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_reduce_rational_inequalities_real_relational():
def OpenInterval(a, b):
return Interval(a, b, True, True)
def LeftOpenInterval(a, b):
return Interval(a, b, True, False)
def RightOpenInterval(a, b):
return Interval(a, b, False, True)
x = Symbol('x', real=True)
assert reduce_rational_inequalities([[(x**2 + 3*x + 2)/(x**2 - 16) >= 0]], x, relational=False) == \
Union(OpenInterval(-oo, -4), Interval(-2, -1), OpenInterval(4, oo))
assert reduce_rational_inequalities([[((-2*x - 10)*(3 - x))/((x**2 + 5)*(x - 2)**2) < 0]], x, relational=False) == \
Union(OpenInterval(-5, 2), OpenInterval(2, 3))
assert reduce_rational_inequalities([[(x + 1)/(x - 5) <= 0]], x, relational=False) == \
RightOpenInterval(-1, 5)
assert reduce_rational_inequalities([[(x**2 + 4*x + 3)/(x - 1) > 0]], x, relational=False) == \
Union(OpenInterval(-3, -1), OpenInterval(1, oo))
assert reduce_rational_inequalities([[(x**2 - 16)/(x - 1)**2 < 0]], x, relational=False) == \
Union(OpenInterval(-4, 1), OpenInterval(1, 4))
assert reduce_rational_inequalities([[(3*x + 1)/(x + 4) >= 1]], x, relational=False) == \
Union(OpenInterval(-oo, -4), RightOpenInterval(S(3)/2, oo))
assert reduce_rational_inequalities([[(x - 8)/x <= 3 - x]], x, relational=False) == \
Union(LeftOpenInterval(-oo, -2), LeftOpenInterval(0, 4))
def test_reduce_rational_inequalities_real_relational():
def OpenInterval(a, b):
return Interval(a, b, True, True)
def LeftOpenInterval(a, b):
return Interval(a, b, True, False)
def RightOpenInterval(a, b):
return Interval(a, b, False, True)
with assuming(Q.real(x)):
assert reduce_rational_inequalities([[(x**2 + 3*x + 2)/(x**2 - 16) >= 0]], x, relational=False) == \
Union(OpenInterval(-oo, -4), Interval(-2, -1), OpenInterval(4, oo))
assert reduce_rational_inequalities([[((-2*x - 10)*(3 - x))/((x**2 + 5)*(x - 2)**2) < 0]], x, relational=False) == \
Union(OpenInterval(-5, 2), OpenInterval(2, 3))
assert reduce_rational_inequalities([[(x + 1)/(x - 5) <= 0]], x, assume=Q.real(x), relational=False) == \
RightOpenInterval(-1, 5)
assert reduce_rational_inequalities([[(x**2 + 4*x + 3)/(x - 1) > 0]], x, assume=Q.real(x), relational=False) == \
Union(OpenInterval(-3, -1), OpenInterval(1, oo))
assert reduce_rational_inequalities([[(x**2 - 16)/(x - 1)**2 < 0]], x, assume=Q.real(x), relational=False) == \
Union(OpenInterval(-4, 1), OpenInterval(1, 4))
assert reduce_rational_inequalities([[(3*x + 1)/(x + 4) >= 1]], x, assume=Q.real(x), relational=False) == \
Union(OpenInterval(-oo, -4), RightOpenInterval(S(3)/2, oo))
assert reduce_rational_inequalities([[(x - 8)/x <= 3 - x]], x, assume=Q.real(x), relational=False) == \
Union(LeftOpenInterval(-oo, -2), LeftOpenInterval(0, 4))
def _has_conflict(rd1, rd2, offset):
"""
:param rd1: rectangular domain from Stencil1 (1D) <- (low, high, stride) (written domain)
:param rd2: rectangular domain from Stencil2 (1D) <- (low, high, stride) (read domain)
:param offset_vector: Offset from the center (relative to rd2), 1D projections
:return: Whether OV from rectangular domain 2 reads from rectangular domain 1
"""
n1 = sympy.Symbol("n1")
n2 = sympy.Symbol("n2")
# Diophantine equations:
# offset(iterationspace1) + n1 * stride(iteration_space1) <- write vectors
diophantine_eq1 = rd1[0] + n1 * rd1[2]
# offset(iterationspace2) + n2 * stride(iteration_space2) + offset_vector <- Read vectors
diophantine_eq2 = rd2[0] + n2 * rd2[2] + offset
# Since sympy solves eq = 0, we want dio_eq1 - dio_eq2 = 0.
eqn = diophantine_eq1 - diophantine_eq2
parameter = sympy.Symbol("t", integer=True)
sat_param = sympy.Symbol("t_0", integer=True)
if not eqn.free_symbols:
return eqn == 0
solutions = dio.diophantine(eqn, parameter) # default parameter is "t"
# print("Sols:", solutions)
for sol in solutions:
if len(eqn.free_symbols) != 2:
if n1 in eqn.free_symbols:
parametric_n1 = sol[0]
parametric_n2 = 0
else:
parametric_n1 = 0
parametric_n2 = sol[0]
else:
(parametric_n1, parametric_n2) = sol
# Solutions is a set of tuples, each containing either a number or parametric expression
# which give conditions on satisfiability.
# are these satisfiable on the original bounds?
# substitute the parametric forms in
substituted_1 = diophantine_eq1.subs({n1: parametric_n1})
substituted_2 = (
rd2[0] + parametric_n2 * rd2[2]
) # we ditch the offset because it's irrelevant since the bounds are based on the center of the stencil
# print(substituted_1, "\t", substituted_2)
# print(rd1[0], rd1[1], rd2[0], rd2[1])
# now do they satisfy the bounds?
satisfactory_interval = ineq.reduce_rational_inequalities(
[[rd1[0] <= substituted_1, rd1[1] > substituted_1, rd2[0] <= substituted_2, rd2[1] > substituted_2]],
sat_param,
relational=False,
)
# print(satisfactory_interval)
if _contains_integer(satisfactory_interval):
return True
return False
def crecimiento(f):
#crecimiento, decrecimiento y puntos singulares
crec = reduce_rational_inequalities([[f.diff() > 0]], x, relational=False)
decrec = reduce_rational_inequalities([[f.diff() < 0]],
x,
relational=False)
sing = []
for t in solve(f.diff()):
if f.diff().diff().subs(x, t) < 0:
tipo = 'máx'
elif f.diff().diff().subs(x, t) > 0:
tipo = 'mín'
else:
tipo = 'inflex'
sing.append([t, tipo])
sol = [crec, decrec, sing]
return sol
def test_reduce_poly_inequalities_complex_relational():
assert reduce_rational_inequalities(
[[Eq(x**2, 0)]], x, relational=True) == Eq(x, 0)
assert reduce_rational_inequalities(
[[Le(x**2, 0)]], x, relational=True) == Eq(x, 0)
assert reduce_rational_inequalities(
[[Lt(x**2, 0)]], x, relational=True) == False
assert reduce_rational_inequalities(
[[Ge(x**2, 0)]], x, relational=True) == And(Lt(-oo, x), Lt(x, oo))
assert reduce_rational_inequalities(
[[Gt(x**2, 0)]], x, relational=True) == \
And(Gt(x, -oo), Lt(x, oo), Ne(x, 0))
assert reduce_rational_inequalities(
[[Ne(x**2, 0)]], x, relational=True) == \
And(Gt(x, -oo), Lt(x, oo), Ne(x, 0))
for one in (S(1), S(1.0)):
inf = one*oo
assert reduce_rational_inequalities(
[[Eq(x**2, one)]], x, relational=True) == \
Or(Eq(x, -one), Eq(x, one))
assert reduce_rational_inequalities(
[[Le(x**2, one)]], x, relational=True) == \
And(And(Le(-one, x), Le(x, one)))
assert reduce_rational_inequalities(
[[Lt(x**2, one)]], x, relational=True) == \
And(And(Lt(-one, x), Lt(x, one)))
assert reduce_rational_inequalities(
[[Ge(x**2, one)]], x, relational=True) == \
And(Or(And(Le(one, x), Lt(x, inf)), And(Le(x, -one), Lt(-inf, x))))
assert reduce_rational_inequalities(
[[Gt(x**2, one)]], x, relational=True) == \
And(Or(And(Lt(-inf, x), Lt(x, -one)), And(Lt(one, x), Lt(x, inf))))
assert reduce_rational_inequalities(
[[Ne(x**2, one)]], x, relational=True) == \
Or(And(Lt(-inf, x), Lt(x, -one)),
And(Lt(-one, x), Lt(x, one)),
And(Lt(one, x), Lt(x, inf)))
def _reduce_inequalities(conditions, var, **kwargs):
try:
return reduce_rational_inequalities(conditions, var, **kwargs)
except PolynomialError:
raise ValueError("Reduction of condition failed %s\n" % conditions[0])
def _reduce_inequalities(conditions, var, **kwargs):
try:
return reduce_rational_inequalities(conditions, var, **kwargs)
except PolynomialError:
raise ValueError("Reduction of condition failed %s\n" % conditions[0])
def test_reduce_poly_inequalities_real_relational():
with assuming(Q.real(x), Q.real(y)):
assert reduce_rational_inequalities(
[[Eq(x**2, 0)]], x, relational=True) == Eq(x, 0)
assert reduce_rational_inequalities(
[[Le(x**2, 0)]], x, relational=True) == Eq(x, 0)
assert reduce_rational_inequalities(
[[Lt(x**2, 0)]], x, relational=True) is False
assert reduce_rational_inequalities(
[[Ge(x**2, 0)]], x, relational=True) is True
assert reduce_rational_inequalities(
[[Gt(x**2, 0)]], x, relational=True) == Or(Lt(x, 0), Gt(x, 0))
assert reduce_rational_inequalities(
[[Ne(x**2, 0)]], x, relational=True) == Or(Lt(x, 0), Gt(x, 0))
assert reduce_rational_inequalities(
[[Eq(x**2, 1)]], x, relational=True) == Or(Eq(x, -1), Eq(x, 1))
assert reduce_rational_inequalities(
[[Le(x**2, 1)]], x, relational=True) == And(Le(-1, x), Le(x, 1))
assert reduce_rational_inequalities(
[[Lt(x**2, 1)]], x, relational=True) == And(Lt(-1, x), Lt(x, 1))
assert reduce_rational_inequalities(
[[Ge(x**2, 1)]], x, relational=True) == Or(Le(x, -1), Ge(x, 1))
assert reduce_rational_inequalities(
[[Gt(x**2, 1)]], x, relational=True) == Or(Lt(x, -1), Gt(x, 1))
assert reduce_rational_inequalities([[Ne(x**2, 1)]], x, relational=True) == Or(
Lt(x, -1), And(Lt(-1, x), Lt(x, 1)), Gt(x, 1))
assert reduce_rational_inequalities(
[[Le(x**2, 1.0)]], x, relational=True) == And(Le(-1.0, x), Le(x, 1.0))
assert reduce_rational_inequalities(
[[Lt(x**2, 1.0)]], x, relational=True) == And(Lt(-1.0, x), Lt(x, 1.0))
assert reduce_rational_inequalities(
[[Ge(x**2, 1.0)]], x, relational=True) == Or(Le(x, -1.0), Ge(x, 1.0))
assert reduce_rational_inequalities(
[[Gt(x**2, 1.0)]], x, relational=True) == Or(Lt(x, -1.0), Gt(x, 1.0))
assert reduce_rational_inequalities([[Ne(x**2, 1.0)]], x, relational=True) == \
Or(Lt(x, -1.0), And(Lt(-1.0, x), Lt(x, 1.0)), Gt(x, 1.0))
def test_reduce_poly_inequalities_real_relational():
with assuming(Q.real(x), Q.real(y)):
assert reduce_rational_inequalities(
[[Eq(x**2, 0)]], x, relational=True) == Eq(x, 0)
assert reduce_rational_inequalities(
[[Le(x**2, 0)]], x, relational=True) == Eq(x, 0)
assert reduce_rational_inequalities(
[[Lt(x**2, 0)]], x, relational=True) == False
assert reduce_rational_inequalities(
[[Ge(x**2, 0)]], x, relational=True) == And(Lt(-oo, x), Lt(x, oo))
assert reduce_rational_inequalities(
[[Gt(x**2, 0)]], x, relational=True) == Or(And(Lt(-oo, x), Lt(x, 0)), And(Lt(0, x), Lt(x, oo)))
assert reduce_rational_inequalities(
[[Ne(x**2, 0)]], x, relational=True) == Or(And(Lt(-oo, x), Lt(x, 0)), And(Lt(0, x), Lt(x, oo)))
assert reduce_rational_inequalities(
[[Eq(x**2, 1)]], x, relational=True) == Or(Eq(x, -1), Eq(x, 1))
assert reduce_rational_inequalities(
[[Le(x**2, 1)]], x, relational=True) == And(Le(-1, x), Le(x, 1))
assert reduce_rational_inequalities(
[[Lt(x**2, 1)]], x, relational=True) == And(Lt(-1, x), Lt(x, 1))
assert reduce_rational_inequalities(
[[Ge(x**2, 1)]], x, relational=True) == Or(And(Le(1, x), Lt(x, oo)), And(Le(x, -1), Lt(-oo, x)))
assert reduce_rational_inequalities(
[[Gt(x**2, 1)]], x, relational=True) == Or(And(Lt(1, x), Lt(x, oo)), And(Lt(x, -1), Lt(-oo, x)))
assert reduce_rational_inequalities([[Ne(x**2, 1)]], x, relational=True) == Or(
And(Lt(-oo, x), Lt(x, -1)), And(Lt(-1, x), Lt(x, 1)), And(Lt(1, x), Lt(x, oo)))
assert reduce_rational_inequalities(
[[Le(x**2, 1.0)]], x, relational=True) == And(Le(-1.0, x), Le(x, 1.0))
assert reduce_rational_inequalities(
[[Lt(x**2, 1.0)]], x, relational=True) == And(Lt(-1.0, x), Lt(x, 1.0))
assert reduce_rational_inequalities(
[[Ge(x**2, 1.0)]], x, relational=True) == Or(And(Lt(Float('-inf'), x), Le(x, -1.0)),
And(Le(1.0, x), Lt(x, Float('+inf'))))
assert reduce_rational_inequalities(
[[Gt(x**2, 1.0)]], x, relational=True) == Or(And(Lt(Float('-inf'), x), Lt(x, -1.0)),
And(Lt(1.0, x), Lt(x, Float('+inf'))))
assert reduce_rational_inequalities([[Ne(x**2, 1.0)]], x, relational=True) == \
Or(And(Lt(-1.0, x), Lt(x, 1.0)), And(Lt(Float('-inf'), x), Lt(x, -1.0)),
And(Lt(1.0, x), Lt(x, Float('+inf'))))
def test_reduce_poly_inequalities_complex_relational():
x = Symbol('x')
cond = Eq(im(x), 0)
assert reduce_rational_inequalities([[Eq(x**2, 0)]], x,
relational=True) == And(
Eq(re(x), 0), cond)
assert reduce_rational_inequalities([[Le(x**2, 0)]], x,
relational=True) == And(
Eq(re(x), 0), cond)
assert reduce_rational_inequalities([[Lt(x**2, 0)]], x,
relational=True) == False
assert reduce_rational_inequalities(
[[Ge(x**2, 0)]], x, relational=True) == And(cond, Lt(-oo, re(x)),
Lt(re(x), oo))
assert reduce_rational_inequalities([[Gt(x**2, 0)]], x, relational=True) == \
And(cond, Or(And(Lt(-oo, re(x)), Lt(re(x), 0)), And(Lt(0, re(x)), Lt(re(x), oo))))
assert reduce_rational_inequalities([[Ne(x**2, 0)]], x, relational=True) == \
And(cond, Or(And(Lt(-oo, re(x)), Lt(re(x), 0)), And(Lt(0, re(x)), Lt(re(x), oo))))
assert reduce_rational_inequalities([[Eq(x**2, 1)]], x, relational=True) == \
And(Or(Eq(re(x), -1), Eq(re(x), 1)), cond)
assert reduce_rational_inequalities([[Le(x**2, 1)]], x, relational=True) == \
And(And(Le(-1, re(x)), Le(re(x), 1)), cond)
assert reduce_rational_inequalities([[Lt(x**2, 1)]], x, relational=True) == \
And(And(Lt(-1, re(x)), Lt(re(x), 1)), cond)
assert reduce_rational_inequalities([[Ge(x**2, 1)]], x, relational=True) == \
And(cond, Or(And(Le(1, re(x)), Lt(re(x), oo)), And(Le(re(x), -1), Lt(-oo, re(x)))))
assert reduce_rational_inequalities([[Gt(x**2, 1)]], x, relational=True) == \
And(cond, Or(And(Lt(-oo, re(x)), Lt(re(x), -1)), And(Lt(1, re(x)), Lt(re(x), oo))))
assert reduce_rational_inequalities([[Ne(x**2, 1)]], x, relational=True) == \
And(cond, Or(And(Lt(-oo, re(x)), Lt(re(x), -1)),
And(Lt(-1, re(x)), Lt(re(x), 1)), And(Lt(1, re(x)), Lt(re(x), oo))))
assert reduce_rational_inequalities([[Le(x**2, 1.0)]], x, relational=True) == \
And(And(Le(-1.0, re(x)), Le(re(x), 1.0)), cond)
assert reduce_rational_inequalities([[Lt(x**2, 1.0)]], x, relational=True) == \
And(And(Lt(-1.0, re(x)), Lt(re(x), 1.0)), cond)
assert reduce_rational_inequalities([[Ge(x**2, 1.0)]], x, relational=True) == \
And(cond, Or(And(Le(1.0, re(x)), re(x) < Float('+inf')),
And(Le(re(x), -1.0), Float('-inf') < re(x))))
assert reduce_rational_inequalities([[Gt(x**2, 1.0)]], x, relational=True) == \
And(cond, Or(And(Float('-inf') < re(x), re(x) < -1.0),
And(Lt(1.0, re(x)), re(x) < Float('+inf'))))
assert reduce_rational_inequalities([[Ne(x**2, 1.0)]], x, relational=True) == \
And(cond, Or(And(Float('-inf') < re(x), Lt(re(x), -1.0)),
And(Lt(-1.0, re(x)), re(x) < 1.0), And(Lt(1.0, re(x)), re(x) < Float('+inf'))))
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_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), Interval(1, oo))
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), Interval(1.0, inf))
assert reduce_rational_inequalities(
[[Gt(x**2, 1.0)]], x, relational=False) == \
Union(Interval(-inf, -1.0, right_open=True),
Interval(1.0, inf, left_open=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))
def test_reduce_poly_inequalities_real_relational():
x = Symbol('x', real=True)
y = Symbol('y', real=True)
assert reduce_rational_inequalities([[Eq(x**2, 0)]], x,
relational=True) == Eq(x, 0)
assert reduce_rational_inequalities([[Le(x**2, 0)]], x,
relational=True) == Eq(x, 0)
assert reduce_rational_inequalities([[Lt(x**2, 0)]], x,
relational=True) == False
assert reduce_rational_inequalities([[Ge(x**2, 0)]], x,
relational=True) == And(
Lt(-oo, x), Lt(x, oo))
assert reduce_rational_inequalities([[Gt(x**2, 0)]], x,
relational=True) == Or(
And(Lt(-oo, x), Lt(x, 0)),
And(Lt(0, x), Lt(x, oo)))
assert reduce_rational_inequalities([[Ne(x**2, 0)]], x,
relational=True) == Or(
And(Lt(-oo, x), Lt(x, 0)),
And(Lt(0, x), Lt(x, oo)))
assert reduce_rational_inequalities([[Eq(x**2, 1)]], x,
relational=True) == Or(
Eq(x, -1), Eq(x, 1))
assert reduce_rational_inequalities([[Le(x**2, 1)]], x,
relational=True) == And(
Le(-1, x), Le(x, 1))
assert reduce_rational_inequalities([[Lt(x**2, 1)]], x,
relational=True) == And(
Lt(-1, x), Lt(x, 1))
assert reduce_rational_inequalities([[Ge(x**2, 1)]], x,
relational=True) == Or(
And(Le(1, x), Lt(x, oo)),
And(Le(x, -1), Lt(-oo, x)))
assert reduce_rational_inequalities([[Gt(x**2, 1)]], x,
relational=True) == Or(
And(Lt(1, x), Lt(x, oo)),
And(Lt(x, -1), Lt(-oo, x)))
assert reduce_rational_inequalities([[Ne(x**2, 1)]], x,
relational=True) == Or(
And(Lt(-oo, x), Lt(x, -1)),
And(Lt(-1, x), Lt(x, 1)),
And(Lt(1, x), Lt(x, oo)))
assert reduce_rational_inequalities([[Le(x**2, 1.0)]], x,
relational=True) == And(
Le(-1.0, x), Le(x, 1.0))
assert reduce_rational_inequalities([[Lt(x**2, 1.0)]], x,
relational=True) == And(
Lt(-1.0, x), Lt(x, 1.0))
assert reduce_rational_inequalities([[Ge(x**2, 1.0)]], x,
relational=True) == Or(
And(Lt(Float('-inf'), x),
Le(x, -1.0)),
And(Le(1.0, x),
Lt(x, Float('+inf'))))
assert reduce_rational_inequalities([[Gt(x**2, 1.0)]], x,
relational=True) == Or(
And(Lt(Float('-inf'), x),
Lt(x, -1.0)),
And(Lt(1.0, x),
Lt(x, Float('+inf'))))
assert reduce_rational_inequalities([[Ne(x**2, 1.0)]], x, relational=True) == \
Or(And(Lt(-1.0, x), Lt(x, 1.0)), And(Lt(Float('-inf'), x), Lt(x, -1.0)),
And(Lt(1.0, x), Lt(x, Float('+inf'))))
def simplify_and(
x: sympy.Basic,
gen: typing.Optional[sympy.Symbol] = None,
extra_conditions: typing.Optional[sympy.Basic] = True) -> sympy.Basic:
"""
Some rules, because SymPy currently does not automatically simplify them...
"""
assert isinstance(x, sympy.Basic), "type x: %r" % type(x)
from sympy.solvers.inequalities import reduce_rational_inequalities
from sympy.core.relational import Relational
syms = []
if gen is not None:
syms.append(gen)
w1 = sympy.Wild("w1")
w2 = sympy.Wild("w2")
for sub_expr in x.find(sympy.Eq(w1, w2)):
m = sub_expr.match(sympy.Eq(w1, w2))
ws_ = m[w1], m[w2]
for w_ in ws_:
if isinstance(w_, sympy.Symbol) and w_ not in syms:
syms.append(w_)
for w_ in x.free_symbols:
if w_ not in syms:
syms.append(w_)
if len(syms) >= 1:
_c = syms[0]
if len(syms) >= 2:
n = syms[1]
else:
n = sympy.Wild("n")
else:
return x
x = x.replace(((_c - 2 * n >= -1) & (_c - 2 * n <= -1)),
sympy.Eq(_c, 2 * n - 1)) # probably not needed anymore...
apply_rules = True
while apply_rules:
apply_rules = False
for and_expr in x.find(sympy.And):
assert isinstance(and_expr, sympy.And)
and_expr_ = reduce_rational_inequalities([and_expr.args], _c)
# print(and_expr, "->", and_expr_)
if and_expr_ != and_expr:
x = x.replace(and_expr, and_expr_)
and_expr = and_expr_
if and_expr == sympy.sympify(False):
continue
if isinstance(and_expr, sympy.Rel):
continue
assert isinstance(and_expr, sympy.And)
and_expr_args = list(and_expr.args)
# for i, part in enumerate(and_expr_args):
# and_expr_args[i] = part.simplify()
if all([
isinstance(part, Relational) and _c in part.free_symbols
for part in and_expr_args
]):
# No equality, as that should have been resolved above.
rel_ops = ["==", ">=", "<="]
if not (_c.is_Integer or _c.assumptions0["integer"]):
rel_ops.extend(["<", ">"])
rhs_by_c = {op: [] for op in rel_ops}
for part in and_expr_args:
assert isinstance(part, Relational)
part = _solve_inequality(part, _c)
assert isinstance(part, Relational)
assert part.lhs == _c
rel_op, rhs = part.rel_op, part.rhs
if _c.is_Integer or _c.assumptions0["integer"]:
if rel_op == "<":
rhs = rhs - 1
rel_op = "<="
elif rel_op == ">":
rhs = rhs + 1
rel_op = ">="
assert rel_op in rhs_by_c, "x: %r, _c: %r, and expr: %r, part %r" % (
x, _c, and_expr, part)
other_rhs = rhs_by_c[rel_op]
assert isinstance(other_rhs, list)
need_to_add = True
for rhs_ in other_rhs:
cmp = Relational.ValidRelationOperator[rel_op](rhs,
rhs_)
if simplify_and(
sympy.And(sympy.Not(cmp),
extra_conditions)) == sympy.sympify(
False): # checks True...
other_rhs.remove(rhs_)
break
elif simplify_and(sympy.And(
cmp,
extra_conditions)) == sympy.sympify(False):
need_to_add = False
break
# else:
# raise NotImplementedError("cannot compare %r in %r; extra cond %r" % (cmp, and_expr, extra_conditions))
if need_to_add:
other_rhs.append(rhs)
if rhs_by_c[">="] and rhs_by_c["<="]:
all_false = False
for lhs in rhs_by_c[">="]:
for rhs in rhs_by_c["<="]:
if sympy.Lt(lhs, rhs) == sympy.sympify(False):
all_false = True
if sympy.Eq(lhs, rhs) == sympy.sympify(True):
rhs_by_c["=="].append(lhs)
if all_false:
x = x.replace(and_expr, False)
continue
if rhs_by_c["=="]:
all_false = False
while len(rhs_by_c["=="]) >= 2:
lhs, rhs = rhs_by_c["=="][:2]
if sympy.Eq(lhs, rhs) == sympy.sympify(False):
all_false = True
break
elif sympy.Eq(lhs, rhs) == sympy.sympify(True):
rhs_by_c["=="].pop(1)
else:
raise NotImplementedError(
"cannot cmp %r == %r. rhs_by_c %r" %
(lhs, rhs, rhs_by_c))
if all_false:
x = x.replace(and_expr, False)
continue
new_parts = [sympy.Eq(_c, rhs_by_c["=="][0])]
for op in rel_ops:
for part in rhs_by_c[op]:
new_parts.append(
Relational.ValidRelationOperator[op](
rhs_by_c["=="][0], part).simplify())
else: # no "=="
new_parts = []
for op in rel_ops:
for part in rhs_by_c[op]:
new_parts.append(
Relational.ValidRelationOperator[op](_c, part))
assert new_parts
and_expr_ = sympy.And(*new_parts)
# print(and_expr, "--->", and_expr_)
x = x.replace(and_expr, and_expr_)
and_expr = and_expr_
# Probably all the remaining hard-coded rules are not needed anymore with the more generic code above...
if sympy.Eq(_c, 2 * n) in and_expr.args:
if (_c - 2 * n <= -1) in and_expr.args:
x = x.replace(and_expr, False)
continue
if sympy.Eq(_c - 2 * n, -1) in and_expr.args:
x = x.replace(and_expr, False)
continue
if (_c - n <= -1) in and_expr.args:
x = x.replace(and_expr, False)
continue
if (_c >= n) in and_expr.args and (_c - n <= -1) in and_expr.args:
x = x.replace(and_expr, False)
continue
if sympy.Eq(_c - 2 * n, -1) in and_expr.args: # assume n>=1
if (_c >= n) in and_expr.args:
x = x.replace(
and_expr,
sympy.And(
*
[arg for arg in and_expr.args
if arg != (_c >= n)]))
apply_rules = True
break
if (_c - n >= -1) in and_expr.args:
x = x.replace(
and_expr,
sympy.And(*[
arg for arg in and_expr.args
if arg != (_c - n >= -1)
]))
apply_rules = True
break
if (_c >= n) in and_expr.args:
if (_c - n >= -1) in and_expr.args:
x = x.replace(
and_expr,
sympy.And(*[
arg for arg in and_expr.args
if arg != (_c - n >= -1)
]))
apply_rules = True
break
if (_c - n >= -1) in and_expr.args and (_c - n <=
-1) in and_expr.args:
args = list(and_expr.args)
args.remove((_c - n >= -1))
args.remove((_c - n <= -1))
args.append(sympy.Eq(_c - n, -1))
if (_c - 2 * n <= -1) in args:
args.remove((_c - 2 * n <= -1))
x = x.replace(and_expr, sympy.And(*args))
apply_rules = True
break
return x
def test_reduce_poly_inequalities_complex_relational():
cond = Eq(im(x), 0)
assert reduce_rational_inequalities([[Eq(x**2, 0)]], x,
relational=True) == And(
Eq(re(x), 0), cond)
assert reduce_rational_inequalities([[Le(x**2, 0)]], x,
relational=True) == And(
Eq(re(x), 0), cond)
assert reduce_rational_inequalities([[Lt(x**2, 0)]], x,
relational=True) is False
assert reduce_rational_inequalities([[Ge(x**2, 0)]], x,
relational=True) == cond
assert reduce_rational_inequalities([[Gt(x**2, 0)]], x,
relational=True) == And(
Or(Lt(re(x), 0), Lt(0, re(x))),
cond)
assert reduce_rational_inequalities([[Ne(x**2, 0)]], x,
relational=True) == And(
Or(Lt(re(x), 0), Lt(0, re(x))),
cond)
assert reduce_rational_inequalities([[Eq(x**2, 1)]], x,
relational=True) == And(
Or(Eq(re(x), -1), Eq(re(x), 1)),
cond)
assert reduce_rational_inequalities([[Le(x**2, 1)]], x,
relational=True) == And(
And(Le(-1, re(x)), Le(re(x), 1)),
cond)
assert reduce_rational_inequalities([[Lt(x**2, 1)]], x,
relational=True) == And(
And(Lt(-1, re(x)), Lt(re(x), 1)),
cond)
assert reduce_rational_inequalities([[Ge(x**2, 1)]], x,
relational=True) == And(
Or(Le(re(x), -1), Le(1, re(x))),
cond)
assert reduce_rational_inequalities([[Gt(x**2, 1)]], x,
relational=True) == And(
Or(Lt(re(x), -1), Lt(1, re(x))),
cond)
assert reduce_rational_inequalities(
[[Ne(x**2, 1)]], x, relational=True) == And(
Or(Lt(re(x), -1), And(Lt(-1, re(x)), Lt(re(x), 1)), Lt(1, re(x))),
cond)
assert reduce_rational_inequalities([[Le(x**2, 1.0)]], x,
relational=True) == And(
And(Le(-1.0, re(x)),
Le(re(x), 1.0)), cond)
assert reduce_rational_inequalities([[Lt(x**2, 1.0)]], x,
relational=True) == And(
And(Lt(-1.0, re(x)),
Lt(re(x), 1.0)), cond)
assert reduce_rational_inequalities([[Ge(x**2, 1.0)]], x,
relational=True) == And(
Or(Le(re(x), -1.0),
Le(1.0, re(x))), cond)
assert reduce_rational_inequalities([[Gt(x**2, 1.0)]], x,
relational=True) == And(
Or(Lt(re(x), -1.0),
Lt(1.0, re(x))), cond)
assert reduce_rational_inequalities([[Ne(x**2, 1.0)]], x,
relational=True) == And(
Or(
Lt(re(x), -1.0),
And(Lt(-1.0, re(x)),
Lt(re(x), 1.0)),
Lt(1.0, re(x))), cond)
def test_reduce_poly_inequalities_real_interval():
with assuming(Q.real(x), Q.real(y)):
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) == Interval(
-oo, oo)
assert reduce_rational_inequalities(
[[Gt(x**2, 0)]], x, relational=False) == FiniteSet(0).complement
assert reduce_rational_inequalities(
[[Ne(x**2, 0)]], x, relational=False) == FiniteSet(0).complement
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),
Interval(1, oo))
assert reduce_rational_inequalities([[Gt(x**2, 1)]],
x,
relational=False) == Interval(
-1, 1).complement
assert reduce_rational_inequalities([[Ne(x**2, 1)]],
x,
relational=False) == FiniteSet(
-1, 1).complement
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),
Interval(1.0, inf))
assert reduce_rational_inequalities([[Gt(x**2, 1.0)]],
x,
relational=False) == Union(
Interval(-inf,
-1.0,
right_open=True),
Interval(1.0,
inf,
left_open=True))
assert reduce_rational_inequalities([[Ne(x**2, 1.0)]],
x,
relational=False) == FiniteSet(
-1.0, 1.0).complement
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))
def test_reduce_poly_inequalities_real_relational():
with assuming(Q.real(x), Q.real(y)):
assert reduce_rational_inequalities([[Eq(x**2, 0)]],
x,
relational=True) == Eq(x, 0)
assert reduce_rational_inequalities([[Le(x**2, 0)]],
x,
relational=True) == Eq(x, 0)
assert reduce_rational_inequalities(
[[Lt(x**2, 0)]], x, relational=True) is False
assert reduce_rational_inequalities(
[[Ge(x**2, 0)]], x, relational=True) is True
assert reduce_rational_inequalities([[Gt(x**2, 0)]],
x,
relational=True) == Or(
Lt(x, 0), Lt(0, x))
assert reduce_rational_inequalities([[Ne(x**2, 0)]],
x,
relational=True) == Or(
Lt(x, 0), Lt(0, x))
assert reduce_rational_inequalities([[Eq(x**2, 1)]],
x,
relational=True) == Or(
Eq(x, -1), Eq(x, 1))
assert reduce_rational_inequalities([[Le(x**2, 1)]],
x,
relational=True) == And(
Le(-1, x), Le(x, 1))
assert reduce_rational_inequalities([[Lt(x**2, 1)]],
x,
relational=True) == And(
Lt(-1, x), Lt(x, 1))
assert reduce_rational_inequalities([[Ge(x**2, 1)]],
x,
relational=True) == Or(
Le(x, -1), Le(1, x))
assert reduce_rational_inequalities([[Gt(x**2, 1)]],
x,
relational=True) == Or(
Lt(x, -1), Lt(1, x))
assert reduce_rational_inequalities([[Ne(x**2, 1)]],
x,
relational=True) == Or(
Lt(x, -1),
And(Lt(-1, x), Lt(x, 1)),
Lt(1, x))
assert reduce_rational_inequalities([[Le(x**2, 1.0)]],
x,
relational=True) == And(
Le(-1.0, x), Le(x, 1.0))
assert reduce_rational_inequalities([[Lt(x**2, 1.0)]],
x,
relational=True) == And(
Lt(-1.0, x), Lt(x, 1.0))
assert reduce_rational_inequalities([[Ge(x**2, 1.0)]],
x,
relational=True) == Or(
Le(x, -1.0), Le(1.0, x))
assert reduce_rational_inequalities([[Gt(x**2, 1.0)]],
x,
relational=True) == Or(
Lt(x, -1.0), Lt(1.0, x))
assert reduce_rational_inequalities([[Ne(x**2, 1.0)]],
x,
relational=True) == Or(
Lt(x, -1.0),
And(Lt(-1.0, x), Lt(x, 1.0)),
Lt(1.0, x))
def test_reduce_poly_inequalities_complex_relational():
assert reduce_rational_inequalities([[Eq(x**2, 0)]], x,
relational=True) == Eq(x, 0)
assert reduce_rational_inequalities([[Le(x**2, 0)]], x,
relational=True) == Eq(x, 0)
assert reduce_rational_inequalities([[Lt(x**2, 0)]], x,
relational=True) == False
assert reduce_rational_inequalities([[Ge(x**2, 0)]], x,
relational=True) == And(
Lt(-oo, x), Lt(x, oo))
assert reduce_rational_inequalities(
[[Gt(x**2, 0)]], x, relational=True) == \
And(Gt(x, -oo), Lt(x, oo), Ne(x, 0))
assert reduce_rational_inequalities(
[[Ne(x**2, 0)]], x, relational=True) == \
And(Gt(x, -oo), Lt(x, oo), Ne(x, 0))
for one in (S(1), S(1.0)):
inf = one * oo
assert reduce_rational_inequalities(
[[Eq(x**2, one)]], x, relational=True) == \
Or(Eq(x, -one), Eq(x, one))
assert reduce_rational_inequalities(
[[Le(x**2, one)]], x, relational=True) == \
And(And(Le(-one, x), Le(x, one)))
assert reduce_rational_inequalities(
[[Lt(x**2, one)]], x, relational=True) == \
And(And(Lt(-one, x), Lt(x, one)))
assert reduce_rational_inequalities(
[[Ge(x**2, one)]], x, relational=True) == \
And(Or(And(Le(one, x), Lt(x, inf)), And(Le(x, -one), Lt(-inf, x))))
assert reduce_rational_inequalities(
[[Gt(x**2, one)]], x, relational=True) == \
And(Or(And(Lt(-inf, x), Lt(x, -one)), And(Lt(one, x), Lt(x, inf))))
assert reduce_rational_inequalities(
[[Ne(x**2, one)]], x, relational=True) == \
Or(And(Lt(-inf, x), Lt(x, -one)),
And(Lt(-one, x), Lt(x, one)),
And(Lt(one, x), Lt(x, inf)))
def test_reduce_poly_inequalities_complex_relational():
cond = Eq(im(x), 0)
assert reduce_rational_inequalities(
[[Eq(x**2, 0)]], x, relational=True) == And(Eq(re(x), 0), cond)
assert reduce_rational_inequalities(
[[Le(x**2, 0)]], x, relational=True) == And(Eq(re(x), 0), cond)
assert reduce_rational_inequalities(
[[Lt(x**2, 0)]], x, relational=True) == False
assert reduce_rational_inequalities(
[[Ge(x**2, 0)]], x, relational=True) == cond
assert reduce_rational_inequalities([[Gt(x**2, 0)]], x, relational=True) == \
And(Or(Lt(re(x), 0), Gt(re(x), 0)), cond)
assert reduce_rational_inequalities([[Ne(x**2, 0)]], x, relational=True) == \
And(Or(Lt(re(x), 0), Gt(re(x), 0)), cond)
assert reduce_rational_inequalities([[Eq(x**2, 1)]], x, relational=True) == \
And(Or(Eq(re(x), -1), Eq(re(x), 1)), cond)
assert reduce_rational_inequalities([[Le(x**2, 1)]], x, relational=True) == \
And(And(Le(-1, re(x)), Le(re(x), 1)), cond)
assert reduce_rational_inequalities([[Lt(x**2, 1)]], x, relational=True) == \
And(And(Lt(-1, re(x)), Lt(re(x), 1)), cond)
assert reduce_rational_inequalities([[Ge(x**2, 1)]], x, relational=True) == \
And(Or(Le(re(x), -1), Ge(re(x), 1)), cond)
assert reduce_rational_inequalities([[Gt(x**2, 1)]], x, relational=True) == \
And(Or(Lt(re(x), -1), Gt(re(x), 1)), cond)
assert reduce_rational_inequalities([[Ne(x**2, 1)]], x, relational=True) == \
And(Or(Lt(re(x), -1), And(Lt(-1, re(x)), Lt(re(x), 1)), Gt(re(x), 1)), cond)
assert reduce_rational_inequalities([[Le(x**2, 1.0)]], x, relational=True) == \
And(And(Le(-1.0, re(x)), Le(re(x), 1.0)), cond)
assert reduce_rational_inequalities([[Lt(x**2, 1.0)]], x, relational=True) == \
And(And(Lt(-1.0, re(x)), Lt(re(x), 1.0)), cond)
assert reduce_rational_inequalities([[Ge(x**2, 1.0)]], x, relational=True) == \
And(Or(Le(re(x), -1.0), Ge(re(x), 1.0)), cond)
assert reduce_rational_inequalities([[Gt(x**2, 1.0)]], x, relational=True) == \
And(Or(Lt(re(x), -1.0), Gt(re(x), 1.0)), cond)
assert reduce_rational_inequalities([[Ne(x**2, 1.0)]], x, relational=True) == \
And(Or(Lt(re(x), -1.0), And(Lt(-1.0, re(x)), Lt(re(x), 1.0)), Gt(re(x), 1.0)), cond)
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_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), Interval(1, oo))
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), Interval(1.0, inf))
assert reduce_rational_inequalities(
[[Gt(x**2, 1.0)]], x, relational=False) == \
Union(Interval(-inf, -1.0, right_open=True),
Interval(1.0, inf, left_open=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))
assert reduce_rational_inequalities([[Lt(x**2, -1.)]], x) is S.false
def test_reduce_poly_inequalities_complex_relational():
cond = Eq(im(x), 0)
assert reduce_rational_inequalities(
[[Eq(x**2, 0)]], x, relational=True) == And(Eq(re(x), 0), cond)
assert reduce_rational_inequalities(
[[Le(x**2, 0)]], x, relational=True) == And(Eq(re(x), 0), cond)
assert reduce_rational_inequalities(
[[Lt(x**2, 0)]], x, relational=True) == False
assert reduce_rational_inequalities(
[[Ge(x**2, 0)]], x, relational=True) == And(cond, Lt(-oo, re(x)), Lt(re(x), oo))
assert reduce_rational_inequalities([[Gt(x**2, 0)]], x, relational=True) == \
And(cond, Or(And(Lt(-oo, re(x)), Lt(re(x), 0)), And(Lt(0, re(x)), Lt(re(x), oo))))
assert reduce_rational_inequalities([[Ne(x**2, 0)]], x, relational=True) == \
And(cond, Or(And(Lt(-oo, re(x)), Lt(re(x), 0)), And(Lt(0, re(x)), Lt(re(x), oo))))
assert reduce_rational_inequalities([[Eq(x**2, 1)]], x, relational=True) == \
And(Or(Eq(re(x), -1), Eq(re(x), 1)), cond)
assert reduce_rational_inequalities([[Le(x**2, 1)]], x, relational=True) == \
And(And(Le(-1, re(x)), Le(re(x), 1)), cond)
assert reduce_rational_inequalities([[Lt(x**2, 1)]], x, relational=True) == \
And(And(Lt(-1, re(x)), Lt(re(x), 1)), cond)
assert reduce_rational_inequalities([[Ge(x**2, 1)]], x, relational=True) == \
And(cond, Or(And(Le(1, re(x)), Lt(re(x), oo)), And(Le(re(x), -1), Lt(-oo, re(x)))))
assert reduce_rational_inequalities([[Gt(x**2, 1)]], x, relational=True) == \
And(cond, Or(And(Lt(-oo, re(x)), Lt(re(x), -1)), And(Lt(1, re(x)), Lt(re(x), oo))))
assert reduce_rational_inequalities([[Ne(x**2, 1)]], x, relational=True) == \
And(cond, Or(And(Lt(-oo, re(x)), Lt(re(x), -1)),
And(Lt(-1, re(x)), Lt(re(x), 1)), And(Lt(1, re(x)), Lt(re(x), oo))))
assert reduce_rational_inequalities([[Le(x**2, 1.0)]], x, relational=True) == \
And(And(Le(-1.0, re(x)), Le(re(x), 1.0)), cond)
assert reduce_rational_inequalities([[Lt(x**2, 1.0)]], x, relational=True) == \
And(And(Lt(-1.0, re(x)), Lt(re(x), 1.0)), cond)
assert reduce_rational_inequalities([[Ge(x**2, 1.0)]], x, relational=True) == \
And(cond, Or(And(Le(1.0, re(x)), re(x) < Float('+inf')),
And(Le(re(x), -1.0), Float('-inf') < re(x))))
assert reduce_rational_inequalities([[Gt(x**2, 1.0)]], x, relational=True) == \
And(cond, Or(And(Float('-inf') < re(x), re(x) < -1.0),
And(Lt(1.0, re(x)), re(x) < Float('+inf'))))
assert reduce_rational_inequalities([[Ne(x**2, 1.0)]], x, relational=True) == \
And(cond, Or(And(Float('-inf') < re(x), Lt(re(x), -1.0)),
And(Lt(-1.0, re(x)), re(x) < 1.0), And(Lt(1.0, re(x)), re(x) < Float('+inf'))))
