gh-36917: `sage.rings`: Modularization fixes
    
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Standard modularization changes:
- Using more lazy imports, more specific imports and similar
- Some try... except for imports of element implementations
- `# needs`
- incidental doctest cosmetics

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This is:
- Cherry-picked from #35095
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URL: #36917
Reported by: Matthias Köppe
Reviewer(s): David Coudert

Author: Release Manager

src/sage/calculus/test_sympy.py

@@ -156,7 +156,7 @@
     <class 'sage.symbolic.expression.Expression'>
     sage: t1, t2
     (omega + x, omega + x)
-    sage: e = sympy.sin(var("y"))+sage.all.cos(sympy.Symbol("x"))
+    sage: e = sympy.sin(var("y")) + sage.functions.trig.cos(sympy.Symbol("x"))
     sage: type(e)
     <class 'sympy.core.add.Add'>
     sage: e
@@ -166,7 +166,7 @@
     <class 'sage.symbolic.expression.Expression'>
     sage: e
     cos(x) + sin(y)
-    sage: e = sage.all.cos(var("y")**3)**4+var("x")**2
+    sage: e = sage.functions.trig.cos(var("y")**3)**4+var("x")**2
     sage: e = e._sympy_()
     sage: e
     x**2 + cos(y**3)**4

src/sage/ext/fast_callable.pyx

@@ -576,7 +576,8 @@ def function_name(fn):
     Given a function, return a string giving a name for the function.
 
     For functions we recognize, we use our standard opcode name for the
-    function (so operator.add becomes 'add', and sage.all.sin becomes 'sin').
+    function (so :func:`operator.add` becomes ``'add'``, and :func:`sage.functions.trig.sin`
+    becomes ``'sin'``).
 
     For functions we don't recognize, we try to come up with a name,
     but the name will be wrapped in braces; this is a signal that

src/sage/groups/generic.py

@@ -117,10 +117,12 @@
 
 from copy import copy
 
+from sage.arith.misc import integer_ceil, integer_floor, xlcm
+from sage.arith.srange import xsrange
+from sage.misc.functional import log
 from sage.misc.misc_c import prod
 import sage.rings.integer_ring as integer_ring
 import sage.rings.integer
-from sage.arith.srange import xsrange
 
 #
 # Lists of names (as strings) which the user may use to identify one
@@ -1396,8 +1398,7 @@ def order_from_bounds(P, bounds, d=None, operation='+',
     if d > 1:
         Q = multiple(P, d, operation=operation)
         lb, ub = bounds
-        bounds = (sage.arith.all.integer_ceil(lb / d),
-                  sage.arith.all.integer_floor(ub / d))
+        bounds = (integer_ceil(lb / d), integer_floor(ub / d))
 
     # Use generic bsgs to find  n=d*m with lb<=n<=ub and n*P=0
 
@@ -1486,7 +1487,7 @@ def merge_points(P1, P2, operation='+',
     if n2.divides(n1):
         return (g1, n1)
 
-    m, k1, k2 = sage.arith.all.xlcm(n1, n2)
+    m, k1, k2 = xlcm(n1, n2)
     m1 = n1 // k1
     m2 = n2 // k2
     g1 = multiple(g1, m1, operation=operation)

src/sage/modular/modform_hecketriangle/hecke_triangle_group_element.py

@@ -117,7 +117,6 @@ def __init__(self, parent, M, check=True, **kwargs):
 
         EXAMPLES::
 
-            sage:
             sage: from sage.modular.modform_hecketriangle.hecke_triangle_groups import HeckeTriangleGroup, HeckeTriangleGroupElement
             sage: lam = PolynomialRing(ZZ, 'lam').gen()
             sage: M = matrix([[-1, 0], [-lam^4 + 5*lam^2 + lam - 5, -1]])

src/sage/rings/bernoulli_mod_p.pyx

@@ -23,6 +23,8 @@ AUTHOR:
 #                  https://www.gnu.org/licenses/
 # ****************************************************************************
 
+from sage.arith.misc import is_prime, primitive_root
+
 cimport sage.rings.fast_arith
 import sage.rings.fast_arith
 cdef sage.rings.fast_arith.arith_int arith_int
@@ -135,12 +137,12 @@ def bernoulli_mod_p(int p):
     if p <= 2:
         raise ValueError("p (=%s) must be a prime >= 3" % p)
 
-    if not sage.arith.all.is_prime(p):
+    if not is_prime(p):
         raise ValueError("p (=%s) must be a prime" % p)
 
     cdef int g, gSqr, gInv, gInvSqr, isOdd
 
-    g = sage.arith.all.primitive_root(p)
+    g = primitive_root(p)
     gInv = arith_int.c_inverse_mod_int(g, p)
     gSqr = ((<llong> g) * g) % p
     gInvSqr = ((<llong> gInv) * gInv) % p
@@ -303,7 +305,7 @@ def bernoulli_mod_p_single(long p, long k):
     if p <= 2:
         raise ValueError("p (=%s) must be a prime >= 3" % p)
 
-    if not sage.arith.all.is_prime(p):
+    if not is_prime(p):
         raise ValueError("p (=%s) must be a prime" % p)
 
     cdef long x = bernmm_bern_modp(p, k)

src/sage/rings/complex_interval.pyx

@@ -290,7 +290,7 @@ cdef class ComplexIntervalFieldElement(FieldElement):
 
         Exact and nearly exact points are still visible::
 
-            sage: # needs sage.plot
+            sage: # needs sage.plot sage.symbolic
             sage: plot(CIF(pi, 1), color='red') + plot(CIF(1, e), color='purple') + plot(CIF(-1, -1))
             Graphics object consisting of 6 graphics primitives
 

src/sage/rings/continued_fraction.py

@@ -208,12 +208,20 @@
 import numbers
 
 import sage.rings.abc
+
+from sage.misc.lazy_import import lazy_import
 from sage.rings.infinity import Infinity
 from sage.rings.integer import Integer
 from sage.rings.integer_ring import ZZ
 from sage.structure.richcmp import rich_to_bool, richcmp_method
 from sage.structure.sage_object import SageObject
 
+lazy_import('sage.combinat.words.abstract_word', 'Word_class')
+lazy_import('sage.combinat.words.finite_word', 'FiniteWord_class')
+lazy_import('sage.combinat.words.infinite_word', 'InfiniteWord_class')
+lazy_import('sage.combinat.words.word', 'Word')
+
+
 ZZ_0 = Integer(0)
 ZZ_1 = Integer(1)
 ZZ_m1 = Integer(-1)
@@ -2490,7 +2498,10 @@ def continued_fraction_list(x, type="std", partial_convergents=False,
 
     cf = None
 
-    from sage.rings.real_mpfr import RealLiteral
+    try:
+        from sage.rings.real_mpfr import RealLiteral
+    except ImportError:
+        RealLiteral = ()
     if isinstance(x, RealLiteral):
         from sage.rings.real_mpfi import RealIntervalField
         x = RealIntervalField(x.prec())(x)
@@ -2661,7 +2672,6 @@ def continued_fraction(x, value=None):
         pass
 
     # input for finite or ultimately periodic partial quotient expansion
-    from sage.combinat.words.finite_word import FiniteWord_class
     if isinstance(x, FiniteWord_class):
         x = list(x)
 
@@ -2675,12 +2685,10 @@ def continued_fraction(x, value=None):
         return ContinuedFraction_periodic(x1, x2)
 
     # input for infinite partial quotient expansion
-    from sage.combinat.words.infinite_word import InfiniteWord_class
     from sage.misc.lazy_list import lazy_list_generic
     if isinstance(x, (lazy_list_generic, InfiniteWord_class)):
         return ContinuedFraction_infinite(x, value)
 
-    from sage.combinat.words.abstract_word import Word_class
     if isinstance(x, Word_class):
         raise ValueError("word with unknown length cannot be converted "
                          "to continued fractions")

src/sage/rings/finite_rings/element_base.pyx

@@ -586,8 +586,8 @@ cdef class FinitePolyExtElement(FiniteRingElement):
             Finite Field in a of size 19^2
             sage: b = a**20
             sage: p = FinitePolyExtElement.charpoly(b, "x", algorithm="pari")
-            sage: q = FinitePolyExtElement.charpoly(b, "x", algorithm="matrix")   # needs sage.modules
-            sage: q == p                                                          # needs sage.modules
+            sage: q = FinitePolyExtElement.charpoly(b, "x", algorithm="matrix")         # needs sage.modules
+            sage: q == p                                                                # needs sage.modules
             True
             sage: p
             x^2 + 15*x + 4

src/sage/rings/finite_rings/element_givaro.pyx

@@ -56,10 +56,11 @@ from cysignals.signals cimport sig_on, sig_off
 
 from cypari2.paridecl cimport *
 
+import sage.arith.misc
+
 from sage.misc.randstate cimport current_randstate
 from sage.rings.finite_rings.element_pari_ffelt cimport FiniteFieldElement_pari_ffelt
 from sage.structure.richcmp cimport richcmp
-import sage.arith.all
 
 from cypari2.gen cimport Gen
 from cypari2.stack cimport clear_stack
@@ -414,9 +415,6 @@ cdef class Cache_givaro(Cache_base):
             # Reduce to pari
             e = e.__pari__()
 
-        elif isinstance(e, sage.libs.gap.element.GapElement_FiniteField):
-            return e.sage(ring=self.parent)
-
         elif isinstance(e, GapElement):
             from sage.libs.gap.libgap import libgap
             return libgap(e).sage(ring=self.parent)
@@ -434,6 +432,13 @@ cdef class Cache_givaro(Cache_base):
             return ret
 
         else:
+            try:
+                from sage.libs.gap.element import GapElement_FiniteField
+            except ImportError:
+                pass
+            else:
+                if isinstance(e, GapElement_FiniteField):
+                    return e.sage(ring=self.parent)
             raise TypeError("unable to coerce %r" % type(e))
 
         cdef GEN t
@@ -1568,7 +1573,7 @@ cdef class FiniteField_givaroElement(FinitePolyExtElement):
                 raise ArithmeticError("Multiplicative order of 0 not defined.")
             n = (self._cache).order_c() - 1
             order = Integer(1)
-            for p, e in sage.arith.all.factor(n):
+            for p, e in sage.arith.misc.factor(n):
                 # Determine the power of p that divides the order.
                 a = self**(n / (p**e))
                 while a != 1:

src/sage/rings/finite_rings/element_ntl_gf2e.pyx

@@ -274,7 +274,6 @@ cdef class Cache_ntl_gf2e(Cache_base):
         cdef FiniteField_ntl_gf2eElement x
         cdef FiniteField_ntl_gf2eElement g
         cdef Py_ssize_t i
-        from sage.libs.gap.element import GapElement_FiniteField
 
         if is_IntegerMod(e):
             e = e.lift()
@@ -334,14 +333,18 @@ cdef class Cache_ntl_gf2e(Cache_base):
             # Reduce to pari
             e = e.__pari__()
 
-        elif isinstance(e, GapElement_FiniteField):
-            return e.sage(ring=self._parent)
-
         elif isinstance(e, GapElement):
             from sage.libs.gap.libgap import libgap
             return libgap(e).sage(ring=self._parent)
 
         else:
+            try:
+                from sage.libs.gap.element import GapElement_FiniteField
+            except ImportError:
+                pass
+            else:
+                if isinstance(e, GapElement_FiniteField):
+                    return e.sage(ring=self._parent)
             raise TypeError("unable to coerce %r" % type(e))
 
         cdef GEN t