def test_construct_domain():
assert construct_domain([1, 2, 3]) == (ZZ, [ZZ(1), ZZ(2), ZZ(3)])
assert construct_domain([1, 2, 3],
field=True) == (QQ, [QQ(1), QQ(2),
QQ(3)])
assert construct_domain([Integer(1), Integer(2),
Integer(3)]) == (ZZ, [ZZ(1), ZZ(2),
ZZ(3)])
assert construct_domain(
[Integer(1), Integer(2), Integer(3)],
field=True) == (QQ, [QQ(1), QQ(2), QQ(3)])
assert construct_domain([Rational(1, 2),
Integer(2)]) == (QQ, [QQ(1, 2), QQ(2)])
assert construct_domain([3.14, 1, Rational(1, 2)
]) == (RR, [RR(3.14), RR(1.0),
RR(0.5)])
assert construct_domain([3.14, sqrt(2)],
extension=None) == (EX, [EX(3.14),
EX(sqrt(2))])
assert construct_domain([3.14, sqrt(2)],
extension=True) == (EX, [EX(3.14),
EX(sqrt(2))])
assert construct_domain([sqrt(2), 3.14],
extension=True) == (EX, [EX(sqrt(2)),
EX(3.14)])
assert construct_domain([1, sqrt(2)],
extension=None) == (EX, [EX(1), EX(sqrt(2))])
assert construct_domain([x, sqrt(x)]) == (EX, [EX(x), EX(sqrt(x))])
assert construct_domain([x, sqrt(x), sqrt(y)
]) == (EX, [EX(x),
EX(sqrt(x)),
EX(sqrt(y))])
alg = QQ.algebraic_field(sqrt(2))
assert (construct_domain(
[7, Rational(1, 2), sqrt(2)],
extension=True) == (alg, [alg(7),
alg(Rational(1, 2)),
alg(sqrt(2))]))
alg = QQ.algebraic_field(sqrt(2) + sqrt(3))
assert (construct_domain([7, sqrt(2), sqrt(3)], extension=True) == (alg, [
alg(7), alg(sqrt(2)), alg(sqrt(3))
]))
dom = ZZ.poly_ring(x)
assert construct_domain([2 * x, 3]) == (dom, [dom(2 * x), dom(3)])
dom = ZZ.poly_ring(x, y)
assert construct_domain([2 * x, 3 * y]) == (dom, [dom(2 * x), dom(3 * y)])
dom = QQ.poly_ring(x)
assert construct_domain([x / 2, 3]) == (dom, [dom(x / 2), dom(3)])
dom = QQ.poly_ring(x, y)
assert construct_domain([x / 2, 3 * y]) == (dom, [dom(x / 2), dom(3 * y)])
dom = RR.poly_ring(x)
assert construct_domain([x / 2, 3.5]) == (dom, [dom(x / 2), dom(3.5)])
dom = RR.poly_ring(x, y)
assert construct_domain([x / 2,
3.5 * y]) == (dom, [dom(x / 2),
dom(3.5 * y)])
dom = ZZ.frac_field(x)
assert construct_domain([2 / x, 3]) == (dom, [dom(2 / x), dom(3)])
dom = ZZ.frac_field(x, y)
assert construct_domain([2 / x, 3 * y]) == (dom, [dom(2 / x), dom(3 * y)])
dom = RR.frac_field(x)
assert construct_domain([2 / x, 3.5]) == (dom, [dom(2 / x), dom(3.5)])
dom = RR.frac_field(x, y)
assert construct_domain([2 / x,
3.5 * y]) == (dom, [dom(2 / x),
dom(3.5 * y)])
assert construct_domain(2) == (ZZ, ZZ(2))
assert construct_domain(Rational(2, 3)) == (QQ, QQ(2, 3))
assert construct_domain({}) == (ZZ, {})
def test_Domain_unify():
F3 = GF(3)
assert unify(F3, F3) == F3
assert unify(F3, ZZ) == F3
assert unify(F3, QQ) == QQ
assert unify(F3, ALG) == ALG
assert unify(F3, RR) == RR
assert unify(F3, CC) == CC
assert unify(F3, ZZ.poly_ring(x)) == F3.poly_ring(x)
assert unify(F3, ZZ.frac_field(x)) == F3.frac_field(x)
assert unify(F3, EX) == EX
assert unify(ZZ, F3) == F3
assert unify(ZZ, ZZ) == ZZ
assert unify(ZZ, QQ) == QQ
assert unify(ZZ, ALG) == ALG
assert unify(ZZ, RR) == RR
assert unify(ZZ, CC) == CC
assert unify(ZZ, ZZ.poly_ring(x)) == ZZ.poly_ring(x)
assert unify(ZZ, ZZ.frac_field(x)) == ZZ.frac_field(x)
assert unify(ZZ, EX) == EX
assert unify(QQ, F3) == QQ
assert unify(QQ, ZZ) == QQ
assert unify(QQ, QQ) == QQ
assert unify(QQ, ALG) == ALG
assert unify(QQ, RR) == RR
assert unify(QQ, CC) == CC
assert unify(QQ, ZZ.poly_ring(x)) == QQ.poly_ring(x)
assert unify(QQ, ZZ.frac_field(x)) == QQ.frac_field(x)
assert unify(QQ, EX) == EX
assert unify(RR, F3) == RR
assert unify(RR, ZZ) == RR
assert unify(RR, QQ) == RR
assert unify(RR, ALG) == RR
assert unify(RR, RR) == RR
assert unify(RR, CC) == CC
assert unify(RR, ZZ.poly_ring(x)) == RR.poly_ring(x)
assert unify(RR, ZZ.frac_field(x)) == RR.frac_field(x)
assert unify(RR, EX) == EX
assert unify(CC, F3) == CC
assert unify(CC, ZZ) == CC
assert unify(CC, QQ) == CC
assert unify(CC, ALG) == CC
assert unify(CC, RR) == CC
assert unify(CC, CC) == CC
assert unify(CC, ZZ.poly_ring(x)) == CC.poly_ring(x)
assert unify(CC, ZZ.frac_field(x)) == CC.frac_field(x)
assert unify(CC, EX) == EX
CC2 = ComplexField(prec=20)
assert unify(CC, CC2) == unify(CC2, CC) == ComplexField(prec=CC.precision,
tol=CC2.tolerance)
RR2 = RealField(prec=20)
assert unify(RR, RR2) == unify(RR2, RR) == RealField(prec=RR.precision,
tol=RR2.tolerance)
assert unify(ZZ.poly_ring(x), F3) == F3.poly_ring(x)
assert unify(ZZ.poly_ring(x), ZZ) == ZZ.poly_ring(x)
assert unify(ZZ.poly_ring(x), QQ) == QQ.poly_ring(x)
assert unify(ZZ.poly_ring(x), ALG) == ALG.poly_ring(x)
assert unify(ZZ.poly_ring(x), RR) == RR.poly_ring(x)
assert unify(ZZ.poly_ring(x), CC) == CC.poly_ring(x)
assert unify(ZZ.poly_ring(x), ZZ.poly_ring(x)) == ZZ.poly_ring(x)
assert unify(ZZ.poly_ring(x), ZZ.frac_field(x)) == ZZ.frac_field(x)
assert unify(ZZ.poly_ring(x), EX) == EX
assert unify(ZZ.frac_field(x), F3) == F3.frac_field(x)
assert unify(ZZ.frac_field(x), ZZ) == ZZ.frac_field(x)
assert unify(ZZ.frac_field(x), QQ) == QQ.frac_field(x)
assert unify(ZZ.frac_field(x), ALG) == ALG.frac_field(x)
assert unify(ZZ.frac_field(x), RR) == RR.frac_field(x)
assert unify(ZZ.frac_field(x), CC) == CC.frac_field(x)
assert unify(ZZ.frac_field(x), ZZ.poly_ring(x)) == ZZ.frac_field(x)
assert unify(ZZ.frac_field(x), ZZ.frac_field(x)) == ZZ.frac_field(x)
assert unify(ZZ.frac_field(x), EX) == EX
assert unify(EX, F3) == EX
assert unify(EX, ZZ) == EX
assert unify(EX, QQ) == EX
assert unify(EX, ALG) == EX
assert unify(EX, RR) == EX
assert unify(EX, CC) == EX
assert unify(EX, ZZ.poly_ring(x)) == EX
assert unify(EX, ZZ.frac_field(x)) == EX
assert unify(EX, EX) == EX
def test_Domain_unify():
F3 = GF(3)
assert unify(F3, F3) == F3
assert unify(F3, ZZ) == F3
assert unify(F3, QQ) == QQ
assert unify(F3, ALG) == ALG
assert unify(F3, RR) == RR
assert unify(F3, CC) == CC
assert unify(F3, ZZ.poly_ring(x)) == F3.poly_ring(x)
assert unify(F3, ZZ.frac_field(x)) == F3.frac_field(x)
assert unify(F3, EX) == EX
assert unify(ZZ, F3) == F3
assert unify(ZZ, ZZ) == ZZ
assert unify(ZZ, QQ) == QQ
assert unify(ZZ, ALG) == ALG
assert unify(ZZ, RR) == RR
assert unify(ZZ, CC) == CC
assert unify(ZZ, ZZ.poly_ring(x)) == ZZ.poly_ring(x)
assert unify(ZZ, ZZ.frac_field(x)) == ZZ.frac_field(x)
assert unify(ZZ, EX) == EX
assert unify(QQ, F3) == QQ
assert unify(QQ, ZZ) == QQ
assert unify(QQ, QQ) == QQ
assert unify(QQ, ALG) == ALG
assert unify(QQ, RR) == RR
assert unify(QQ, CC) == CC
assert unify(QQ, ZZ.poly_ring(x)) == QQ.poly_ring(x)
assert unify(QQ, ZZ.frac_field(x)) == QQ.frac_field(x)
assert unify(QQ, EX) == EX
assert unify(RR, F3) == RR
assert unify(RR, ZZ) == RR
assert unify(RR, QQ) == RR
assert unify(RR, ALG) == RR
assert unify(RR, RR) == RR
assert unify(RR, CC) == CC
assert unify(RR, ZZ.poly_ring(x)) == RR.poly_ring(x)
assert unify(RR, ZZ.frac_field(x)) == RR.frac_field(x)
assert unify(RR, EX) == EX
assert unify(CC, F3) == CC
assert unify(CC, ZZ) == CC
assert unify(CC, QQ) == CC
assert unify(CC, ALG) == CC
assert unify(CC, RR) == CC
assert unify(CC, CC) == CC
assert unify(CC, ZZ.poly_ring(x)) == CC.poly_ring(x)
assert unify(CC, ZZ.frac_field(x)) == CC.frac_field(x)
assert unify(CC, EX) == EX
CC2 = ComplexField(prec=20)
assert unify(CC, CC2) == unify(CC2, CC) == ComplexField(prec=CC.precision,
tol=CC2.tolerance)
RR2 = RealField(prec=20)
assert unify(RR, RR2) == unify(RR2, RR) == RealField(prec=RR.precision,
tol=RR2.tolerance)
assert unify(ZZ.poly_ring(x), F3) == F3.poly_ring(x)
assert unify(ZZ.poly_ring(x), ZZ) == ZZ.poly_ring(x)
assert unify(ZZ.poly_ring(x), QQ) == QQ.poly_ring(x)
assert unify(ZZ.poly_ring(x), ALG) == ALG.poly_ring(x)
assert unify(ZZ.poly_ring(x), RR) == RR.poly_ring(x)
assert unify(ZZ.poly_ring(x), CC) == CC.poly_ring(x)
assert unify(ZZ.poly_ring(x), ZZ.poly_ring(x)) == ZZ.poly_ring(x)
assert unify(ZZ.poly_ring(x), ZZ.frac_field(x)) == ZZ.frac_field(x)
assert unify(ZZ.poly_ring(x), EX) == EX
assert unify(ZZ.frac_field(x), F3) == F3.frac_field(x)
assert unify(ZZ.frac_field(x), ZZ) == ZZ.frac_field(x)
assert unify(ZZ.frac_field(x), QQ) == QQ.frac_field(x)
assert unify(ZZ.frac_field(x), ALG) == ALG.frac_field(x)
assert unify(ZZ.frac_field(x), RR) == RR.frac_field(x)
assert unify(ZZ.frac_field(x), CC) == CC.frac_field(x)
assert unify(ZZ.frac_field(x), ZZ.poly_ring(x)) == ZZ.frac_field(x)
assert unify(ZZ.frac_field(x), ZZ.frac_field(x)) == ZZ.frac_field(x)
assert unify(ZZ.frac_field(x), EX) == EX
assert unify(EX, F3) == EX
assert unify(EX, ZZ) == EX
assert unify(EX, QQ) == EX
assert unify(EX, ALG) == EX
assert unify(EX, RR) == EX
assert unify(EX, CC) == EX
assert unify(EX, ZZ.poly_ring(x)) == EX
assert unify(EX, ZZ.frac_field(x)) == EX
assert unify(EX, EX) == EX
def test_Domain__contains__():
assert (0 in EX) is True
assert (0 in ZZ) is True
assert (0 in QQ) is True
assert (0 in RR) is True
assert (0 in CC) is True
assert (0 in ALG) is True
assert (0 in ZZ.poly_ring(x, y)) is True
assert (0 in QQ.poly_ring(x, y)) is True
assert (0 in RR.poly_ring(x, y)) is True
assert (-7 in EX) is True
assert (-7 in ZZ) is True
assert (-7 in QQ) is True
assert (-7 in RR) is True
assert (-7 in CC) is True
assert (-7 in ALG) is True
assert (-7 in ZZ.poly_ring(x, y)) is True
assert (-7 in QQ.poly_ring(x, y)) is True
assert (-7 in RR.poly_ring(x, y)) is True
assert (17 in EX) is True
assert (17 in ZZ) is True
assert (17 in QQ) is True
assert (17 in RR) is True
assert (17 in CC) is True
assert (17 in ALG) is True
assert (17 in ZZ.poly_ring(x, y)) is True
assert (17 in QQ.poly_ring(x, y)) is True
assert (17 in RR.poly_ring(x, y)) is True
assert (-Rational(1, 7) in EX) is True
assert (-Rational(1, 7) in ZZ) is False
assert (-Rational(1, 7) in QQ) is True
assert (-Rational(1, 7) in RR) is True
assert (-Rational(1, 7) in CC) is True
assert (-Rational(1, 7) in ALG) is True
assert (-Rational(1, 7) in ZZ.poly_ring(x, y)) is False
assert (-Rational(1, 7) in QQ.poly_ring(x, y)) is True
assert (-Rational(1, 7) in RR.poly_ring(x, y)) is True
assert (Rational(3, 5) in EX) is True
assert (Rational(3, 5) in ZZ) is False
assert (Rational(3, 5) in QQ) is True
assert (Rational(3, 5) in RR) is True
assert (Rational(3, 5) in CC) is True
assert (Rational(3, 5) in ALG) is True
assert (Rational(3, 5) in ZZ.poly_ring(x, y)) is False
assert (Rational(3, 5) in QQ.poly_ring(x, y)) is True
assert (Rational(3, 5) in RR.poly_ring(x, y)) is True
assert (3.0 in EX) is True
assert (3.0 in ZZ) is True
assert (3.0 in QQ) is True
assert (3.0 in RR) is True
assert (3.0 in CC) is True
assert (3.0 in ALG) is True
assert (3.0 in ZZ.poly_ring(x, y)) is True
assert (3.0 in QQ.poly_ring(x, y)) is True
assert (3.0 in RR.poly_ring(x, y)) is True
assert (3.14 in EX) is True
assert (3.14 in ZZ) is False
assert (3.14 in QQ) is True
assert (3.14 in RR) is True
assert (3.14 in CC) is True
assert (3.14 in ALG) is True
assert (3.14 in ZZ.poly_ring(x, y)) is False
assert (3.14 in QQ.poly_ring(x, y)) is True
assert (3.14 in RR.poly_ring(x, y)) is True
assert (oo in EX) is True
assert (oo in ZZ) is False
assert (oo in QQ) is False
assert (oo in RR) is True
assert (oo in CC) is True
assert (oo in ALG) is False
assert (oo in ZZ.poly_ring(x, y)) is False
assert (oo in QQ.poly_ring(x, y)) is False
assert (oo in RR.poly_ring(x, y)) is True
assert (-oo in EX) is True
assert (-oo in ZZ) is False
assert (-oo in QQ) is False
assert (-oo in RR) is True
assert (-oo in CC) is True
assert (-oo in ALG) is False
assert (-oo in ZZ.poly_ring(x, y)) is False
assert (-oo in QQ.poly_ring(x, y)) is False
assert (-oo in RR.poly_ring(x, y)) is True
assert (sqrt(7) in EX) is True
assert (sqrt(7) in ZZ) is False
assert (sqrt(7) in QQ) is False
assert (sqrt(7) in RR) is True
assert (sqrt(7) in CC) is True
assert (sqrt(7) in ALG) is False
assert (sqrt(7) in ZZ.poly_ring(x, y)) is False
assert (sqrt(7) in QQ.poly_ring(x, y)) is False
assert (sqrt(7) in RR.poly_ring(x, y)) is True
assert (2 * sqrt(3) + 1 in EX) is True
assert (2 * sqrt(3) + 1 in ZZ) is False
assert (2 * sqrt(3) + 1 in QQ) is False
assert (2 * sqrt(3) + 1 in RR) is True
assert (2 * sqrt(3) + 1 in CC) is True
assert (2 * sqrt(3) + 1 in ALG) is True
assert (2 * sqrt(3) + 1 in ZZ.poly_ring(x, y)) is False
assert (2 * sqrt(3) + 1 in QQ.poly_ring(x, y)) is False
assert (2 * sqrt(3) + 1 in RR.poly_ring(x, y)) is True
assert (sin(1) in EX) is True
assert (sin(1) in ZZ) is False
assert (sin(1) in QQ) is False
assert (sin(1) in RR) is True
assert (sin(1) in CC) is True
assert (sin(1) in ALG) is False
assert (sin(1) in ZZ.poly_ring(x, y)) is False
assert (sin(1) in QQ.poly_ring(x, y)) is False
assert (sin(1) in RR.poly_ring(x, y)) is True
assert (x**2 + 1 in EX) is True
assert (x**2 + 1 in ZZ) is False
assert (x**2 + 1 in QQ) is False
assert (x**2 + 1 in RR) is False
assert (x**2 + 1 in CC) is False
assert (x**2 + 1 in ALG) is False
assert (x**2 + 1 in ZZ.poly_ring(x)) is True
assert (x**2 + 1 in QQ.poly_ring(x)) is True
assert (x**2 + 1 in RR.poly_ring(x)) is True
assert (x**2 + 1 in ZZ.poly_ring(x, y)) is True
assert (x**2 + 1 in QQ.poly_ring(x, y)) is True
assert (x**2 + 1 in RR.poly_ring(x, y)) is True
assert (x**2 + y**2 in EX) is True
assert (x**2 + y**2 in ZZ) is False
assert (x**2 + y**2 in QQ) is False
assert (x**2 + y**2 in RR) is False
assert (x**2 + y**2 in CC) is False
assert (x**2 + y**2 in ALG) is False
assert (x**2 + y**2 in ZZ.poly_ring(x)) is False
assert (x**2 + y**2 in QQ.poly_ring(x)) is False
assert (x**2 + y**2 in RR.poly_ring(x)) is False
assert (x**2 + y**2 in ZZ.poly_ring(x, y)) is True
assert (x**2 + y**2 in QQ.poly_ring(x, y)) is True
assert (x**2 + y**2 in RR.poly_ring(x, y)) is True
assert (Rational(3, 2) * x / (y + 1) - z in QQ.poly_ring(x, y, z)) is False
def test_Domain__contains__():
assert (0 in EX) is True
assert (0 in ZZ) is True
assert (0 in QQ) is True
assert (0 in RR) is True
assert (0 in CC) is True
assert (0 in ALG) is True
assert (0 in ZZ.poly_ring(x, y)) is True
assert (0 in QQ.poly_ring(x, y)) is True
assert (0 in RR.poly_ring(x, y)) is True
assert (-7 in EX) is True
assert (-7 in ZZ) is True
assert (-7 in QQ) is True
assert (-7 in RR) is True
assert (-7 in CC) is True
assert (-7 in ALG) is True
assert (-7 in ZZ.poly_ring(x, y)) is True
assert (-7 in QQ.poly_ring(x, y)) is True
assert (-7 in RR.poly_ring(x, y)) is True
assert (17 in EX) is True
assert (17 in ZZ) is True
assert (17 in QQ) is True
assert (17 in RR) is True
assert (17 in CC) is True
assert (17 in ALG) is True
assert (17 in ZZ.poly_ring(x, y)) is True
assert (17 in QQ.poly_ring(x, y)) is True
assert (17 in RR.poly_ring(x, y)) is True
assert (-Rational(1, 7) in EX) is True
assert (-Rational(1, 7) in ZZ) is False
assert (-Rational(1, 7) in QQ) is True
assert (-Rational(1, 7) in RR) is True
assert (-Rational(1, 7) in CC) is True
assert (-Rational(1, 7) in ALG) is True
assert (-Rational(1, 7) in ZZ.poly_ring(x, y)) is False
assert (-Rational(1, 7) in QQ.poly_ring(x, y)) is True
assert (-Rational(1, 7) in RR.poly_ring(x, y)) is True
assert (Rational(3, 5) in EX) is True
assert (Rational(3, 5) in ZZ) is False
assert (Rational(3, 5) in QQ) is True
assert (Rational(3, 5) in RR) is True
assert (Rational(3, 5) in CC) is True
assert (Rational(3, 5) in ALG) is True
assert (Rational(3, 5) in ZZ.poly_ring(x, y)) is False
assert (Rational(3, 5) in QQ.poly_ring(x, y)) is True
assert (Rational(3, 5) in RR.poly_ring(x, y)) is True
assert (3.0 in EX) is True
assert (3.0 in ZZ) is True
assert (3.0 in QQ) is True
assert (3.0 in RR) is True
assert (3.0 in CC) is True
assert (3.0 in ALG) is True
assert (3.0 in ZZ.poly_ring(x, y)) is True
assert (3.0 in QQ.poly_ring(x, y)) is True
assert (3.0 in RR.poly_ring(x, y)) is True
assert (3.14 in EX) is True
assert (3.14 in ZZ) is False
assert (3.14 in QQ) is True
assert (3.14 in RR) is True
assert (3.14 in CC) is True
assert (3.14 in ALG) is True
assert (3.14 in ZZ.poly_ring(x, y)) is False
assert (3.14 in QQ.poly_ring(x, y)) is True
assert (3.14 in RR.poly_ring(x, y)) is True
assert (oo in EX) is True
assert (oo in ZZ) is False
assert (oo in QQ) is False
assert (oo in RR) is True
assert (oo in CC) is True
assert (oo in ALG) is False
assert (oo in ZZ.poly_ring(x, y)) is False
assert (oo in QQ.poly_ring(x, y)) is False
assert (oo in RR.poly_ring(x, y)) is True
assert (-oo in EX) is True
assert (-oo in ZZ) is False
assert (-oo in QQ) is False
assert (-oo in RR) is True
assert (-oo in CC) is True
assert (-oo in ALG) is False
assert (-oo in ZZ.poly_ring(x, y)) is False
assert (-oo in QQ.poly_ring(x, y)) is False
assert (-oo in RR.poly_ring(x, y)) is True
assert (sqrt(7) in EX) is True
assert (sqrt(7) in ZZ) is False
assert (sqrt(7) in QQ) is False
assert (sqrt(7) in RR) is True
assert (sqrt(7) in CC) is True
assert (sqrt(7) in ALG) is False
assert (sqrt(7) in ZZ.poly_ring(x, y)) is False
assert (sqrt(7) in QQ.poly_ring(x, y)) is False
assert (sqrt(7) in RR.poly_ring(x, y)) is True
assert (2*sqrt(3) + 1 in EX) is True
assert (2*sqrt(3) + 1 in ZZ) is False
assert (2*sqrt(3) + 1 in QQ) is False
assert (2*sqrt(3) + 1 in RR) is True
assert (2*sqrt(3) + 1 in CC) is True
assert (2*sqrt(3) + 1 in ALG) is True
assert (2*sqrt(3) + 1 in ZZ.poly_ring(x, y)) is False
assert (2*sqrt(3) + 1 in QQ.poly_ring(x, y)) is False
assert (2*sqrt(3) + 1 in RR.poly_ring(x, y)) is True
assert (sin(1) in EX) is True
assert (sin(1) in ZZ) is False
assert (sin(1) in QQ) is False
assert (sin(1) in RR) is True
assert (sin(1) in CC) is True
assert (sin(1) in ALG) is False
assert (sin(1) in ZZ.poly_ring(x, y)) is False
assert (sin(1) in QQ.poly_ring(x, y)) is False
assert (sin(1) in RR.poly_ring(x, y)) is True
assert (x**2 + 1 in EX) is True
assert (x**2 + 1 in ZZ) is False
assert (x**2 + 1 in QQ) is False
assert (x**2 + 1 in RR) is False
assert (x**2 + 1 in CC) is False
assert (x**2 + 1 in ALG) is False
assert (x**2 + 1 in ZZ.poly_ring(x)) is True
assert (x**2 + 1 in QQ.poly_ring(x)) is True
assert (x**2 + 1 in RR.poly_ring(x)) is True
assert (x**2 + 1 in ZZ.poly_ring(x, y)) is True
assert (x**2 + 1 in QQ.poly_ring(x, y)) is True
assert (x**2 + 1 in RR.poly_ring(x, y)) is True
assert (x**2 + y**2 in EX) is True
assert (x**2 + y**2 in ZZ) is False
assert (x**2 + y**2 in QQ) is False
assert (x**2 + y**2 in RR) is False
assert (x**2 + y**2 in CC) is False
assert (x**2 + y**2 in ALG) is False
assert (x**2 + y**2 in ZZ.poly_ring(x)) is False
assert (x**2 + y**2 in QQ.poly_ring(x)) is False
assert (x**2 + y**2 in RR.poly_ring(x)) is False
assert (x**2 + y**2 in ZZ.poly_ring(x, y)) is True
assert (x**2 + y**2 in QQ.poly_ring(x, y)) is True
assert (x**2 + y**2 in RR.poly_ring(x, y)) is True
assert (Rational(3, 2)*x/(y + 1) - z in QQ.poly_ring(x, y, z)) is False