# Should a theorem be numbered by where it is first stated or where it is proven?

Suppose I am writing a paper in which an important lemma has a proof which is either long and unenlightening or requires additional background of the reader (or both). Thus, to avoid disrupting the narrative flow, I want to postpone the proof of the lemma to a later section. So I have a lemma in section 3 (say) whose proof is postponed to section 7 (say). And in section 7, I restate the lemma before proving it.

Now I can think of three ways to assign numbers to the lemma:

1. The statement in section 3 and the restatement in section 7 each get their own number. Thus we have Lemma 3.4 and Lemma 7.2 which are identical.
2. The restatement in section 7 gets the same number as the original statement in section 3. Thus we have only Lemma 3.4, which is stated in section 3 and then restated with the same number and proven in section 7.
3. The statement in section 3 gets the same number as the restatement in section 7. Thus we have only Lemma 7.2, which is stated and proven in section 7, and also stated with a forward reference in section 3.

(Some questions on tex.stackexchange here and here explain how to accomplish the latter two options technically.)

Which of these options is preferable?

Edit: Note that the first statement of the lemma is in section 3. This is not the introduction (at least, I've never read a paper where section 3 was the introduction). The situation of stating one or more of the paper's main theorems in the introduction is similar, but not quite the same.

• I'd say it depends on the text, but please do not refer backwards to the introduction, even if you state some of your results there. (There are many readers who skip the introduction because it is the hardest to understand part of many papers.) – darij grinberg May 20 '14 at 19:13
• @darijgrinberg, I've edited to clarify that in this case, the initial statement is not in the introduction. (I'm very sad to hear that such readers exist; I would say that the introduction is the most important part of many of my papers.) – Mike Shulman May 20 '14 at 23:09
• Then you are one of the few authors who write their introduction for readers rather than for reviewers. Most introductions I see are really long abstracts, written to convince that the work is substantial and involving more complicated material than appears anywhere in the actual paper. – darij grinberg May 21 '14 at 6:57

It depends on what you want for your reader, who will be referred to it a number of times, I suppose. I would rather choose the second option, because I assume you explain why it is so important when it first appears, so the reader should be constantly referred to that occurrence, and not to the cumbersome demonstration.

• I agree with this. The lemma should be named after the section of its primary appearance, so the question reduces to whether Section 3 or Section 7 is primary. But the way Mike described the situation seems to make it clear that Section 3 is the primary appearance, and its appearance in Section 7 is akin to an appearance in an appendix. – Timothy Chow May 21 '14 at 17:45

There is a fourth way: give the Lemma a name out of the given numbering scheme, e.g., call it Lemma A.

• The problem with this is that when I refer to "Lemma A" in section 5, the reader has no idea where to look to find it. – Mike Shulman May 20 '14 at 23:00
• You can say "Lemma A (stated on page \pageref{lem:A})". – André Henriques May 21 '14 at 4:09

I tend to use Option 1, with the restatement listed as

\begin{lemma}[Restatement of Lemma 3.1]
xxxx
\end{lemma}


Alternatively, "For the convenience of the reader, we restate Lemma 3.1."

But I'm not sure how good a question this is for MO, since it certainly doesn't have a specific answer; you'll undoubtedly receive different opinions. Also, specific journals may have their own preferred styles for doing this, in which case that's the one to use.

• What's the point of giving the restatement its own number (say 7.2?) Wouldn't it be rather confusing to ever refer to "Lemma 7.2" if Lemma 7.2 is just a restatement of Lemma 3.1? Why not "\begin{lemma*}[Restatement of Lemma 3.1]" instead? – Trevor Wilson May 20 '14 at 22:27
• @TrevorWilson Well, if I want someone to easily look towards the lemma and it's proof, referring to Lemma 7.2 will direct the reader to the right section, i.e., to the statement and proof. (And even if I don't do that in the paper, maybe someday someone will want to in another article.) Relabeling it as Lemma 3.1 is confusing because the numbering is out of order, and (just my preference) I don't like un-numbered statements. – Joe Silverman May 20 '14 at 22:49
• I was imagining saying something right before Lemma 3.1 like "the following result will be proved in Section 7." However, I can see how what you are saying applies if Section 7 is long and the restatement of Lemma 3.1 doesn't come at the beginning of it. – Trevor Wilson May 20 '14 at 22:54
• @TrevorWilson, as Andre pointed out in a comment on Jeff's answer, even if section 7 is long, one could instead say "the following result will be proved in section 7 on page \pageref{lemma7.2}." – Mike Shulman May 21 '14 at 4:14
• I completely agree with Trevor Wilson. You give a theorem a number for the sake of later reference, but if a later reference is already easy and possible, why bother? – Delio Mugnolo May 21 '14 at 4:26

I would give only one of them a number. The other one would not be numbered at all, but would refer to the numbered version.

Depending on the text at hand, I would thus choose one of:

In §3:

A very important result is:

3.x Lemma. Every regular doodad is a widgit.

In §7:

We recall Lemma 3.x:

Lemma. Every regular doodad is a widgit.

Proof. Proof goes here.

or

In §3:

A very important result is:

Lemma. Every regular doodad is a widgit.

In §7:

We now prove the important result announced in §3:

7.x Lemma. Every regular doodad is a widgit.

Proof. Proof goes here.

• Ok, that's a minor modification, but the real question is, which of them would you do? You say "depending on the text at hand", but what exactly would it depend on for you? – Mike Shulman May 20 '14 at 23:03
• Why do you want there to be one that's right and one that's wrong. I think that it's better to have the two options side by side, along with the understanding that neither is better than the other. Of course, in your paper, you'll have to chose between the two: you'll have to break the symmetry. – André Henriques May 21 '14 at 4:12
• @AndréHenriques Exactly: I have to choose between the two, and I want advice on how to make that choice. – Mike Shulman May 21 '14 at 12:55
• Whichever one you choose, your referee will prefer the other one and let you know it. Then after you change it, or don't, you'll find out the journal you submitted it to has an unwritten house rule about exactly these situations and you'll have to choose that way. – Zack Wolske May 21 '14 at 17:03

I like Option 3. The question, when choosing between Options 2 and 3, seems to me to be whether you want a reader looking for this theorem to look in the intro or at the proof. This could either be a reader chasing a reference internal to your paper, or coming from some other paper which cites [Shulman, Lemma 7.2]. Either way, it seems to me you want him or her to go to Section 7.

I see no reason not to number the theorem in the intro -- why not tell the reader where to find the proof?

The exception I'd make is if the intro gives a less precise, more user friendly, version of the theorem. Then I'd give them separate numbers, so that they can be cited separately.

• I've edited the question to clarify that in this case, the initial statement is not in the introduction. Do you still prefer option 3? If so, can you explain why you would want a reader to be referred to section 7 rather than section 3? – Mike Shulman May 20 '14 at 23:10

Clearly, having two identical lemmata with different numbers is just plain silly and confusing, so option 1 is out.

Option 2 is probably the easiest and most natural choice, but it does have one disadvantage: readers looking for the proof of the lemma (rather than just a statement of it) will first have to flip back to section 3, before finding out that the actual proof is in section 7. That said, it's an issue you could easily live with, especially if your paper is not absurdly long. After all, people do this all the time in applied math journals, where long or complicated proofs are routinely delegated to appendices.

The problem with option 3 is that, when you first introduce the lemma in section 3, you basically have two choices: either number it as 7.2 (which looks jarring; theorem numbers are supposed to be consecutive) or leave it unnumbered (in which case sequential readers will be confused when you suddenly start referring to a "lemma 7.2", even though they haven't seen any such lemma yet).

That said, if you really want to go with option 3, this is how I'd implement it:

"We will now apply a lemma which will be proven later in Section 7. For convenience, we will restate this lemma here.

Restatement of Lemma 7.2. If $a > 0$ and $b < 0$, then $a - b > 0$.

Proof. See Section 7 (page 123).

By applying Lemma 7.2 to equation (3.14), we see that..."

A variant of this, which might be worth considering, if the lemma in question is indeed important and used in several place, would be to give the lemma a name, as in:

"We will now apply a lemma which will be proven later in Section 7. For convenience, we will restate this lemma here.

Restatement of Lemma 7.2 (The Sign Lemma). If $a > 0$ and $b < 0$, then $a - b > 0$.

Proof. See Section 7 (page 123).

By applying the sign lemma (Lemma 7.2) to equation (3.14), we see that..."

The advantage of naming your lemma is that it signals to the reader that this lemma will be reused later, and that they should actually stop and remember what it says. By also referring to the lemma by its number, you still ensure that any readers who somehow managed to miss the initial statement can still find it by flipping to section 7.

(Also, as a bonus, if the lemma really is important and useful, you might find other authors picking up the name when they cite your work.)

Just keep the proof where it belongs, after the statement of the lemma. Prior to the statement, you could say something to the effect of "The reader may wish to postpone reading the long and technical proof of the next lemma, and proceed to [ ... ], since understanding the details will not assist in following the continuing argument".

I would avoid altogether restating the lemma in Section 7. This introduces some redundancy and can eventually lead to inaccuracies and confusion: what if at some stage you modify one of the instances of the lemma, but forget to modify the second one?

• Using the thmtools package for LaTeX, the inaccuracies will not happen. – jmc May 20 '14 at 20:42
• And I find it very confusing to try to read a proof without knowing what it's a proof of. – Mike Shulman May 20 '14 at 23:02