def test_issue_10285():
assert FiniteSet(-x - 1).intersect(Interval.Ropen(1, 2)) == FiniteSet(x).intersect(Interval.Lopen(-3, -2))
eq = -x - 2 * (-x - y)
s = signsimp(eq)
ivl = Interval.open(0, 1)
assert FiniteSet(eq).intersect(ivl) == FiniteSet(s).intersect(ivl)
assert FiniteSet(-eq).intersect(ivl) == FiniteSet(s).intersect(Interval.open(-1, 0))
eq -= 1
ivl = Interval.Lopen(1, oo)
assert FiniteSet(eq).intersect(ivl) == FiniteSet(s).intersect(Interval.Lopen(2, oo))
def test_reduce_rational_inequalities_real_relational():
assert reduce_rational_inequalities([], x) == False
assert reduce_rational_inequalities(
[[(x**2 + 3*x + 2)/(x**2 - 16) >= 0]], x, relational=False) == \
Union(Interval.open(-oo, -4), Interval(-2, -1), Interval.open(4, oo))
assert reduce_rational_inequalities(
[[((-2*x - 10)*(3 - x))/((x**2 + 5)*(x - 2)**2) < 0]], x,
relational=False) == \
Union(Interval.open(-5, 2), Interval.open(2, 3))
assert reduce_rational_inequalities([[(x + 1)/(x - 5) <= 0]], x,
relational=False) == \
Interval.Ropen(-1, 5)
assert reduce_rational_inequalities([[(x**2 + 4*x + 3)/(x - 1) > 0]], x,
relational=False) == \
Union(Interval.open(-3, -1), Interval.open(1, oo))
assert reduce_rational_inequalities([[(x**2 - 16)/(x - 1)**2 < 0]], x,
relational=False) == \
Union(Interval.open(-4, 1), Interval.open(1, 4))
assert reduce_rational_inequalities([[(3*x + 1)/(x + 4) >= 1]], x,
relational=False) == \
Union(Interval.open(-oo, -4), Interval.Ropen(S(3)/2, oo))
assert reduce_rational_inequalities([[(x - 8)/x <= 3 - x]], x,
relational=False) == \
Union(Interval.Lopen(-oo, -2), Interval.Lopen(0, 4))
# issue sympy/sympy#10237
assert reduce_rational_inequalities([[x < oo, x >= 0, -oo < x]],
x,
relational=False) == Interval(0, oo)
def test_reduce_rational_inequalities_real_relational():
assert reduce_rational_inequalities([], x) == False
assert reduce_rational_inequalities(
[[(x**2 + 3*x + 2)/(x**2 - 16) >= 0]], x, relational=False) == \
Union(Interval.open(-oo, -4), Interval(-2, -1), Interval.open(4, oo))
assert reduce_rational_inequalities(
[[((-2*x - 10)*(3 - x))/((x**2 + 5)*(x - 2)**2) < 0]], x,
relational=False) == \
Union(Interval.open(-5, 2), Interval.open(2, 3))
assert reduce_rational_inequalities([[(x + 1)/(x - 5) <= 0]], x,
relational=False) == \
Interval.Ropen(-1, 5)
assert reduce_rational_inequalities([[(x**2 + 4*x + 3)/(x - 1) > 0]], x,
relational=False) == \
Union(Interval.open(-3, -1), Interval.open(1, oo))
assert reduce_rational_inequalities([[(x**2 - 16)/(x - 1)**2 < 0]], x,
relational=False) == \
Union(Interval.open(-4, 1), Interval.open(1, 4))
assert reduce_rational_inequalities([[(3*x + 1)/(x + 4) >= 1]], x,
relational=False) == \
Union(Interval.open(-oo, -4), Interval.Ropen(S(3)/2, oo))
assert reduce_rational_inequalities([[(x - 8)/x <= 3 - x]], x,
relational=False) == \
Union(Interval.Lopen(-oo, -2), Interval.Lopen(0, 4))
# issue sympy/sympy#10237
assert reduce_rational_inequalities(
[[x < oo, x >= 0, -oo < x]], x, relational=False) == Interval(0, oo)
def test_issue_10285():
assert FiniteSet(-x - 1).intersect(Interval.Ropen(1, 2)) == \
FiniteSet(x).intersect(Interval.Lopen(-3, -2))
eq = -x - 2 * (-x - y)
s = signsimp(eq)
ivl = Interval.open(0, 1)
assert FiniteSet(eq).intersect(ivl) == FiniteSet(s).intersect(ivl)
assert FiniteSet(-eq).intersect(ivl) == \
FiniteSet(s).intersect(Interval.open(-1, 0))
eq -= 1
ivl = Interval.Lopen(1, oo)
assert FiniteSet(eq).intersect(ivl) == \
FiniteSet(s).intersect(Interval.Lopen(2, oo))
def test_solve_abs():
assert solveset_real(Abs(x) - 2, x) == FiniteSet(-2, 2)
assert solveset_real(Abs(x + 3) - 2 * Abs(x - 3), x) == FiniteSet(1, 9)
assert solveset_real(2 * Abs(x) - Abs(x - 1), x) == FiniteSet(-1, Rational(1, 3))
assert solveset_real(Abs(x - 7) - 8, x) == FiniteSet(-S(1), S(15))
# issue 9565. Note: solveset_real does not solve this as it is
# solveset's job to handle Relationals
assert solveset(Abs((x - 1) / (x - 5)) <= S(1) / 3, domain=S.Reals) == Interval(-1, 2)
# issue #10069
eq = abs(1 / (x - 1)) - 1 > 0
u = Union(Interval.open(0, 1), Interval.open(1, 2))
assert solveset_real(eq, x) == u
assert solveset(eq, x, domain=S.Reals) == u
raises(ValueError, lambda: solveset(abs(x) - 1, x))
def test_solve_abs():
assert solveset_real(Abs(x) - 2, x) == FiniteSet(-2, 2)
assert solveset_real(Abs(x + 3) - 2*Abs(x - 3), x) == \
FiniteSet(1, 9)
assert solveset_real(2*Abs(x) - Abs(x - 1), x) == \
FiniteSet(-1, Rational(1, 3))
assert solveset_real(Abs(x - 7) - 8, x) == FiniteSet(-S(1), S(15))
# issue 9565
assert solveset_real(Abs((x - 1)/(x - 5)) <= S(1)/3, x) == Interval(-1, 2)
# issue #10069
eq = abs(1/(x - 1)) - 1 > 0
u = Union(Interval.open(0, 1), Interval.open(1, 2))
assert solveset_real(eq, x) == u
assert solveset(eq, x, domain=S.Reals) == u
raises(ValueError, lambda: solveset(abs(x) - 1, x))
def test_solve_abs():
assert solveset_real(Abs(x) - 2, x) == FiniteSet(-2, 2)
assert solveset_real(Abs(x + 3) - 2*Abs(x - 3), x) == \
FiniteSet(1, 9)
assert solveset_real(2*Abs(x) - Abs(x - 1), x) == \
FiniteSet(-1, Rational(1, 3))
assert solveset_real(Abs(x - 7) - 8, x) == FiniteSet(-S(1), S(15))
# issue 9565. Note: solveset_real does not solve this as it is
# solveset's job to handle Relationals
assert solveset(Abs((x - 1)/(x - 5)) <= S(1)/3, domain=S.Reals
) == Interval(-1, 2)
# issue #10069
assert solveset_real(abs(1/(x - 1)) - 1 > 0, x) == \
ConditionSet(x, Eq((1 - Abs(x - 1))/Abs(x - 1) > 0, 0),
S.Reals)
assert solveset(abs(1/(x - 1)) - 1 > 0, x, domain=S.Reals
) == Union(Interval.open(0, 1), Interval.open(1, 2))
def test_solve_abs():
assert solveset_real(Abs(x) - 2, x) == FiniteSet(-2, 2)
assert solveset_real(Abs(x + 3) - 2*Abs(x - 3), x) == \
FiniteSet(1, 9)
assert solveset_real(2*Abs(x) - Abs(x - 1), x) == \
FiniteSet(-1, Rational(1, 3))
assert solveset_real(Abs(x - 7) - 8, x) == FiniteSet(-S(1), S(15))
# issue 9565. Note: solveset_real does not solve this as it is
# solveset's job to handle Relationals
assert solveset(Abs((x - 1)/(x - 5)) <= S(1)/3, domain=S.Reals
) == Interval(-1, 2)
# issue #10069
assert solveset_real(abs(1/(x - 1)) - 1 > 0, x) == \
ConditionSet(x, Eq((1 - Abs(x - 1))/Abs(x - 1) > 0, 0),
S.Reals)
assert solveset(abs(1/(x - 1)) - 1 > 0, x, domain=S.Reals
) == Union(Interval.open(0, 1), Interval.open(1, 2))
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.open(pi*Rational(5, 6), 2*pi))
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.Ropen(pi*Rational(3, 2), 2*pi))
assert isolve(tan(x) < S.One, x, relational=False) == \
Union(Interval.Ropen(0, pi/4), Interval.open(pi/2, pi))
assert isolve(sin(x) <= S.Zero, x, relational=False) == \
Union(FiniteSet(S.Zero), Interval.Ropen(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_issue_8715():
eq = x + 1 / x > -2 + 1 / x
assert solveset(eq, x, S.Reals) == \
(Interval.open(-2, oo) - FiniteSet(0))
assert solveset(eq.subs(x,log(x)), x, S.Reals) == \
Interval.open(exp(-2), oo) - FiniteSet(1)
def test_issue_8715():
eq = x + 1 / x > -2 + 1 / x
assert solveset(eq, x, S.Reals) == (Interval.open(-2, oo) - FiniteSet(0))
assert solveset(eq.subs(x, log(x)), x, S.Reals) == Interval.open(exp(-2), oo) - FiniteSet(1)
def test_find_real_domain():
from sympy.abc import t as my_t
from sympy import Union, Interval, oo, Intersection, Abs
assert_equals(find_real_domain(1 / my_t, my_t),
Union(Interval.open(-oo, 0), Interval.open(0, oo)))
assert_equals(find_real_domain(Abs(my_t), my_t), S.Reals)
def test_Union_contains():
assert zoo not in Union(
Interval.open(-oo, 0), Interval.open(0, oo))
def inequation_to_interval(inequation):
"""
Devuelve un intervalo a partir de una inequación
Parte del supuesto de que de un lado de la inecuación
se encuentra "x", sin operaciones.
"""
args = inequation.args
less = [LessThan, StrictLessThan]
greater = [GreaterThan, StrictGreaterThan]
if inequation.__class__ in greater + less: # [StrictGreaterThan, GreaterThan, StrictLessThan, LessThan]
inequation = _adapt(inequation)
args = inequation.args
a = args[0]
b = args[1]
if inequation.__class__ in greater:
if inequation.__class__ == StrictGreaterThan:
# Ejemplo: x > b
# Intervalo: (b, +oo)
return Interval.open(b, oo)
elif inequation.__class__ == GreaterThan:
# Ejemplo: x >= b
# Intervalo: [b, +oo)
return Interval(b, oo) # Cerrado por izquierda
elif inequation.__class__ in less:
if inequation.__class__ == StrictLessThan:
# Ejemplo: x < b
# Intervalo: (-oo, b)
return Interval.open(-oo, b)
elif inequation.__class__ == LessThan:
# Ejemplo: x <= b
# Intervalo: (-oo, b]
return Interval(-oo, b) # Cerrado por derecha
elif inequation.__class__ == And:
# El primer valor en args siempre es una inequación
_args = []
for arg in args:
_args.append(_adapt(arg))
args = _args
d = {1: [], 2: []}
for arg in args:
# FIXME: Y si hay otro And o un Or ?
if arg.__class__ in less:
d[1].append(inequation_to_interval(arg))
elif arg.__class__ in greater:
d[2].append(inequation_to_interval(arg))
if d[1]:
__less = True
_less = d[1][0]
for _i in d[1][1:]:
_less = Interval.union(_less, _i)
else:
__less = False
_less = EmptySet()
if d[2]:
__greater = True
_greater = d[2][0]
for _i in d[2][1:]:
_greater = Interval.union(_greater, _i)
else:
__greater = False
_greater = EmptySet()
if __greater and __less:
return _less.intersect(_greater)
elif __less and not __greater:
return _less
elif __greater and not __less:
return _greater
elif inequation.__class__ == Or:
interval = inequation_to_interval(args[0])
for arg in args[1:]:
interval = Union(interval, inequation_to_interval(arg))
return interval
elif inequation.__class__ == BooleanTrue:
# Por ejemplo: 4 > 2, todos los x pertenecientes
# a los reales cumplen esta inecuación
return REALS
elif inequation.__class__ == BooleanFalse:
# Por ejemplo: 2 > 4, ningún x perteneciente a los
# reales cumple con esta inecuación
return EmptySet()
else:
print inequation, inequation.__class__