4
$\begingroup$

In a text there will often be a few important results that are usually called Theorems. Intermediate statements are called Lemmas, and statements that follow immediately from previous results are called Corollaries.

There is a problem here, however. Especially in the context of theorem provers: A Theorem $A$ is often proved using induction. In order to perform this induction, the statement of the theorem needs to be strengthened to $B$ (otherwise the induction will not 'go through'). The original statement $A$ of the theorem will then follow trivially from $B$. We now have three possible naming conventions for this:

  • Call $B$ a Lemma and $A$ a Theorem. This is attractive, because $B$ is the main statement of interest. The problem is, however, that the proof of $A$ is mainly done in $B$, which is now called an 'unimportant' Lemma.

  • Call $B$ a Theorem and $A$ a Corollary. This is attractive because $B$ follows immediately from $A$, hence a Corollary. Also, it makes clear that the proof of $B$ contains a lot of important steps. However, when looking over the text quickly, one might miss the main statement $A$, because it is phrased as a simple consequence of $B$.

  • A third option is to not have $B$ at all but only state Theorem $A$, and then perform the strengthening of the statement 'inline' in the proof via a cut. This is also inconvenient because it hides the technique used to prove $B$, which may be very important.

I would be interested in how this is commonly solved.

$\endgroup$
  • $\begingroup$ Why not call the subordinate, but stronger, statement a "proposition"? And the main result a theorem (but state in its proof that it's a corollary of the proposition)? $\endgroup$ – Robert Wolfe Jan 15 '18 at 15:11
  • $\begingroup$ I do not personally like using propositions. The meaning of a proposition is ambiguous. It can present a fact that is more important than a lemma but less important than a theorem (this seems to be the meaning you are using). Alternatively, it can also state an arbitrary mathematical sentence that may or may not be true (a conjecture). Because of this ambiguity, I don't really like using it. $\endgroup$ – Lasse Jan 15 '18 at 15:38
  • $\begingroup$ I've never seen "proposition" used to state a conjecture. I've always seen it used in the first sense. In most textbooks (at least in the US), you'll find that most authors either use "proposition" to mark the majority of their results and list some important results as a "theorem" or exclusively use "theorem" to write all results. I prefer the former approach. But it's your paper. $\endgroup$ – Robert Wolfe Jan 15 '18 at 16:13
  • $\begingroup$ Well, the word itself kind of implies this second meaning. One is "proposing" a mathematical sentence, of which it is not yet known wheter or not it is true. But I agree that the first meaning is most often used. $\endgroup$ – Lasse Jan 15 '18 at 16:47
  • $\begingroup$ Crossposted to MathOverflow: mathoverflow.net/questions/290773/… Please don’t do that, as it leads to duplicated effort. $\endgroup$ – Jeremy Rickard Jan 15 '18 at 17:07
2
$\begingroup$

I would use option 2, but I would warn the reader in the introduction of the paper, by writing something along these lines:

The main result of this paper states that

[statement of Corollary A]

This result is actually obtained as a corollary of a stronger, but more technical, result (Theorem B).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.