User blog comment:Mh314159/Natural number recursion - first 4 rule sets/@comment-35470197-20191019143654/@comment-35470197-20191020045812

> Well, now I'm a little discouraged because some of this is beyond me.

Sorry about it. It is perhaps due to my lack of the description on what to do.

> I will say that I am calling A the "class" and S the "index" but intend A⟨S⟩ to be a function and x to be the argument of the function.

I think that A is a map which assigns to each non-empty strings S of natural numbers a map A⟨S⟩ from the set of natural numbers to itself. If so, then you can state "I define a map \(A \langle S \rangle \colon \mathbb{N} \to \mathbb{N}\) indexed by non-empty strings \(S\) of natural numbers". Is it what you intend?

> I thought this was standard, but perhaps it is not because I needed to specify A⟨S⟩ as a function.

The issue was that you set the rule \(A \langle 0, 1 \rangle^2(x) = A \langle A \langle 1,0 \rangle^2(x), 0 \rangle^2(x)\). If \(A \langle S \rangle\) is just a function, then this recursion is ill-defined because you directly use the expression itself. Such a term rewriting is only allowed to a function symbol. That is why I guessed that you are using a function symbol instead of a function.

If you use the current rule in the comment, i.e. \(A \langle S \rangle\) is a function but not a function symbol and \(A \langle S \rangle^m\) is the usual composite of \(A \langle S \rangle\), then it is no longer an issue. You do not have to clarify the meanings of the superscript.