def test_SymmetricDifference():
assert str(SymmetricDifference(Interval(2, 3), Interval(3, 4), evaluate=False)) == \
'SymmetricDifference([2, 3], [3, 4])'
def test_Intersection_as_relational():
assert (Intersection(Interval(0, 1), FiniteSet(2),
evaluate=False).as_relational(x) ==
And(And(Le(0, x), Le(x, 1)), Eq(x, 2)))
def test_product_basic():
H, T = 'H', 'T'
unit_line = Interval(0, 1)
d6 = FiniteSet(1, 2, 3, 4, 5, 6)
d4 = FiniteSet(1, 2, 3, 4)
coin = FiniteSet(H, T)
square = unit_line * unit_line
assert (0, 0) in square
assert 0 not in square
assert (H, T) in coin ** 2
assert (.5, .5, .5) in square * unit_line
assert (H, 3, 3) in coin*d6*d6
HH, TT = sympify(H), sympify(T)
assert set(coin**2) == {(HH, HH), (HH, TT), (TT, HH), (TT, TT)}
assert (d4*d4).is_subset(d6*d6)
assert (square.complement(Interval(-oo, oo)*Interval(-oo, oo)) ==
Union((Interval(-oo, 0, True, True) +
Interval(1, oo, True, True))*Interval(-oo, oo),
Interval(-oo, oo)*(Interval(-oo, 0, True, True) +
Interval(1, oo, True, True))))
assert (Interval(-5, 5)**3).is_subset(Interval(-10, 10)**3)
assert not (Interval(-10, 10)**3).is_subset(Interval(-5, 5)**3)
assert not (Interval(-5, 5)**2).is_subset(Interval(-10, 10)**3)
assert (Interval(.2, .5)*FiniteSet(.5)).is_subset(square) # segment in square
assert len(coin*coin*coin) == 8
assert len(S.EmptySet*S.EmptySet) == 0
assert len(S.EmptySet*coin) == 0
pytest.raises(TypeError, lambda: len(coin*Interval(0, 2)))
def test_Interval_is_right_unbounded():
assert Interval(3, 4).is_right_unbounded is False
assert Interval(3, oo).is_right_unbounded is True
assert Interval(3, Float("+inf")).is_right_unbounded is True
def test_union():
assert Union(Interval(1, 2), Interval(2, 3)) == Interval(1, 3)
assert Union(Interval(1, 2), Interval(2, 3, True)) == Interval(1, 3)
assert Union(Interval(1, 3), Interval(2, 4)) == Interval(1, 4)
assert Union(Interval(1, 2), Interval(1, 3)) == Interval(1, 3)
assert Union(Interval(1, 3), Interval(1, 2)) == Interval(1, 3)
assert Union(Interval(1, 3, False, True), Interval(1, 2)) == \
Interval(1, 3, False, True)
assert Union(Interval(1, 3), Interval(1, 2, False, True)) == Interval(1, 3)
assert Union(Interval(1, 2, True), Interval(1, 3)) == Interval(1, 3)
assert Union(Interval(1, 2, True), Interval(1, 3, True)) == \
Interval(1, 3, True)
assert Union(Interval(1, 2, True), Interval(1, 3, True, True)) == \
Interval(1, 3, True, True)
assert Union(Interval(1, 2, True, True), Interval(1, 3, True)) == \
Interval(1, 3, True)
assert Union(Interval(1, 3), Interval(2, 3)) == Interval(1, 3)
assert Union(Interval(1, 3, False, True), Interval(2, 3)) == \
Interval(1, 3)
assert Union(Interval(1, 2, False, True), Interval(2, 3, True)) != \
Interval(1, 3)
assert Union(Interval(1, 2), S.EmptySet) == Interval(1, 2)
assert Union(S.EmptySet) == S.EmptySet
assert Union(Interval(0, 1), [FiniteSet(1.0/n) for n in range(1, 10)]) == \
Interval(0, 1)
assert Interval(1, 2).union(Interval(2, 3)) == \
Interval(1, 2) + Interval(2, 3)
assert Interval(1, 2).union(Interval(2, 3)) == Interval(1, 3)
assert Union(Set()) == Set()
assert FiniteSet(1) + FiniteSet(2) + FiniteSet(3) == FiniteSet(1, 2, 3)
assert FiniteSet('ham') + FiniteSet('eggs') == FiniteSet('ham', 'eggs')
assert FiniteSet(1, 2, 3) + S.EmptySet == FiniteSet(1, 2, 3)
assert FiniteSet(1, 2, 3) & FiniteSet(2, 3, 4) == FiniteSet(2, 3)
assert FiniteSet(1, 2, 3) | FiniteSet(2, 3, 4) == FiniteSet(1, 2, 3, 4)
assert S.EmptySet | FiniteSet(x, FiniteSet(y, z)) == \
FiniteSet(x, FiniteSet(y, z))
# Test that Intervals and FiniteSets play nicely
assert Interval(1, 3) + FiniteSet(2) == Interval(1, 3)
assert Interval(1, 3, True, True) + FiniteSet(3) == \
Interval(1, 3, True, False)
X = Interval(1, 3) + FiniteSet(5)
Y = Interval(1, 2) + FiniteSet(3)
XandY = X.intersection(Y)
assert 2 in X and 3 in X and 3 in XandY
assert XandY.is_subset(X) and XandY.is_subset(Y)
pytest.raises(TypeError, lambda: Union(1, 2, 3))
assert X.is_iterable is False
Z = Union(FiniteSet(1, 2)*FiniteSet(3, 4), FiniteSet(1, 2, 3, 4))
assert Z.is_iterable
# issue sympy/sympy#7843
assert Union(S.EmptySet, FiniteSet(-sqrt(-I), sqrt(-I))) == FiniteSet(-sqrt(-I), sqrt(-I))
assert Union(ProductSet(FiniteSet(1), FiniteSet(2)), Interval(0, 1)).is_Union
assert Union(ProductSet(FiniteSet(1), FiniteSet(2)),
ProductSet(FiniteSet(1), FiniteSet(2), FiniteSet(3))).is_Union
assert list(Union(FiniteSet(1, 2), FiniteSet(3, 4), evaluate=False)) == [1, 3, 2, 4]
pytest.raises(TypeError, lambda: iter(Union(FiniteSet(1, 2), Interval(0, 1))))
assert (Union(FiniteSet(E), FiniteSet(pi), evaluate=False).evalf() ==
FiniteSet(Float('2.7182818284590451', prec=15),
Float('3.1415926535897931', prec=15)))
def test_contains():
assert Interval(0, 2).contains(1) is S.true
assert Interval(0, 2).contains(3) is S.false
assert Interval(0, 2, True, False).contains(0) is S.false
assert Interval(0, 2, True, False).contains(2) is S.true
assert Interval(0, 2, False, True).contains(0) is S.true
assert Interval(0, 2, False, True).contains(2) is S.false
assert Interval(0, 2, True, True).contains(0) is S.false
assert Interval(0, 2, True, True).contains(2) is S.false
# issue sympy/sympy#10326
assert S.Reals.contains(oo) is S.false
assert S.Reals.contains(-oo) is S.false
assert Interval(-oo, oo, True).contains(oo) is S.true
assert Interval(-oo, oo).contains(-oo) is S.true
bad = [EmptySet(), FiniteSet(1), Interval(1, 2), S.ComplexInfinity,
S.ImaginaryUnit, S.Infinity, S.NaN, S.NegativeInfinity]
assert all(i not in Interval(0, 5) for i in bad)
assert FiniteSet(1, 2, 3).contains(2) is S.true
assert FiniteSet(1, 2, x).contains(x) is S.true
# issue sympy/sympy#8197
assert isinstance(FiniteSet(b).contains(-a), Contains)
assert isinstance(FiniteSet(b).contains(a), Contains)
assert isinstance(FiniteSet(a).contains(1), Contains)
pytest.raises(TypeError, lambda: 1 in FiniteSet(a))
# issue sympy/sympy#8209
rad1 = Integer(4)
rad2 = Pow(2, 2, evaluate=False)
s1 = FiniteSet(rad1)
s2 = FiniteSet(rad2)
assert s1 - s2 == S.EmptySet
items = [1, 2, S.Infinity, Symbol('ham'), -1.1]
fset = FiniteSet(*items)
assert all(item in fset for item in items)
assert all(fset.contains(item) is S.true for item in items)
assert Union(Interval(0, 1), Interval(2, 5)).contains(3) is S.true
assert Union(Interval(0, 1), Interval(2, 5)).contains(6) is S.false
assert Union(Interval(0, 1), FiniteSet(2, 5)).contains(3) is S.false
assert S.EmptySet.contains(1) is S.false
assert FiniteSet(RootOf(x**3 + x - 1, 0)).contains(S.Infinity) is S.false
assert RootOf(x**5 + x**3 + 1, 0) in S.Reals
assert not RootOf(x**5 + x**3 + 1, 1) in S.Reals
def test_is_number():
assert Interval(0, 1).is_number is False
assert Set().is_number is False
def test_complement():
assert Interval(0, 1).complement(S.Reals) == \
Union(Interval(-oo, 0, True, True), Interval(1, oo, True, True))
assert Interval(0, 1, True, False).complement(S.Reals) == \
Union(Interval(-oo, 0, True, False), Interval(1, oo, True, True))
assert Interval(0, 1, False, True).complement(S.Reals) == \
Union(Interval(-oo, 0, True, True), Interval(1, oo, False, True))
assert Interval(0, 1, True, True).complement(S.Reals) == \
Union(Interval(-oo, 0, True, False), Interval(1, oo, False, True))
assert S.UniversalSet.complement(S.EmptySet) == S.EmptySet
assert S.UniversalSet.complement(S.Reals) == S.EmptySet
assert S.UniversalSet.complement(S.UniversalSet) == S.EmptySet
assert S.EmptySet.complement(S.Reals) == S.Reals
assert Union(Interval(0, 1), Interval(2, 3)).complement(S.Reals) == \
Union(Interval(-oo, 0, True, True), Interval(1, 2, True, True),
Interval(3, oo, True, True))
assert FiniteSet(0).complement(S.Reals) == \
Union(Interval(-oo, 0, True, True), Interval(0, oo, True, True))
assert (FiniteSet(5) + Interval(S.NegativeInfinity,
0)).complement(S.Reals) == \
Interval(0, 5, True, True) + Interval(5, S.Infinity, True, True)
assert FiniteSet(1, 2, 3).complement(S.Reals) == \
Interval(S.NegativeInfinity, 1, True, True) + \
Interval(1, 2, True, True) + Interval(2, 3, True, True) +\
Interval(3, S.Infinity, True, True)
assert FiniteSet(x).complement(S.Reals) == Complement(S.Reals, FiniteSet(x))
assert FiniteSet(0, x).complement(S.Reals) == Complement(Interval(-oo, 0, True, True) +
Interval(0, oo, True, True),
FiniteSet(x), evaluate=False)
square = Interval(0, 1) * Interval(0, 1)
notsquare = square.complement(S.Reals*S.Reals)
assert all(pt in square for pt in [(0, 0), (.5, .5), (1, 0), (1, 1)])
assert not any(
pt in notsquare for pt in [(0, 0), (.5, .5), (1, 0), (1, 1)])
assert not any(pt in square for pt in [(-1, 0), (1.5, .5), (10, 10)])
assert all(pt in notsquare for pt in [(-1, 0), (1.5, .5), (10, 10)])
def test_intersection():
pytest.raises(TypeError, lambda: Intersection(1))
pytest.raises(TypeError, lambda: Intersection())
assert Interval(0, 2).intersection(Interval(1, 2)) == Interval(1, 2)
assert Interval(0, 2).intersection(Interval(1, 2, True)) == \
Interval(1, 2, True)
assert Interval(0, 2, True).intersection(Interval(1, 2)) == \
Interval(1, 2, False, False)
assert Interval(0, 2, True, True).intersection(Interval(1, 2)) == \
Interval(1, 2, False, True)
assert Interval(0, 2).intersection(Union(Interval(0, 1), Interval(2, 3))) == \
Union(Interval(0, 1), Interval(2, 2))
x = Symbol('x')
assert Intersection(Interval(0, 1), FiniteSet(x)).is_iterable
assert not Intersection(Interval(0, 1), Interval(0, x)).is_iterable
assert FiniteSet(1, 2, x).intersection(FiniteSet(x)) == FiniteSet(x)
assert FiniteSet('ham', 'eggs').intersection(FiniteSet('ham')) == \
FiniteSet('ham')
assert FiniteSet(1, 2, 3, 4, 5).intersection(S.EmptySet) == S.EmptySet
assert Interval(0, 5).intersection(FiniteSet(1, 3)) == FiniteSet(1, 3)
assert Interval(0, 1, True, True).intersection(FiniteSet(1)) == S.EmptySet
assert Union(Interval(0, 1), Interval(2, 3)).intersection(Interval(1, 2)) == \
Union(Interval(1, 1), Interval(2, 2))
assert Union(Interval(0, 1), Interval(2, 3)).intersection(Interval(0, 2)) == \
Union(Interval(0, 1), Interval(2, 2))
assert Union(Interval(0, 1), Interval(2, 3)).intersection(Interval(1, 2, True, True)) == \
S.EmptySet
assert Union(Interval(0, 1), Interval(2, 3)).intersection(S.EmptySet) == \
S.EmptySet
assert Union(Interval(0, 5), FiniteSet('ham')).intersection(FiniteSet(2, 3, 4, 5, 6)) == \
Union(FiniteSet(2, 3, 4, 5), Intersection(FiniteSet(6), Union(Interval(0, 5), FiniteSet('ham'))))
# issue sympy/sympy#8217
assert Intersection(FiniteSet(x), FiniteSet(y)) == \
Intersection(FiniteSet(x), FiniteSet(y), evaluate=False)
assert FiniteSet(x).intersection(S.Reals) == \
Intersection(S.Reals, FiniteSet(x), evaluate=False)
# tests for the intersection alias
assert Interval(0, 5).intersection(FiniteSet(1, 3)) == FiniteSet(1, 3)
assert Interval(0, 1, True, True).intersection(FiniteSet(1)) == S.EmptySet
assert Union(Interval(0, 1), Interval(2, 3)).intersection(Interval(1, 2)) == \
Union(Interval(1, 1), Interval(2, 2))
assert ProductSet(FiniteSet(1),
FiniteSet(2)).intersection(Interval(1, 2)).is_Intersection
# iterable
i = Intersection(FiniteSet(1, 2, 3), Interval(2, 5), evaluate=False)
assert i.is_iterable
assert set(i) == {Integer(2), Integer(3)}
# challenging intervals
x = Symbol('x', extended_real=True)
i = Intersection(Interval(0, 3), Interval(x, 6))
assert (5 in i) is False
pytest.raises(TypeError, lambda: 2 in i)
# Singleton special cases
assert Intersection(Interval(0, 1), S.EmptySet) == S.EmptySet
assert Intersection(S.Reals, Interval(-oo, x, True)) == Interval(-oo, x, True)
# Products
line = Interval(0, 5)
i = Intersection(line**2, line**3, evaluate=False)
assert (2, 2) not in i
assert (2, 2, 2) not in i
pytest.raises(ValueError, lambda: list(i))
assert (Intersection(Intersection(S.Integers, S.Naturals, evaluate=False),
S.Reals, evaluate=False) ==
Intersection(S.Integers, S.Naturals, S.Reals, evaluate=False))
assert (imageset(Lambda(x, x**2),
Intersection(FiniteSet(1, 2), FiniteSet(2, 3),
evaluate=False)) == FiniteSet(4))
def test_difference():
assert Interval(1, 3) - Interval(1, 2) == Interval(2, 3, True)
assert Interval(1, 3) - Interval(2, 3) == Interval(1, 2, False, True)
assert Interval(1, 3, True) - Interval(2, 3) == Interval(1, 2, True, True)
assert Interval(1, 3, True) - Interval(2, 3, True) == \
Interval(1, 2, True, False)
assert Interval(0, 2) - FiniteSet(1) == \
Union(Interval(0, 1, False, True), Interval(1, 2, True, False))
assert FiniteSet(1, 2, 3) - FiniteSet(2) == FiniteSet(1, 3)
assert (FiniteSet('ham', 'eggs') - FiniteSet('eggs') ==
Complement(FiniteSet('ham'), FiniteSet('eggs')))
assert FiniteSet(1, 2, 3, 4) - Interval(2, 10, True, False) == \
FiniteSet(1, 2)
assert FiniteSet(1, 2, 3, 4) - S.EmptySet == FiniteSet(1, 2, 3, 4)
assert Union(Interval(0, 2), FiniteSet(2, 3, 4)) - Interval(1, 3) == \
Union(Interval(0, 1, False, True), FiniteSet(4))
assert -1 in S.Reals - S.Naturals
def test_interval_arguments():
assert Interval(0, oo).right_open is false
assert Interval(-oo, 0).left_open is false
assert Interval(oo, -oo) == S.EmptySet
assert isinstance(Interval(1, 1), FiniteSet)
e = Sum(x, (x, 1, 3))
assert isinstance(Interval(e, e), FiniteSet)
assert Interval(1, 0) == S.EmptySet
assert Interval(1, 1).measure == 0
assert Interval(1, 1, False, True) == S.EmptySet
assert Interval(1, 1, True, False) == S.EmptySet
assert Interval(1, 1, True, True) == S.EmptySet
assert isinstance(Interval(0, Symbol('a')), Interval)
assert Interval(Symbol('a', extended_real=True, positive=True), 0) == S.EmptySet
pytest.raises(ValueError, lambda: Interval(0, S.ImaginaryUnit))
pytest.raises(ValueError, lambda: Interval(0, Symbol('z', extended_real=False)))
pytest.raises(NotImplementedError, lambda: Interval(0, 1, And(x, y)))
pytest.raises(NotImplementedError, lambda: Interval(0, 1, False, And(x, y)))
pytest.raises(NotImplementedError, lambda: Interval(0, 1, z, And(x, y)))
def test_reduce_poly_inequalities_real_interval():
assert reduce_rational_inequalities([[Eq(x**2, 0)]], x,
relational=False) == FiniteSet(0)
assert reduce_rational_inequalities([[Le(x**2, 0)]], x,
relational=False) == FiniteSet(0)
assert reduce_rational_inequalities([[Lt(x**2, 0)]], x,
relational=False) == S.EmptySet
assert reduce_rational_inequalities(
[[Ge(x**2, 0)]], x, relational=False) == \
S.Reals if x.is_extended_real else Interval(-oo, oo)
assert reduce_rational_inequalities(
[[Gt(x**2, 0)]], x, relational=False) == \
FiniteSet(0).complement(S.Reals)
assert reduce_rational_inequalities(
[[Ne(x**2, 0)]], x, relational=False) == \
FiniteSet(0).complement(S.Reals)
assert reduce_rational_inequalities([[Eq(x**2, 1)]], x,
relational=False) == FiniteSet(-1, 1)
assert reduce_rational_inequalities([[Le(x**2, 1)]], x,
relational=False) == Interval(-1, 1)
assert reduce_rational_inequalities([[Lt(x**2, 1)]], x,
relational=False) == Interval(
-1, 1, True, True)
assert reduce_rational_inequalities(
[[Ge(x**2, 1)]], x, relational=False) == \
Union(Interval(-oo, -1, True), Interval(1, oo, False, True))
assert reduce_rational_inequalities(
[[Gt(x**2, 1)]], x, relational=False) == \
Interval(-1, 1).complement(S.Reals)
assert reduce_rational_inequalities(
[[Ne(x**2, 1)]], x, relational=False) == \
FiniteSet(-1, 1).complement(S.Reals)
assert reduce_rational_inequalities([[Eq(x**2, 1.0)]], x,
relational=False) == FiniteSet(
-1.0, 1.0).evalf()
assert reduce_rational_inequalities([[Le(x**2, 1.0)]], x,
relational=False) == Interval(
-1.0, 1.0)
assert reduce_rational_inequalities([[Lt(x**2, 1.0)]], x,
relational=False) == Interval(
-1.0, 1.0, True, True)
assert reduce_rational_inequalities(
[[Ge(x**2, 1.0)]], x, relational=False) == \
Union(Interval(-inf, -1.0, True), Interval(1.0, inf, False, True))
assert reduce_rational_inequalities(
[[Gt(x**2, 1.0)]], x, relational=False) == \
Union(Interval(-inf, -1.0, True, True),
Interval(1.0, inf, True, True))
assert reduce_rational_inequalities([[Ne(
x**2, 1.0)]], x, relational=False) == \
FiniteSet(-1.0, 1.0).complement(S.Reals)
s = sqrt(2)
assert reduce_rational_inequalities(
[[Lt(x**2 - 1, 0), Gt(x**2 - 1, 0)]], x,
relational=False) == S.EmptySet
assert reduce_rational_inequalities(
[[Le(x**2 - 1, 0), Ge(x**2 - 1, 0)]], x,
relational=False) == FiniteSet(-1, 1)
assert reduce_rational_inequalities(
[[Le(x**2 - 2, 0), Ge(x**2 - 1, 0)]], x,
relational=False) == Union(Interval(-s, -1, False, False),
Interval(1, s, False, False))
assert reduce_rational_inequalities(
[[Le(x**2 - 2, 0), Gt(x**2 - 1, 0)]], x,
relational=False) == Union(Interval(-s, -1, False, True),
Interval(1, s, True, False))
assert reduce_rational_inequalities(
[[Lt(x**2 - 2, 0), Ge(x**2 - 1, 0)]], x,
relational=False) == Union(Interval(-s, -1, True, False),
Interval(1, s, False, True))
assert reduce_rational_inequalities(
[[Lt(x**2 - 2, 0), Gt(x**2 - 1, 0)]], x,
relational=False) == Union(Interval(-s, -1, True, True),
Interval(1, s, True, True))
assert reduce_rational_inequalities(
[[Lt(x**2 - 2, 0), Ne(x**2 - 1, 0)]], x,
relational=False) == Union(Interval(-s, -1, True, True),
Interval(-1, 1, True, True),
Interval(1, s, True, True))
# issue sympy/sympy#10237
assert reduce_rational_inequalities([[x < oo, x >= 0, -oo < x]],
x,
relational=False) == Interval(
0, oo, False, True)
assert reduce_rational_inequalities([[Eq(
(x + 1) / (x**2 - 1), 0)]], x) is false
def test_sample():
z = Symbol('z')
Z = ContinuousRV(z, exp(-z), set=Interval(0, oo))
assert sample(Z) in Z.pspace.domain.set
sym, val = list(Z.pspace.sample().items())[0]
assert sym == Z and val in Interval(0, oo)
def test_piecewise():
# In each case, test eval() the lambdarepr() to make sure there are a
# correct number of parentheses. It will give a SyntaxError if there aren't.
h = "lambda x: "
p = Piecewise((x, True), evaluate=False)
l = lambdarepr(p)
eval(h + l)
assert l == "((x) if (True) else None)"
p = Piecewise((x, x < 0))
l = lambdarepr(p)
eval(h + l)
assert l == "((x) if (x < 0) else None)"
p = Piecewise((1, x < 1), (2, x < 2), (0, True))
l = lambdarepr(p)
eval(h + l)
assert l == "((1) if (x < 1) else (((2) if (x < 2) else " \
"(((0) if (True) else None)))))"
p = Piecewise(
(1, x < 1),
(2, x < 2),
)
l = lambdarepr(p)
eval(h + l)
assert l == "((1) if (x < 1) else (((2) if (x < 2) else None)))"
p = Piecewise(
(x, x < 1),
(x**2, Interval(3, 4, True, False).contains(x)),
(0, True),
)
l = lambdarepr(p)
eval(h + l)
assert l == "((x) if (x < 1) else (((x**2) if (((x <= 4) and " \
"(x > 3))) else (((0) if (True) else None)))))"
p = Piecewise((x**2, x < 0), (x, Interval(0, 1, False, True).contains(x)),
(2 - x, x >= 1), (0, True))
l = lambdarepr(p)
eval(h + l)
assert l == "((x**2) if (x < 0) else (((x) if (((x >= 0) and (x < 1))) " \
"else (((-x + 2) if (x >= 1) else (((0) if (True) else None)))))))"
p = Piecewise(
(x**2, x < 0),
(x, Interval(0, 1, False, True).contains(x)),
(2 - x, x >= 1),
)
l = lambdarepr(p)
eval(h + l)
assert l == "((x**2) if (x < 0) else (((x) if (((x >= 0) and " \
"(x < 1))) else (((-x + 2) if (x >= 1) else None)))))"
p = Piecewise((1, x >= 1), (2, x >= 2), (3, x >= 3), (4, x >= 4),
(5, x >= 5), (6, True))
l = lambdarepr(p)
eval(h + l)
assert l == (
"((1) if (x >= 1) else (((2) if (x >= 2) else (((3) if "
"(x >= 3) else (((4) if (x >= 4) else (((5) if (x >= 5) else (((6) if "
"(True) else None)))))))))))")
p = Piecewise((1, x <= 1), (2, x <= 2), (3, x <= 3), (4, x <= 4),
(5, x <= 5), (6, True))
l = lambdarepr(p)
eval(h + l)
assert l == "((1) if (x <= 1) else (((2) if (x <= 2) else (((3) if " \
"(x <= 3) else (((4) if (x <= 4) else (((5) if (x <= 5) else (((6) if " \
"(True) else None)))))))))))"
p = Piecewise((1, x > 1), (2, x > 2), (3, x > 3), (4, x > 4), (5, x > 5),
(6, True))
l = lambdarepr(p)
eval(h + l)
assert l == (
"((1) if (x > 1) else (((2) if (x > 2) else (((3) if "
"(x > 3) else (((4) if (x > 4) else (((5) if (x > 5) else (((6) if "
"(True) else None)))))))))))")
p = Piecewise((1, x < 1), (2, x < 2), (3, x < 3), (4, x < 4), (5, x < 5),
(6, True))
l = lambdarepr(p)
eval(h + l)
assert l == "((1) if (x < 1) else (((2) if (x < 2) else (((3) if " \
"(x < 3) else (((4) if (x < 4) else (((5) if (x < 5) else (((6) if " \
"(True) else None)))))))))))"
def test_interval_evalf():
assert (Interval(E, pi).evalf() ==
Interval(Float('2.7182818284590451', prec=15),
Float('3.1415926535897931', prec=15), false, false))
def test_is_disjoint():
assert Interval(0, 2).is_disjoint(Interval(1, 2)) is False
assert Interval(0, 2).is_disjoint(Interval(3, 4)) is True
assert Interval(0, 2).isdisjoint(Interval(3, 4)) is True
def test_is_proper_superset():
assert Interval(0, 1).is_proper_superset(Interval(0, 2)) is False
assert Interval(0, 3).is_proper_superset(Interval(0, 2)) is True
assert FiniteSet(1, 2, 3).is_proper_superset(S.EmptySet) is True
pytest.raises(ValueError, lambda: Interval(0, 1).is_proper_superset(0))
def test_ProductSet_of_single_arg_is_arg():
assert ProductSet(Interval(0, 1)) == Interval(0, 1)
def test_interval_symbolic():
e = Interval(0, 1)
assert e.contains(x) == And(0 <= x, x <= 1)
pytest.raises(TypeError, lambda: x in e)
e = Interval(0, 1, True, True)
assert e.contains(x) == And(0 < x, x < 1)
def test_pow_args():
pytest.raises(ValueError, lambda: Interval(0, 1)**-1)
pytest.raises(TypeError, lambda: ProductSet(2))
def test_Interval_is_left_unbounded():
assert Interval(3, 4).is_left_unbounded is False
assert Interval(-oo, 3).is_left_unbounded is True
assert Interval(Float("-inf"), 3).is_left_unbounded is True
def test_interval_subs():
a = Symbol('a', extended_real=True)
assert Interval(0, a).subs(a, 2) == Interval(0, 2)
assert Interval(a, 0).subs(a, 2) == S.EmptySet
def test_Interval_as_relational():
x = Symbol('x')
assert Interval(-1, 2, False, False).as_relational(x) == \
And(Le(-1, x), Le(x, 2))
assert Interval(-1, 2, True, False).as_relational(x) == \
And(Lt(-1, x), Le(x, 2))
assert Interval(-1, 2, False, True).as_relational(x) == \
And(Le(-1, x), Lt(x, 2))
assert Interval(-1, 2, True, True).as_relational(x) == \
And(Lt(-1, x), Lt(x, 2))
assert Interval(-oo, 2, right_open=False).as_relational(x) == And(Le(-oo, x), Le(x, 2))
assert Interval(-oo, 2, right_open=True).as_relational(x) == And(Le(-oo, x), Lt(x, 2))
assert Interval(-2, oo, left_open=False).as_relational(x) == And(Le(-2, x), Le(x, oo))
assert Interval(-2, oo, left_open=True).as_relational(x) == And(Lt(-2, x), Le(x, oo))
assert Interval(-oo, oo).as_relational(x) == And(Le(-oo, x), Le(x, oo))
x = Symbol('x', extended_real=True)
y = Symbol('y', extended_real=True)
assert Interval(x, y).as_relational(x) == (x <= y)
assert Interval(y, x).as_relational(x) == (y <= x)
def test_interval_to_mpi():
assert Interval(0, 1).to_mpi() == mpi(0, 1)
assert Interval(0, 1, True, False).to_mpi() == mpi(0, 1)
assert isinstance(Interval(0, 1).to_mpi(), type(mpi(0, 1)))
def test_Union_as_relational():
assert (Interval(0, 1) + FiniteSet(2)).as_relational(x) == \
Or(And(Le(0, x), Le(x, 1)), Eq(x, 2))
assert (Interval(0, 1, True, True) + FiniteSet(1)).as_relational(x) == \
And(Lt(0, x), Le(x, 1))
def test_measure():
a = Symbol('a', extended_real=True)
assert Interval(1, 3).measure == 2
assert Interval(0, a).measure == a
assert Interval(1, a).measure == a - 1
assert Union(Interval(1, 2), Interval(3, 4)).measure == 2
assert Union(Interval(1, 2), Interval(3, 4), FiniteSet(5, 6, 7)).measure \
== 2
assert FiniteSet(1, 2, oo, a, -oo, -5).measure == 0
assert S.EmptySet.measure == 0
square = Interval(0, 10) * Interval(0, 10)
offsetsquare = Interval(5, 15) * Interval(5, 15)
band = Interval(-oo, oo) * Interval(2, 4)
assert square.measure == offsetsquare.measure == 100
assert (square + offsetsquare).measure == 175 # there is some overlap
assert (square - offsetsquare).measure == 75
assert (square * FiniteSet(1, 2, 3)).measure == 0
assert (square.intersection(band)).measure == 20
assert (square + band).measure == oo
assert (band * FiniteSet(1, 2, 3)).measure == nan
def test_EmptySet():
assert S.EmptySet.as_relational(x) is false
assert S.EmptySet.intersection(S.UniversalSet) == S.EmptySet
assert S.EmptySet.boundary == S.EmptySet
assert Interval(0, 1).symmetric_difference(S.EmptySet) == Interval(0, 1)
assert Interval(1, 2).intersection(S.EmptySet) == S.EmptySet
def test_is_superset():
assert Interval(0, 1).is_superset(Interval(0, 2)) is False
assert Interval(0, 3).is_superset(Interval(0, 2))
assert FiniteSet(1, 2).is_superset(FiniteSet(1, 2, 3, 4)) is False
assert FiniteSet(4, 5).is_superset(FiniteSet(1, 2, 3, 4)) is False
assert FiniteSet(1).is_superset(Interval(0, 2)) is False
assert FiniteSet(1, 2).is_superset(Interval(0, 2, True, True)) is False
assert (Interval(1, 2) + FiniteSet(3)).is_superset(
(Interval(0, 2, False, True) + FiniteSet(2, 3))) is False
assert Interval(3, 4).is_superset(Union(Interval(0, 1), Interval(2, 5))) is False
assert FiniteSet(1, 2, 3, 4).is_superset(Interval(0, 5)) is False
assert S.EmptySet.is_superset(FiniteSet(1, 2, 3)) is False
assert Interval(0, 1).is_superset(S.EmptySet) is True
assert S.EmptySet.is_superset(S.EmptySet) is True
pytest.raises(ValueError, lambda: S.EmptySet.is_superset(1))
# tests for the issuperset alias
assert Interval(0, 1).issuperset(S.EmptySet) is True
assert S.EmptySet.issuperset(S.EmptySet) is True
def test_Union_of_ProductSets_shares():
line = Interval(0, 2)
points = FiniteSet(0, 1, 2)
assert Union(line * line, line * points) == line * line
def test_sympyissue_9956():
assert Union(Interval(-oo, oo), FiniteSet(1)) == Interval(-oo, oo)
assert Interval(-oo, oo).contains(1) is true
