replacement - Replace custom functions, leave built in functions untouched?

I have three expressions a[x, y], b[x, y], c[x, y] that act as placeholders for functions of two variables x,y. Consider the following substitution:

a[x, y]/(b[x, y] c[x, y]) /. f_[x1_, y1_] :> f[2 x1, 3 y1]

a[2 x, 3 y]/(64 b[x, y]^3 c[x, y]^3)

In the output we see that the numerator expression was substituted properly, but in the denominator the pattern f_ registered for the head Power instead of looking for my own expressions. Of course I can fix this by:

a[x, y]/(b[x, y] c[x, y]) /. a[x1_, y1_] :> a[2 x1, 3 y1] /.b[x1_, y1_] :> b[2 x1, 3 y1] /. c[x1_, y1_] :> c[2 x1, 3 y1]

a[2 x, 3 y]/(b[2 x, 3 y] c[2 x, 3 y])

which gives the desired output. But this amounts to writing three times as many substitution directives and is therefore inconvenient. To fix the first example, I tried using /. f_Symbol[x1_, y1_] :> f[2 x1, 3 y1] or /. f_[x1_, y1_]/;Head[f]===Symbol :> f[2 x1, 3 y1], but this does not correct it. Is there a way to write a proper substitution that works with headers and does not act on built in functions? Thanks for any suggestions.

EDIT:

Just noticed that Head[Power] actually returns Symbol, which is kind of weird. I would have expected it to return e.g. Function, or Directive, or something along the lines. (If one unprotects and clears the Power function, then I would again expect Head[Power] to return Symbol of course. But maybe that's just me...)

Answer

The best method I am aware of to handle this kind of problem is to filter by context.⁽¹⁾

SetAttributes[user, HoldFirst]
user[s_Symbol] := Context@Unevaluated@s =!= "System`";


a[x, y]/(b[x, y] c[x, y]) /. f_?user[x1_, y1_] :> f[2 x1, 3 y1]

a[2 x, 3 y]/(b[2 x, 3 y] c[2 x, 3 y])

One could include other contexts in the exclusion besides System, or use the inverse and test only for user symbols existing in the "Global`" context. Without additional examples my example is as specific as I can make it.

Regarding the unusual evaluation of the ? operator (PatternTest) please see:

Comments

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