def test_naturals0():
N = S.Naturals0
assert 0 in N
assert -1 not in N
assert next(iter(N)) == 0
assert not N.contains(oo)
assert N.contains(sin(x)) == Contains(sin(x), N)
def test_Rationals():
assert S.Integers.is_subset(S.Rationals)
assert S.Naturals.is_subset(S.Rationals)
assert S.Naturals0.is_subset(S.Rationals)
assert S.Rationals.is_subset(S.Reals)
assert S.Rationals.inf is -oo
assert S.Rationals.sup is oo
it = iter(S.Rationals)
assert [next(it) for i in range(12)] == [
0, 1, -1, S.Half, 2,
Rational(-1, 2), -2,
Rational(1, 3), 3,
Rational(-1, 3), -3,
Rational(2, 3)
]
assert Basic() not in S.Rationals
assert S.Half in S.Rationals
assert S.Rationals.contains(0.5) == Contains(0.5,
S.Rationals,
evaluate=False)
assert 2 in S.Rationals
r = symbols('r', rational=True)
assert r in S.Rationals
raises(TypeError, lambda: x in S.Rationals)
# issue #18134:
assert S.Rationals.boundary == S.Reals
assert S.Rationals.closure == S.Reals
assert S.Rationals.is_open == False
assert S.Rationals.is_closed == False
def test_ComplexRegion_contains():
r = Symbol('r', real=True)
# contains in ComplexRegion
a = Interval(2, 3)
b = Interval(4, 6)
c = Interval(7, 9)
c1 = ComplexRegion(a*b)
c2 = ComplexRegion(Union(a*b, c*a))
assert 2.5 + 4.5*I in c1
assert 2 + 4*I in c1
assert 3 + 4*I in c1
assert 8 + 2.5*I in c2
assert 2.5 + 6.1*I not in c1
assert 4.5 + 3.2*I not in c1
assert c1.contains(x) == Contains(x, c1, evaluate=False)
assert c1.contains(r) == False
assert c2.contains(x) == Contains(x, c2, evaluate=False)
assert c2.contains(r) == False
r1 = Interval(0, 1)
theta1 = Interval(0, 2*S.Pi)
c3 = ComplexRegion(r1*theta1, polar=True)
assert (0.5 + I*Rational(6, 10)) in c3
assert (S.Half + I*Rational(6, 10)) in c3
assert (S.Half + .6*I) in c3
assert (0.5 + .6*I) in c3
assert I in c3
assert 1 in c3
assert 0 in c3
assert 1 + I not in c3
assert 1 - I not in c3
assert c3.contains(x) == Contains(x, c3, evaluate=False)
assert c3.contains(r + 2*I) == Contains(
r + 2*I, c3, evaluate=False) # is in fact False
assert c3.contains(1/(1 + r**2)) == Contains(
1/(1 + r**2), c3, evaluate=False) # is in fact True
r2 = Interval(0, 3)
theta2 = Interval(pi, 2*pi, left_open=True)
c4 = ComplexRegion(r2*theta2, polar=True)
assert c4.contains(0) == True
assert c4.contains(2 + I) == False
assert c4.contains(-2 + I) == False
assert c4.contains(-2 - I) == True
assert c4.contains(2 - I) == True
assert c4.contains(-2) == False
assert c4.contains(2) == True
assert c4.contains(x) == Contains(x, c4, evaluate=False)
assert c4.contains(3/(1 + r**2)) == Contains(
3/(1 + r**2), c4, evaluate=False) # is in fact True
raises(ValueError, lambda: ComplexRegion(r1*theta1, polar=2))
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_Range_symbolic():
# symbolic Range
xr = Range(x, x + 4, 5)
sr = Range(x, y, t)
i = Symbol('i', integer=True)
ip = Symbol('i', integer=True, positive=True)
ipr = Range(ip)
inr = Range(0, -ip, -1)
ir = Range(i, i + 19, 2)
ir2 = Range(i, i*8, 3*i)
i = Symbol('i', integer=True)
inf = symbols('inf', infinite=True)
raises(ValueError, lambda: Range(inf))
raises(ValueError, lambda: Range(inf, 0, -1))
raises(ValueError, lambda: Range(inf, inf, 1))
raises(ValueError, lambda: Range(1, 1, inf))
# args
assert xr.args == (x, x + 5, 5)
assert sr.args == (x, y, t)
assert ir.args == (i, i + 20, 2)
assert ir2.args == (i, 10*i, 3*i)
# reversed
raises(ValueError, lambda: xr.reversed)
raises(ValueError, lambda: sr.reversed)
assert ipr.reversed.args == (ip - 1, -1, -1)
assert inr.reversed.args == (-ip + 1, 1, 1)
assert ir.reversed.args == (i + 18, i - 2, -2)
assert ir2.reversed.args == (7*i, -2*i, -3*i)
# contains
assert inf not in sr
assert inf not in ir
assert 0 in ipr
assert 0 in inr
raises(TypeError, lambda: 1 in ipr)
raises(TypeError, lambda: -1 in inr)
assert .1 not in sr
assert .1 not in ir
assert i + 1 not in ir
assert i + 2 in ir
raises(TypeError, lambda: x in xr) # XXX is this what contains is supposed to do?
raises(TypeError, lambda: 1 in sr) # XXX is this what contains is supposed to do?
# iter
raises(ValueError, lambda: next(iter(xr)))
raises(ValueError, lambda: next(iter(sr)))
assert next(iter(ir)) == i
assert next(iter(ir2)) == i
assert sr.intersect(S.Integers) == sr
assert sr.intersect(FiniteSet(x)) == Intersection({x}, sr)
raises(ValueError, lambda: sr[:2])
raises(ValueError, lambda: xr[0])
raises(ValueError, lambda: sr[0])
# len
assert len(ir) == ir.size == 10
assert len(ir2) == ir2.size == 3
raises(ValueError, lambda: len(xr))
raises(ValueError, lambda: xr.size)
raises(ValueError, lambda: len(sr))
raises(ValueError, lambda: sr.size)
# bool
assert bool(Range(0)) == False
assert bool(xr)
assert bool(ir)
assert bool(ipr)
assert bool(inr)
raises(ValueError, lambda: bool(sr))
raises(ValueError, lambda: bool(ir2))
# inf
raises(ValueError, lambda: xr.inf)
raises(ValueError, lambda: sr.inf)
assert ipr.inf == 0
assert inr.inf == -ip + 1
assert ir.inf == i
raises(ValueError, lambda: ir2.inf)
# sup
raises(ValueError, lambda: xr.sup)
raises(ValueError, lambda: sr.sup)
assert ipr.sup == ip - 1
assert inr.sup == 0
assert ir.inf == i
raises(ValueError, lambda: ir2.sup)
# getitem
raises(ValueError, lambda: xr[0])
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
assert ir2[:2] == Range(i, 7*i, 3*i)
assert ir2[0] == i
assert ir2[-2] == 4*i
assert ir2[-1] == 7*i
raises(ValueError, lambda: Range(i)[-1])
assert ipr[0] == ipr.inf == 0
assert ipr[-1] == ipr.sup == ip - 1
assert inr[0] == inr.sup == 0
assert inr[-1] == inr.inf == -ip + 1
raises(ValueError, lambda: ipr[-2])
assert ir.inf == i
assert ir.sup == i + 18
raises(ValueError, lambda: Range(i).inf)
# as_relational
assert ir.as_relational(x) == ((x >= i) & (x <= i + 18) &
Eq(Mod(-i + x, 2), 0))
assert ir2.as_relational(x) == Eq(
Mod(-i + x, 3*i), 0) & (((x >= i) & (x <= 7*i) & (3*i >= 1)) |
((x <= i) & (x >= 7*i) & (3*i <= -1)))
assert Range(i, i + 1).as_relational(x) == Eq(x, i)
assert sr.as_relational(z) == Eq(
Mod(t, 1), 0) & Eq(Mod(x, 1), 0) & Eq(Mod(-x + z, t), 0
) & (((z >= x) & (z <= -t + y) & (t >= 1)) |
((z <= x) & (z >= -t + y) & (t <= -1)))
assert xr.as_relational(z) == Eq(z, x) & Eq(Mod(x, 1), 0)
# symbols can clash if user wants (but it must be integer)
assert xr.as_relational(x) == Eq(Mod(x, 1), 0)
# 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))