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():
raises(NotImplementedError, lambda: reduce_inequalities(Ge(sin(x) + x, 1)))
raises(NotImplementedError,
lambda: reduce_inequalities(Ge(x**2 * y + y, 1)))
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(
S(-1) < x, x < oo)
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_inequalities_multivariate():
assert reduce_inequalities([Ge(x**2, 1), Ge(y**2, 1)]) == \
And(And(Or(Le(re(x), -1), Le(1, re(x))), Eq(im(x), 0)),
And(Or(Le(re(y), -1), Le(1, re(y))), Eq(im(y), 0)))
def test_reduce_poly_inequalities_complex_relational():
cond = Eq(im(x), 0)
assert reduce_poly_inequalities([[Eq(x**2, 0)]], x,
relational=True) == And(
Eq(re(x), 0), cond)
assert reduce_poly_inequalities([[Le(x**2, 0)]], x,
relational=True) == And(
Eq(re(x), 0), cond)
assert reduce_poly_inequalities([[Lt(x**2, 0)]], x,
relational=True) == False
assert reduce_poly_inequalities([[Ge(x**2, 0)]], x,
relational=True) == cond
assert reduce_poly_inequalities([[Gt(x**2, 0)]], x,
relational=True) == And(
Or(Lt(re(x), 0), Lt(0, re(x))), cond)
assert reduce_poly_inequalities([[Ne(x**2, 0)]], x,
relational=True) == And(
Or(Lt(re(x), 0), Lt(0, re(x))), cond)
assert reduce_poly_inequalities([[Eq(x**2, 1)]], x,
relational=True) == And(
Or(Eq(re(x), -1), Eq(re(x), 1)), cond)
assert reduce_poly_inequalities([[Le(x**2, 1)]], x,
relational=True) == And(
And(Le(-1, re(x)), Le(re(x), 1)), cond)
assert reduce_poly_inequalities([[Lt(x**2, 1)]], x,
relational=True) == And(
And(Lt(-1, re(x)), Lt(re(x), 1)), cond)
assert reduce_poly_inequalities([[Ge(x**2, 1)]], x,
relational=True) == And(
Or(Le(re(x), -1), Le(1, re(x))), cond)
assert reduce_poly_inequalities([[Gt(x**2, 1)]], x,
relational=True) == And(
Or(Lt(re(x), -1), Lt(1, re(x))), cond)
assert reduce_poly_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_poly_inequalities([[Eq(x**2, 1.0)]], x,
relational=True).evalf() == And(
Or(Eq(re(x), -1.0), Eq(re(x), 1.0)),
cond)
assert reduce_poly_inequalities([[Le(x**2, 1.0)]], x,
relational=True) == And(
And(Le(-1.0, re(x)), Le(re(x), 1.0)),
cond)
assert reduce_poly_inequalities([[Lt(x**2, 1.0)]], x,
relational=True) == And(
And(Lt(-1.0, re(x)), Lt(re(x), 1.0)),
cond)
assert reduce_poly_inequalities([[Ge(x**2, 1.0)]], x,
relational=True) == And(
Or(Le(re(x), -1.0), Le(1.0, re(x))),
cond)
assert reduce_poly_inequalities([[Gt(x**2, 1.0)]], x,
relational=True) == And(
Or(Lt(re(x), -1.0), Lt(1.0, re(x))),
cond)
assert reduce_poly_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():
global_assumptions.add(Q.real(x))
global_assumptions.add(Q.real(y))
assert reduce_poly_inequalities([[Eq(x**2, 0)]], x,
relational=False) == FiniteSet(0)
assert reduce_poly_inequalities([[Le(x**2, 0)]], x,
relational=False) == FiniteSet(0)
assert reduce_poly_inequalities([[Lt(x**2, 0)]], x,
relational=False) == S.EmptySet
assert reduce_poly_inequalities([[Ge(x**2, 0)]], x,
relational=False) == Interval(-oo, oo)
assert reduce_poly_inequalities(
[[Gt(x**2, 0)]], x, relational=False) == FiniteSet(0).complement
assert reduce_poly_inequalities(
[[Ne(x**2, 0)]], x, relational=False) == FiniteSet(0).complement
assert reduce_poly_inequalities([[Eq(x**2, 1)]], x,
relational=False) == FiniteSet(-1, 1)
assert reduce_poly_inequalities([[Le(x**2, 1)]], x,
relational=False) == Interval(-1, 1)
assert reduce_poly_inequalities([[Lt(x**2, 1)]], x,
relational=False) == Interval(
-1, 1, True, True)
assert reduce_poly_inequalities([[Ge(x**2, 1)]], x,
relational=False) == Union(
Interval(-oo, -1), Interval(1, oo))
assert reduce_poly_inequalities([[Gt(x**2, 1)]], x,
relational=False) == Interval(-1,
1).complement
assert reduce_poly_inequalities(
[[Ne(x**2, 1)]], x, relational=False) == FiniteSet(-1, 1).complement
assert reduce_poly_inequalities([[Eq(x**2, 1.0)]],
x, relational=False) == FiniteSet(
-1.0, 1.0).evalf()
assert reduce_poly_inequalities([[Le(x**2, 1.0)]], x,
relational=False) == Interval(-1.0, 1.0)
assert reduce_poly_inequalities([[Lt(x**2, 1.0)]], x,
relational=False) == Interval(
-1.0, 1.0, True, True)
assert reduce_poly_inequalities([[Ge(x**2, 1.0)]], x,
relational=False) == Union(
Interval(-inf, -1.0),
Interval(1.0, inf))
assert reduce_poly_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_poly_inequalities([[Ne(x**2, 1.0)]],
x, relational=False) == FiniteSet(
-1.0, 1.0).complement
s = sqrt(2)
assert reduce_poly_inequalities(
[[Lt(x**2 - 1, 0), Gt(x**2 - 1, 0)]], x,
relational=False) == S.EmptySet
assert reduce_poly_inequalities(
[[Le(x**2 - 1, 0), Ge(x**2 - 1, 0)]], x,
relational=False) == FiniteSet(-1, 1)
assert reduce_poly_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_poly_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_poly_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_poly_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_poly_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))
global_assumptions.remove(Q.real(x))
global_assumptions.remove(Q.real(y))