# Should there be a theorem or proposition that uses lemma?

I was thinking the following: suppose that one writes an article or lecture notes where he or she uses some lemma. I have understood that every lemma should be used somewhere in the document. But is it enough if I use some lemma in an example, or should there be a theorem or proposition that uses that particular lemma?

• You'd have better luck at mathstackexchange: math.stackexchange.com
– WAS
Nov 7, 2015 at 20:34
• Welcome to Academia SE. I am somewhat puzzled by the motivation of your question. Why would any sane person impose such a restriction? And even if somebody would, you could just rebrand your lemma and call it theorem. Nov 7, 2015 at 20:40
• @Wrzlprmft: Surely there is no such restriction; I think the question is just asking for style advice. Nov 7, 2015 at 21:50
• I've had lectures where things were given exactly one of four labels: Definition, Theorem, Corollary, Example. Some people don't use lemma. Some that do don't use Proposition; and that goes for virtually any combination of names. Nov 7, 2015 at 23:40

But is it enough if I use some lemma in an example, or should there be a theorem or proposition that uses that particular lemma?

I don't think it really matters. An example would be fine for most readers, while a few really formal people might disapprove. However, even someone who disapproves wouldn't consider this seriously problematic. I.e., they wouldn't think less of you or negatively evaluate your paper because of this. The worst case scenario is that someone complains about it in a referee report, and even then you wouldn't have to follow their advice.

The typical convention that I have seen is for a lemma to be a theorem that is only of interest as an "intermediate" result on the way to another theorem. Well-organized presentations thus often use the lemma/theorem distinction to highlight which results are most significant within the material.

There are, however, exceptions to this typical practice. For example, Urysohn's Lemma is an important result in topology that is nonetheless referred to as a lemma. I suspect that such examples typically appear for historical reasons (i.e., the result was originally presented as a lemma but turned out to be more important than the associated theorem), though I do not know for certain.

• The Yoneda Lemma is another example of an extremely important, powerful, and ubiquitous result that goes by a humble name. Accordding to the wikipedia page "The Yoneda lemma was introduced but not proved in a 1954 paper by Nobuo Yoneda. Yoshiki Kinoshita stated in 1996 that the term "Yoneda lemma" was coined by Saunders Mac Lane following an interview he had with Yoneda." Nov 7, 2015 at 23:33

As far as I understand, a lemma is a mathematical statement that does not have an aesthetic value of its own, and exists only to serve a technical role in proofs of other more interesting statements. The usage of a lemma need not appear in the same document, or in the form of a theorem. For example, it would make total sense for a paper to say "the main result of this paper is a lemma that we think would be useful for proving some fancy theorems in the future. However we still don't know how to use the lemma, and reserve the fancy theorems for future work"...

The question you should ask yourself is whether this is a technical statement that has no internal beauty but can be very useful for other proofs. If this is the case, you can call it a Lemma. However if it is aesthetic and/or unravels some deep truth, it should be a theorem (if its proof is long or complicated) or a proposition (if its proof is short and elegant).

• I don't think any paper actually would say that, though. Mathematicians still have to "sell" their work--to convince referees and editors that their paper should be published. If the name of one of the authors sells the paper on its own, that may be warranted, but for most people I think it's in their own best interest to describe their main results as "Theorem", even if they're convinced it's just some dinky little lemma-class step on a way to something "actually interesting" (and a lot of us think that about or results no matter what!). +1 because this is the "role" of "lemma" I learned. Nov 7, 2015 at 23:37
• The usage I'm familiar with is opposite to the one you describe. To me, a "lemma" is a result of central importance—a strong branch on the tree of math that supports many twigs, leaves, and flowers. Lemmas can be beautiful: Bézout's lemma, Burnside's lemma, Schur's lemma, the Yoneda lemma, and the Borel–Cantelli lemma are some of the prettiest results I know. On the other hand, lemmas don't have to be beautiful: I admit that I find Euclid's lemma and the tube lemma somewhat mundane, though extremely useful. [cont. ...] Nov 8, 2015 at 2:18
• [... cont.] A "theorem," by contrast, is just a beautiful result; it doesn't have to have any importance larger than itself. Leaves and flowers are theorems; twigs and branches can be too. Nov 8, 2015 at 2:18
• On second thought, Vectornaut is right. A Lemma can be beautiful or just a bunch of technicalities - what defines it is the fact that it supports proving other results. Nov 9, 2015 at 11:44