Trac #31756: regression: conversion of Mathematica's Sqrt to Sage fails

This works with Sage 9.2, but fails with 9.3.rc4:

{{{
sage: mathematica('Sqrt[x]').sage()
Sqrt(x)
}}}

The result is an uppercase symbolic function, but should be the
lowercase function `sqrt` in Sage.

A doctest in interfaces/mathematica.py now fails:
{{{
File "src/sage/interfaces/mathematica.py", line 776, in
sage.interfaces.mathematica.MathematicaElement._sage_
Failed example:
    m.sage()                          # optional - mathematica
Expected:
    (cos(1/x) - 1)^2*sin(sqrt(-x^2 + 1))
Got:
    (cos(1/x) - 1)^2*sin(Sqrt(-x^2 + 1))
}}}

This is also reproducible with `mathematica_free`, see
[https://groups.google.com/g/sage-release/c/rjM44KX8pbc/m/8xYVp77iBgAJ
this post on sage-release].

This also affects the Gamma function:
{{{
File "src/sage/functions/gamma.py", line 731, in
sage.functions.gamma._mathematica_gamma
Failed example:
    gamma(4/3)._mathematica_().sage()       # indirect doctest, optional
- mathematica
Expected:
    gamma(4/3)
Got:
    Gamma(4/3)
}}}

URL: https://trac.sagemath.org/31756
Reported by: gh-mwageringel
Ticket author(s): Markus Wageringel
Reviewer(s): Emmanuel Charpentier

src/sage/calculus/calculus.py

@@ -421,6 +421,7 @@
 
 from sage.misc.lazy_import import lazy_import
 lazy_import('sage.interfaces.maxima_lib', 'maxima')
+from types import FunctionType
 
 
 ########################################################
@@ -2057,6 +2058,34 @@ def _inverse_laplace_latex_(self, *args):
 maxima_hyper = re.compile(r"\%f\[\d+,\d+\]")  # matches %f[m,n]
 
 
+def _is_function(v):
+    r"""
+    Return whether a symbolic element is a function, not a variable.
+
+    TESTS::
+
+        sage: from sage.calculus.calculus import _is_function
+        sage: _is_function(x)
+        False
+        sage: _is_function(sin)
+        True
+
+    Check that :trac:`31756` is fixed::
+
+        sage: from sage.libs.pynac.pynac import symbol_table
+        sage: _is_function(symbol_table['mathematica']['Gamma'])
+        True
+
+        sage: from sage.libs.pynac.pynac import register_symbol
+        sage: foo = lambda x: x^2 + 1
+        sage: register_symbol(foo, dict(mathematica='Foo'))  # optional - mathematica
+        sage: mathematica('Foo[x]').sage()                   # optional - mathematica
+        x^2 + 1
+    """
+    # note that Sage variables are callable, so we only check the type
+    return isinstance(v, Function) or isinstance(v, FunctionType)
+
+
 def symbolic_expression_from_maxima_string(x, equals_sub=False, maxima=maxima):
     r"""
     Given a string representation of a Maxima expression, parse it and
@@ -2144,9 +2173,9 @@ def symbolic_expression_from_maxima_string(x, equals_sub=False, maxima=maxima):
         +Infinity
     """
     var_syms = {k: v for k, v in symbol_table.get('maxima', {}).items()
-                if not isinstance(v,Function)}
+                if not _is_function(v)}
     function_syms = {k: v for k, v in symbol_table.get('maxima', {}).items()
-                     if isinstance(v,Function)}
+                     if _is_function(v)}
 
     if not len(x):
         raise RuntimeError("invalid symbolic expression -- ''")
@@ -2402,9 +2431,9 @@ def symbolic_expression_from_string(s, syms={}, accept_sequence=False):
     """
     parse_func = SR_parser.parse_sequence if accept_sequence else SR_parser.parse_expression
     parser_make_var.set_names({k: v for k, v in syms.items()
-                               if not isinstance(v,Function)})
+                               if not _is_function(v)})
     parser_make_function.set_names({k: v for k, v in syms.items()
-                                    if isinstance(v,Function)})
+                                    if _is_function(v)})
     return parse_func(s)