non-numeric non-symbolic BuiltinFunction arguments

[rwst]

Symbolic expressions can be seen as programs for applying functions to numerics and symbolics. Non-numeric non-symbolics (NNNS, most of the rest of Sage) are usually not handled by symbolic functions, although they often allow the same operations and many of the functions. The manual workaround is to evaluate the expression "on x" and then to substitute x with the respective NNNS. We propose that BuiltinFunction automates this process.

Part of this is already duplicated in rings.arith but the need arises with symbolic functions returning polynomials, e.g. #16813. At the moment BuiltinFunction simply converts the NNNS argument, losing the type information.

Practically, BuiltinFunction should eval with x, try substitution, and throw an error if the substitution leads to unknown methods of the NNNS. I.e.

sage: elliptic_f (x, 1)
log(tan(1/4*pi + 1/2*x))
sage: elliptic_f (polygen(CC), 1)
...
AttributeError: cannot apply function to argument x: argument does not know how to do log
(coming from this exception)
AttributeError: 'Polynomial_generic_dense_field' object has no attribute 'log'

Depends on #20312

CC: @jdemeyer

Component: symbolics

Issue created by migration from https://trac.sagemath.org/ticket/18832

[rwst]

Description changed:

--- 
+++ 
@@ -1,3 +1,13 @@
 Symbolic expressions can be seen as programs for applying functions to numerics and symbolics. Non-numeric non-symbolics (NNNS, most of the rest of Sage) are usually not handled by symbolic functions, although they often allow the same operations and many of the functions. The manual workaround is to evaluate the expression "on `x`" and then to substitute `x` with the respective NNNS. We propose that `BuiltinFunction` automates this process.
 
 Part of this is already duplicated in `rings.arith` but the need arises with symbolic functions returning polynomials, e.g. #16813. At the moment `BuiltinFunction` simply converts the NNNS argument, losing the type information.
+
+Practically, `BuiltinFunction` should eval with `x`, try substitution, and throw an error if the substitution leads to unknown methods of the NNNS. I.e. 
+
+```
+sage: elliptic_f (x, 1)
+log(tan(1/4*pi + 1/2*x))
+sage: elliptic_f (polygen(CC), 1)
+...
+AttributeError: 'Polynomial_generic_dense_field' object has no attribute 'log'
+```

[rwst]

Description changed:

--- 
+++ 
@@ -9,5 +9,7 @@
 log(tan(1/4*pi + 1/2*x))
 sage: elliptic_f (polygen(CC), 1)
 ...
+AttributeError: cannot apply function to argument x: argument does not know how to do log
+(coming from this exception)
 AttributeError: 'Polynomial_generic_dense_field' object has no attribute 'log'
 ```

[rwst]

Dependencies: #20312

[rwst]

comment:4

The example is not the best.