How to require that a function argument be a pure function or a function-style replacement rule?

Question

I'm trying to write a function whose first argument must be another function, defined either as a pure function or a function-style replacement rule. I don't know how to specify a condition on the argument to achieve that. I know I can use g[func_Function,…] to require pure functions, but functions defined via replacement rules don't have a useful Head.

Here's an example:

f[x_] := x^2 - 1;
g[func_?MatchQ[#, _[___]] &, a_, b_] := func[a + b];
g[f, 1, 2]
(* g[f, 1, 2] *)

Checking the Head of f doesn't seem helpful:

Head[f] (* ==> Symbol *)
Head[f[x]] (* ==> Plus *)

How can I constrain the arguments to g such that it will accept either a pure function or a function defined with :=?

Answer 1

You may Condition the function pattern such that it matches for a parameter of Head Function or on the DownValues of the symbol when they have the pattern of that matches a basic function define by SetDelayed.

As pointed out in the OP comments there are many forms that a function can take and the solution below will not cover them all. However, it may be enough to cover the cases within your particular project.

With

ClearAll[g]
g[func_, a_, b_] /; 
  MatchQ[func, _Function] \[Or] 
   Length@Cases[DownValues[func][[All, 1]], \[FormalP] : HoldPattern[_[_Pattern .., ___]] :> 
       Hold@\[FormalP], 2] > 0 :=
 func[a + b]

and

f[x_] := x^2 - 1

Then

g[f, 1, 2]

8

g[# + 1 &, 1, 2]

4

g[y, 1, 2]

g[y, 1, 2]

Hope this helps.