(0, pi / 2)
assert normalize_theta_set(Interval.Ropen(
4 * pi, pi * Rational(9, 2))) == Interval.Ropen(0, pi / 2)
assert normalize_theta_set(Interval.open(3*pi, 5*pi)) == \
Union(Interval.Ropen(0, pi), Interval.open(pi, 2*pi))
# FiniteSet
assert normalize_theta_set(FiniteSet(0, pi, 3 * pi)) == FiniteSet(0, pi)
assert normalize_theta_set(FiniteSet(0, pi / 2, pi,
2 * pi)) == FiniteSet(0, pi / 2, pi)
assert normalize_theta_set(FiniteSet(0, -pi / 2, -pi,
-2 * pi)) == FiniteSet(
0, pi, pi * Rational(3, 2))
assert normalize_theta_set(FiniteSet(pi*Rational(-3, 2), pi/2)) == \
FiniteSet(pi/2)
assert normalize_theta_set(FiniteSet(2 * pi)) == FiniteSet(0)
# Unions
assert normalize_theta_set(Union(Interval(0, pi/3), Interval(pi/2, pi))) == \
Union(Interval(0, pi/3), Interval(pi/2, pi))
assert normalize_theta_set(Union(Interval(0, pi), Interval(2*pi, pi*Rational(7, 3)))) == \
Interval(0, pi)
# ValueError for non-real sets
raises(ValueError, lambda: normalize_theta_set(S.Complexes))
# NotImplementedError for subset of reals
raises(NotImplementedError, lambda: normalize_theta_set(Interval(0, 1)))
# NotImplementedError without pi as coefficient
raises(NotImplementedError,
lambda: normalize_theta_set(Interval(1, 2 * pi)))
raises(NotImplementedError,
lambda: normalize_theta_set(Interval(2 * pi, 10)))
raises(NotImplementedError,
lambda: normalize_theta_set(FiniteSet(0, 3, 3 * pi)))
def test_codomain():
x = Symbol('x', real=True)
assert codomain(x, Interval(-1, 1), x) == Interval(-1, 1)
assert codomain(x, Interval(0, 1, True, True), x) == \
Interval(0, 1, True, True)
assert codomain(x, Interval(1, 2, True, False), x) == Interval(1, 2, True, False)
assert codomain(x, Interval(1, 2, False, True), x) == Interval(1, 2, False, True)
assert codomain(x**2, Interval(-1, 1), x) == Interval(0, 1)
assert codomain(x**3, Interval(0, 1), x) == Interval(0, 1)
assert codomain(x/(x**2 - 4), Interval(3, 4), x) == Interval(S(1)/3, S(3)/5)
assert codomain(1, Interval(-1, 4), x) == FiniteSet(1)
assert codomain(x, Interval(-oo, oo), x) == S.Reals
assert codomain(1/x**2, FiniteSet(1, 2, -1, 0), x) == FiniteSet(1, S(1)/4)
assert codomain(x, FiniteSet(1, -1, 3, 5), x) == FiniteSet(-1, 1, 3, 5)
assert codomain(x**2 - x, FiniteSet(1, -1, 3, 5, -oo), x) == \
FiniteSet(0, 2, 6, 20, oo)
assert codomain(x**2/(x - 4), FiniteSet(4), x) == S.EmptySet
assert codomain(x**2 - x, FiniteSet(S(1)/2, -oo, oo, 2), x) == \
FiniteSet(S(-1)/4, 2, oo)
assert codomain(x**2, Interval(-1, 1, True, True), x) == Interval(0, 1, False, True)
assert codomain(x**2, Interval(-1, 1, False, True), x) == Interval(0, 1)
assert codomain(x**2, Interval(-1, 1, True, False), x) == Interval(0, 1)
assert codomain(1/x, Interval(0, 1), x) == Interval(1, oo)
assert codomain(1/x, Interval(-1, 1), x) == Union(Interval(-oo, -1), Interval(1, oo))
assert codomain(1/x**2, Interval(-1, 1), x) == Interval(1, oo)
assert codomain(1/x**2, Interval(-1, 1, True, False), x) == Interval(1, oo)
assert codomain(1/x**2, Interval(-1, 1, True, True), x) == \
Interval(1, oo, True, True)
assert codomain(1/x**2, Interval(-1, 1, False, True), x) == Interval(1, oo)
assert codomain(1/x, Interval(1, 2), x) == Interval(S(1)/2, 1)
assert codomain(1/x**2, Interval(-2, -1, True, True), x) == \
Interval(S(1)/4, 1, True, True)
assert codomain(x**2/(x - 4), Interval(-oo, oo), x) == \
Complement(S.Reals, Interval(0, 16, True, True))
assert codomain(x**2/(x - 4), Interval(3, 4), x) == Interval(-oo, -9)
assert codomain(-x**2/(x - 4), Interval(3, 4), x) == Interval(9, oo)
assert codomain((x**2 - x)/(x**3 - 1), S.Reals, x) == Interval(-1, S(1)/3, False, True)
assert codomain(-x**2 + 1/x, S.Reals, x) == S.Reals
assert codomain(x**2 - 1/x, S.Reals, x) == S.Reals
assert codomain(x**2, Union(Interval(1, 2), FiniteSet(3)), x) == \
Union(Interval(1, 4), FiniteSet(9))
assert codomain(x/(x**2 - 4), Union(Interval(-oo, 1), Interval(0, oo)), x) == S.Reals
assert codomain(x, Union(Interval(-1, 1), FiniteSet(-oo)), x) == \
Union(Interval(-1, 1), FiniteSet(-oo))
assert codomain(x**2 - x, Interval(1, oo), x) == Interval(0, oo)
raises(ValueError, lambda: codomain(sqrt(x), Interval(-1, 2), x))
def _set_pow(x, z): # noqa:F811
return FiniteSet(S.One)
def test_issue_16871():
assert ImageSet(Lambda(x, x), FiniteSet(1)) == {1}
assert ImageSet(Lambda(x, x - 3), S.Integers).intersection(
S.Integers) is S.Integers
def symbols(self):
return FiniteSet(*[val.symbol for val in self.rs_space_dict.keys()])
def solve_univariate_inequality(expr,
gen,
relational=True,
domain=S.Reals,
continuous=False):
"""Solves a real univariate inequality.
Parameters
==========
expr : Relational
The target inequality
gen : Symbol
The variable for which the inequality is solved
relational : bool
A Relational type output is expected or not
domain : Set
The domain over which the equation is solved
continuous: bool
True if expr is known to be continuous over the given domain
(and so continuous_domain() doesn't need to be called on it)
Raises
======
NotImplementedError
The solution of the inequality cannot be determined due to limitation
in :func:`sympy.solvers.solveset.solvify`.
Notes
=====
Currently, we cannot solve all the inequalities due to limitations in
:func:`sympy.solvers.solveset.solvify`. Also, the solution returned for trigonometric inequalities
are restricted in its periodic interval.
See Also
========
sympy.solvers.solveset.solvify: solver returning solveset solutions with solve's output API
Examples
========
>>> from sympy.solvers.inequalities import solve_univariate_inequality
>>> from sympy import Symbol, sin, Interval, S
>>> x = Symbol('x')
>>> solve_univariate_inequality(x**2 >= 4, x)
((2 <= x) & (x < oo)) | ((x <= -2) & (-oo < x))
>>> solve_univariate_inequality(x**2 >= 4, x, relational=False)
Union(Interval(-oo, -2), Interval(2, oo))
>>> domain = Interval(0, S.Infinity)
>>> solve_univariate_inequality(x**2 >= 4, x, False, domain)
Interval(2, oo)
>>> solve_univariate_inequality(sin(x) > 0, x, relational=False)
Interval.open(0, pi)
"""
from sympy import im
from sympy.calculus.util import continuous_domain, periodicity, function_range
from sympy.solvers.solvers import denoms
from sympy.solvers.solveset import solvify, solveset
# This keeps the function independent of the assumptions about `gen`.
# `solveset` makes sure this function is called only when the domain is
# real.
_gen = gen
_domain = domain
if gen.is_extended_real is False:
rv = S.EmptySet
return rv if not relational else rv.as_relational(_gen)
elif gen.is_extended_real is None:
gen = Dummy("gen", extended_real=True)
try:
expr = expr.xreplace({_gen: gen})
except TypeError:
raise TypeError(
filldedent("""
When gen is real, the relational has a complex part
which leads to an invalid comparison like I < 0.
"""))
rv = None
if expr is S.true:
rv = domain
elif expr is S.false:
rv = S.EmptySet
else:
e = expr.lhs - expr.rhs
period = periodicity(e, gen)
if period == S.Zero:
e = expand_mul(e)
const = expr.func(e, 0)
if const is S.true:
rv = domain
elif const is S.false:
rv = S.EmptySet
elif period is not None:
frange = function_range(e, gen, domain)
rel = expr.rel_op
if rel == "<" or rel == "<=":
if expr.func(frange.sup, 0):
rv = domain
elif not expr.func(frange.inf, 0):
rv = S.EmptySet
elif rel == ">" or rel == ">=":
if expr.func(frange.inf, 0):
rv = domain
elif not expr.func(frange.sup, 0):
rv = S.EmptySet
inf, sup = domain.inf, domain.sup
if sup - inf is S.Infinity:
domain = Interval(0, period, False, True)
if rv is None:
n, d = e.as_numer_denom()
try:
if gen not in n.free_symbols and len(e.free_symbols) > 1:
raise ValueError
# this might raise ValueError on its own
# or it might give None...
solns = solvify(e, gen, domain)
if solns is None:
# in which case we raise ValueError
raise ValueError
except (ValueError, NotImplementedError):
# replace gen with generic x since it's
# univariate anyway
raise NotImplementedError(
filldedent("""
The inequality, %s, cannot be solved using
solve_univariate_inequality.
""" % expr.subs(gen, Symbol("x"))))
expanded_e = expand_mul(e)
def valid(x):
# this is used to see if gen=x satisfies the
# relational by substituting it into the
# expanded form and testing against 0, e.g.
# if expr = x*(x + 1) < 2 then e = x*(x + 1) - 2
# and expanded_e = x**2 + x - 2; the test is
# whether a given value of x satisfies
# x**2 + x - 2 < 0
#
# expanded_e, expr and gen used from enclosing scope
v = expanded_e.subs(gen, expand_mul(x))
try:
r = expr.func(v, 0)
except TypeError:
r = S.false
if r in (S.true, S.false):
return r
if v.is_extended_real is False:
return S.false
else:
v = v.n(2)
if v.is_comparable:
return expr.func(v, 0)
# not comparable or couldn't be evaluated
raise NotImplementedError(
"relationship did not evaluate: %s" % r)
singularities = []
for d in denoms(expr, gen):
singularities.extend(solvify(d, gen, domain))
if not continuous:
domain = continuous_domain(expanded_e, gen, domain)
include_x = "=" in expr.rel_op and expr.rel_op != "!="
try:
discontinuities = set(domain.boundary -
FiniteSet(domain.inf, domain.sup))
# remove points that are not between inf and sup of domain
critical_points = FiniteSet(
*(solns + singularities +
list(discontinuities))).intersection(
Interval(
domain.inf,
domain.sup,
domain.inf not in domain,
domain.sup not in domain,
))
if all(r.is_number for r in critical_points):
reals = _nsort(critical_points, separated=True)[0]
else:
sifted = sift(critical_points,
lambda x: x.is_extended_real)
if sifted[None]:
# there were some roots that weren't known
# to be real
raise NotImplementedError
try:
reals = sifted[True]
if len(reals) > 1:
reals = list(sorted(reals))
except TypeError:
raise NotImplementedError
except NotImplementedError:
raise NotImplementedError(
"sorting of these roots is not supported")
# If expr contains imaginary coefficients, only take real
# values of x for which the imaginary part is 0
make_real = S.Reals
if im(expanded_e) != S.Zero:
check = True
im_sol = FiniteSet()
try:
a = solveset(im(expanded_e), gen, domain)
if not isinstance(a, Interval):
for z in a:
if (z not in singularities and valid(z)
and z.is_extended_real):
im_sol += FiniteSet(z)
else:
start, end = a.inf, a.sup
for z in _nsort(critical_points + FiniteSet(end)):
valid_start = valid(start)
if start != end:
valid_z = valid(z)
pt = _pt(start, z)
if (pt not in singularities
and pt.is_extended_real and valid(pt)):
if valid_start and valid_z:
im_sol += Interval(start, z)
elif valid_start:
im_sol += Interval.Ropen(start, z)
elif valid_z:
im_sol += Interval.Lopen(start, z)
else:
im_sol += Interval.open(start, z)
start = z
for s in singularities:
im_sol -= FiniteSet(s)
except (TypeError):
im_sol = S.Reals
check = False
if isinstance(im_sol, EmptySet):
raise ValueError(
filldedent("""
%s contains imaginary parts which cannot be
made 0 for any value of %s satisfying the
inequality, leading to relations like I < 0.
""" % (expr.subs(gen, _gen), _gen)))
make_real = make_real.intersect(im_sol)
sol_sets = [S.EmptySet]
start = domain.inf
if valid(start) and start.is_finite:
sol_sets.append(FiniteSet(start))
for x in reals:
end = x
if valid(_pt(start, end)):
sol_sets.append(Interval(start, end, True, True))
if x in singularities:
singularities.remove(x)
else:
if x in discontinuities:
discontinuities.remove(x)
_valid = valid(x)
else: # it's a solution
_valid = include_x
if _valid:
sol_sets.append(FiniteSet(x))
start = end
end = domain.sup
if valid(end) and end.is_finite:
sol_sets.append(FiniteSet(end))
if valid(_pt(start, end)):
sol_sets.append(Interval.open(start, end))
if im(expanded_e) != S.Zero and check:
rv = (make_real).intersect(_domain)
else:
rv = Intersection((Union(*sol_sets)), make_real,
_domain).subs(gen, _gen)
return rv if not relational else rv.as_relational(_gen)
def symbols(self):
return FiniteSet(
*[sym for domain in self.domains for sym in domain.symbols])
def _intersect(self, other):
from sympy.solvers.diophantine import diophantine
if self.base_set is S.Integers:
g = None
if isinstance(other, ImageSet) and other.base_set is S.Integers:
g = other.lamda.expr
m = other.lamda.variables[0]
elif other is S.Integers:
m = g = Dummy('x')
if g is not None:
f = self.lamda.expr
n = self.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:
return
# since 'a' < 'b', select soln for n
nsol = solns[0][0]
t = nsol.free_symbols.pop()
return imageset(Lambda(n, f.subs(a, nsol.subs(t, n))),
S.Integers)
if other == S.Reals:
from sympy.solvers.solveset import solveset_real
from sympy.core.function import expand_complex
if len(self.lamda.variables) > 1:
return None
f = self.lamda.expr
n = self.lamda.variables[0]
n_ = Dummy(n.name, real=True)
f_ = f.subs(n, n_)
re, im = f_.as_real_imag()
im = expand_complex(im)
return imageset(Lambda(n_, re),
self.base_set.intersect(solveset_real(im, n_)))
elif isinstance(other, Interval):
from sympy.solvers.solveset import (invert_real, invert_complex,
solveset)
f = self.lamda.expr
n = self.lamda.variables[0]
base_set = self.base_set
new_inf, new_sup = None, None
if f.is_real:
inverter = invert_real
else:
inverter = invert_complex
g1, h1 = inverter(f, other.inf, n)
g2, h2 = inverter(f, other.sup, n)
if all(isinstance(i, FiniteSet) for i in (h1, h2)):
if g1 == n:
if len(h1) == 1:
new_inf = h1.args[0]
if g2 == n:
if len(h2) == 1:
new_sup = h2.args[0]
# TODO: Design a technique to handle multiple-inverse
# functions
# Any of the new boundary values cannot be determined
if any(i is None for i in (new_sup, new_inf)):
return
range_set = S.EmptySet
if all(i.is_real for i in (new_sup, new_inf)):
new_interval = Interval(new_inf, new_sup)
range_set = base_set._intersect(new_interval)
else:
if other.is_subset(S.Reals):
solutions = solveset(f, n, S.Reals)
if not isinstance(range_set, (ImageSet, ConditionSet)):
range_set = solutions._intersect(other)
else:
return
if range_set is S.EmptySet:
return S.EmptySet
elif isinstance(range_set,
Range) and range_set.size is not S.Infinity:
range_set = FiniteSet(*list(range_set))
if range_set is not None:
return imageset(Lambda(n, f), range_set)
return
else:
return
def symbols(self):
return FiniteSet(self.symbol)
def test_fun():
assert (FiniteSet(
*ImageSet(Lambda(x, sin(pi * x / 4)), Range(-10, 11))) == FiniteSet(
-1, -sqrt(2) / 2, 0,
sqrt(2) / 2, 1))
def test_Range_set():
empty = Range(0)
assert Range(5) == Range(0, 5) == Range(0, 5, 1)
r = Range(10, 20, 2)
assert 12 in r
assert 8 not in r
assert 11 not in r
assert 30 not in r
assert list(Range(0, 5)) == list(range(5))
assert list(Range(5, 0, -1)) == list(range(5, 0, -1))
assert Range(5, 15).sup == 14
assert Range(5, 15).inf == 5
assert Range(15, 5, -1).sup == 15
assert Range(15, 5, -1).inf == 6
assert Range(10, 67, 10).sup == 60
assert Range(60, 7, -10).inf == 10
assert len(Range(10, 38, 10)) == 3
assert Range(0, 0, 5) == empty
assert Range(oo, oo, 1) == empty
assert Range(oo, 1, 1) == empty
assert Range(-oo, 1, -1) == empty
assert Range(1, oo, -1) == empty
assert Range(1, -oo, 1) == empty
assert Range(1, -4, oo) == empty
assert Range(1, -4, -oo) == Range(1, 2)
assert Range(1, 4, oo) == Range(1, 2)
assert Range(-oo, oo).size == oo
assert Range(oo, -oo, -1).size == oo
raises(ValueError, lambda: Range(-oo, oo, 2))
raises(ValueError, lambda: Range(x, pi, y))
raises(ValueError, lambda: Range(x, y, 0))
assert 5 in Range(0, oo, 5)
assert -5 in Range(-oo, 0, 5)
assert oo not in Range(0, oo)
ni = symbols('ni', integer=False)
assert ni not in Range(oo)
u = symbols('u', integer=None)
assert Range(oo).contains(u) is not False
inf = symbols('inf', infinite=True)
assert inf not in Range(-oo, oo)
raises(ValueError, lambda: Range(0, oo, 2)[-1])
raises(ValueError, lambda: Range(0, -oo, -2)[-1])
assert Range(-oo, 1, 1)[-1] is S.Zero
assert Range(oo, 1, -1)[-1] == 2
assert inf not in Range(oo)
inf = symbols('inf', infinite=True)
assert inf not in Range(oo)
assert Range(-oo, 1, 1)[-1] is S.Zero
assert Range(oo, 1, -1)[-1] == 2
assert Range(1, 10, 1)[-1] == 9
assert all(i.is_Integer for i in Range(0, -1, 1))
it = iter(Range(-oo, 0, 2))
raises(TypeError, lambda: next(it))
assert empty.intersect(S.Integers) == empty
assert Range(-1, 10, 1).intersect(S.Integers) == Range(-1, 10, 1)
assert Range(-1, 10, 1).intersect(S.Naturals) == Range(1, 10, 1)
assert Range(-1, 10, 1).intersect(S.Naturals0) == Range(0, 10, 1)
# test slicing
assert Range(1, 10, 1)[5] == 6
assert Range(1, 12, 2)[5] == 11
assert Range(1, 10, 1)[-1] == 9
assert Range(1, 10, 3)[-1] == 7
raises(ValueError, lambda: Range(oo, 0, -1)[1:3:0])
raises(ValueError, lambda: Range(oo, 0, -1)[:1])
raises(ValueError, lambda: Range(1, oo)[-2])
raises(ValueError, lambda: Range(-oo, 1)[2])
raises(IndexError, lambda: Range(10)[-20])
raises(IndexError, lambda: Range(10)[20])
raises(ValueError, lambda: Range(2, -oo, -2)[2:2:0])
assert Range(2, -oo, -2)[2:2:2] == empty
assert Range(2, -oo, -2)[:2:2] == Range(2, -2, -4)
raises(ValueError, lambda: Range(-oo, 4, 2)[:2:2])
assert Range(-oo, 4, 2)[::-2] == Range(2, -oo, -4)
raises(ValueError, lambda: Range(-oo, 4, 2)[::2])
assert Range(oo, 2, -2)[::] == Range(oo, 2, -2)
assert Range(-oo, 4, 2)[:-2:-2] == Range(2, 0, -4)
assert Range(-oo, 4, 2)[:-2:2] == Range(-oo, 0, 4)
raises(ValueError, lambda: Range(-oo, 4, 2)[:0:-2])
raises(ValueError, lambda: Range(-oo, 4, 2)[:2:-2])
assert Range(-oo, 4, 2)[-2::-2] == Range(0, -oo, -4)
raises(ValueError, lambda: Range(-oo, 4, 2)[-2:0:-2])
raises(ValueError, lambda: Range(-oo, 4, 2)[0::2])
assert Range(oo, 2, -2)[0::] == Range(oo, 2, -2)
raises(ValueError, lambda: Range(-oo, 4, 2)[0:-2:2])
assert Range(oo, 2, -2)[0:-2:] == Range(oo, 6, -2)
raises(ValueError, lambda: Range(oo, 2, -2)[0:2:])
raises(ValueError, lambda: Range(-oo, 4, 2)[2::-1])
assert Range(-oo, 4, 2)[-2::2] == Range(0, 4, 4)
assert Range(oo, 0, -2)[-10:0:2] == empty
raises(ValueError, lambda: Range(oo, 0, -2)[-10:10:2])
raises(ValueError, lambda: Range(oo, 0, -2)[0::-2])
assert Range(oo, 0, -2)[0:-4:-2] == empty
assert Range(oo, 0, -2)[:0:2] == empty
raises(ValueError, lambda: Range(oo, 0, -2)[:1:-1])
# test empty Range
assert Range(x, x, y) == empty
assert empty.reversed == empty
assert 0 not in empty
assert list(empty) == []
assert len(empty) == 0
assert empty.size is S.Zero
assert empty.intersect(FiniteSet(0)) is S.EmptySet
assert bool(empty) is False
raises(IndexError, lambda: empty[0])
assert empty[:0] == empty
raises(NotImplementedError, lambda: empty.inf)
raises(NotImplementedError, lambda: empty.sup)
AB = [None] + list(range(12))
for R in [
Range(1, 10),
Range(1, 10, 2),
]:
r = list(R)
for a, b, c in cartes(AB, AB, [-3, -1, None, 1, 3]):
for reverse in range(2):
r = list(reversed(r))
R = R.reversed
result = list(R[a:b:c])
ans = r[a:b:c]
txt = ('\n%s[%s:%s:%s] = %s -> %s' % (R, a, b, c, result, ans))
check = ans == result
assert check, txt
assert Range(1, 10, 1).boundary == Range(1, 10, 1)
for r in (Range(1, 10, 2), Range(1, oo, 2)):
rev = r.reversed
assert r.inf == rev.inf and r.sup == rev.sup
assert r.step == -rev.step
builtin_range = range
raises(TypeError, lambda: Range(builtin_range(1)))
assert S(builtin_range(10)) == Range(10)
assert S(builtin_range(1000000000000)) == Range(1000000000000)
# test Range.as_relational
assert Range(1,
4).as_relational(x) == (x >= 1) & (x <= 3) & Eq(x, floor(x))
assert Range(oo, 1,
-2).as_relational(x) == (x >= 3) & (x < oo) & Eq(x, floor(x))
def elements(self):
return FiniteSet(*self)
def test_Range_symbolic():
# symbolic Range
sr = Range(x, y, t)
i = Symbol('i', integer=True)
ip = Symbol('i', integer=True, positive=True)
ir = Range(i, i + 20, 2)
inf = symbols('inf', infinite=True)
# args
assert sr.args == (x, y, t)
assert ir.args == (i, i + 20, 2)
# reversed
raises(ValueError, lambda: sr.reversed)
assert ir.reversed == Range(i + 18, i - 2, -2)
# contains
assert inf not in sr
assert inf not in ir
assert .1 not in sr
assert .1 not in ir
assert i + 1 not in ir
assert i + 2 in ir
raises(TypeError,
lambda: 1 in sr) # XXX is this what contains is supposed to do?
# iter
raises(ValueError, lambda: next(iter(sr)))
assert next(iter(ir)) == i
assert sr.intersect(S.Integers) == sr
assert sr.intersect(FiniteSet(x)) == Intersection({x}, sr)
raises(ValueError, lambda: sr[:2])
raises(ValueError, lambda: sr[0])
raises(ValueError, lambda: sr.as_relational(x))
# len
assert len(ir) == ir.size == 10
raises(ValueError, lambda: len(sr))
raises(ValueError, lambda: sr.size)
# bool
assert bool(ir) == bool(sr) == True
# getitem
raises(ValueError, lambda: sr[0])
raises(ValueError, lambda: sr[-1])
raises(ValueError, lambda: sr[:2])
assert ir[:2] == Range(i, i + 4, 2)
assert ir[0] == i
assert ir[-2] == i + 16
assert ir[-1] == i + 18
raises(ValueError, lambda: Range(i)[-1])
assert Range(ip)[-1] == ip - 1
assert ir.inf == i
assert ir.sup == i + 18
assert Range(ip).inf == 0
assert Range(ip).sup == ip - 1
raises(ValueError, lambda: Range(i).inf)
# as_relational
raises(ValueError, lambda: sr.as_relational(x))
assert ir.as_relational(x) == (x >= i) & Eq(x, floor(x)) & (x <= i + 18)
assert Range(i, i + 1).as_relational(x) == Eq(x, i)
# contains() for symbolic values (issue #18146)
e = Symbol('e', integer=True, even=True)
o = Symbol('o', integer=True, odd=True)
assert Range(5).contains(i) == And(i >= 0, i <= 4)
assert Range(1).contains(i) == Eq(i, 0)
assert Range(-oo, 5, 1).contains(i) == (i <= 4)
assert Range(-oo, oo).contains(i) == True
assert Range(0, 8, 2).contains(i) == Contains(i, Range(0, 8, 2))
assert Range(0, 8, 2).contains(e) == And(e >= 0, e <= 6)
assert Range(0, 8, 2).contains(2 * i) == And(2 * i >= 0, 2 * i <= 6)
assert Range(0, 8, 2).contains(o) == False
assert Range(1, 9, 2).contains(e) == False
assert Range(1, 9, 2).contains(o) == And(o >= 1, o <= 7)
assert Range(8, 0, -2).contains(o) == False
assert Range(9, 1, -2).contains(o) == And(o >= 3, o <= 9)
assert Range(-oo, 8, 2).contains(i) == Contains(i, Range(-oo, 8, 2))
def test_DiagramGrid():
# Set up some objects and morphisms.
A = Object("A")
B = Object("B")
C = Object("C")
D = Object("D")
E = Object("E")
f = NamedMorphism(A, B, "f")
g = NamedMorphism(B, C, "g")
h = NamedMorphism(D, A, "h")
k = NamedMorphism(D, B, "k")
# A one-morphism diagram.
d = Diagram([f])
grid = DiagramGrid(d)
assert grid.width == 2
assert grid.height == 1
assert grid[0, 0] == A
assert grid[0, 1] == B
assert grid.morphisms == {f: FiniteSet()}
# A triangle.
d = Diagram([f, g], {g * f: "unique"})
grid = DiagramGrid(d)
assert grid.width == 2
assert grid.height == 2
assert grid[0, 0] == A
assert grid[0, 1] == B
assert grid[1, 0] == C
assert grid[1, 1] is None
assert grid.morphisms == {f: FiniteSet(), g: FiniteSet(),
g * f: FiniteSet("unique")}
# A triangle with a "loop" morphism.
l_A = NamedMorphism(A, A, "l_A")
d = Diagram([f, g, l_A])
grid = DiagramGrid(d)
assert grid.width == 2
assert grid.height == 2
assert grid[0, 0] == A
assert grid[0, 1] == B
assert grid[1, 0] is None
assert grid[1, 1] == C
assert grid.morphisms == {f: FiniteSet(), g: FiniteSet(), l_A: FiniteSet()}
# A simple diagram.
d = Diagram([f, g, h, k])
grid = DiagramGrid(d)
assert grid.width == 3
assert grid.height == 2
assert grid[0, 0] == A
assert grid[0, 1] == B
assert grid[0, 2] == D
assert grid[1, 0] is None
assert grid[1, 1] == C
assert grid[1, 2] is None
assert grid.morphisms == {f: FiniteSet(), g: FiniteSet(), h: FiniteSet(),
k: FiniteSet()}
assert str(grid) == '[[Object("A"), Object("B"), Object("D")], ' \
'[None, Object("C"), None]]'
# A chain of morphisms.
f = NamedMorphism(A, B, "f")
g = NamedMorphism(B, C, "g")
h = NamedMorphism(C, D, "h")
k = NamedMorphism(D, E, "k")
d = Diagram([f, g, h, k])
grid = DiagramGrid(d)
assert grid.width == 3
assert grid.height == 3
assert grid[0, 0] == A
assert grid[0, 1] == B
assert grid[0, 2] is None
assert grid[1, 0] is None
assert grid[1, 1] == C
assert grid[1, 2] == D
assert grid[2, 0] is None
assert grid[2, 1] is None
assert grid[2, 2] == E
assert grid.morphisms == {f: FiniteSet(), g: FiniteSet(), h: FiniteSet(),
k: FiniteSet()}
# A square.
f = NamedMorphism(A, B, "f")
g = NamedMorphism(B, D, "g")
h = NamedMorphism(A, C, "h")
k = NamedMorphism(C, D, "k")
d = Diagram([f, g, h, k])
grid = DiagramGrid(d)
assert grid.width == 2
assert grid.height == 2
assert grid[0, 0] == A
assert grid[0, 1] == B
assert grid[1, 0] == C
assert grid[1, 1] == D
assert grid.morphisms == {f: FiniteSet(), g: FiniteSet(), h: FiniteSet(),
k: FiniteSet()}
# A strange diagram which resulted from a typo when creating a
# test for five lemma, but which allowed to stop one extra problem
# in the algorithm.
A = Object("A")
B = Object("B")
C = Object("C")
D = Object("D")
E = Object("E")
A_ = Object("A'")
B_ = Object("B'")
C_ = Object("C'")
D_ = Object("D'")
E_ = Object("E'")
f = NamedMorphism(A, B, "f")
g = NamedMorphism(B, C, "g")
h = NamedMorphism(C, D, "h")
i = NamedMorphism(D, E, "i")
# These 4 morphisms should be between primed objects.
j = NamedMorphism(A, B, "j")
k = NamedMorphism(B, C, "k")
l = NamedMorphism(C, D, "l")
m = NamedMorphism(D, E, "m")
o = NamedMorphism(A, A_, "o")
p = NamedMorphism(B, B_, "p")
q = NamedMorphism(C, C_, "q")
r = NamedMorphism(D, D_, "r")
s = NamedMorphism(E, E_, "s")
d = Diagram([f, g, h, i, j, k, l, m, o, p, q, r, s])
grid = DiagramGrid(d)
assert grid.width == 3
assert grid.height == 4
assert grid[0, 0] is None
assert grid[0, 1] == A
assert grid[0, 2] == A_
assert grid[1, 0] == C
assert grid[1, 1] == B
assert grid[1, 2] == B_
assert grid[2, 0] == C_
assert grid[2, 1] == D
assert grid[2, 2] == D_
assert grid[3, 0] is None
assert grid[3, 1] == E
assert grid[3, 2] == E_
morphisms = {}
for m in [f, g, h, i, j, k, l, m, o, p, q, r, s]:
morphisms[m] = FiniteSet()
assert grid.morphisms == morphisms
# A cube.
A1 = Object("A1")
A2 = Object("A2")
A3 = Object("A3")
A4 = Object("A4")
A5 = Object("A5")
A6 = Object("A6")
A7 = Object("A7")
A8 = Object("A8")
# The top face of the cube.
f1 = NamedMorphism(A1, A2, "f1")
f2 = NamedMorphism(A1, A3, "f2")
f3 = NamedMorphism(A2, A4, "f3")
f4 = NamedMorphism(A3, A4, "f3")
# The bottom face of the cube.
f5 = NamedMorphism(A5, A6, "f5")
f6 = NamedMorphism(A5, A7, "f6")
f7 = NamedMorphism(A6, A8, "f7")
f8 = NamedMorphism(A7, A8, "f8")
# The remaining morphisms.
f9 = NamedMorphism(A1, A5, "f9")
f10 = NamedMorphism(A2, A6, "f10")
f11 = NamedMorphism(A3, A7, "f11")
f12 = NamedMorphism(A4, A8, "f11")
d = Diagram([f1, f2, f3, f4, f5, f6, f7, f8, f9, f10, f11, f12])
grid = DiagramGrid(d)
assert grid.width == 4
assert grid.height == 3
assert grid[0, 0] is None
assert grid[0, 1] == A5
assert grid[0, 2] == A6
assert grid[0, 3] is None
assert grid[1, 0] is None
assert grid[1, 1] == A1
assert grid[1, 2] == A2
assert grid[1, 3] is None
assert grid[2, 0] == A7
assert grid[2, 1] == A3
assert grid[2, 2] == A4
assert grid[2, 3] == A8
morphisms = {}
for m in [f1, f2, f3, f4, f5, f6, f7, f8, f9, f10, f11, f12]:
morphisms[m] = FiniteSet()
assert grid.morphisms == morphisms
# A line diagram.
A = Object("A")
B = Object("B")
C = Object("C")
D = Object("D")
E = Object("E")
f = NamedMorphism(A, B, "f")
g = NamedMorphism(B, C, "g")
h = NamedMorphism(C, D, "h")
i = NamedMorphism(D, E, "i")
d = Diagram([f, g, h, i])
grid = DiagramGrid(d, layout="sequential")
assert grid.width == 5
assert grid.height == 1
assert grid[0, 0] == A
assert grid[0, 1] == B
assert grid[0, 2] == C
assert grid[0, 3] == D
assert grid[0, 4] == E
assert grid.morphisms == {f: FiniteSet(), g: FiniteSet(), h: FiniteSet(),
i: FiniteSet()}
# Test the transposed version.
grid = DiagramGrid(d, layout="sequential", transpose=True)
assert grid.width == 1
assert grid.height == 5
assert grid[0, 0] == A
assert grid[1, 0] == B
assert grid[2, 0] == C
assert grid[3, 0] == D
assert grid[4, 0] == E
assert grid.morphisms == {f: FiniteSet(), g: FiniteSet(), h: FiniteSet(),
i: FiniteSet()}
# A pullback.
m1 = NamedMorphism(A, B, "m1")
m2 = NamedMorphism(A, C, "m2")
s1 = NamedMorphism(B, D, "s1")
s2 = NamedMorphism(C, D, "s2")
f1 = NamedMorphism(E, B, "f1")
f2 = NamedMorphism(E, C, "f2")
g = NamedMorphism(E, A, "g")
d = Diagram([m1, m2, s1, s2, f1, f2], {g: "unique"})
grid = DiagramGrid(d)
assert grid.width == 3
assert grid.height == 2
assert grid[0, 0] == A
assert grid[0, 1] == B
assert grid[0, 2] == E
assert grid[1, 0] == C
assert grid[1, 1] == D
assert grid[1, 2] is None
morphisms = {g: FiniteSet("unique")}
for m in [m1, m2, s1, s2, f1, f2]:
morphisms[m] = FiniteSet()
assert grid.morphisms == morphisms
# Test the pullback with sequential layout, just for stress
# testing.
grid = DiagramGrid(d, layout="sequential")
assert grid.width == 5
assert grid.height == 1
assert grid[0, 0] == D
assert grid[0, 1] == B
assert grid[0, 2] == A
assert grid[0, 3] == C
assert grid[0, 4] == E
assert grid.morphisms == morphisms
# Test a pullback with object grouping.
grid = DiagramGrid(d, groups=FiniteSet(E, FiniteSet(A, B, C, D)))
assert grid.width == 3
assert grid.height == 2
assert grid[0, 0] == E
assert grid[0, 1] == A
assert grid[0, 2] == B
assert grid[1, 0] is None
assert grid[1, 1] == C
assert grid[1, 2] == D
assert grid.morphisms == morphisms
# Five lemma, actually.
A = Object("A")
B = Object("B")
C = Object("C")
D = Object("D")
E = Object("E")
A_ = Object("A'")
B_ = Object("B'")
C_ = Object("C'")
D_ = Object("D'")
E_ = Object("E'")
f = NamedMorphism(A, B, "f")
g = NamedMorphism(B, C, "g")
h = NamedMorphism(C, D, "h")
i = NamedMorphism(D, E, "i")
j = NamedMorphism(A_, B_, "j")
k = NamedMorphism(B_, C_, "k")
l = NamedMorphism(C_, D_, "l")
m = NamedMorphism(D_, E_, "m")
o = NamedMorphism(A, A_, "o")
p = NamedMorphism(B, B_, "p")
q = NamedMorphism(C, C_, "q")
r = NamedMorphism(D, D_, "r")
s = NamedMorphism(E, E_, "s")
d = Diagram([f, g, h, i, j, k, l, m, o, p, q, r, s])
grid = DiagramGrid(d)
assert grid.width == 5
assert grid.height == 3
assert grid[0, 0] is None
assert grid[0, 1] == A
assert grid[0, 2] == A_
assert grid[0, 3] is None
assert grid[0, 4] is None
assert grid[1, 0] == C
assert grid[1, 1] == B
assert grid[1, 2] == B_
assert grid[1, 3] == C_
assert grid[1, 4] is None
assert grid[2, 0] == D
assert grid[2, 1] == E
assert grid[2, 2] is None
assert grid[2, 3] == D_
assert grid[2, 4] == E_
morphisms = {}
for m in [f, g, h, i, j, k, l, m, o, p, q, r, s]:
morphisms[m] = FiniteSet()
assert grid.morphisms == morphisms
# Test the five lemma with object grouping.
grid = DiagramGrid(d, FiniteSet(
FiniteSet(A, B, C, D, E), FiniteSet(A_, B_, C_, D_, E_)))
assert grid.width == 6
assert grid.height == 3
assert grid[0, 0] == A
assert grid[0, 1] == B
assert grid[0, 2] is None
assert grid[0, 3] == A_
assert grid[0, 4] == B_
assert grid[0, 5] is None
assert grid[1, 0] is None
assert grid[1, 1] == C
assert grid[1, 2] == D
assert grid[1, 3] is None
assert grid[1, 4] == C_
assert grid[1, 5] == D_
assert grid[2, 0] is None
assert grid[2, 1] is None
assert grid[2, 2] == E
assert grid[2, 3] is None
assert grid[2, 4] is None
assert grid[2, 5] == E_
assert grid.morphisms == morphisms
# Test the five lemma with object grouping, but mixing containers
# to represent groups.
grid = DiagramGrid(d, [(A, B, C, D, E), {A_, B_, C_, D_, E_}])
assert grid.width == 6
assert grid.height == 3
assert grid[0, 0] == A
assert grid[0, 1] == B
assert grid[0, 2] is None
assert grid[0, 3] == A_
assert grid[0, 4] == B_
assert grid[0, 5] is None
assert grid[1, 0] is None
assert grid[1, 1] == C
assert grid[1, 2] == D
assert grid[1, 3] is None
assert grid[1, 4] == C_
assert grid[1, 5] == D_
assert grid[2, 0] is None
assert grid[2, 1] is None
assert grid[2, 2] == E
assert grid[2, 3] is None
assert grid[2, 4] is None
assert grid[2, 5] == E_
assert grid.morphisms == morphisms
# Test the five lemma with object grouping and hints.
grid = DiagramGrid(d, {
FiniteSet(A, B, C, D, E): {"layout": "sequential",
"transpose": True},
FiniteSet(A_, B_, C_, D_, E_): {"layout": "sequential",
"transpose": True}},
transpose=True)
assert grid.width == 5
assert grid.height == 2
assert grid[0, 0] == A
assert grid[0, 1] == B
assert grid[0, 2] == C
assert grid[0, 3] == D
assert grid[0, 4] == E
assert grid[1, 0] == A_
assert grid[1, 1] == B_
assert grid[1, 2] == C_
assert grid[1, 3] == D_
assert grid[1, 4] == E_
assert grid.morphisms == morphisms
# A two-triangle disconnected diagram.
f = NamedMorphism(A, B, "f")
g = NamedMorphism(B, C, "g")
f_ = NamedMorphism(A_, B_, "f")
g_ = NamedMorphism(B_, C_, "g")
d = Diagram([f, g, f_, g_], {g * f: "unique", g_ * f_: "unique"})
grid = DiagramGrid(d)
assert grid.width == 4
assert grid.height == 2
assert grid[0, 0] == A
assert grid[0, 1] == B
assert grid[0, 2] == A_
assert grid[0, 3] == B_
assert grid[1, 0] == C
assert grid[1, 1] is None
assert grid[1, 2] == C_
assert grid[1, 3] is None
assert grid.morphisms == {f: FiniteSet(), g: FiniteSet(), f_: FiniteSet(),
g_: FiniteSet(), g * f: FiniteSet("unique"),
g_ * f_: FiniteSet("unique")}
# A two-morphism disconnected diagram.
f = NamedMorphism(A, B, "f")
g = NamedMorphism(C, D, "g")
d = Diagram([f, g])
grid = DiagramGrid(d)
assert grid.width == 4
assert grid.height == 1
assert grid[0, 0] == A
assert grid[0, 1] == B
assert grid[0, 2] == C
assert grid[0, 3] == D
assert grid.morphisms == {f: FiniteSet(), g: FiniteSet()}
# Test a one-object diagram.
f = NamedMorphism(A, A, "f")
d = Diagram([f])
grid = DiagramGrid(d)
assert grid.width == 1
assert grid.height == 1
assert grid[0, 0] == A
# Test a two-object disconnected diagram.
g = NamedMorphism(B, B, "g")
d = Diagram([f, g])
grid = DiagramGrid(d)
assert grid.width == 2
assert grid.height == 1
assert grid[0, 0] == A
assert grid[0, 1] == B