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))