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I am writing an academic paper for submission to a journal. One of my co-authors wrote the following:

Theorem Statement of the theorem

Proof of theorem We first show the following result

Lemma Statement of lemma used to prove the theorem

However, I think that it is more natural to present things in the following way:

Theorem Statement of the theorem

We first show the following result which is used to prove the theorem:

Lemma Statement of lemma used to prove the theorem

Proof of theorem blah blah

I understand that this is a subjective question so I am happy to mark this as community wiki if most people feel that it belongs there.

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    $\begingroup$ I more often see the order lemma, theorem, proof. $\endgroup$
    – JRN
    Commented Aug 1, 2013 at 23:54
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    $\begingroup$ I agree that the second way is better. The first way leaves the question unanswered whether the lemma is only formulated in the context of the theorem, in the context of its proof (which may be more restrictive, e. g., due to WLOG assumptions or cases) or in the context of the surrounding paper. In the second way, it is clear that the context is that of the surrounding paper. $\endgroup$
    – darij grinberg
    Commented Aug 2, 2013 at 0:07
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    $\begingroup$ Obviously which is more natural / easier to read depends a lot on the specific context (for example, the relative length of the proof of the theorem and/or lemma, and whether the lemma is really just a "claim" on the way to proving the theorem or rather more independent). $\endgroup$
    – John Pardon
    Commented Aug 2, 2013 at 0:07
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    $\begingroup$ +1 for asking such a question, although I understand the close voters: such a question might be better at math.stackexchange. $\endgroup$
    – darij grinberg
    Commented Aug 2, 2013 at 0:08
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    $\begingroup$ I think this question is too localized for MO as phrased currently, but that doesn't make MSE a better fit. A better way to phrase it might be, "What are good ways to present proofs of theorems requiring auxiliary lemmas?" Certainly, though, this question is of interest to research mathematicians, and is specifically about mathematical writing. $\endgroup$
    – Noah Schweber
    Commented Aug 2, 2013 at 0:19

1 Answer 1

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The guiding principle should be: think of your readers. Nesting results inside the proofs of other results can make things confusing. I am looking right now at a paper that has three instances of that, and one of them is truly egregious: the bracket diagram is like this: $((((()))(())))$.

Maybe one nested lemma is ok, but if I could avoid it I would just put the lemma before, and say something like: "The following lemma will be used in the proof of Theorem 1." If you really want to state the theorem before the lemma, then your second option is better.

OK, I'll stop grinding this axe now.

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    $\begingroup$ In the introduction/statement of the main results, one can always state the most important theorems in any order one sees fit. So, there is hardly a single reason why to break the logical flow of things later when all the lemmas and theorems are presented rigorously with their proofs. If anything, it makes following the rigor only more difficult. Putting lemmas and their proofs inside other proofs is just bad style. $\endgroup$
    – M.G.
    Commented Aug 2, 2013 at 12:10

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