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Suppose that I am writing a proof to a theorem that has the following structure:

Let $x \in X$, then "some statement involving $x$".

I would usually start the proof by repeating the conditions of the theorem, that is, I would usually start:

Let $x \in X$. Then...

Let's assume that the proof follows directly after the theorem, there is no intermittent text.

One of the maxims of good writing is to avoid unnecessary repetitions. This leads me to my question:

Is it necessary to repeat the conditions of a theorem in its proof, in the sense that it improves legibility?


Some of my thoughts so far.

Unconditional reasons to repeat the conditions of a theorem in its proof:

  1. It is logically necessary. A proof should be readable as a self-sufficient statement independently from the statement of the theorem.
  2. It helps the reader distinguish the conditions of the theorem from the statement of the theorem.

Conditional reasons to repeat the conditions of a proof:

  1. If the theorem statement is long, the the conditions were presumably written down some time ago, hence one eliminates the necessity of skipping back and forth in the text.
  2. If the theorem proves several related but separable claims then it helps the reader identify which point you are about to prove.
  3. In the case that the theorem reads something along the lines of:

    If $x \in X$, then .... If moreover $x \in Y \subseteq X$, then ....

It establishes if you are talking about $x \in X$, or about the more special case of $x \in Y$.

Refutations:

  1. This is simply false, a proof need not be independently readable.
  2. If the theorem statement is well written, then this should be already clear.
  3. Reading mathematics is already non-linear, the attempt to force it to be linear is at best futile and at worst counter-productive.
  4. In this case the reader would be better served by an itemized list of the separate claims and an itemized list of proofs.
  5. I have no refutation for this point.

In conclusion, I think that 5. is the most convincing reason to repeat the conditions of a theorem in its proof. This tempts me to do this always for a consistency in style, but I am not sure if this is misguided.

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    $\begingroup$ Regarding a related issue, many years ago (over 30) when I first read the Paul Halmos article How to write mathematics, I noticed (among many other things) his interesting observation that one should not introduce notation in the statement of a theorem that is only needed to prove the theorem, such as "Theorem: Every point $x$ in a bop-bee quasi-uniform space $X$ has a proactive neighborhood." There is no need to include $x$ or $X$ into the statement of this theorem. Interestingly, I see this advice violated in just about every math paper I look at . . . $\endgroup$ – Dave L. Renfro Jun 8 '17 at 15:13
  • $\begingroup$ @Dave - This is always a delightful read. Could you point me towards the section in question? $\endgroup$ – Peter Jun 8 '17 at 15:33
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    $\begingroup$ See the 2nd paragraph of Section 16. Resist Symbols. This doesn't appear to specifically target statements of theorems, but it certainly includes what I was talking about. There might be another place where he discusses more specifically the use of unnecessary notation in theorem statements, but possibly this paragraph in Section 16 is what I'm remembering. (moments later) Yep, this is what I was thinking of. Note the part that says "probably a preparation for the proof". $\endgroup$ – Dave L. Renfro Jun 8 '17 at 15:57
  • $\begingroup$ "but possibly this paragraph in Section 16 is what I'm remembering" What I meant to say is "but possibly the second paragraph in Section 16 is what I'm remembering". $\endgroup$ – Dave L. Renfro Jun 8 '17 at 16:03
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I think that you are allowed to repeat anything you want as long as the Proof remains correct and as clear as possible. i suggest you the book of proofs to learn how to structure a good proof.

In conclusion, the more your proof is clean, the better

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  • $\begingroup$ I understand that a proof should as clear as possible (while still being correct of course), the question is if my proposed proof writing "rule" makes proofs clearer or not. I do not have access to the book "book of proof", and in any case I would prefer a self-contained answer to a reference. $\endgroup$ – Peter Jun 8 '17 at 14:37

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