How do you set attributes on SubValues?

Question

SubValues, as discussed in a previous question, are declared as follows

f[x_][y_] := {ToString[Unevaluated[x]], ToString[Unevaluated[y]]}

But, attempting to use SetAttributes on f only affects the DownValues of f during evaluation, not the SubValues. In other words, if HoldAll is set on f, then only x, in the above code, is held. In code,

SetAttributes[f, HoldAll]
f[ 1 + 2 ][ 3 + 4 ]
(*
==> { "1 + 2", "7" }
*)

Attempting to use SetAttributes on f[x] results in the error

SetAttributes::sym: "Argument f[x] at position 1 is expected to be a symbol."

and, similarly, for f[x_] simply because neither are symbols.

A work around is not to set a SubValue directly, but, instead, return a pure function and use the third argument to set the attribute, as follows

SetAttributes[g, HoldAll]
g[x_] := Function[{y}, 
          {ToString[Unevaluated[x]], ToString[Unevaluated[y]]},
          {HoldAll}
         ]
g[ 1 + 2 ][ 3 + 4 ]

(*
==> {"1 + 2", "3 + 4"}
*)

But, SubValues[g] returns an empty list, indicating that while equivalent, this construct is not processed in the same manner.

So, how does one set the attributes on f such that the SubValues are affected during evaluation?

Answer 1

Your question really is about how to make attributes of f affect also the evaluation of other groups of elements, like y and z in f[x___][y___][z___]. To my knowledge, you can not do it other than using tricks like returning a pure function and the like.

This is because, the only tool you have to intercept the stages of evaluation sequence when y and z are evaluated, is the fact the heads are evaluated first. So, anything you can do to divert the evaluation from its standard form (regarding y and z), must be related to evaluation of f[x], in particular substituting it by something like a pure function. Once you pass that stage of head evaluation, you have no more control of how y and z will be evaluated, as far as I know.

Generally, I see only a few possibilities to imitate this:

• return a pure function with relevant attributes (as discussed in the linked answer)
• return an auxiliary symbol with relevant attributes (similar to the first route)
• play with evaluation stack. An example of this last possibility can be found in my answer here

Here is another example with Stack, closer to those used in the question:

ClearAll[f];
f := 
  With[{stack = Stack[_]},
   With[{fcallArgs =
      Cases[stack, HoldForm[f[x_][y_]] :>
         {ToString[Unevaluated[x]], ToString[Unevaluated[y]]}]},
      (First@fcallArgs &) & /; fcallArgs =!= {}]];

And:

In[34]:= f[1 + 2][3 + 4] // InputForm
Out[34]//InputForm=  {"1 + 2", "3 + 4"}

Perhaps, there are other ways I am not aware of. The general conclusion I made for myself from considering cases like this is that the extent to which one can manipulate evaluation sequence is large but limited, and once you run into a limitation like this, it is best to reconsider the design and find some other approach to the problem, since things will quickly get quite complex and go out of control.