def test_reduce_piecewise_inequalities():
e = abs(x - 5) < 3
ans = And(Lt(2, x), Lt(x, 8))
assert reduce_inequalities(e) == ans
assert reduce_inequalities(e, x) == ans
assert reduce_inequalities(abs(x - 5)) == Eq(x, 5)
assert reduce_inequalities(
abs(2*x + 3) >= 8) == Or(And(Le(Rational(5, 2), x), Lt(x, oo)),
And(Le(x, -Rational(11, 2)), Lt(-oo, x)))
assert reduce_inequalities(abs(x - 4) + abs(
3*x - 5) < 7) == And(Lt(Rational(1, 2), x), Lt(x, 4))
assert reduce_inequalities(abs(x - 4) + abs(3*abs(x) - 5) < 7) == \
Or(And(Integer(-2) < x, x < -1), And(Rational(1, 2) < x, x < 4))
nr = Symbol('nr', extended_real=False)
pytest.raises(TypeError, lambda: reduce_inequalities(abs(nr - 5) < 3))
# sympy/sympy#10198
assert reduce_inequalities(-1 + 1/abs(1/x - 1) < 0) == \
Or(And(Lt(0, x), x < Rational(1, 2)), And(-oo < x, x < 0))
# sympy/sympy#10255
assert reduce_inequalities(Piecewise((1, x < 1), (3, True)) > 1) == \
And(Le(1, x), x < oo)
assert reduce_inequalities(Piecewise((x**2, x < 0), (2*x, x >= 0)) < 1) == \
And(Lt(-1, x), x < Rational(1, 2))
def test_reduce_piecewise_inequalities():
e = abs(x - 5) < 3
ans = And(Lt(2, x), Lt(x, 8))
assert reduce_inequalities(e) == ans
assert reduce_inequalities(e, x) == ans
assert reduce_inequalities(abs(x - 5)) == Eq(x, 5)
assert reduce_inequalities(
abs(2*x + 3) >= 8) == Or(And(Le(Rational(5, 2), x), Lt(x, oo)),
And(Le(x, -Rational(11, 2)), Lt(-oo, x)))
assert reduce_inequalities(abs(x - 4) + abs(
3*x - 5) < 7) == And(Lt(Rational(1, 2), x), Lt(x, 4))
assert reduce_inequalities(abs(x - 4) + abs(3*abs(x) - 5) < 7) == \
Or(And(Integer(-2) < x, x < -1), And(Rational(1, 2) < x, x < 4))
nr = Symbol('nr', extended_real=False)
pytest.raises(TypeError, lambda: reduce_inequalities(abs(nr - 5) < 3))
# sympy/sympy#10198
assert reduce_inequalities(-1 + 1/abs(1/x - 1) < 0) == \
Or(And(S.Zero < x, x < S.Half), And(-oo < x, x < S.Zero))
# sympy/sympy#10255
assert reduce_inequalities(Piecewise((1, x < 1), (3, True)) > 1) == \
And(S.One <= x, x < oo)
assert reduce_inequalities(Piecewise((x**2, x < 0), (2*x, x >= 0)) < 1) == \
And(-S.One < x, x < S.Half)
def test_issue_8235():
assert reduce_inequalities(x**2 - 1 < 0) == \
And(Integer(-1) < x, x < Integer(1))
assert reduce_inequalities(x**2 - 1 <= 0) == \
And(Integer(-1) <= x, x <= 1)
assert reduce_inequalities(x**2 - 1 > 0) == \
Or(And(-oo < x, x < -1), And(x < oo, Integer(1) < x))
assert reduce_inequalities(x**2 - 1 >= 0) == \
Or(And(-oo < x, x <= Integer(-1)), And(Integer(1) <= x, x < oo))
eq = x**8 + x - 9 # we want RootOf solns here
sol = solve(eq >= 0)
tru = Or(And(RootOf(eq, 1) <= x, x < oo), And(-oo < x, x <= RootOf(eq, 0)))
assert sol == tru
# recast vanilla as real
assert solve(sqrt((-x + 1)**2) < 1) == And(Integer(0) < x, x < 2)
def test_sympyissue_8235():
assert reduce_inequalities(x**2 - 1 < 0) == \
And(Integer(-1) < x, x < Integer(1))
assert reduce_inequalities(x**2 - 1 <= 0) == \
And(Integer(-1) <= x, x <= 1)
assert reduce_inequalities(x**2 - 1 > 0) == \
Or(And(-oo < x, x < -1), And(x < oo, Integer(1) < x))
assert reduce_inequalities(x**2 - 1 >= 0) == \
Or(And(-oo < x, x <= Integer(-1)), And(Integer(1) <= x, x < oo))
eq = x**8 + x - 9 # we want RootOf solns here
sol = reduce_inequalities(eq >= 0)
tru = Or(And(RootOf(eq, 1) <= x, x < oo), And(-oo < x, x <= RootOf(eq, 0)))
assert sol == tru
# recast vanilla as real
assert reduce_inequalities(sqrt((-x + 1)**2) < 1) == And(Integer(0) < x, x < 2)
def test_reduce_inequalities_boolean():
assert reduce_inequalities(
[Eq(x**2, 0), True]) == Eq(x, 0)
assert reduce_inequalities([Eq(x**2, 0), False]) is S.false
def test_reduce_inequalities_general():
assert reduce_inequalities(Ge(sqrt(2)*x, 1)) == And(sqrt(2)/2 <= x, x < oo)
assert reduce_inequalities(PurePoly(x + 1, x) > 0) == And(Integer(-1) < x, x < oo)
def test_reduce_inequalities_general():
assert reduce_inequalities(Ge(sqrt(2)*x, 1)) == And(sqrt(2)/2 <= x, x < oo)
assert reduce_inequalities(PurePoly(x + 1, x) > 0) == And(Integer(-1) < x, x < oo)
def test_sympyissue_10196():
assert reduce_inequalities(x**2 >= 0)
assert reduce_inequalities(x**2 < 0) is S.false
def test_reduce_inequalities_boolean():
assert reduce_inequalities(
[Eq(x**2, 0), True]) == Eq(x, 0)
assert reduce_inequalities([Eq(x**2, 0), False]) is false
def test_sympyissue_6343():
eq = -3*x**2/2 - 45*x/4 + Rational(33, 2) > 0
assert reduce_inequalities(eq) == \
And(x < -Rational(15, 4) + sqrt(401)/4, -sqrt(401)/4 - Rational(15, 4) < x)
def test_slow_general_univariate():
r = RootOf(x**5 - x**2 + 1, 0)
assert reduce_inequalities(sqrt(x) + 1/root(x, 3) > 1) == \
Or(And(Integer(0) < x, x < r**6), And(r**6 < x, x < oo))
def test_reduce_inequalities_multivariate():
assert reduce_inequalities([Ge(x**2, 1), Ge(y**2, 1)]) == And(
Or(And(Le(1, x), Lt(x, oo)), And(Le(x, -1), Lt(-oo, x))),
Or(And(Le(1, y), Lt(y, oo)), And(Le(y, -1), Lt(-oo, y))))
def test_hacky_inequalities():
y = Symbol('y', real=True)
assert reduce_inequalities(x + y < 1, symbols=[x]) == And(-oo < x, x < -y + 1)
assert reduce_inequalities(x + y >= 1, symbols=[x]) == And(-y + 1 <= x, x < oo)
def test_sympyissue_6343():
eq = -3*x**2/2 - 45*x/4 + Rational(33, 2) > 0
assert reduce_inequalities(eq) == \
And(x < -Rational(15, 4) + sqrt(401)/4, -sqrt(401)/4 - Rational(15, 4) < x)
def test_sympyissue_10203():
y = Symbol('y', extended_real=True)
assert reduce_inequalities(Eq(0, x - y), symbols=[x]) == Eq(x, y)
assert reduce_inequalities(Ne(0, x - y), symbols=[x]) == \
Or(And(-oo < x, x < y), And(x < oo, y < x))
def test_hacky_inequalities():
y = Symbol('y', real=True)
assert reduce_inequalities(x + y < 1, symbols=[x]) == And(-oo < x, x < -y + 1)
assert reduce_inequalities(x + y >= 1, symbols=[x]) == And(-y + 1 <= x, x < oo)
def test_reduce_inequalities_errors():
pytest.raises(NotImplementedError, lambda: reduce_inequalities(Ge(sin(x) + x, 1)))
pytest.raises(NotImplementedError, lambda: reduce_inequalities(Ge(x**2*y + y, 1)))
def test_reduce_inequalities_multivariate():
assert reduce_inequalities([Ge(x**2, 1), Ge(y**2, 1)]) == And(
Or(And(Le(1, x), Lt(x, oo)), And(Le(x, -1), Lt(-oo, x))),
Or(And(Le(1, y), Lt(y, oo)), And(Le(y, -1), Lt(-oo, y))))
def test_reduce_inequalities_errors():
pytest.raises(NotImplementedError, lambda: reduce_inequalities(Ge(sin(x) + x, 1)))
pytest.raises(NotImplementedError, lambda: reduce_inequalities(Ge(x**2*y + y, 1)))
def test_sympyissue_5526():
assert reduce_inequalities(Integer(0) <=
x + Integral(y**2, (y, 1, 3)) - 1, [x]) == \
And(-Integral(y**2, (y, 1, 3)) + 1 <= x, x < oo)
def test_sympyissue_10203():
y = Symbol('y', extended_real=True)
assert reduce_inequalities(Eq(0, x - y), symbols=[x]) == Eq(x, y)
assert reduce_inequalities(Ne(0, x - y), symbols=[x]) == \
Or(And(-oo < x, x < y), And(x < oo, y < x))
def test_slow_general_univariate():
r = RootOf(x**5 - x**2 + 1, 0)
assert reduce_inequalities(sqrt(x) + 1/root(x, 3) > 1) == \
Or(And(Integer(0) < x, x < r**6), And(r**6 < x, x < oo))
def test_sympyissue_5526():
assert reduce_inequalities(Integer(0) <=
x + Integral(y**2, (y, 1, 3)) - 1, [x]) == \
And(-Integral(y**2, (y, 1, 3)) + 1 <= x, x < oo)
def test_sympyissue_8545():
eq = 1 - x - abs(1 - x)
ans = And(Lt(1, x), Lt(x, oo))
assert reduce_piecewise_inequality(eq, '<', x) == ans
eq = 1 - x - sqrt((1 - x)**2)
assert reduce_inequalities(eq < 0) == ans
def test_sympyissue_8545():
eq = 1 - x - abs(1 - x)
ans = And(Lt(1, x), Lt(x, oo))
assert reduce_piecewise_inequality(eq, '<', x) == ans
eq = 1 - x - sqrt((1 - x)**2)
assert reduce_inequalities(eq < 0) == ans
def test_sympyissue_10196():
assert reduce_inequalities(x**2 >= 0)
assert reduce_inequalities(x**2 < 0) is false
def test_sympyissue_10268():
assert reduce_inequalities(log(x) < 300) == And(-oo < x, x < E**300)
def test_sympyissue_10268():
assert reduce_inequalities(log(x) < 300) == And(-oo < x, x < E**300)
