User blog comment:Mh314159/Natural number recursion - first 4 rule sets/@comment-35470197-20191019143654/@comment-35470197-20191019235020

> Do you agree that this makes the most sense?

If it is just a composite, then it makes sense. However, according to your comment, I guess that it is not a composite. Then it does not make sense. Remember that your A⟨S⟩ is just a function symbol, and hence the assignment of a natural number to the expression A⟨S⟩(x) is given by an omitted actual function, say, F. Since omitting F causes the confusion of two different result F(A⟨0,1⟩^2(x)) and F(A⟨0,1⟩(F(A⟨0,1⟩(x)))), you need to be careful to clarify the domain of F (even if you omit F), i.e. what expressions you intend to apply the whole rules.

The current clarification of conditions for each rule is good. It will be still better if you clarify conditions for the whole system.

> m, x, j represent natural numbers

> S = sequence of two or more natural numbers

Since you wrote "non-zero natural number" in 1, I guess that you use the convention that 0 is a natural number. Then x can be 0, which looks not intended.

Also, declaration of the range of the symbol is not sufficient. Unless you precisely specify what expressions you intend to apply the whole rules, it is ambiguous whether superscripts are portion of expressions or not, because A⟨S⟩ is not a function but is a functon symbol according to the previous version. Or does it change in the current definition?

It is good to clarify whether you are defining A⟨S⟩ as a function or a function symbol. For a function, superscripts usually mean composition. For a function symbol, it is ambiguous, i.e. you need a rule to solve superscripts.

To summarise:
 * 1) It is good to clarify whether you are defining a function A⟨S⟩ or a term rewrting process (F in the sense above) applicable to a function symbol A⟨S⟩.
 * 2) If A⟨S⟩(x) is a function, you need to specify the domain, i.e. the range of S and x (conditions not only for each case classification, but also for the whole case).
 * 3) If A⟨S⟩(x) is a function symbol, you need to specify the domain of the term rewriting process, i.e. the range of S and x (conditions not only for each rule, but also for the whole rule), and specify the meanings (how to solve) superscripts.