def _richcmp_(self, other, op):
r"""
Compare self and other (where the coercion model has already ensured
that self and other have the same parent). Hecke operators on the same
space compare as equal if and only if their matrices are equal, so we
check if the indices are the same and if not we compute the matrices
(which is potentially expensive).
EXAMPLES::
sage: M = ModularSymbols(Gamma0(7), 4)
sage: m = M.hecke_operator(3)
sage: m == m
True
sage: m == 2*m
False
sage: m == M.hecke_operator(5)
False
These last two tests involve a coercion::
sage: m == m.matrix_form()
True
sage: m == m.matrix()
False
"""
if not isinstance(other, HeckeOperator):
if isinstance(other, HeckeAlgebraElement_matrix):
return richcmp(self.matrix_form(), other, op)
else:
raise RuntimeError("Bug in coercion code") # can't get here
if self.__n == other.__n:
return rich_to_bool(op, 0)
return richcmp(self.matrix(), other.matrix(), op)
def _richcmp_(self, other, op):
r"""
Compare this element with ``other`` for the comparison
operator ``op``.
TESTS::
sage: R.<t> = QQ[]
sage: sigma = R.hom([t+1])
sage: der = R.derivation(1, twist=sigma)
sage: S.<delta> = R['delta', der]
sage: K = S.fraction_field()
sage: P = K.random_element()
sage: Q = K.random_element()
sage: D = K.random_element()
sage: (P*D) / (Q*D) == P/Q
True
"""
if self.parent()._simplification:
return richcmp((self._numerator, self._denominator),
(other._numerator, other._denominator), op)
if op == op_EQ or op == op_NE:
_, U, V = self._denominator.left_xlcm(other._denominator)
return richcmp(U * self._numerator, V * other._numerator, op)
return NotImplemented
def _richcmp_(self, other, op):
r"""
Compare self and other (where the coercion model has already ensured
that self and other have the same parent). Hecke operators on the same
space compare as equal if and only if their matrices are equal, so we
check if the indices are the same and if not we compute the matrices
(which is potentially expensive).
EXAMPLES::
sage: M = ModularSymbols(Gamma0(7), 4)
sage: m = M.hecke_operator(3)
sage: m == m
True
sage: m == 2*m
False
sage: m == M.hecke_operator(5)
False
These last two tests involve a coercion::
sage: m == m.matrix_form()
True
sage: m == m.matrix()
False
"""
if not isinstance(other, HeckeOperator):
if isinstance(other, HeckeAlgebraElement_matrix):
return richcmp(self.matrix_form(), other, op)
else:
raise RuntimeError("Bug in coercion code") # can't get here
if self.__n == other.__n:
return rich_to_bool(op, 0)
return richcmp(self.matrix(), other.matrix(), op)
def __richcmp__(self, other, op):
r"""
Standard comparison function for Kodaira Symbols.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.kodaira_symbol import KodairaSymbol_class
sage: KS1 = KodairaSymbol_class(15); KS1
I11
sage: KS2 = KodairaSymbol_class(-34); KS2
I30*
sage: KS1 < KS2
True
sage: KS2 < KS1
False
::
sage: Klist = [KodairaSymbol_class(i) for i in [-10..10] if i!=0]
sage: Klist.sort()
sage: Klist
[I0,
I0*,
I1,
I1*,
I2,
I2*,
I3,
I3*,
I4,
I4*,
I5,
I5*,
I6,
I6*,
II,
II*,
III,
III*,
IV,
IV*]
"""
if isinstance(other, KodairaSymbol_class):
if (self._n == "generic"
and not other._n is None) or (other._n == "generic"
and not self._n is None):
return richcmp(self._starred, other._starred, op)
return richcmp(self._str, other._str, op)
else:
return NotImplemented
def __richcmp__(self, other, op):
r"""
Standard comparison function for Kodaira Symbols.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.kodaira_symbol import KodairaSymbol_class
sage: KS1 = KodairaSymbol_class(15); KS1
I11
sage: KS2 = KodairaSymbol_class(-34); KS2
I30*
sage: KS1 < KS2
True
sage: KS2 < KS1
False
::
sage: Klist = [KodairaSymbol_class(i) for i in [-10..10] if i!=0]
sage: Klist.sort()
sage: Klist
[I0,
I0*,
I1,
I1*,
I2,
I2*,
I3,
I3*,
I4,
I4*,
I5,
I5*,
I6,
I6*,
II,
II*,
III,
III*,
IV,
IV*]
"""
if isinstance(other, KodairaSymbol_class):
if (self._n == "generic" and not other._n is None) or (other._n == "generic" and not self._n is None):
return richcmp(self._starred, other._starred, op)
return richcmp(self._str, other._str, op)
else:
return NotImplemented
def _richcmp_(self, other, op):
r"""
Compare this element with ``other``.
TESTS::
sage: R = ZpLC(2)
sage: x = R(1, 5)
sage: y = R(128, 10)
sage: z = x + y
sage: x
1 + O(2^5)
sage: z
1 + O(2^5)
sage: x == z # Indirect doctest
False
sage: z - x
2^7 + O(2^10)
"""
if (self - other).is_zero():
return rich_to_bool(op, 0)
else:
return richcmp(QQ(self.lift()), QQ(other.lift()), op)
def __richcmp__(self, other, op):
"""
Intervals are sorted by lower bound, then upper bound
OUTPUT:
`-1`, `0`, or `+1` depending on how the intervals compare.
EXAMPLES::
sage: I1 = RealSet.open_closed(1, 3)[0]; I1
(1, 3]
sage: I2 = RealSet.open_closed(0, 5)[0]; I2
(0, 5]
sage: I1 > I2
True
sage: sorted([I1, I2])
[(0, 5], (1, 3]]
TESTS:
Check if a bug in sorting is fixed (:trac:`17714`)::
sage: RealSet((0, 1),[1, 1],(1, 2))
(0, 2)
"""
x = (self._lower, not self._lower_closed, self._upper, self._upper_closed)
y = (other._lower, not other._lower_closed, other._upper, other._upper_closed)
return richcmp(x, y, op)
def __richcmp__(self, right, op):
r"""
Compare ``self`` and ``right``.
If ``right`` is not a :class:`Set_object`, return ``NotImplemented``.
If ``right`` is also a :class:`Set_object`, returns comparison
on the underlying objects.
.. NOTE::
If `X < Y` is true this does *not* necessarily mean
that `X` is a subset of `Y`. Also, any two sets can be
compared still, but the result need not be meaningful
if they are not equal.
EXAMPLES::
sage: Set(ZZ) == Set(QQ)
False
sage: Set(ZZ) < Set(QQ)
True
sage: Primes() == Set(QQ)
False
The following is random, illustrating that comparison of
sets is not the subset relation, when they are not equal::
sage: Primes() < Set(QQ) # random
True or False
"""
if not isinstance(right, Set_object):
return NotImplemented
return richcmp(self.__object, right.__object, op)
def __richcmp__(self, J, op):
"""
Compare the Jacobian self to `J`. If `J` is a Jacobian, then
self and `J` are equal if and only if their curves are equal.
EXAMPLES::
sage: from sage.schemes.jacobians.abstract_jacobian import Jacobian
sage: P2.<x, y, z> = ProjectiveSpace(QQ, 2)
sage: J1 = Jacobian(Curve(x^3 + y^3 + z^3))
sage: J1 == J1
True
sage: J1 == P2
False
sage: J1 != P2
True
sage: J2 = Jacobian(Curve(x + y + z))
sage: J1 == J2
False
sage: J1 != J2
True
"""
if not is_Jacobian(J):
return NotImplemented
return richcmp(self.curve(), J.curve(), op)
def __richcmp__(self, other, op):
"""
Compare ``self`` to ``other``.
EXAMPLES::
sage: M = ModularSymbols(11)
sage: s = M.2.modular_symbol_rep()[0][1]
sage: t = M.0.modular_symbol_rep()[0][1]
sage: s, t
({-1/9, 0}, {Infinity, 0})
sage: s < t
True
sage: t > s
True
sage: s == s
True
sage: t == t
True
"""
if not isinstance(other, ModularSymbol):
return NotImplemented
return richcmp((self.__space, -self.__i, self.__alpha, self.__beta),
(other.__space,-other.__i,other.__alpha,other.__beta),
op)
def _richcmp_(self, other, op):
"""
Rich comparison of ``self`` to ``other``.
EXAMPLES::
sage: F.<x,y> = FreeAlgebra(ZZ)
sage: H = F.hochschild_complex(F)
sage: a = H({0: x - y,
....: 1: H.module(1).basis().an_element(),
....: 2: H.module(2).basis().an_element()})
sage: a == 3*a
False
sage: a + a == 2*a
True
sage: a == H.zero()
False
sage: a != 3*a
True
sage: a + a != 2*a
False
sage: a != H.zero()
True
"""
return richcmp(self._vec, other._vec, op)
def _richcmp_(left, right, op):
"""
Compare ``left`` and ``right``.
INPUT:
- ``right`` -- a :class:`FormalSum` with the same parent
- ``op`` -- a comparison operator
EXAMPLES::
sage: a = FormalSum([(1,3),(2,5)]); a
3 + 2*5
sage: b = FormalSum([(1,3),(2,7)]); b
3 + 2*7
sage: a != b
True
sage: a_QQ = FormalSum([(1,3),(2,5)],parent=FormalSums(QQ))
sage: a == a_QQ # a is coerced into FormalSums(QQ)
True
sage: a == 0 # 0 is coerced into a.parent()(0)
False
"""
return richcmp(left._data, right._data, op)
def __richcmp__(self, other, op):
"""
Intervals are sorted by lower bound, then upper bound
OUTPUT:
`-1`, `0`, or `+1` depending on how the intervals compare.
EXAMPLES::
sage: I1 = RealSet.open_closed(1, 3)[0]; I1
(1, 3]
sage: I2 = RealSet.open_closed(0, 5)[0]; I2
(0, 5]
sage: cmp(I1, I2)
1
sage: sorted([I1, I2])
[(0, 5], (1, 3]]
TESTS:
Check if a bug in sorting is fixed (:trac:`17714`)::
sage: RealSet((0, 1),[1, 1],(1, 2))
(0, 2)
"""
x = (self._lower, not self._lower_closed, self._upper, self._upper_closed)
y = (other._lower, not other._lower_closed, other._upper, other._upper_closed)
return richcmp(x, y, op)
def _richcmp_(self, other, op):
r"""
Compare this morphism to ``other``.
.. NOTE::
This implementation assumes that this morphism is defined by its
domain and codomain. Morphisms for which this is not true must
override this implementation.
EXAMPLES::
sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x*y + 4*x^3)
sage: f = L.space_of_differentials().coerce_map_from(K.space_of_differentials())
sage: K.<x> = FunctionField(QQbar); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x*y + 4*x^3)
sage: g = L.space_of_differentials().coerce_map_from(K.space_of_differentials())
sage: f == g
False
sage: f == f
True
"""
if type(self) != type(other):
return NotImplemented
from sage.structure.richcmp import richcmp
return richcmp((self.domain(), self.codomain()),
(other.domain(), other.codomain()), op)
def _richcmp_(self, other, op):
"""
Compare two linear expressions.
INPUT:
- ``other`` -- another linear expression (will be enforced by
the coercion framework)
EXAMPLES::
sage: from sage.geometry.linear_expression import LinearExpressionModule
sage: L.<x> = LinearExpressionModule(QQ)
sage: x == L([0, 1])
True
sage: x == x + 1
False
sage: M.<x> = LinearExpressionModule(ZZ)
sage: L.gen(0) == M.gen(0) # because there is a conversion
True
sage: L.gen(0) == L(M.gen(0)) # this is the conversion
True
sage: x == 'test'
False
"""
return richcmp((self._coeffs, self._const),
(other._coeffs, other._const), op)
def _richcmp_(self, other, op):
"""
Compare two free monoid elements with the same parents.
The ordering is first by increasing length, then lexicographically
on the underlying word.
EXAMPLES::
sage: S = FreeMonoid(3, 'a')
sage: (x,y,z) = S.gens()
sage: x * y < y * x
True
sage: a = FreeMonoid(5, 'a').gens()
sage: x = a[0]*a[1]*a[4]**3
sage: x < x
False
sage: x == x
True
sage: x >= x*x
False
"""
m = sum(i for x, i in self._element_list)
n = sum(i for x, i in other._element_list)
if m != n:
return richcmp_not_equal(m, n, op)
v = tuple([x for x, i in self._element_list for j in range(i)])
w = tuple([x for x, i in other._element_list for j in range(i)])
return richcmp(v, w, op)
def __richcmp__(self, J, op):
"""
Compare the Jacobian self to `J`. If `J` is a Jacobian, then
self and `J` are equal if and only if their curves are equal.
EXAMPLES::
sage: from sage.schemes.jacobians.abstract_jacobian import Jacobian
sage: P2.<x, y, z> = ProjectiveSpace(QQ, 2)
sage: J1 = Jacobian(Curve(x^3 + y^3 + z^3))
sage: J1 == J1
True
sage: J1 == P2
False
sage: J1 != P2
True
sage: J2 = Jacobian(Curve(x + y + z))
sage: J1 == J2
False
sage: J1 != J2
True
"""
if not is_Jacobian(J):
return NotImplemented
return richcmp(self.curve(), J.curve(), op)
def __richcmp__(self, other, op):
"""
Intervals are sorted by lower bound, then upper bound
OUTPUT:
`-1`, `0`, or `+1` depending on how the intervals compare.
EXAMPLES::
sage: I1 = RealSet.open_closed(1, 3); I1
(1, 3]
sage: I2 = RealSet.open_closed(0, 5); I2
(0, 5]
sage: cmp(I1, I2)
1
sage: sorted([I1, I2])
[(0, 5], (1, 3]]
sage: I1 == I1
True
"""
if not isinstance(other, RealSet):
return NotImplemented
# note that the interval representation is normalized into a
# unique form
return richcmp(self._intervals, other._intervals, op)
def __richcmp__(self, other, op):
"""
Intervals are sorted by lower bound, then upper bound
OUTPUT:
`-1`, `0`, or `+1` depending on how the intervals compare.
EXAMPLES::
sage: I1 = RealSet.open_closed(1, 3); I1
(1, 3]
sage: I2 = RealSet.open_closed(0, 5); I2
(0, 5]
sage: I1 > I2
True
sage: sorted([I1, I2])
[(0, 5], (1, 3]]
sage: I1 == I1
True
"""
if not isinstance(other, RealSet):
return NotImplemented
# note that the interval representation is normalized into a
# unique form
return richcmp(self._intervals, other._intervals, op)
def __richcmp__(self, other, op):
"""
Compare ``self`` and ``other``.
INPUT:
- ``other`` -- anything.
OUTPUT:
Two faces test equal if and only if they are faces of the same
(not just isomorphic) polyhedron and their generators have the
same indices.
EXAMPLES::
sage: square = polytopes.hypercube(2)
sage: f = square.faces(1)
sage: matrix(4,4, lambda i,j: ZZ(f[i] <= f[j]))
[1 1 1 1]
[0 1 1 1]
[0 0 1 1]
[0 0 0 1]
sage: matrix(4,4, lambda i,j: ZZ(f[i] == f[j])) == 1
True
"""
if not isinstance(other, PolyhedronFace):
return NotImplemented
if self._polyhedron is not other._polyhedron:
return NotImplemented
return richcmp(self._ambient_Vrepresentation_indices,
other._ambient_Vrepresentation_indices, op)
def _richcmp_(self, other, op):
r"""
Compare ``self`` to ``other``.
Eta products are compared according to their rdicts.
EXAMPLES::
sage: EtaProduct(2, {2:24,1:-24}) == 1
False
sage: EtaProduct(6, {1:-24, 2:24}) == EtaProduct(6, {1:-24, 2:24})
True
sage: EtaProduct(6, {1:-24, 2:24}) == EtaProduct(6, {1:24, 2:-24})
False
sage: EtaProduct(6, {1:-24, 2:24}) < EtaProduct(6, {1:-24, 2:24, 3:24, 6:-24})
True
sage: EtaProduct(6, {1:-24, 2:24, 3:24, 6:-24}) < EtaProduct(6, {1:-24, 2:24})
False
"""
if op in [op_EQ, op_NE]:
test = (self._N == other._N and
self._rdict == other._rdict)
return test == (op == op_EQ)
return richcmp((self._N, sorted(self._rdict.items())),
(other._N, sorted(other._rdict.items())), op)
def _richcmp_(left, right, op):
"""
Compare ``left`` and ``right``.
INPUT:
- ``right`` -- a :class:`FormalSum` with the same parent
- ``op`` -- a comparison operator
EXAMPLES::
sage: a = FormalSum([(1,3),(2,5)]); a
3 + 2*5
sage: b = FormalSum([(1,3),(2,7)]); b
3 + 2*7
sage: a != b
True
sage: a_QQ = FormalSum([(1,3),(2,5)],parent=FormalSums(QQ))
sage: a == a_QQ # a is coerced into FormalSums(QQ)
True
sage: a == 0 # 0 is coerced into a.parent()(0)
False
"""
return richcmp(left._data, right._data, op)
def __richcmp__(self, other, op):
r"""
Rich comparison for class functions.
Compares groups and then the values of the class function on the
conjugacy classes.
EXAMPLES::
sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]])
sage: chi = G.character([1, 1, 1, 1, 1, 1, 1])
sage: H = PermutationGroup([[(1,2,3),(4,5)]])
sage: xi = H.character([1, 1, 1, 1, 1, 1])
sage: chi == chi
True
sage: xi == xi
True
sage: xi == chi
False
sage: chi < xi
False
sage: xi < chi
True
"""
if isinstance(other, ClassFunction_libgap):
return richcmp((self._group, self.values()),
(other._group, other.values()), op)
else:
return NotImplemented
def _richcmp_(self, right, op):
"""
Comparison of ``self`` and ``right``.
EXAMPLES::
sage: V = span([[1/2,1,1],[3/2,2,1],[0,0,1]],ZZ)
sage: W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1, 4*V.2])
sage: Q = V/W
sage: phi = Q.hom([Q.0,Q.0 + 2*Q.1])
sage: phi.im_gens()
((1, 0), (1, 2))
sage: phi.im_gens() is phi.im_gens()
True
sage: phi == phi
True
sage: psi = Q.hom([Q.0,Q.0 - 2*Q.1])
sage: phi == psi
False
sage: psi = Q.hom([Q.0,Q.0 - 2*Q.1])
sage: phi < psi
True
sage: psi >= phi
True
sage: psi = Q.hom([Q.0,Q.0 + 2*Q.1])
sage: phi == psi
True
"""
a = (self.domain(), self.codomain())
b = (right.domain(), right.codomain())
if a != b:
return (op == op_NE)
return richcmp(self.im_gens(), right.im_gens(), op)
def __richcmp__(self, other, op):
r"""
Rich comparison for class functions.
Compares groups and then the values of the class function on the
conjugacy classes.
EXAMPLES::
sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]])
sage: chi = G.character([1, 1, 1, 1, 1, 1, 1])
sage: H = PermutationGroup([[(1,2,3),(4,5)]])
sage: xi = H.character([1, 1, 1, 1, 1, 1])
sage: chi == chi
True
sage: xi == xi
True
sage: xi == chi
False
sage: chi < xi
False
sage: xi < chi
True
"""
if isinstance(other, ClassFunction_libgap):
return richcmp((self._group, self.values()),
(other._group, other.values()), op)
else:
return NotImplemented
def __richcmp__(self, other, op):
r"""
Only quotients by the *same* ring and same ideal (with the same
generators!!) are considered equal.
EXAMPLES::
sage: R.<x,y> = PolynomialRing(QQ)
sage: S = R.quotient_ring(x^2 + y^2)
sage: S == R.quotient_ring(x^2 + y^2)
True
The ideals `(x^2 + y^2)` and `(-x^2-y^2)` are
equal, but since the generators are different, the corresponding
quotient rings are not equal::
sage: R.ideal(x^2+y^2) == R.ideal(-x^2 - y^2)
True
sage: R.quotient_ring(x^2 + y^2) == R.quotient_ring(-x^2 - y^2)
False
"""
if not isinstance(other, QuotientRing_nc):
return NotImplemented
return richcmp((self.cover_ring(), self.defining_ideal().gens()),
(other.cover_ring(), other.defining_ideal().gens()), op)
def __richcmp__(self, other, op) -> bool:
"""
EXAMPLES::
sage: a = CuspFamily(16, 4, "1"); a
(c_{4,1})
sage: b = CuspFamily(16, 4, "2"); b
(c_{4,2})
sage: c = CuspFamily(8, 8); c
(0)
sage: a == a
True
sage: a == b
False
sage: a != b
True
sage: a == c
False
sage: a < c
False
sage: a > c
True
sage: a != "foo"
True
"""
if not isinstance(other, CuspFamily):
return NotImplemented
return richcmp(self.__tuple, other.__tuple, op)
def _richcmp_(self, other, op):
"""
Compare ``self`` and ``other``.
TESTS::
sage: A = ArtinGroup(['B',3])
sage: x = A([1, 2, 1])
sage: y = A([2, 1, 2])
sage: x == y
True
sage: x < y^(-1)
True
sage: A([]) == A.one()
True
sage: x = A([2, 3, 2, 3])
sage: y = A([3, 2, 3, 2])
sage: x == y
True
sage: x < y^(-1)
True
"""
if self.Tietze() == other.Tietze():
return rich_to_bool(op, 0)
nfself = [i.Tietze() for i in self.left_normal_form()]
nfother = [i.Tietze() for i in other.left_normal_form()]
return richcmp(nfself, nfother, op)
def _richcmp_(self, other, op):
"""
Rich comparison.
EXAMPLES::
sage: F = FreeAbelianMonoid(5, 'abcde')
sage: F(1)
1
sage: a, b, c, d, e = F.gens()
sage: x = a^2 * b^3
sage: F(1) < x
True
sage: x > b
True
sage: x <= a^4
True
sage: x != a*b
True
sage: a*b == b*a
True
sage: x > a^3*b^2
False
"""
return richcmp(self._element_vector, other._element_vector, op)
def _richcmp_(self, right, op):
"""
Comparison of ``self`` and ``right``.
EXAMPLES::
sage: V = span([[1/2,1,1],[3/2,2,1],[0,0,1]],ZZ)
sage: W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1, 4*V.2])
sage: Q = V/W
sage: phi = Q.hom([Q.0,Q.0 + 2*Q.1])
sage: phi.im_gens()
((1, 0), (1, 2))
sage: phi.im_gens() is phi.im_gens()
True
sage: phi == phi
True
sage: psi = Q.hom([Q.0,Q.0 - 2*Q.1])
sage: phi == psi
False
sage: psi = Q.hom([Q.0,Q.0 - 2*Q.1])
sage: phi < psi
True
sage: psi >= phi
True
sage: psi = Q.hom([Q.0,Q.0 + 2*Q.1])
sage: phi == psi
True
"""
a = (self.domain(), self.codomain())
b = (right.domain(), right.codomain())
if a != b:
return (op == op_NE)
return richcmp(self.im_gens(), right.im_gens(), op)
def _richcmp_(self, other, op):
r"""
Compare this map to ``other``.
.. NOTE::
This implementation assumes that this isomorphism is defined by its
domain and codomain. Isomorphisms for which this is not true must
override this implementation.
EXAMPLES::
sage: K = QQ['x'].fraction_field()
sage: L = K.function_field()
sage: f = K.coerce_map_from(L)
sage: K = QQbar['x'].fraction_field()
sage: L = K.function_field()
sage: g = K.coerce_map_from(L)
sage: f == g
False
sage: f == f
True
"""
if type(self) != type(other):
return NotImplemented
from sage.structure.richcmp import richcmp
return richcmp((self.domain(), self.codomain()),
(other.domain(), other.codomain()), op)
def _richcmp_(self, other, op):
"""
Compare two linear expressions.
INPUT:
- ``other`` -- another linear expression (will be enforced by
the coercion framework)
EXAMPLES::
sage: from sage.geometry.linear_expression import LinearExpressionModule
sage: L.<x> = LinearExpressionModule(QQ)
sage: x == L([0, 1])
True
sage: x == x + 1
False
sage: M.<x> = LinearExpressionModule(ZZ)
sage: L.gen(0) == M.gen(0) # because there is a conversion
True
sage: L.gen(0) == L(M.gen(0)) # this is the conversion
True
sage: x == 'test'
False
"""
return richcmp((self._coeffs, self._const),
(other._coeffs, other._const), op)
def __richcmp__(self, other, op):
"""
Compare torsion subgroups.
INPUT:
- ``other`` -- an object
If other is a torsion subgroup, the abelian varieties are compared.
Otherwise, the generic behavior for finite abelian variety
subgroups is used.
EXAMPLES::
sage: G = J0(11).rational_torsion_subgroup(); H = J0(13).rational_torsion_subgroup()
sage: G == G
True
sage: G < H # since 11 < 13
True
sage: G > H
False
"""
if isinstance(other, RationalTorsionSubgroup):
return richcmp(self.abelian_variety(), other.abelian_variety(), op)
return FiniteSubgroup.__richcmp__(self, other, op)
def _richcmp_(self, other, op):
"""
Rich comparison of ``self`` to ``other``.
EXAMPLES::
sage: F.<x,y> = FreeAlgebra(ZZ)
sage: H = F.hochschild_complex(F)
sage: a = H({0: x - y,
....: 1: H.module(1).basis().an_element(),
....: 2: H.module(2).basis().an_element()})
sage: a == 3*a
False
sage: a + a == 2*a
True
sage: a == H.zero()
False
sage: a != 3*a
True
sage: a + a != 2*a
False
sage: a != H.zero()
True
"""
return richcmp(self._vec, other._vec, op)
def _richcmp_(self, other, op):
"""
Rich comparison for equal parents.
TESTS::
sage: R.<x,y,z> = QQ[]
sage: W = DifferentialWeylAlgebra(R)
sage: dx,dy,dz = W.differentials()
sage: dy*(x^3-y*z)*dx == -z*dx + x^3*dx*dy - y*z*dx*dy
True
sage: W.zero() == 0
True
sage: W.one() == 1
True
sage: x == 1
False
sage: x + 1 == 1
False
sage: W(x^3 - y*z) == x^3 - y*z
True
sage: W.<x,y,z> = DifferentialWeylAlgebra(QQ)
sage: dx,dy,dz = W.differentials()
sage: dx != dy
True
sage: W.one() != 1
False
"""
return richcmp(self.__monomials, other.__monomials, op)
def __richcmp__(self, right, op):
r"""
Compare ``self`` and ``right``.
If ``right`` is not a :class:`Set_object`, return ``NotImplemented``.
If ``right`` is also a :class:`Set_object`, returns comparison
on the underlying objects.
.. NOTE::
If `X < Y` is true this does *not* necessarily mean
that `X` is a subset of `Y`. Also, any two sets can be
compared still, but the result need not be meaningful
if they are not equal.
EXAMPLES::
sage: Set(ZZ) == Set(QQ)
False
sage: Set(ZZ) < Set(QQ)
True
sage: Primes() == Set(QQ)
False
The following is random, illustrating that comparison of
sets is not the subset relation, when they are not equal::
sage: Primes() < Set(QQ) # random
True or False
"""
if not isinstance(right, Set_object):
return NotImplemented
return richcmp(self.__object, right.__object, op)
def __richcmp__(self, other, op):
"""
Standard comparison function.
The ordering is just lexicographic on the tuple `(u,r,s,t)`.
.. NOTE::
In a list of automorphisms, there is no guarantee that the
identity will be first!
EXAMPLES::
sage: from sage.schemes.elliptic_curves.weierstrass_morphism import baseWI
sage: baseWI(1,2,3,4) == baseWI(1,2,3,4)
True
sage: baseWI(1,2,3,4) != baseWI(1,2,3,4)
False
sage: baseWI(1,2,3,4) < baseWI(1,2,3,5)
True
sage: baseWI(1,2,3,4) > baseWI(1,2,3,4)
False
It will never return equality if ``other`` is of another type::
sage: baseWI() == 1
False
"""
if not isinstance(other, baseWI):
return (op == op_NE)
return richcmp(self.tuple(), other.tuple(), op)
def __richcmp__(self, other, op):
"""
Rich comparison.
EXAMPLES::
sage: ct1 = CartanType(['A',1],['B',2])
sage: ct2 = CartanType(['B',2],['A',1])
sage: ct3 = CartanType(['A',4])
sage: ct1 == ct1
True
sage: ct1 == ct2
False
sage: ct1 == ct3
False
TESTS:
Check that :trac:`20418` is fixed::
sage: ct = CartanType(["A2", "B2"])
sage: ct == (1, 2, 1)
False
"""
if isinstance(other, CartanType_simple):
return rich_to_bool(op, 1)
if not isinstance(other, CartanType):
return NotImplemented
return richcmp(self._types, other._types, op)
def _richcmp_(self, other, op):
r"""
Comparison of self and other.
EXAMPLES::
sage: p1 = HyperbolicPlane().UHP().get_point(1 + I)
sage: p2 = HyperbolicPlane().UHP().get_point(2 + I)
sage: p1 == p2
False
sage: p1 == p1
True
sage: p1 = HyperbolicPlane().PD().get_point(0)
sage: p2 = HyperbolicPlane().PD().get_point(1/2 + 2*I/3)
sage: p1 == p2
False
sage: p1 == p1
True
sage: p1 = HyperbolicPlane().KM().get_point((0,0))
sage: p2 = HyperbolicPlane().KM().get_point((0, 1/2))
sage: p1 == p2
False
sage: p1 = HyperbolicPlane().HM().get_point((0,0,1))
sage: p2 = HyperbolicPlane().HM().get_point((0,0,1/1))
sage: p1 == p2
True
"""
if not(isinstance(other, HyperbolicPoint)
or self.parent() is other.parent()):
return op == op_NE
# bool is required to convert symbolic (in)equalities
return bool(richcmp(self._coordinates, other._coordinates, op))
def _richcmp_(self, other, op):
"""
Rich comparison of morphisms.
EXAMPLES::
sage: V = ZZ^2
sage: phi = V.hom([3*V.0, 2*V.1])
sage: psi = V.hom([5*V.0, 5*V.1])
sage: id = V.hom([V.0, V.1])
sage: phi == phi
True
sage: phi == psi
False
sage: psi == End(V)(5)
True
sage: psi == 5 * id
True
sage: psi == 5 # no coercion
False
sage: id == End(V).identity()
True
"""
if not isinstance(other, MatrixMorphism) or op not in (op_EQ, op_NE):
# Generic comparison
return sage.categories.morphism.Morphism._richcmp_(self, other, op)
return richcmp(self.matrix(), other.matrix(), op)
def _richcmp_(self, other, op):
"""
Compare two free monoid elements with the same parents.
The ordering is first by increasing length, then lexicographically
on the underlying word.
EXAMPLES::
sage: S = FreeMonoid(3, 'a')
sage: (x,y,z) = S.gens()
sage: x * y < y * x
True
sage: a = FreeMonoid(5, 'a').gens()
sage: x = a[0]*a[1]*a[4]**3
sage: x < x
False
sage: x == x
True
sage: x >= x*x
False
"""
m = sum(i for x, i in self._element_list)
n = sum(i for x, i in other._element_list)
if m != n:
return richcmp_not_equal(m, n, op)
v = tuple([x for x, i in self._element_list for j in range(i)])
w = tuple([x for x, i in other._element_list for j in range(i)])
return richcmp(v, w, op)
def _richcmp_(self, other, op):
"""
Compare ``self`` and ``other``.
TESTS::
sage: A = ArtinGroup(['B',3])
sage: x = A([1, 2, 1])
sage: y = A([2, 1, 2])
sage: x == y
True
sage: x < y^(-1)
True
sage: A([]) == A.one()
True
sage: x = A([2, 3, 2, 3])
sage: y = A([3, 2, 3, 2])
sage: x == y
True
sage: x < y^(-1)
True
"""
if self.Tietze() == other.Tietze():
return rich_to_bool(op, 0)
nfself = [i.Tietze() for i in self.left_normal_form()]
nfother = [i.Tietze() for i in other.left_normal_form()]
return richcmp(nfself, nfother, op)
def _richcmp_(self, other, op):
"""
Rich comparison of morphisms.
EXAMPLES::
sage: V = ZZ^2
sage: phi = V.hom([3*V.0, 2*V.1])
sage: psi = V.hom([5*V.0, 5*V.1])
sage: id = V.hom([V.0, V.1])
sage: phi == phi
True
sage: phi == psi
False
sage: psi == End(V)(5)
True
sage: psi == 5 * id
True
sage: psi == 5 # no coercion
False
sage: id == End(V).identity()
True
"""
if not isinstance(other, MatrixMorphism) or op not in (op_EQ, op_NE):
# Generic comparison
return sage.categories.morphism.Morphism._richcmp_(self, other, op)
return richcmp(self.matrix(), other.matrix(), op)
def __richcmp__(self, other, op):
"""
Rich comparison.
EXAMPLES::
sage: ct1 = CartanType(['A',1],['B',2])
sage: ct2 = CartanType(['B',2],['A',1])
sage: ct3 = CartanType(['A',4])
sage: ct1 == ct1
True
sage: ct1 == ct2
False
sage: ct1 == ct3
False
TESTS:
Check that :trac:`20418` is fixed::
sage: ct = CartanType(["A2", "B2"])
sage: ct == (1, 2, 1)
False
"""
if isinstance(other, CartanType_simple):
return rich_to_bool(op, 1)
if not isinstance(other, CartanType):
return NotImplemented
return richcmp(self._types, other._types, op)
def _richcmp_(self, other, op):
"""
Rich comparison.
EXAMPLES::
sage: F = FreeAbelianMonoid(5, 'abcde')
sage: F(1)
1
sage: a, b, c, d, e = F.gens()
sage: x = a^2 * b^3
sage: F(1) < x
True
sage: x > b
True
sage: x <= a^4
True
sage: x != a*b
True
sage: a*b == b*a
True
sage: x > a^3*b^2
False
"""
return richcmp(self._element_vector, other._element_vector, op)
def __richcmp__(self, other, op):
"""
Standard comparison function.
The ordering is just lexicographic on the tuple `(u,r,s,t)`.
.. NOTE::
In a list of automorphisms, there is no guarantee that the
identity will be first!
EXAMPLES::
sage: from sage.schemes.elliptic_curves.weierstrass_morphism import baseWI
sage: baseWI(1,2,3,4) == baseWI(1,2,3,4)
True
sage: baseWI(1,2,3,4) < baseWI(1,2,3,5)
True
sage: baseWI(1,2,3,4) > baseWI(1,2,3,4)
False
It will never return equality if other is of another type::
sage: baseWI() == 1
False
"""
if not isinstance(other, baseWI):
return (op == op_NE)
return richcmp(self.tuple(), other.tuple(), op)
def __richcmp__(self, other, op):
r"""
Only quotients by the *same* ring and same ideal (with the same
generators!!) are considered equal.
EXAMPLES::
sage: R.<x,y> = PolynomialRing(QQ)
sage: S = R.quotient_ring(x^2 + y^2)
sage: S == R.quotient_ring(x^2 + y^2)
True
The ideals `(x^2 + y^2)` and `(-x^2-y^2)` are
equal, but since the generators are different, the corresponding
quotient rings are not equal::
sage: R.ideal(x^2+y^2) == R.ideal(-x^2 - y^2)
True
sage: R.quotient_ring(x^2 + y^2) == R.quotient_ring(-x^2 - y^2)
False
"""
if not isinstance(other, QuotientRing_nc):
return NotImplemented
return richcmp((self.cover_ring(), self.defining_ideal().gens()),
(other.cover_ring(), other.defining_ideal().gens()), op)
def _richcmp_(self, other, op):
"""
EXAMPLES::
sage: C = crystals.FastRankTwo(['A',2],shape=[2,1])
sage: D = crystals.FastRankTwo(['B',2],shape=[2,1])
sage: C(0) == C(0)
True
sage: C(1) == C(0)
False
sage: C(0) == D(0)
False
sage: C = crystals.FastRankTwo(['A',2],shape=[2,1])
sage: D = crystals.FastRankTwo(['B',2],shape=[2,1])
sage: C(0) != C(0)
False
sage: C(1) != C(0)
True
sage: C(0) != D(0)
True
sage: C = crystals.FastRankTwo(['A',2],shape=[2,1])
sage: C(1) < C(2)
True
sage: C(2) < C(1)
False
sage: C(2) > C(1)
True
sage: C(1) <= C(1)
True
"""
return richcmp(self.value, other.value, op)
def __richcmp__(self, other, op):
"""
Compare ``self`` to ``other``.
EXAMPLES::
sage: M = ModularSymbols(11)
sage: s = M.2.modular_symbol_rep()[0][1]
sage: t = M.0.modular_symbol_rep()[0][1]
sage: s, t
({-1/9, 0}, {Infinity, 0})
sage: s < t
True
sage: t > s
True
sage: s == s
True
sage: t == t
True
"""
if not isinstance(other, ModularSymbol):
return NotImplemented
return richcmp((self.__space, -self.__i, self.__alpha, self.__beta),
(other.__space,-other.__i,other.__alpha,other.__beta),
op)
def _richcmp_(self, other, op):
r"""
Comparison of self and other.
EXAMPLES::
sage: p1 = HyperbolicPlane().UHP().get_point(1 + I)
sage: p2 = HyperbolicPlane().UHP().get_point(2 + I)
sage: p1 == p2
False
sage: p1 == p1
True
sage: p1 = HyperbolicPlane().PD().get_point(0)
sage: p2 = HyperbolicPlane().PD().get_point(1/2 + 2*I/3)
sage: p1 == p2
False
sage: p1 == p1
True
sage: p1 = HyperbolicPlane().KM().get_point((0,0))
sage: p2 = HyperbolicPlane().KM().get_point((0, 1/2))
sage: p1 == p2
False
sage: p1 = HyperbolicPlane().HM().get_point((0,0,1))
sage: p2 = HyperbolicPlane().HM().get_point((0,0,1/1))
sage: p1 == p2
True
"""
if not (isinstance(other, HyperbolicPoint)
or self.parent() is other.parent()):
return op == op_NE
# bool is required to convert symbolic (in)equalities
return bool(richcmp(self._coordinates, other._coordinates, op))
def __richcmp__(self, other, op):
"""
Compare ``self`` and ``other``.
INPUT:
- ``other`` -- anything.
OUTPUT:
Two faces test equal if and only if they are faces of the same
(not just isomorphic) polyhedron and their generators have the
same indices.
EXAMPLES::
sage: square = polytopes.hypercube(2)
sage: f = square.faces(1)
sage: matrix(4,4, lambda i,j: ZZ(f[i] <= f[j]))
[1 1 1 0]
[0 1 0 0]
[0 1 1 0]
[1 1 1 1]
sage: matrix(4,4, lambda i,j: ZZ(f[i] == f[j])) == 1
True
"""
if not isinstance(other, PolyhedronFace):
return NotImplemented
if self._polyhedron is not other._polyhedron:
return NotImplemented
return richcmp(self._ambient_Vrepresentation_indices,
other._ambient_Vrepresentation_indices, op)
def _richcmp_(self, other, op):
"""
Rich comparison for equal parents.
TESTS::
sage: R.<x,y,z> = QQ[]
sage: W = DifferentialWeylAlgebra(R)
sage: dx,dy,dz = W.differentials()
sage: dy*(x^3-y*z)*dx == -z*dx + x^3*dx*dy - y*z*dx*dy
True
sage: W.zero() == 0
True
sage: W.one() == 1
True
sage: x == 1
False
sage: x + 1 == 1
False
sage: W(x^3 - y*z) == x^3 - y*z
True
sage: W.<x,y,z> = DifferentialWeylAlgebra(QQ)
sage: dx,dy,dz = W.differentials()
sage: dx != dy
True
sage: W.one() != 1
False
"""
return richcmp(self.__monomials, other.__monomials, op)
def __richcmp__(self, other, op):
r"""
Check whether ``self`` is equal to ``other``.
.. TODO::
This overwrites the equality check of
:class:`~sage.structure.list_clone.ClonableArray`
in order to circumvent the coercion framework.
Eventually this should be solved more elegantly.
For now, two elements are compared by their defining lists.
"""
if isinstance(other, SuperPartition):
return richcmp(list(self), list(other), op)
else:
return richcmp(list(self), other, op)
def _richcmp_(self, other, op):
"""
EXAMPLES:
For == operator::
sage: A = crystals.KirillovReshetikhin(['C',2,1], 1,2).affinization()
sage: S = A.subcrystal(max_depth=2)
sage: sorted(S)
[[[1, 1]](-1),
[[1, 2]](-1),
[](0),
[[1, 1]](0),
[[1, 2]](0),
[[1, -2]](0),
[[2, 2]](0),
[](1),
[[2, -1]](1),
[[-2, -1]](1),
[[-1, -1]](1),
[[-1, -1]](2)]
For != operator::
sage: ([(i,j) for i in range(len(S)) for j in range(len(S)) if S[i]!=S[j]]
....: == [(i,j) for i in range(len(S)) for j in range(len(S)) if
....: S[i].value!=S[j].value])
True
For < operator::
sage: ([(i,j) for i in range(len(S)) for j in range(len(S)) if S[i]<S[j]]
....: == [(i,j) for i in range(len(S)) for j in range(len(S)) if
....: S[i].value<S[j].value])
True
For <= operator::
sage: ([(i,j) for i in range(len(S)) for j in range(len(S)) if S[i]<=S[j]]
....: == [(i,j) for i in range(len(S)) for j in range(len(S)) if
....: S[i].value<=S[j].value])
True
For > operator::
sage: ([(i,j) for i in range(len(S)) for j in range(len(S)) if S[i]>S[j]]
....: == [(i,j) for i in range(len(S)) for j in range(len(S)) if
....: S[i].value>S[j].value])
True
For >= operator::
sage: ([(i,j) for i in range(len(S)) for j in range(len(S)) if S[i]>=S[j]]
....: == [(i,j) for i in range(len(S)) for j in range(len(S)) if
....: S[i].value>=S[j].value])
True
"""
return richcmp(self.value, other.value, op)
def _richcmp_(self, other, op):
r"""
Compare self and other.
TESTS::
sage: x = QQ['x'].gen()
sage: f = x^5 - x
sage: H = HyperellipticCurve(f); H
Hyperelliptic Curve over Rational Field defined by y^2 = x^5 - x
sage: J = H.jacobian()(QQ); J
Set of rational points of Jacobian of Hyperelliptic Curve over
Rational Field defined by y^2 = x^5 - x
The following point is 2-torsion::
sage: P = J(H.lift_x(-1)); P
(x + 1, y)
sage: 0 == 2 * P # indirect doctest
True
sage: P == P
True
::
sage: Q = J(H.lift_x(-1))
sage: Q == P
True
::
sage: 2 == Q
False
sage: P == False
False
Let's verify the same "points" on different schemes are not equal::
sage: x = QQ['x'].gen()
sage: f = x^5 + x
sage: H2 = HyperellipticCurve(f)
sage: J2 = H2.jacobian()(QQ)
::
sage: P1 = J(H.lift_x(0)); P1
(x, y)
sage: P2 = J2(H2.lift_x(0)); P2
(x, y)
sage: P1 == P2
False
"""
if self.scheme() != other.scheme():
return op == op_NE
# since divisors are internally represented as Mumford divisors,
# comparing polynomials is well-defined
return richcmp(self.__polys, other.__polys, op)
def _richcmp_(self, right, op):
r"""
Compare two points in products of projective spaces.
INPUT:
- ``other`` -- another point
OUTPUT:
boolean
EXAMPLES::
sage: T = ProductProjectiveSpaces([2, 2], ZZ, 'x')
sage: P = T([1, 2, 3, 4, 5, 6])
sage: Q = T([2, 4, 6, 4, 5, 6])
sage: P == Q
True
EXAMPLES::
sage: T = ProductProjectiveSpaces([1, 1, 1], ZZ, 'x')
sage: P = T([1, 2, 3, 4, 5, 6])
sage: Q = T([2, 4, 6, 4, 1, 0])
sage: P != Q
True
EXAMPLES::
sage: T = ProductProjectiveSpaces([1, 1, 1], GF(5), 'x')
sage: P = T([3, 2, 3, 4, 1, 0])
sage: Q = T([1, 2, 3, 4, 3, 1])
sage: P > Q
True
::
sage: T = ProductProjectiveSpaces([1, 1, 1], GF(5), 'x')
sage: P = T([1, 2, 3, 4, 1, 0])
sage: Q = T([1, 2, 3, 4, 3, 0])
sage: P == Q
True
::
sage: T = ProductProjectiveSpaces([1, 1, 1], GF(5), 'x')
sage: P = T([1, 2, 3, 4, 1, 0])
sage: Q = T([1, 2, 3, 4, 3, 1])
sage: P < Q
True
"""
#needed for Digraph
if not isinstance(right, (ProductProjectiveSpaces_point_ring)):
return NotImplemented
else:
return richcmp(self._points, right._points, op)
def _richcmp_(self, right, op):
"""
Compare the cusps ``self`` and ``right``.
Comparison is as for rational numbers, except with the cusp oo
greater than everything but itself.
The ordering in comparison is only really meaningful for infinity
or elements that coerce to the rationals.
EXAMPLES::
sage: Cusp(2/3) == Cusp(oo)
False
sage: Cusp(2/3) < Cusp(oo)
True
sage: Cusp(2/3)> Cusp(oo)
False
sage: Cusp(2/3) > Cusp(5/2)
False
sage: Cusp(2/3) < Cusp(5/2)
True
sage: Cusp(2/3) == Cusp(5/2)
False
sage: Cusp(oo) == Cusp(oo)
True
sage: 19/3 < Cusp(oo)
True
sage: Cusp(oo) < 19/3
False
sage: Cusp(2/3) < Cusp(11/7)
True
sage: Cusp(11/7) < Cusp(2/3)
False
sage: 2 < Cusp(3)
True
"""
if not self.__b:
s = Infinity
else:
s = self._rational_()
if not right.__b:
o = Infinity
else:
o = right._rational_()
return richcmp(s, o, op)
def _richcmp_(self, right, op):
"""
Compare ``self`` with ``right``.
EXAMPLES::
sage: F = GF(3).algebraic_closure()
sage: F.gen(2) == F.gen(3)
False
"""
x, y = self.parent()._to_common_subfield(self, right)
return richcmp(x, y, op)
