BUG: Incorrect return type when differentiating an Action with respect to a Cofunction

[jrmaddison]

Describe the bug
The derivative of an Action may not be aBaseForm subclass.

Steps to Reproduce

from firedrake import *

import ufl

mesh = UnitIntervalMesh(2)
space = FunctionSpace(mesh, "Lagrange", 1)

test = TestFunction(space)
u = Function(space, name="u")
v = Cofunction(space.dual(), name="v")
w = Cofunction(space.dual(), name="w")

F = v(u)
dF = ufl.algorithms.expand_derivatives(
    ufl.derivative(F, v))
print(f"{type(dF)=}")  # Function, should be Action

F = w(u)
dF = ufl.algorithms.expand_derivatives(
    ufl.derivative(F, v))
print(f"{type(dF)=}")  # int, should be ZeroBaseForm

Expected behavior
The result of ufl.algorithms.expand_derivative(ufl.derivative(form, ...)) where form is a BaseForm should always be a BaseForm.

Error message

This leads to difficulties when constructing tangent-linear right-hand-sides, e.g.

from firedrake import *

import ufl

mesh = UnitIntervalMesh(2)
space = FunctionSpace(mesh, "Lagrange", 1)

test = TestFunction(space)
u = Function(space, name="u")
v = Cofunction(space.dual(), name="v")

form = v(u)

tlm_v = Cofunction(space.dual(), name="tlm_v")
ufl.action(ufl.algorithms.expand_derivatives(ufl.derivative(form, v)), tlm_v)

leads to the error

Traceback (most recent call last):
  File "/home/maddison/test.py", line 15, in <module>
    ufl.action(ufl.algorithms.expand_derivatives(ufl.derivative(form, v)), tlm_v)
  File "/home/maddison/build/firedrake/firedrake/src/ufl/ufl/formoperators.py", line 109, in action
    form = as_form(form)
  File "/home/maddison/build/firedrake/firedrake/src/ufl/ufl/form.py", line 688, in as_form
    raise ValueError(f"Unable to convert object to a UFL form: {ufl_err_str(form)}")
ValueError: Unable to convert object to a UFL form: <Coefficient id=127176171859712>

Additional Info
The natural solution would be for Functions to be BaseForms with one Coargument argument, but I can imagine that might cause a lot of problems.

[connorjward]

@nbouziani what are your thoughts?

Regarding the proposed inheritance change I think that that would be extremely disruptive in UFL, even if it is technically the right approach.

[jrmaddison]

Using a Function for the simplified derivative is also complicated by #3346.

[jrmaddison]

After some digging I think the relevant lines are 66-70 at https://github.com/FEniCS/ufl/blob/0f2cb2199476f9aaf98bbd942a22aa472ce5c639/ufl/action.py#L66. Removing these fixes the first case (after an update to _check_function_spaces).

I also see that this code has a double initialization bug -- if the returned right or left are themselves Actions then Action.__init__ will be called for them again (https://docs.python.org/3/reference/datamodel.html#object.__new__).