# Terminating each branch of a proof with $\square$

My question is a variation on this one. I have a proof which divides at the top level into a number of mutually exclusive cases, with further partitioning within that.

Is it reasonable to place a $\square$ at the end of each case which terminates a branch of the proof? In other words, is it permissible (stylistically) to insert $\square$ at any lexical point where all the pending proof obligations have been discharged, or is it wiser to insist that the reader "pop the stack" to the top level, and have a single terminating $\square$?

These two possibilities are illustrated below.

There are three possibilities:

• [Proof case 1] $\square$
• [Proof case 2] $\square$
• [Proof case 3] $\square$

vs. the following:

There are three possibilities:

• [Proof case 1]
• [Proof case 2]
• [Proof case 3]

$\square$

The advantage of the former is that it makes it clear that there is no pending context left to consume (no enclosing case has a "tail"), but I'm not experienced enough to know whether it's reasonable.

• As I understand the $\square$ symbol, it stands for Q.E.D., and thus should only come at the very end of a proof. Obviously this is a matter of convention, but that is how I have seen it. – Elliot G Mar 11 '16 at 8:34
• Q.E.E. is latin for "thus it is proven." so it doesn't really make sense to use $\square$ unless you have finished the proof (in my opinion). – Elliot G Mar 11 '16 at 8:36

## 4 Answers

The latter case, where one tombstone symbol is used for the end of the proof is the right way. The first one is confusing because you think the proof has terminated at inopportune points.

• Ok, the consensus seems clear enough! Thanks to everyone who answered/commented. – Roly Mar 11 '16 at 8:38

Only use $\square$ when what you set out to prove has been shown. Therefore, it should be placed at the end only.

You could summarize by saying something akin to

This concludes the third and final possibility, which therefore concludes the proof. $\;\;\;\;\;\square$

If you need to prove some substantial results and then combine them to prove your main theorem, it is best to organize the narrative so that the sub-results stand alone as lemmas prior to the statement and proof of the theorem. Thus: Lemma 1 . Proof of Lemma 1 $\square$ . . . Lemma 5 . Proof of Lemma 5 $\square$ . Theorem 1 . Proof of Theorem 1 (citing Lemmas 1 to 5) $\square$. The end of every proof should be marked by the square symbol, which should not be used otherwise.

• Agreed. My question would apply even to simple proofs, though. The question is really about whether "control" (to use a possibly unhelpful programming analogy) is allowed to "flow off" the end of a branch, or whether it must always pop back to the top level. (Can I put "return x" anywhere in a function definition, or must there only be a single top-level "return".) It seems the consensus is the latter. – Roly Mar 11 '16 at 10:02

In my books I use the following signs for the begin and end of a proof: $$\bigr|^{\!\!-\!-}\qquad {\mathstrut}_{-\!-}\!\!|$$ (of course with more beautiful typography, that I cannot produce here). Since these right angles act like parentheses they can be nested, so that a subproof has its own such enclosure.