Trac #32777: Allow some imports of RLF to fail

The `RealLazyField` implementation is tied closely to `real_mpfr`.

We modify some imports for the purpose of modularization, in particular
'''sagemath-polyhedra''' (#32432)

In particular for `... is RLF` tests:
{{{
$ git grep 'is RLF'
src/sage/rings/complex_double.pyx:        if S is ZZ or S is QQ or S is
RDF or S is RLF:
src/sage/rings/number_field/number_field_base.pyx:        if codomain is
AA or codomain is RLF:
src/sage/rings/real_double.pyx:        if S is ZZ or S is QQ or S is
RLF:
src/sage/schemes/plane_conics/con_number_field.py:                if not
(isinstance(p, Map) and p.category_for().is_subcategory(Rings())) or
p.codomain() is AA or p.codomain() is RLF:
}}}

URL: https://trac.sagemath.org/32777
Reported by: mkoeppe
Ticket author(s): Matthias Koeppe
Reviewer(s): Travis Scrimshaw

src/sage/groups/misc_gps/argument_groups.py

@@ -1825,16 +1825,10 @@ def create_key_and_extra_args(self,
             sage: ArgumentGroup('Arg_CC') is ArgumentGroup(domain=CC)  # indirect doctest
             True
         """
-        from sage.rings.complex_arb import ComplexBallField
-        from sage.rings.complex_mpfr import ComplexField_class
-        from sage.rings.complex_interval_field import ComplexIntervalField_class
         from sage.rings.integer_ring import ZZ
         from sage.misc.misc import exactly_one_is_true
         from sage.rings.qqbar import AA
         from sage.rings.rational_field import QQ
-        from sage.rings.real_arb import RealBallField
-        from sage.rings.real_mpfr import RealField_class
-        from sage.rings.real_mpfi import RealIntervalField_class
 
         if not exactly_one_is_true(
                 (data is not None,

src/sage/rings/number_field/number_field_base.pyx

@@ -361,8 +361,16 @@ cdef class NumberField(Field):
         if self._gen_approx is not None or self._embedding is None:
             return
 
-        from sage.rings.qqbar import AA
-        from sage.rings.real_lazy import RLF
+        try:
+            from sage.rings.qqbar import AA
+        except ImportError:
+            AA = None
+
+        try:
+            from sage.rings.real_lazy import RLF
+        except ImportError:
+            RLF = None
+
         codomain = self._embedding.codomain()
         if codomain is AA or codomain is RLF:
             self._gen_approx = []

src/sage/rings/real_double.pyx

@@ -353,7 +353,11 @@ cdef class RealDoubleField_class(sage.rings.abc.RealDoubleField):
             return ToRDF(S)
 
         from .rational_field import QQ
-        from .real_lazy import RLF
+        try:
+            from .real_lazy import RLF
+        except ImportError:
+            RLF = None
+
         if S is ZZ or S is QQ or S is RLF:
             return ToRDF(S)
 
@@ -369,10 +373,14 @@ cdef class RealDoubleField_class(sage.rings.abc.RealDoubleField):
             else:
                 return None
 
-        from .real_mpfr import RR
-        connecting = RR._internal_coerce_map_from(S)
-        if connecting is not None:
-            return ToRDF(RR) * connecting
+        try:
+            from .real_mpfr import RR
+        except ImportError:
+            pass
+        else:
+            connecting = RR._internal_coerce_map_from(S)
+            if connecting is not None:
+                return ToRDF(RR) * connecting
 
     def _magma_init_(self, magma):
         r"""

src/sage/schemes/plane_conics/con_number_field.py

@@ -21,8 +21,8 @@
 #                  https://www.gnu.org/licenses/
 # ****************************************************************************
 
-from sage.rings.all import (AA, RLF, PolynomialRing)
 from sage.rings.rational_field import is_RationalField
+from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
 from .con_field import ProjectiveConic_field
 
 
@@ -343,6 +343,9 @@ def is_locally_solvable(self, p):
             if self._local_obstruction is None:
                 from sage.categories.map import Map
                 from sage.categories.all import Rings
+                from sage.rings.qqbar import AA
+                from sage.rings.real_lazy import RLF
+
                 if not (isinstance(p, Map) and p.category_for().is_subcategory(Rings())) or p.codomain() is AA or p.codomain() is RLF:
                     self._local_obstruction = p
             return False
@@ -387,6 +390,7 @@ def local_obstructions(self, finite=True, infinite=True, read_cache=True):
             if read_cache and self._infinite_obstructions is not None:
                 obs0 = self._infinite_obstructions
             else:
+                from sage.rings.qqbar import AA
                 for b in B.embeddings(AA):
                     if not self.is_locally_solvable(b):
                         obs0.append(b)

src/sage/schemes/projective/projective_morphism.py

@@ -77,8 +77,6 @@
 import sage.rings.abc
 from sage.rings.integer import Integer
 from sage.rings.algebraic_closure_finite_field import AlgebraicClosureFiniteField_generic
-from sage.rings.complex_mpfr import ComplexField_class
-from sage.rings.complex_interval_field import ComplexIntervalField_class
 from sage.rings.finite_rings.finite_field_constructor import is_FiniteField
 from sage.rings.finite_rings.finite_field_constructor import is_PrimeFiniteField
 from sage.rings.finite_rings.finite_field_constructor import GF
@@ -89,8 +87,6 @@
 from sage.rings.qqbar import QQbar, number_field_elements_from_algebraics
 from sage.rings.quotient_ring import QuotientRing_generic
 from sage.rings.rational_field import QQ
-from sage.rings.real_mpfr import RealField_class
-from sage.rings.real_mpfi import RealIntervalField_class
 
 from sage.schemes.generic.morphism import SchemeMorphism_polynomial