User blog comment:C7X/CNF Comparison Algorithm/@comment-35470197-20200110045759/@comment-35470197-20200110055743

If you used Σ in the definition of T and T as the domain of Σ, then it is a circular logic. On the other hand, you defined T without using Σ, and after that you defined Σ using T. It is not a circular logic. If you want to use Σ in the definition of T, you need to define a wider set of strings as the domain of Σ.

Also, you need to be careful that even if you have a=Σ_{i=0}^{n}b_i for a,b_0,…,b_n∈T, the sequence b_0,…,b_n is not unique unless you require them to be additively indecomposable. If you extend Σ in order to use it in the definition of T, then it will be more difficult to formulate "additively indecomposable".