def test_reduce_poly_inequalities_real_relational():
global_assumptions.add(Q.real(x))
global_assumptions.add(Q.real(y))
assert reduce_poly_inequalities([[Eq(x ** 2, 0)]], x, relational=True) == Eq(x, 0)
assert reduce_poly_inequalities([[Le(x ** 2, 0)]], x, relational=True) == Eq(x, 0)
assert reduce_poly_inequalities([[Lt(x ** 2, 0)]], x, relational=True) == False
assert reduce_poly_inequalities([[Ge(x ** 2, 0)]], x, relational=True) == True
assert reduce_poly_inequalities([[Gt(x ** 2, 0)]], x, relational=True) == Or(Lt(x, 0), Lt(0, x))
assert reduce_poly_inequalities([[Ne(x ** 2, 0)]], x, relational=True) == Or(Lt(x, 0), Lt(0, x))
assert reduce_poly_inequalities([[Eq(x ** 2, 1)]], x, relational=True) == Or(Eq(x, -1), Eq(x, 1))
assert reduce_poly_inequalities([[Le(x ** 2, 1)]], x, relational=True) == And(Le(-1, x), Le(x, 1))
assert reduce_poly_inequalities([[Lt(x ** 2, 1)]], x, relational=True) == And(Lt(-1, x), Lt(x, 1))
assert reduce_poly_inequalities([[Ge(x ** 2, 1)]], x, relational=True) == Or(Le(x, -1), Le(1, x))
assert reduce_poly_inequalities([[Gt(x ** 2, 1)]], x, relational=True) == Or(Lt(x, -1), Lt(1, x))
assert reduce_poly_inequalities([[Ne(x ** 2, 1)]], x, relational=True) == Or(
Lt(x, -1), And(Lt(-1, x), Lt(x, 1)), Lt(1, x)
)
assert (
reduce_poly_inequalities([[Eq(x ** 2, 1.0)]], x, relational=True).evalf() == Or(Eq(x, -1.0), Eq(x, 1.0)).evalf()
)
assert reduce_poly_inequalities([[Le(x ** 2, 1.0)]], x, relational=True) == And(Le(-1.0, x), Le(x, 1.0))
assert reduce_poly_inequalities([[Lt(x ** 2, 1.0)]], x, relational=True) == And(Lt(-1.0, x), Lt(x, 1.0))
assert reduce_poly_inequalities([[Ge(x ** 2, 1.0)]], x, relational=True) == Or(Le(x, -1.0), Le(1.0, x))
assert reduce_poly_inequalities([[Gt(x ** 2, 1.0)]], x, relational=True) == Or(Lt(x, -1.0), Lt(1.0, x))
assert reduce_poly_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)
)
global_assumptions.remove(Q.real(x))
global_assumptions.remove(Q.real(y))
def test_reduce_poly_inequalities_real_relational():
global_assumptions.add(Q.real(x))
global_assumptions.add(Q.real(y))
assert reduce_poly_inequalities([[Eq(x**2, 0)]], x, relational=True) == Eq(x, 0)
assert reduce_poly_inequalities([[Le(x**2, 0)]], x, relational=True) == Eq(x, 0)
assert reduce_poly_inequalities([[Lt(x**2, 0)]], x, relational=True) == False
assert reduce_poly_inequalities([[Ge(x**2, 0)]], x, relational=True) == True
assert reduce_poly_inequalities([[Gt(x**2, 0)]], x, relational=True) == Or(Lt(x, 0), Lt(0, x))
assert reduce_poly_inequalities([[Ne(x**2, 0)]], x, relational=True) == Or(Lt(x, 0), Lt(0, x))
assert reduce_poly_inequalities([[Eq(x**2, 1)]], x, relational=True) == Or(Eq(x, -1), Eq(x, 1))
assert reduce_poly_inequalities([[Le(x**2, 1)]], x, relational=True) == And(Le(-1, x), Le(x, 1))
assert reduce_poly_inequalities([[Lt(x**2, 1)]], x, relational=True) == And(Lt(-1, x), Lt(x, 1))
assert reduce_poly_inequalities([[Ge(x**2, 1)]], x, relational=True) == Or(Le(x, -1), Le(1, x))
assert reduce_poly_inequalities([[Gt(x**2, 1)]], x, relational=True) == Or(Lt(x, -1), Lt(1, x))
assert reduce_poly_inequalities([[Ne(x**2, 1)]], x, relational=True) == Or(Lt(x, -1), And(Lt(-1, x), Lt(x, 1)), Lt(1, x))
assert reduce_poly_inequalities([[Eq(x**2, 1.0)]], x, relational=True).evalf() == Or(Eq(x, -1.0), Eq(x, 1.0)).evalf()
assert reduce_poly_inequalities([[Le(x**2, 1.0)]], x, relational=True) == And(Le(-1.0, x), Le(x, 1.0))
assert reduce_poly_inequalities([[Lt(x**2, 1.0)]], x, relational=True) == And(Lt(-1.0, x), Lt(x, 1.0))
assert reduce_poly_inequalities([[Ge(x**2, 1.0)]], x, relational=True) == Or(Le(x, -1.0), Le(1.0, x))
assert reduce_poly_inequalities([[Gt(x**2, 1.0)]], x, relational=True) == Or(Lt(x, -1.0), Lt(1.0, x))
assert reduce_poly_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))
global_assumptions.remove(Q.real(x))
global_assumptions.remove(Q.real(y))
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))
def test_reduce_poly_inequalities_real_interval():
global_assumptions.add(x_assume)
global_assumptions.add(y_assume)
assert reduce_poly_inequalities([[Eq(x**2, 0)]], x, relational=False) == [Interval(0, 0)]
assert reduce_poly_inequalities([[Le(x**2, 0)]], x, relational=False) == [Interval(0, 0)]
assert reduce_poly_inequalities([[Lt(x**2, 0)]], x, relational=False) == []
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) == [Interval(-oo, 0, right_open=True), Interval(0, oo, left_open=True)]
assert reduce_poly_inequalities([[Ne(x**2, 0)]], x, relational=False) == [Interval(-oo, 0, right_open=True), Interval(0, oo, left_open=True)]
assert reduce_poly_inequalities([[Eq(x**2, 1)]], x, relational=False) == [Interval(-1,-1), Interval(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) == [Interval(-oo, -1), Interval(1, oo)]
assert reduce_poly_inequalities([[Gt(x**2, 1)]], x, relational=False) == [Interval(-oo, -1, right_open=True), Interval(1, oo, left_open=True)]
assert reduce_poly_inequalities([[Ne(x**2, 1)]], x, relational=False) == [Interval(-oo, -1, right_open=True), Interval(-1, 1, True, True), Interval(1, oo, left_open=True)]
assert reduce_poly_inequalities([[Eq(x**2, 1.0)]], x, relational=False) == [Interval(-1.0,-1.0), Interval(1.0, 1.0)]
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) == [Interval(-inf, -1.0), Interval(1.0, inf)]
assert reduce_poly_inequalities([[Gt(x**2, 1.0)]], x, relational=False) == [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) == [Interval(-inf, -1.0, right_open=True), Interval(-1.0, 1.0, True, True), Interval(1.0, inf, left_open=True)]
s = sqrt(2)
assert reduce_poly_inequalities([[Lt(x**2 - 1, 0), Gt(x**2 - 1, 0)]], x, relational=False) == []
assert reduce_poly_inequalities([[Le(x**2 - 1, 0), Ge(x**2 - 1, 0)]], x, relational=False) == [Interval(-1,-1), Interval(1, 1)]
assert reduce_poly_inequalities([[Le(x**2 - 2, 0), Ge(x**2 - 1, 0)]], x, relational=False) == [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) == [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) == [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) == [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) == [Interval(-s, -1, True, True), Interval(-1, 1, True, True), Interval(1, s, True, True)]
global_assumptions.remove(x_assume)
global_assumptions.remove(y_assume)
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))