gh-40341: Set cython directive `binding=True`
    
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description below. -->
<!-- ^ For example, instead of "Fixes #12345" use "Introduce new method
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example, "Fixes #12345". -->

Continuation of #26254. This is needed for better integration of Cython
functions and thereby unlocks a few further documentation improvements
(#27578, #30884, #31309, ...).

Moreover, `binding=True` is the default in Cython 3.

### 📝 Checklist


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- [ ] The title is concise and informative.
- [ ] The description explains in detail what this PR is about.
- [ ] I have linked a relevant issue or discussion.
- [ ] I have created tests covering the changes.
- [ ] I have updated the documentation and checked the documentation
preview.

### ⌛ Dependencies

- #40357
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-->
<!-- - #12345: short description why this is a dependency -->
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URL: #40341
Reported by: Tobias Diez
Reviewer(s): David Coudert

Author: Release Manager

src/doc/en/prep/Programming.rst

@@ -109,7 +109,7 @@ It is very important to keep in the parentheses.
    ::
 
        sage: A.det # Won't work
-       <built-in method det of sage.matrix.matrix_integer_dense.Matrix_integer_dense object at ...>
+       <bound method Matrix.det of ...>
 
 This is so useful because we can use the 'tab' key, remember!
 

src/doc/en/prep/Symbolics-and-Basic-Plotting.rst

@@ -178,7 +178,7 @@ This is a good place for a few reminders of basic help.
 ::
 
     sage: z.simplify
-    <built-in method simplify of sage.symbolic.expression.Expression object at ...>
+    <bound method Expression.simplify of -((x + 1)*sqrt(x - 1) - (x - 1)^(3/2))/sqrt((x + 1)*(x - 1))>
 
 Finally, recall that you can get nicely typeset versions of the output
 in several ways.

src/meson.build

@@ -219,7 +219,7 @@ add_project_arguments('-I', meson.current_build_dir(), language: 'cython')
 # Add global compiler flags
 add_project_arguments('-X auto_pickle=False', language: 'cython')
 add_project_arguments('-X autotestdict=False', language: 'cython')
-add_project_arguments('-X binding=False', language: 'cython')
+add_project_arguments('-X binding=True', language: 'cython')
 add_project_arguments('-X c_api_binop_methods=True', language: 'cython')
 add_project_arguments('-X cdivision=True', language: 'cython')
 add_project_arguments('-X cpow=True', language: 'cython')

src/sage/arith/misc.py

@@ -2236,9 +2236,9 @@ def get_gcd(order):
     EXAMPLES::
 
         sage: sage.arith.misc.get_gcd(4000)
-        <built-in method gcd_int of sage.rings.fast_arith.arith_int object at ...>
+        <bound method arith_int.gcd_int of <sage.rings.fast_arith.arith_int object at ...>
         sage: sage.arith.misc.get_gcd(400000)
-        <built-in method gcd_longlong of sage.rings.fast_arith.arith_llong object at ...>
+        <bound method arith_llong.gcd_longlong of <sage.rings.fast_arith.arith_llong object at ...>
         sage: sage.arith.misc.get_gcd(4000000000)
         <function gcd at ...>
     """
@@ -2258,9 +2258,9 @@ def get_inverse_mod(order):
     EXAMPLES::
 
         sage: sage.arith.misc.get_inverse_mod(6000)
-        <built-in method inverse_mod_int of sage.rings.fast_arith.arith_int object at ...>
+        <bound method arith_int.inverse_mod_int of <sage.rings.fast_arith.arith_int object at ...>
         sage: sage.arith.misc.get_inverse_mod(600000)
-        <built-in method inverse_mod_longlong of sage.rings.fast_arith.arith_llong object at ...>
+        <bound method arith_llong.inverse_mod_longlong of <sage.rings.fast_arith.arith_llong object at ...>
         sage: sage.arith.misc.get_inverse_mod(6000000000)
         <function inverse_mod at ...>
     """

src/sage/categories/modules.py

@@ -769,7 +769,8 @@ def base_ring(self):
                 This ``base_ring`` method is actually overridden by
                 :meth:`sage.structure.category_object.CategoryObject.base_ring`::
 
-                    sage: H.base_ring.__module__                                        # needs sage.modules
+                    sage: H.base_ring.__module__
+                    'sage.structure.category_object'
 
                 Here we call it directly::
 

src/sage/categories/primer.py

@@ -931,6 +931,7 @@ class SubcategoryMethods:
 to use Sage's introspection tools to recover where it's implemented::
 
     sage: x._repr_.__module__
+    'sage.structure.element_wrapper'
     sage: sage.misc.sageinspect.sage_getfile(x._repr_)
     '.../sage/structure/element_wrapper.pyx'
 

src/sage/combinat/sf/classical.py

@@ -43,7 +43,7 @@ def init():
         sage: sage.combinat.sf.classical.conversion_functions = {}
         sage: init()
         sage: sage.combinat.sf.classical.conversion_functions[('Schur', 'powersum')]
-        <built-in function t_SCHUR_POWSYM_symmetrica>
+        <cyfunction t_SCHUR_POWSYM_symmetrica at ...>
 
     The following checks if the bug described in :issue:`15312` is fixed. ::
 

src/sage/graphs/base/static_sparse_backend.pyx

@@ -1509,42 +1509,6 @@ cdef class StaticSparseBackend(CGraphBackend):
                     yield self._vertex_to_labels[u]
                     seen.add(u)
 
-    def add_vertex(self, v):
-        r"""
-        Addition of vertices is not available on an immutable graph.
-
-        EXAMPLES::
-
-            sage: g = DiGraph(graphs.PetersenGraph(), data_structure='static_sparse')
-            sage: g.add_vertex(1)
-            Traceback (most recent call last):
-            ...
-            TypeError: this graph is immutable and so cannot be changed
-            sage: g.add_vertices([1,2,3])
-            Traceback (most recent call last):
-            ...
-            TypeError: this graph is immutable and so cannot be changed
-        """
-        (<StaticSparseCGraph> self._cg).add_vertex(v)
-
-    def del_vertex(self, v):
-        r"""
-        Removal of vertices is not available on an immutable graph.
-
-        EXAMPLES::
-
-            sage: g = DiGraph(graphs.PetersenGraph(), data_structure='static_sparse')
-            sage: g.delete_vertex(1)
-            Traceback (most recent call last):
-            ...
-            TypeError: this graph is immutable and so cannot be changed
-            sage: g.delete_vertices([1,2,3])
-            Traceback (most recent call last):
-            ...
-            TypeError: this graph is immutable and so cannot be changed
-        """
-        (<StaticSparseCGraph> self._cg).del_vertex(v)
-
 
 @cython.binding(True)
 def _run_it_on_static_instead(f):

src/sage/graphs/strongly_regular_db.pyx

@@ -1161,7 +1161,7 @@ def is_RSHCD(int v, int k, int l, int mu):
 
         sage: from sage.graphs.strongly_regular_db import is_RSHCD
         sage: t = is_RSHCD(64,27,10,12); t                                              # needs sage.combinat sage.modules
-        [<built-in function SRG_from_RSHCD>, 64, 27, 10, 12]
+        [<cyfunction SRG_from_RSHCD at ...>, 64, 27, 10, 12]
         sage: g = t[0](*t[1:]); g                                                       # needs sage.combinat sage.modules
         Graph on 64 vertices
         sage: g.is_strongly_regular(parameters=True)                                    # needs sage.combinat sage.modules

src/sage/interfaces/axiom.py

@@ -179,36 +179,49 @@
 import os
 import re
 
-import sage.interfaces.abc
+from pexpect import EOF
 
-from .expect import Expect, ExpectElement, FunctionElement, ExpectFunction
+import sage.interfaces.abc
 from sage.env import DOT_SAGE
-from pexpect import EOF
-from sage.misc.multireplace import multiple_replace
+from sage.interfaces.expect import (
+    Expect,
+    ExpectElement,
+    ExpectFunction,
+    FunctionElement,
+)
 from sage.interfaces.tab_completion import ExtraTabCompletion
 from sage.misc.instancedoc import instancedoc
+from sage.misc.multireplace import multiple_replace
 from sage.structure.richcmp import rich_to_bool
 
-
 # The Axiom commands ")what thing det" ")show Matrix" and ")display
 # op det" commands, gives a list of all identifiers that begin in
 # a certain way.  This could maybe be useful somehow... (?)  Also
 # axiom has a lot a lot of ways for getting documentation from the
 # system -- this could also be useful.
 
+
 class PanAxiom(ExtraTabCompletion, Expect):
     """
     Interface to a PanAxiom interpreter.
     """
-    def __init__(self, name='axiom', command='axiom -nox -noclef',
-                 script_subdirectory=None, logfile=None,
-                 server=None, server_tmpdir=None,
-                 init_code=[')lisp (si::readline-off)']):
+
+    def __init__(
+        self,
+        name="axiom",
+        command="axiom -nox -noclef",
+        script_subdirectory=None,
+        logfile=None,
+        server=None,
+        server_tmpdir=None,
+        init_code=[")lisp (si::readline-off)"],
+    ):
         """
         Create an instance of the Axiom interpreter.
 
         TESTS::
 
+            sage: from sage.interfaces.axiom import axiom
             sage: axiom == loads(dumps(axiom))
             True
         """
@@ -829,8 +842,8 @@ def _sage_(self):
             return self._sage_domain()
 
         if type == "Float":
-            from sage.rings.real_mpfr import RealField
             from sage.rings.integer_ring import ZZ
+            from sage.rings.real_mpfr import RealField
             prec = max(self.mantissa().length()._sage_(), 53)
             R = RealField(prec)
             x, e, b = self.unparsed_input_form().lstrip('float(').rstrip(')').split(',')