When should we repeat a statement of a theorem before proof in a paper?

Question

When should we repeat a statement of a theorem before proof?

Suppose we have a theorem in a paper at the introduction or at the beginning of a section. However, a proof is given, say, six pages later after a series of definitions, lemmas and propositions. It would be convenient for the readers to repeat a theorem before we give proofs to recall hypothesis and notations (especially those who read an electronic copy to see in a screen). Then, however, the paper becomes redundant. Should we repeat?

More pragmatic question: Are there any guideline/style of journal papers/books mention when to repeat or not?

---

Added: Although I asked when to repeat, how to repeat can be found, for example, in section 6.10.1(b) in this style guide.

Answer 1

Unless, you are facing a particular nasty page limit, those few lines spent repeating the theorem should not be a concern to you. Thus, the only concern you should have is what benefits the reader. Here, redundancy is not bad as such, but only if it annoys the reader and wastes their time. Therefore, you have to compromise between the following:

• Readers whom the redundant information helps. In most cases, you have to expect that readers have not memorised the entire theorem yet, and this is clearly information that is relevant to reading the proof. Of course, readers can flip back to the theorem, but that’s a major annoyance (even if you can put the pages next to each other).

  Even if the theorem equals the title of your paper and is something like:

  Every frasmotic contrafibularity is a compunctuous pericobobulation.

  and you spend the entirety of the space between first stating the theorem and proving it with talking about the four terms involved, you cannot expect the reader to remember that, e.g., Theorem 2 was this titular theorem. Even if you refer to it as titular theorem, readers may need more time to understand this than to read the short theorem again.

• Readers who are annoyed by the repetition. – In the situation you describe, I would expect that only very few readers do not need the theorem repeated to them. If the formatting is any reasonable, those should notice very quickly that you are restating the theorem so they can skip it.

Therefore, in most cases (including the one you describe), I would conclude that for the average reader the usefulness of restating the theorem outweighs the annoyance.

Answer 2

Apart from externally imposed page limits and such, I think it makes mathematics papers much more readable to repeat key points as relevant, rather than requiring that the reader flip back and forth (even if this is made somewhat easier in certain computer-reading situations). For that matter, I'd advocate explanatory comments in addition to otherwise-cryptic internal cross references, e.g., not "by theorem [3.7]", but, perhaps, "by the long exact sequence relating X and Y from theorem [3.7]"...

The notion that redundancy is undesirable is, I think, misguided, although perhaps understandable. Ordinary language has substantial redundancy, and this gives it stability (e.g., in the face of most typos or grammatical errors). In contrast, highly-compressed (in the information-theoretic sense) but long-ish mathematical writing can be very brittle and sensitive to typos and/or mis-reading.

If we were all infallible logical automatons, maybe the information-theoretic issues would be different, and I do know that many people like the mythology (for mathematics) about behaving as though we were, indeed, immortal and infallible and so on, but, in practice, many of us aren't.

Answer 3

I remember asking this question once. The professional I consulted is very rigorous with his work (he was a book editor at Springer). He replied that, in general, high quality paper should be read from top to bottom without looking up, and be as accurate and austere as possible. His answer at first seemed rather vague to me, but believe me, that's practically what it's all about: communicating as well as possible.

As an early interpretation, if you need to give a theorem at the beginning and then prove it at the end, you must take care that the reader does not have to read the theorem again. The only thing that changes is the elegance with which you achieve it.

For example, if your paper deals with a problem you solve, it is better:

• Initiate a brief introduction to the problem (taking the opportunity to introduce references).
• Formulate the theorem as a problem, conjecture, or something unresolved: state things as you found them. Remember, it's not a good idea to rush through everything at first; the trick is to induce the reader to want to read your work to the end more than anything else in the world.
• If your development is very long, you can also add an "outline" of how you have structured your paper to reach the solution, so the reader will not get lost in the tangle of reasoning.
• In each development, explain in human words what you are going to do or the results you get; you can take advantage of this to show how you advance in the "outline" of the paper you gave before.

In this way the reader will be imbued with the problem and will keep it in mind. Then, after all the propositions, lemmas, etc., enunciate the final theorem with its proof.

Your question is ultimately about how you structure the paper for the best possible communication; whether or not to use a theorem or problem or both to enunciate and develop its content is pure strategy. For the structuring of the paper, there are several interesting books. One that served me well is Writing Science.

To see more specific strategies, I strongly recommend that you decide beforehand which journal you will publish in, and review how the other authors have structured their work. Look at a lot of them, so you get the average idea. It is also very good to review high-impact journals; although it is difficult for you to understand their content, focus on understanding the structure of the explanation, which is more important (of course, you can take advantage of this to read your references in detail).

---

If you still want to keep your theorem at the beginning and your proof at the end, and if you write in LaTeX then you can call the proof environment with the optional parameter [Proof of Theorem <enumeration-here>].

Answer 4

I don't think it is generally necessary to repeat statements of theorems.

The theorem has already been stated, so repeating it verbatim is unnecessary and not useful. I appreciate that proofs and theorem statements may appear in distinct parts of the paper. E.g., a theorem may appear in the main body and the proof may appear in the appendix. This isn't problematic for the reader, because they can simply switch between the two. (For publisher printed manuscripts, this is a little tricky. But, for other versions (electronic, personally printed, ...), this is easy.) Indeed, I regularly have multiple copies of the same PDF open for this purpose.

Sometimes, you might state a theorem, introduce some lemmas, and then prove the theorem using those lemmas. In this case, I advocate a style similar to that proposed by Andreas Blass, namely, recap some details of the theorem statement and explain how the lemmas.