User blog comment:Mh314159/Natural number recursion - first 4 rule sets/@comment-35470197-20191019143654/@comment-35470197-20191020135843

> Is it a map instead of a function because of the indexing?

You can call \(A\) a function, because the terminology of a function depends on authors unlike a map. The issue was not the difference between a map and a function, but the difference of a function and a function symbol. (A function symbol is an expression but not a map, while a function is a map but not an expression.)

> I think the rules are now free of ambiguity and can now be recursed all the way to the end result which is a single natural number via the successor function.

I think so, because you are not using the rule \(A \langle 0,1 \rangle^2(x) = A \langle A \langle 1,0 \rangle^2(x),0 \rangle^2(x)\) now. I will analyse it later.