User blog comment:P進大好きbot/Kyodaisuutan System/@comment-35870936-20181003040133/@comment-35470197-20181004022737

I also defined the syntax sugar \(\lim\) in the submission to the contest. I think that you know the difference of the layers of the base theory in which we define terms and the meta theory in which we define syntax. For example, \(1\) is the syntax sugar of the successor of \(0\), but the applying the syntax sugar does not change the number of rules in the base theory.

> how about this

Of course, it is allowed. However, your definition is a formula which is not atomic, because it contain three or four relation symbols after carefully expanding the syntax sugar in the usual way in \(\epsilon \delta\) logic.

So it is better to put \(\lim\) in the beginning of the right hand side so that the resulting formula is still atomic.

> (note that b and c can be negative, thus solving your "(undefined)x0" objection)

Ah, it is a cool solution :O

Not only negative integers can be substituted, but the values then are clearly unniquely defined as \(0\). It is good. I should have considered the technique. I badly persisted to set the starting point of the recursion, but your technique helps us to avoid setting it.

> Why are you editing your post instead of replying? That's a very strange way to have a conversation...

Oh, is it strange? Sorry. I thought that it were better to edit it if there has not been any reply yet. I will be more carful about it. (Anyway, when are we expected to use EDIT?)