def test_trig_inequalities():
# all the inequalities are solved in a periodic interval.
assert isolve(sin(x) < S.Half, x, relational=False) == Union(
Interval(0, pi / 6, False, True),
Interval(pi * Rational(5, 6), 2 * pi, True, False),
)
assert isolve(sin(x) > S.Half, x, relational=False) == Interval(
pi / 6, pi * Rational(5, 6), True, True
)
assert isolve(cos(x) < S.Zero, x, relational=False) == Interval(
pi / 2, pi * Rational(3, 2), True, True
)
assert isolve(cos(x) >= S.Zero, x, relational=False) == Union(
Interval(0, pi / 2), Interval(pi * Rational(3, 2), 2 * pi)
)
assert isolve(tan(x) < S.One, x, relational=False) == Union(
Interval.Ropen(0, pi / 4), Interval.Lopen(pi / 2, pi)
)
assert isolve(sin(x) <= S.Zero, x, relational=False) == Union(
FiniteSet(S.Zero), Interval(pi, 2 * pi)
)
assert isolve(sin(x) <= S.One, x, relational=False) == S.Reals
assert isolve(cos(x) < S(-2), x, relational=False) == S.EmptySet
assert isolve(sin(x) >= S.NegativeOne, x, relational=False) == S.Reals
assert isolve(cos(x) > S.One, x, relational=False) == S.EmptySet
def test_solve_univariate_inequality():
assert isolve(x**2 >= 4, x, relational=False) == Union(Interval(-oo, -2),
Interval(2, oo))
assert isolve(x**2 >= 4, x) == Or(And(Le(2, x), Lt(x, oo)), And(Le(x, -2),
Lt(-oo, x)))
assert isolve((x - 1)*(x - 2)*(x - 3) >= 0, x, relational=False) == \
Union(Interval(1, 2), Interval(3, oo))
assert isolve((x - 1)*(x - 2)*(x - 3) >= 0, x) == \
Or(And(Le(1, x), Le(x, 2)), And(Le(3, x), Lt(x, oo)))
# issue 2785:
assert isolve(x**3 - 2*x - 1 > 0, x, relational=False) == \
Union(Interval(-1, -sqrt(5)/2 + S(1)/2, True, True),
Interval(S(1)/2 + sqrt(5)/2, oo, True, True))
# issue 2794:
assert isolve(x**3 - x**2 + x - 1 > 0, x, relational=False) == \
Interval(1, oo, True)
# numerical testing in valid() is needed
assert isolve(x**7 - x - 2 > 0, x) == \
And(rootof(x**7 - x - 2, 0) < x, x < oo)
# handle numerator and denominator; although these would be handled as
# rational inequalities, these test confirm that the right thing is done
# when the domain is EX (e.g. when 2 is replaced with sqrt(2))
assert isolve(1/(x - 2) > 0, x) == And(S(2) < x, x < oo)
den = ((x - 1)*(x - 2)).expand()
assert isolve((x - 1)/den <= 0, x) == \
Or(And(-oo < x, x < 1), And(S(1) < x, x < 2))
n = Dummy('n')
raises(NotImplementedError, lambda: isolve(Abs(x) <= n, x, relational=False))
def test_solve_univariate_inequality():
assert isolve(x**2 >= 4, x, relational=False) == Union(Interval(-oo, -2),
Interval(2, oo))
assert isolve(x**2 >= 4, x) == Or(And(Le(2, x), Lt(x, oo)), And(Le(x, -2),
Lt(-oo, x)))
assert isolve((x - 1)*(x - 2)*(x - 3) >= 0, x, relational=False) == \
Union(Interval(1, 2), Interval(3, oo))
assert isolve((x - 1)*(x - 2)*(x - 3) >= 0, x) == \
Or(And(Le(1, x), Le(x, 2)), And(Le(3, x), Lt(x, oo)))
# issue 2785:
assert isolve(x**3 - 2*x - 1 > 0, x, relational=False) == \
Union(Interval(-1, -sqrt(5)/2 + S(1)/2, True, True),
Interval(S(1)/2 + sqrt(5)/2, oo, True, True))
# issue 2794:
assert isolve(x**3 - x**2 + x - 1 > 0, x, relational=False) == \
Interval(1, oo, True)
# numerical testing in valid() is needed
assert isolve(x**7 - x - 2 > 0, x) == \
And(rootof(x**7 - x - 2, 0) < x, x < oo)
# handle numerator and denominator; although these would be handled as
# rational inequalities, these test confirm that the right thing is done
# when the domain is EX (e.g. when 2 is replaced with sqrt(2))
assert isolve(1/(x - 2) > 0, x) == And(S(2) < x, x < oo)
den = ((x - 1)*(x - 2)).expand()
assert isolve((x - 1)/den <= 0, x) == \
Or(And(-oo < x, x < 1), And(S(1) < x, x < 2))
n = Dummy('n')
raises(NotImplementedError, lambda: isolve(Abs(x) <= n, x, relational=False))
def test_integer_domain_relational_isolve():
dom = FiniteSet(0, 3)
x = Symbol('x', zero=False)
assert isolve((x - 1) * (x - 2) * (x - 4) < 0, x, domain=dom) == Eq(x, 3)
x = Symbol('x')
assert isolve(x + 2 < 0, x, domain=S.Integers) ==\
(x <= -3) & (x > -oo) & Eq(x, floor(x))
assert isolve(2*x + 3 > 0, x, domain=S.Integers) ==\
(x >= -1) & (x < oo) & Eq(x, floor(x))
assert isolve((x**2 + 3*x - 2) < 0, x, domain=S.Integers) ==\
(x >= -3) & (x <= 0) & Eq(x, floor(x))
assert isolve((x**2 + 3*x - 2) > 0, x, domain=S.Integers) ==\
((x >= 1) & (x < oo) & Eq(x, floor(x))) | ((x <= -4) & (x > -oo) & Eq(x, floor(x)))
def test_trig_inequalities():
# all the inequalities are solved in a periodic interval.
assert isolve(sin(x) < S.Half, x, relational=False) == \
Union(Interval(0, pi/6, False, True), Interval(5*pi/6, 2*pi, True, False))
assert isolve(sin(x) > S.Half, x, relational=False) == \
Interval(pi/6, 5*pi/6, True, True)
assert isolve(cos(x) < S.Zero, x, relational=False) == \
Interval(pi/2, 3*pi/2, True, True)
assert isolve(cos(x) >= S.Zero, x, relational=False) == \
Union(Interval(0, pi/2), Interval(3*pi/2, 2*pi))
assert isolve(tan(x) < S.One, x, relational=False) == \
Union(Interval.Ropen(0, pi/4), Interval.Lopen(pi/2, pi))
assert isolve(sin(x) <= S.Zero, x, relational=False) == \
Union(FiniteSet(S(0)), Interval(pi, 2*pi))
assert isolve(sin(x) <= S(1), x, relational=False) == S.Reals
assert isolve(cos(x) < S(-2), x, relational=False) == S.EmptySet
assert isolve(sin(x) >= S(-1), x, relational=False) == S.Reals
assert isolve(cos(x) > S(1), x, relational=False) == S.EmptySet
def test_trig_inequalities():
# all the inequalities are solved in a periodic interval.
assert isolve(sin(x) < S.Half, x, relational=False) == \
Union(Interval(0, pi/6, False, True), Interval(5*pi/6, 2*pi, True, False))
assert isolve(sin(x) > S.Half, x, relational=False) == \
Interval(pi/6, 5*pi/6, True, True)
assert isolve(cos(x) < S.Zero, x, relational=False) == \
Interval(pi/2, 3*pi/2, True, True)
assert isolve(cos(x) >= S.Zero, x, relational=False) == \
Union(Interval(0, pi/2), Interval(3*pi/2, 2*pi))
assert isolve(tan(x) < S.One, x, relational=False) == \
Union(Interval.Ropen(0, pi/4), Interval.Lopen(pi/2, pi))
assert isolve(sin(x) <= S.Zero, x, relational=False) == \
Union(FiniteSet(S(0)), Interval(pi, 2*pi))
assert isolve(sin(x) <= S(1), x, relational=False) == S.Reals
assert isolve(cos(x) < S(-2), x, relational=False) == S.EmptySet
assert isolve(sin(x) >= S(-1), x, relational=False) == S.Reals
assert isolve(cos(x) > S(1), x, relational=False) == S.EmptySet
def test_solve_univariate_inequality():
x = Symbol('x', real=True)
assert isolve(x**2 >= 4, x, relational=False) == Union(Interval(-oo, -2), Interval(2, oo))
assert isolve(x**2 >= 4, x) == Or(And(Le(2, x), Lt(x, oo)), And(Le(x, -2), Lt(-oo, x)))
assert isolve((x - 1)*(x - 2)*(x - 3) >= 0, x, relational=False) == \
Union(Interval(1, 2), Interval(3, oo))
assert isolve((x - 1)*(x - 2)*(x - 3) >= 0, x) == \
Or(And(Le(1, x), Le(x, 2)), And(Le(3, x), Lt(x, oo)))
# issue 2785:
assert isolve(x**3 - 2*x - 1 > 0, x, relational=False) == \
Union(Interval(-1, -sqrt(5)/2 + S(1)/2, True, True),
Interval(S(1)/2 + sqrt(5)/2, oo, True, True))
# issue 2794:
assert isolve(x**3 - x**2 + x - 1 > 0, x, relational=False) == \
Interval(1, oo, True)
def test_solve_univariate_inequality():
x = Symbol('x', real=True)
assert isolve(x**2 >= 4,
x, relational=False) == Union(Interval(-oo, -2),
Interval(2, oo))
assert isolve(x**2 >= 4, x) == Or(And(Le(2, x), Lt(x, oo)),
And(Le(x, -2), Lt(-oo, x)))
assert isolve((x - 1)*(x - 2)*(x - 3) >= 0, x, relational=False) == \
Union(Interval(1, 2), Interval(3, oo))
assert isolve((x - 1)*(x - 2)*(x - 3) >= 0, x) == \
Or(And(Le(1, x), Le(x, 2)), And(Le(3, x), Lt(x, oo)))
# issue 2785:
assert isolve(x**3 - 2*x - 1 > 0, x, relational=False) == \
Union(Interval(-1, -sqrt(5)/2 + S(1)/2, True, True),
Interval(S(1)/2 + sqrt(5)/2, oo, True, True))
# issue 2794:
assert isolve(x**3 - x**2 + x - 1 > 0, x, relational=False) == \
Interval(1, oo, True)
def test_isolve_Sets():
n = Dummy('n')
assert isolve(Abs(x) <= n, x, relational=False) == \
Piecewise((S.EmptySet, n < 0), (Interval(-n, n), True))
def test_issue_8974():
assert isolve(-oo < x, x) == And(-oo < x, x < oo)
assert isolve(oo > x, x) == And(-oo < x, x < oo)
def test_issue_9954():
assert isolve(x**2 >= 0, x, relational=False) == S.Reals
assert isolve(x**2 >= 0, x, relational=True) == S.Reals.as_relational(x)
assert isolve(x**2 < 0, x, relational=False) == S.EmptySet
assert isolve(x**2 < 0, x, relational=True) == S.EmptySet.as_relational(x)
def test_solve_univariate_inequality():
assert isolve(x**2 >= 4,
x, relational=False) == Union(Interval(-oo, -2),
Interval(2, oo))
assert isolve(x**2 >= 4, x) == Or(And(Le(2, x), Lt(x, oo)),
And(Le(x, -2), Lt(-oo, x)))
assert isolve((x - 1)*(x - 2)*(x - 3) >= 0, x, relational=False) == \
Union(Interval(1, 2), Interval(3, oo))
assert isolve((x - 1)*(x - 2)*(x - 3) >= 0, x) == \
Or(And(Le(1, x), Le(x, 2)), And(Le(3, x), Lt(x, oo)))
assert isolve((x - 1)*(x - 2)*(x - 4) < 0, x, domain = FiniteSet(0, 3)) == \
Or(Eq(x, 0), Eq(x, 3))
# issue 2785:
assert isolve(x**3 - 2*x - 1 > 0, x, relational=False) == \
Union(Interval(-1, -sqrt(5)/2 + S(1)/2, True, True),
Interval(S(1)/2 + sqrt(5)/2, oo, True, True))
# issue 2794:
assert isolve(x**3 - x**2 + x - 1 > 0, x, relational=False) == \
Interval(1, oo, True)
#issue 13105
assert isolve((x + I) * (x + 2 * I) < 0, x) == Eq(x, 0)
assert isolve(((x - 1) * (x - 2) + I) * ((x - 1) * (x - 2) + 2 * I) < 0,
x) == Or(Eq(x, 1), Eq(x, 2))
assert isolve(
(((x - 1) * (x - 2) + I) * ((x - 1) * (x - 2) + 2 * I)) / (x - 2) > 0,
x) == Eq(x, 1)
raises(ValueError, lambda: isolve((x**2 - 3 * x * I + 2) / x < 0, x))
# numerical testing in valid() is needed
assert isolve(x**7 - x - 2 > 0, x) == \
And(rootof(x**7 - x - 2, 0) < x, x < oo)
# handle numerator and denominator; although these would be handled as
# rational inequalities, these test confirm that the right thing is done
# when the domain is EX (e.g. when 2 is replaced with sqrt(2))
assert isolve(1 / (x - 2) > 0, x) == And(S(2) < x, x < oo)
den = ((x - 1) * (x - 2)).expand()
assert isolve((x - 1)/den <= 0, x) == \
Or(And(-oo < x, x < 1), And(S(1) < x, x < 2))
n = Dummy('n')
raises(NotImplementedError,
lambda: isolve(Abs(x) <= n, x, relational=False))
c1 = Dummy("c1", positive=True)
raises(NotImplementedError, lambda: isolve(n / c1 < 0, c1))
n = Dummy('n', negative=True)
assert isolve(n / c1 > -2, c1) == (-n / 2 < c1)
assert isolve(n / c1 < 0, c1) == True
assert isolve(n / c1 > 0, c1) == False
zero = cos(1)**2 + sin(1)**2 - 1
raises(NotImplementedError, lambda: isolve(x**2 < zero, x))
raises(NotImplementedError, lambda: isolve(x**2 < zero * I, x))
raises(NotImplementedError, lambda: isolve(1 / (x - y) < 2, x))
raises(NotImplementedError, lambda: isolve(1 / (x - y) < 0, x))
raises(ValueError, lambda: isolve(x - I < 0, x))
zero = x**2 + x - x * (x + 1)
assert isolve(zero < 0, x, relational=False) is S.EmptySet
assert isolve(zero <= 0, x, relational=False) is S.Reals
# make sure iter_solutions gets a default value
raises(NotImplementedError,
lambda: isolve(Eq(cos(x)**2 + sin(x)**2, 1), x))
def test_isolve_Sets():
n = Dummy('n')
assert isolve(Abs(x) <= n, x, relational=False) == \
Piecewise((S.EmptySet, n < 0), (Interval(-n, n), True))
def test_issue_8974():
assert isolve(-oo < x, x) == And(-oo < x, x < oo)
assert isolve(oo > x, x) == And(-oo < x, x < oo)
def test_issue_9954():
assert isolve(x**2 >= 0, x, relational=False) == S.Reals
assert isolve(x**2 >= 0, x, relational=True) == S.Reals.as_relational(x)
assert isolve(x**2 < 0, x, relational=False) == S.EmptySet
assert isolve(x**2 < 0, x, relational=True) == S.EmptySet.as_relational(x)
def test_solve_univariate_inequality():
assert isolve(x**2 >= 4, x, relational=False) == Union(Interval(-oo, -2),
Interval(2, oo))
assert isolve(x**2 >= 4, x) == Or(And(Le(2, x), Lt(x, oo)), And(Le(x, -2),
Lt(-oo, x)))
assert isolve((x - 1)*(x - 2)*(x - 3) >= 0, x, relational=False) == \
Union(Interval(1, 2), Interval(3, oo))
assert isolve((x - 1)*(x - 2)*(x - 3) >= 0, x) == \
Or(And(Le(1, x), Le(x, 2)), And(Le(3, x), Lt(x, oo)))
assert isolve((x - 1)*(x - 2)*(x - 4) < 0, x, domain = FiniteSet(0, 3)) == \
Or(Eq(x, 0), Eq(x, 3))
# issue 2785:
assert isolve(x**3 - 2*x - 1 > 0, x, relational=False) == \
Union(Interval(-1, -sqrt(5)/2 + S(1)/2, True, True),
Interval(S(1)/2 + sqrt(5)/2, oo, True, True))
# issue 2794:
assert isolve(x**3 - x**2 + x - 1 > 0, x, relational=False) == \
Interval(1, oo, True)
#issue 13105
assert isolve((x + I)*(x + 2*I) < 0, x) == Eq(x, 0)
assert isolve(((x - 1)*(x - 2) + I)*((x - 1)*(x - 2) + 2*I) < 0, x) == Or(Eq(x, 1), Eq(x, 2))
assert isolve((((x - 1)*(x - 2) + I)*((x - 1)*(x - 2) + 2*I))/(x - 2) > 0, x) == Eq(x, 1)
raises (ValueError, lambda: isolve((x**2 - 3*x*I + 2)/x < 0, x))
# numerical testing in valid() is needed
assert isolve(x**7 - x - 2 > 0, x) == \
And(rootof(x**7 - x - 2, 0) < x, x < oo)
# handle numerator and denominator; although these would be handled as
# rational inequalities, these test confirm that the right thing is done
# when the domain is EX (e.g. when 2 is replaced with sqrt(2))
assert isolve(1/(x - 2) > 0, x) == And(S(2) < x, x < oo)
den = ((x - 1)*(x - 2)).expand()
assert isolve((x - 1)/den <= 0, x) == \
Or(And(-oo < x, x < 1), And(S(1) < x, x < 2))
n = Dummy('n')
raises(NotImplementedError, lambda: isolve(Abs(x) <= n, x, relational=False))
c1 = Dummy("c1", positive=True)
raises(NotImplementedError, lambda: isolve(n/c1 < 0, c1))
n = Dummy('n', negative=True)
assert isolve(n/c1 > -2, c1) == (-n/2 < c1)
assert isolve(n/c1 < 0, c1) == True
assert isolve(n/c1 > 0, c1) == False
zero = cos(1)**2 + sin(1)**2 - 1
raises(NotImplementedError, lambda: isolve(x**2 < zero, x))
raises(NotImplementedError, lambda: isolve(
x**2 < zero*I, x))
raises(NotImplementedError, lambda: isolve(1/(x - y) < 2, x))
raises(NotImplementedError, lambda: isolve(1/(x - y) < 0, x))
raises(TypeError, lambda: isolve(x - I < 0, x))
# make sure iter_solutions gets a default value
raises(NotImplementedError, lambda: isolve(
Eq(cos(x)**2 + sin(x)**2, 1), x))
def test_solve_univariate_inequality():
assert isolve(x**2 >= 4, x, relational=False) == Union(Interval(-oo, -2),
Interval(2, oo))
assert isolve(x**2 >= 4, x) == Or(And(Le(2, x), Lt(x, oo)), And(Le(x, -2),
Lt(-oo, x)))
assert isolve((x - 1)*(x - 2)*(x - 3) >= 0, x, relational=False) == \
Union(Interval(1, 2), Interval(3, oo))
assert isolve((x - 1)*(x - 2)*(x - 3) >= 0, x) == \
Or(And(Le(1, x), Le(x, 2)), And(Le(3, x), Lt(x, oo)))
# issue 2785:
assert isolve(x**3 - 2*x - 1 > 0, x, relational=False) == \
Union(Interval(-1, -sqrt(5)/2 + S(1)/2, True, True),
Interval(S(1)/2 + sqrt(5)/2, oo, True, True))
# issue 2794:
assert isolve(x**3 - x**2 + x - 1 > 0, x, relational=False) == \
Interval(1, oo, True)
# XXX should be limited in domain, e.g. between 0 and 2*pi
assert isolve(sin(x) < S.Half, x) == \
Or(And(-oo < x, x < pi/6), And(5*pi/6 < x, x < oo))
assert isolve(sin(x) > S.Half, x) == And(pi/6 < x, x < 5*pi/6)
# numerical testing in valid() is needed
assert isolve(x**7 - x - 2 > 0, x) == \
And(rootof(x**7 - x - 2, 0) < x, x < oo)
# handle numerator and denominator; although these would be handled as
# rational inequalities, these test confirm that the right thing is done
# when the domain is EX (e.g. when 2 is replaced with sqrt(2))
assert isolve(1/(x - 2) > 0, x) == And(S(2) < x, x < oo)
den = ((x - 1)*(x - 2)).expand()
assert isolve((x - 1)/den <= 0, x) == \
Or(And(-oo < x, x < 1), And(S(1) < x, x < 2))
def test_solve_univariate_inequality():
assert isolve(x**2 >= 4,
x, relational=False) == Union(Interval(-oo, -2),
Interval(2, oo))
assert isolve(x**2 >= 4, x) == Or(And(Le(2, x), Lt(x, oo)),
And(Le(x, -2), Lt(-oo, x)))
assert isolve((x - 1)*(x - 2)*(x - 3) >= 0, x, relational=False) == \
Union(Interval(1, 2), Interval(3, oo))
assert isolve((x - 1)*(x - 2)*(x - 3) >= 0, x) == \
Or(And(Le(1, x), Le(x, 2)), And(Le(3, x), Lt(x, oo)))
# issue 2785:
assert isolve(x**3 - 2*x - 1 > 0, x, relational=False) == \
Union(Interval(-1, -sqrt(5)/2 + S(1)/2, True, True),
Interval(S(1)/2 + sqrt(5)/2, oo, True, True))
# issue 2794:
assert isolve(x**3 - x**2 + x - 1 > 0, x, relational=False) == \
Interval(1, oo, True)
# XXX should be limited in domain, e.g. between 0 and 2*pi
assert isolve(sin(x) < S.Half, x) == \
Or(And(-oo < x, x < pi/6), And(5*pi/6 < x, x < oo))
assert isolve(sin(x) > S.Half, x) == And(pi / 6 < x, x < 5 * pi / 6)
# numerical testing in valid() is needed
assert isolve(x**7 - x - 2 > 0, x) == \
And(RootOf(x**7 - x - 2, 0) < x, x < oo)
# handle numerator and denominator; although these would be handled as
# rational inequalities, these test confirm that the right thing is done
# when the domain is EX (e.g. when 2 is replaced with sqrt(2))
assert isolve(1 / (x - 2) > 0, x) == And(S(2) < x, x < oo)
den = ((x - 1) * (x - 2)).expand()
assert isolve((x - 1)/den <= 0, x) == \
Or(And(-oo < x, x < 1), And(S(1) < x, x < 2))
