Index notation in derivation rules

Question

I need some advice on a (fairly rudimentary) notation. Consider the derivation rule

(1) $\qquad\qquad\qquad\qquad \dfrac{k_0, ..., k_n}{A}$.

While it's possible that $k_0 = k_n$, is it possible that $n = 0$? In other words, is

(2) $\qquad\qquad\qquad\qquad \dfrac{k_0}{A}$

a legitimate possible interpretation of (1), for the case that $n = 0$?

The reason for me asking this is that I'm using a notation like (1) and want to preclude the possibility that $n = 0$. I wrote

(3) $\qquad\qquad\qquad\qquad \dfrac{k_0, ..., k_{n+1}}{A}$,

but thought it may be just obfuscating and overthinking. Note that commas are optional in the assumptions

Answer 1

In general, a finite sequence of $n \in \mathbb{N}$ elements in some set $K$ is commonly denoted by $(k_1, \dots, k_n)$. Note that this notation includes the case of $n = 0$ elements, where you get the empty sequence $(\ )$; and the case of $n = 1$ elements, where you get the sequence $(k_1)$. A common notation that forces a finite sequence to not be empty is $(k_0, \dots, k_n)$ (or $(k_1, \dots, k_{n+1})$, respectively) with $n \in \mathbb{N}$, because in this way, for the minimal case $n = 0$ you get the sequence $(k_0)$ (or $(k_1)$, respectively).

Thus, in accordance with that, a natural notation to force a finite sequence to have at least two elements is $(k_0, \dots, k_{n+1})$ with $n \in \mathbb{N}$: in this way, for the minimal case $n = 0$ you get the sequence $(k_0, k_1)$. This notation is not so common because it is not so common to talk about finite sequences with at least two elements, but it is perfectly legitimate and coherent with the well-established notations I mentioned in paragraph above. In my opinion, everybody will understand, especially if you say explicitly what do you mean the first time you write something of the form $(k_0, \dots, k_{n+1})$, just to avoid that the reader thinks that there is a typo.

Therefore, if you want to talk about an inference rule with at least two premises, your notation

(3) $\qquad\qquad\qquad\qquad \dfrac{k_0, ..., k_{n+1}}{A}$

is, in my opinion, the best choice, the easiest one to understand and perfectly consistent with well-known notations.

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Another notation for a finite sequence of at least two elements is $(k_0, \dots, k_n)$ with $n \in \mathbb{N}^+$, where $\mathbb{N}^+ = \{1, 2, \dots \}$ (which corresponds to your solution (2)). But I dislike this notation because it forces to change the index set from $\mathbb{N} = \{0, 1, 2, \dots\}$ to $\mathbb{N}^+$, which is misleading especially if you are in a context where you work with the set $\mathbb{N}$ and you implicitly or explicitly assume that in general the index set is $\mathbb{N}$.