gh-35450: Replace remaining uses of `is_SymbolicEquation`, `is_SymbolicVariable`
    
<!-- Please provide a concise, informative and self-explanatory title.
-->
<!-- Don't put issue numbers in the title. Put it in the Description
below. -->
<!-- For example, instead of "Fixes #12345", use "Add a new method to
multiply two integers" -->

### 📚 Description

<!-- Describe your changes here in detail. -->
<!-- Why is this change required? What problem does it solve? -->
<!-- If this PR resolves an open issue, please link to it here. For
example "Fixes #12345". -->
<!-- If your change requires a documentation PR, please link it
appropriately. -->


- Follow up from #32665
- Part of #32414

### 📝 Checklist

<!-- Put an `x` in all the boxes that apply. It should be `[x]` not `[x
]`. -->

- [x] The title is concise, informative, and self-explanatory.
- [x] 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 accordingly.

### ⌛ Dependencies

<!-- List all open PRs that this PR logically depends on
- #12345: short description why this is a dependency
- #34567: ...
-->

<!-- If you're unsure about any of these, don't hesitate to ask. We're
here to help! -->
    
URL: #35450
Reported by: Matthias Köppe
Reviewer(s):

Author: Release Manager

src/sage/calculus/calculus.py

@@ -424,9 +424,9 @@
 
 from sage.misc.latex import latex
 from sage.misc.parser import Parser, LookupNameMaker
-
-from sage.symbolic.ring import var, SR, is_SymbolicVariable
-from sage.symbolic.expression import Expression, symbol_table
+from sage.structure.element import Expression
+from sage.symbolic.ring import var, SR
+from sage.symbolic.expression import symbol_table
 from sage.symbolic.function import Function
 from sage.symbolic.function_factory import function_factory
 from sage.symbolic.integration.integral import (indefinite_integral,
@@ -613,7 +613,7 @@ def symbolic_sum(expression, v, a, b, algorithm='maxima', hold=False):
        the summation the result might not be convertible into a Sage
        expression.
     """
-    if not is_SymbolicVariable(v):
+    if not (isinstance(v, Expression) and v.is_symbol()):
         if isinstance(v, str):
             v = var(v)
         else:
@@ -879,7 +879,7 @@ def symbolic_product(expression, v, a, b, algorithm='maxima', hold=False):
         (-1)^n*factorial(n)^2
 
     """
-    if not is_SymbolicVariable(v):
+    if not (isinstance(v, Expression) and v.is_symbol()):
         if isinstance(v, str):
             v = var(v)
         else:

src/sage/calculus/desolvers.py

@@ -77,12 +77,11 @@
 
 from sage.interfaces.maxima import Maxima
 from sage.misc.lazy_import import lazy_import
-lazy_import("sage.plot.all", "line")
-from sage.symbolic.expression import is_SymbolicEquation
-from sage.symbolic.ring import SR, is_SymbolicVariable
-from sage.calculus.functional import diff
 from sage.misc.functional import N
 from sage.rings.real_mpfr import RealField
+from sage.structure.element import Expression
+
+from .functional import diff
 
 
 maxima = Maxima()
@@ -550,9 +549,9 @@ def desolve(de, dvar, ics=None, ivar=None, show_method=False, contrib_ode=False,
     - Robert Bradshaw (10-2008)
     - Robert Marik (10-2009)
     """
-    if is_SymbolicEquation(de):
+    if isinstance(de, Expression) and de.is_relational():
         de = de.lhs() - de.rhs()
-    if is_SymbolicVariable(dvar):
+    if isinstance(dvar, Expression) and dvar.is_symbol():
         raise ValueError("You have to declare dependent variable as a function evaluated at the independent variable, eg. y=function('y')(x)")
     # for backwards compatibility
     if isinstance(dvar, list):
@@ -600,8 +599,8 @@ def sanitize_var(exprs):
     if show_method:
         maxima_method=P("method")
 
-    if (ics is not None):
-        if not is_SymbolicEquation(soln.sage()):
+    if ics is not None:
+        if not (isinstance(soln.sage(), Expression) and soln.sage().is_relational()):
             if not show_method:
                 maxima_method=P("method")
             raise NotImplementedError("Unable to use initial condition for this equation (%s)."%(str(maxima_method).strip()))
@@ -649,8 +648,8 @@ def sanitize_var(exprs):
             if str(soln).strip() == 'false':
                 raise NotImplementedError("Maxima was unable to solve this BVP. Remove the initial condition to get the general solution.")
 
-    soln=soln.sage()
-    if is_SymbolicEquation(soln) and soln.lhs() == dvar:
+    soln = soln.sage()
+    if isinstance(soln, Expression) and soln.is_relational() and soln.lhs() == dvar:
         # Remark: Here we do not check that the right hand side does not depend on dvar.
         # This probably will not happen for solutions obtained via ode2, anyway.
         soln = soln.rhs()
@@ -796,9 +795,9 @@ def desolve_laplace(de, dvar, ics=None, ivar=None):
     #return maxima(cmd).rhs()._maxima_init_()
 
     ## verbatim copy from desolve - begin
-    if is_SymbolicEquation(de):
+    if isinstance(de, Expression) and de.is_relational():
         de = de.lhs() - de.rhs()
-    if is_SymbolicVariable(dvar):
+    if isinstance(dvar, Expression) and dvar.is_symbol():
         raise ValueError("You have to declare dependent variable as a function evaluated at the independent variable, eg. y=function('y')(x)")
     # for backwards compatibility
     if isinstance(dvar, list):
@@ -959,7 +958,7 @@ def desolve_system(des, vars, ics=None, ivar=None, algorithm="maxima"):
         return desolve_laplace(des[0], vars[0], ics=ics, ivar=ivar)
     ivars = set([])
     for i, de in enumerate(des):
-        if not is_SymbolicEquation(de):
+        if not (isinstance(de, Expression) and de.is_relational()):
             des[i] = de == 0
         ivars = ivars.union(set(de.variables()))
     if ivar is None:
@@ -1206,6 +1205,8 @@ def eulers_method_2x2_plot(f,g, t0, x0, y0, h, t1):
         sage: f = lambda z : z[2]; g = lambda z : -sin(z[1])
         sage: P = eulers_method_2x2_plot(f,g, 0.0, 0.75, 0.0, 0.1, 1.0)
     """
+    from sage.plot.line import line
+
     n = int((1.0)*(t1-t0)/h)
     t00 = t0
     x00 = x0
@@ -1409,11 +1410,11 @@ def desolve_rk4_inner(de, dvar):
                     YMIN = t
             return plot_slope_field(de, (ivar,XMIN,XMAX), (dvar,YMIN,YMAX))+R
 
-    if not is_SymbolicVariable(dvar):
+    if not (isinstance(dvar, Expression) and dvar.is_symbol()):
         from sage.symbolic.ring import SR
         from sage.calculus.all import diff
         from sage.symbolic.relation import solve
-        if is_SymbolicEquation(de):
+        if isinstance(de, Expression) and de.is_relational():
             de = de.lhs() - de.rhs()
         # consider to add warning if the solution is not unique
         de=solve(de,diff(dvar,ivar),solution_dict=True)
@@ -1706,7 +1707,7 @@ def Dfun(y, t):
             mxhnil=mxhnil, mxordn=mxordn, mxords=mxords, printmessg=printmessg)
         return sol
 
-    if is_SymbolicVariable(dvars):
+    if isinstance(dvars, Expression) and dvars.is_symbol():
         dvars = [dvars]
 
     if not isinstance(des, (list, tuple)):
@@ -1719,6 +1720,7 @@ def Dfun(y, t):
         if len(ivars)==1:
             return desolve_odeint_inner(next(iter(ivars)))
         elif not ivars:
+            from sage.symbolic.ring import SR
             with SR.temp_var() as ivar:
                 return desolve_odeint_inner(ivar)
         else:

src/sage/calculus/functional.py

@@ -26,8 +26,8 @@
     -a^3 + 3*a^2*x - 3*a*x^2 + x^3
 """
 
-from .calculus import SR
-from sage.symbolic.expression import Expression
+from sage.structure.element import Expression
+
 
 def simplify(f):
     r"""
@@ -147,6 +147,7 @@ def derivative(f, *args, **kwds):
     except AttributeError:
         pass
     if not isinstance(f, Expression):
+        from sage.symbolic.ring import SR
         f = SR(f)
     return f.derivative(*args, **kwds)
 
@@ -296,6 +297,7 @@ def integral(f, *args, **kwds):
         pass
 
     if not isinstance(f, Expression):
+        from sage.symbolic.ring import SR
         f = SR(f)
     return f.integral(*args, **kwds)
 
@@ -349,6 +351,7 @@ def limit(f, dir=None, taylor=False, **argv):
         -limit((erf(x) - 1)*e^(x^2), x, +Infinity)
     """
     if not isinstance(f, Expression):
+        from sage.symbolic.ring import SR
         f = SR(f)
     return f.limit(dir=dir, taylor=taylor, **argv)
 
@@ -392,6 +395,7 @@ def taylor(f, *args):
         (x - 1)*(y + 1)^3 - 3*(x - 1)*(y + 1)^2 + (y + 1)^3 + 3*(x - 1)*(y + 1) - 3*(y + 1)^2 - x + 3*y + 3
     """
     if not isinstance(f, Expression):
+        from sage.symbolic.ring import SR
         f = SR(f)
     return f.taylor(*args)
 

src/sage/calculus/functions.py

@@ -2,9 +2,8 @@
 Calculus functions
 """
 from sage.matrix.constructor import matrix
-from sage.structure.element import is_Matrix
-from sage.structure.element import is_Vector
-from sage.symbolic.ring import is_SymbolicVariable
+from sage.structure.element import is_Matrix, is_Vector, Expression
+
 from .functional import diff
 
 
@@ -82,7 +81,7 @@ def wronskian(*args):
         # a 1x1 Wronskian is just its argument
         return args[0]
     else:
-        if is_SymbolicVariable(args[-1]):
+        if isinstance(args[-1], Expression) and args[-1].is_symbol():
             # if last argument is a variable, peel it off and
             # differentiate the other args
             v = args[-1]

src/sage/calculus/var.pyx

@@ -384,10 +384,12 @@ def clear_vars():
         sage: k
         15
     """
+    from sage.structure.element import Expression
+
     G = globals()
-    from sage.symbolic.ring import is_SymbolicVariable
     for i in list(range(65, 65 + 26)) + list(range(97, 97 + 26)):
-        if chr(i) in G and is_SymbolicVariable(G[chr(i)]):
+        chr_i = chr(i)
+        if chr_i in G and isinstance(G[chr_i], Expression) and G[chr_i].is_symbol():
             # We check to see if there is a corresponding pyobject
             # associated with the expression.  This will work for
             # constants which we want to keep, but will fail for

src/sage/interfaces/maxima_lib.py

@@ -114,6 +114,7 @@
 #                  https://www.gnu.org/licenses/
 # ****************************************************************************
 
+from sage.structure.element import Expression
 from sage.symbolic.ring import SR
 
 from sage.libs.ecl import EclObject, ecl_eval
@@ -1615,9 +1616,8 @@ def sr_to_max(expr):
         # For that, we should change the API of the functions there
         # (we need to have access to op, not only to expr.operands()
         if isinstance(op, FDerivativeOperator):
-            from sage.symbolic.ring import is_SymbolicVariable
             args = expr.operands()
-            if (not all(is_SymbolicVariable(v) for v in args) or
+            if (not all(isinstance(v, Expression) and v.is_symbol() for v in args) or
                 len(args) != len(set(args))):
                 # An evaluated derivative of the form f'(1) is not a
                 # symbolic variable, yet we would like to treat it

src/sage/symbolic/assumptions.py

@@ -75,7 +75,7 @@
 from sage.rings.rational_field import QQ
 from sage.rings.real_mpfr import RR
 from sage.rings.cc import CC
-from sage.symbolic.ring import is_SymbolicVariable
+from sage.structure.element import Expression
 from sage.structure.unique_representation import UniqueRepresentation
 
 # #30074: We use the keys of a dict to store the assumptions.
@@ -427,7 +427,8 @@ def preprocess_assumptions(args):
         if isinstance(x, str):
             del args[i]
             last = x
-        elif ((not hasattr(x, 'assume') or is_SymbolicVariable(x))
+        elif ((not hasattr(x, 'assume')
+               or (isinstance(x, Expression) and x.is_symbol()))
               and last is not None):
             args[i] = GenericDeclaration(x, last)
         else:

src/sage/symbolic/callable.py

@@ -490,11 +490,11 @@ def create_key(self, args, check=True):
             (x, y)
         """
         if check:
-            from sage.symbolic.ring import is_SymbolicVariable
+            from sage.structure.element import Expression
             if len(args) == 1 and isinstance(args[0], (list, tuple)):
                 args, = args
             for arg in args:
-                if not is_SymbolicVariable(arg):
+                if not (isinstance(arg, Expression) and arg.is_symbol()):
                     raise TypeError("must construct a function with a tuple (or list) of variables")
             args = tuple(args)
         return args

src/sage/symbolic/expression.pyx

@@ -613,7 +613,7 @@ def _subs_make_dict(s):
     """
     if isinstance(s, dict):
         return s
-    elif is_SymbolicEquation(s):
+    elif isinstance(s, Expression) and s.is_relational():
         if s.operator() is not operator.eq:
             msg = "can only substitute equality, not inequalities; got {}"
             raise TypeError(msg.format(s))
@@ -692,7 +692,7 @@ def _subs_fun_make_dict(s):
     """
     if isinstance(s, dict):
         return dict((k, v) if not isinstance(k, Expression) else (k.operator(), v.function(*k.operands())) for k, v in s.items())
-    elif is_SymbolicEquation(s):
+    elif isinstance(s, Expression) and s.is_relational():
         if s.operator() is not operator.eq:
             msg = "can only substitute equality, not inequalities; got {}"
             raise TypeError(msg.format(s))
@@ -6810,12 +6810,11 @@ cdef class Expression(Expression_abc):
         # we override type checking in CallableSymbolicExpressionRing,
         # since it checks for old SymbolicVariable's
         # and do the check here instead
-        from sage.symbolic.callable import CallableSymbolicExpressionRing
-        from sage.symbolic.ring import is_SymbolicVariable
         for i in args:
-            if not is_SymbolicVariable(i):
+            if not (isinstance(i, Expression) and i.is_symbol()):
                 break
         else:
+            from sage.symbolic.callable import CallableSymbolicExpressionRing
             R = CallableSymbolicExpressionRing(args, check=False)
             return R(self)
         raise TypeError(f"must construct a function with symbolic variables as arguments, got {args}.")
@@ -12881,7 +12880,6 @@ cdef class Expression(Expression_abc):
             sage: plot(f,0,1)
             Graphics object consisting of 1 graphics primitive
         """
-        from sage.symbolic.ring import is_SymbolicVariable
         from sage.plot.plot import plot
 
         # see if the user passed a variable in.
@@ -12890,7 +12888,7 @@ cdef class Expression(Expression_abc):
         else:
             param = None
             for i, arg in enumerate(args):
-                if is_SymbolicVariable(arg):
+                if isinstance(arg, Expression) and arg.is_symbol():
                     param = arg
                     args = args[:i] + args[i+1:]
                     break

src/sage/symbolic/expression_conversions.py

@@ -579,11 +579,10 @@ def derivative(self, ex, operator):
         """
         # This code should probably be moved into the interface
         # object in a nice way.
-        from sage.symbolic.ring import is_SymbolicVariable
         if self.name_init != "_maxima_init_":
             raise NotImplementedError
         args = ex.operands()
-        if (not all(is_SymbolicVariable(v) for v in args) or
+        if (not all(isinstance(v, Expression) and v.is_symbol() for v in args) or
             len(args) != len(set(args))):
             # An evaluated derivative of the form f'(1) is not a
             # symbolic variable, yet we would like to treat it like
@@ -1103,12 +1102,11 @@ def derivative(self, ex, operator):
              ,1,1,2
 
         """
-        from sage.symbolic.ring import is_SymbolicVariable
         args = ex.operands()  # the arguments the derivative is evaluated at
         params = operator.parameter_set()
         params_set = set(params)
         mult = ",".join(str(params.count(i)) for i in params_set)
-        if (not all(is_SymbolicVariable(v) for v in args) or
+        if (not all(isinstance(v, Expression) and v.is_symbol() for v in args) or
             len(args) != len(set(args))):
             # An evaluated derivative of the form f'(1) is not a
             # symbolic variable, yet we would like to treat it like