# How to organize lots of assumptions/equations in a math paper? (I need to refer to them in my main theorem)

I'm writing a paper. It has a lot of lemmas that are used to prove a theorem. I put those lemmas first and then my put theorem at the end.

In the lemmas, I introduce assumptions (they're displayed equations that I label so I can recall them later) that are needed to prove the lemma.

Then at the end of the paper, when I get to my theorem, I say "Let equations (1), (5), (21), (24) and (45) be true. Then this wonderful result is true: blah blah blah".

Now a referee for my paper said that it's not nice to do this because he/she has to go back to find (1) and then (5) and then (21) etc.

What's a good practice or the best practice for this? I could collect all of the assumptions right at the start and then refer to them separately later in the lemmas, which doesn't look nice, but then the theorem would look nice because I could say "Let equations (1)-(5) be true..." Another reason against doing this is that the assumptions are different in nature so it doesn't seem natural to put them together, especially if there are like 10 of them.

Does anyone have any advice or examples?

• "In the proof of this theorem, we use equations 1 (see page 2), 4 (page 8), and 45 (page 12)". Mar 17 at 22:43

Our main theorem requires the following assumptions:

• A1:
• A2:
• ...

(and here comments about what they mean and are they necessary etc. if such comments are appropriate.

In the main theorem itself: If the assumptions A1-A5 hold, we have results.

(Please have the main theorem in the introduction so that it can be quickly found when scanning the paper, if this is at all feasible.)

In the lemmata: Suppose A4 and let us define some entities. Then we have a result.

# The benefits

• All the assumptions are written in one place. This makes it easy for someone using your result as a black box or just wanting to see if it can be used as such, and makes it much easier to compare assumptions of different results. A reader knows where to find all the assumptions.
• The main theorem follows the assumptions. No or little page-flipping required.
• You have an obvious place for discussing if a given assumption is actually required, good or inconvenient for modelling or computational purposes, sharp, could maybe be avoided, has a restatement, has or has not been used before, etc.

However, this might not always be possible. Maybe one of the assumptions is very technical or requires notation developed later in the paper. You can still have it in the introduction or together with the other assumptions, but as a reference:

A4 is a technical assumption that we state at the beginning of section 3.

Possibly you could lessen the problem by also mentioning this in your introductory paragraphs, such as saying something along the lines of "Our main result, Theorem 6, will follow from (1), (5), (21), (24), and (45)." If the following applies, even better would be "... and (45), which have independent interest" (or "which are of", or "which we believe have", etc.).

If there aren't too many of them, you might consider naming the assumptions, rather than just numbering them. If the names reveal the intention of the assumption it might read better. It might also lead to better insights overall.

Even if you could only name the majority of them then the paper might read better. Some assumptions have common names, such as the Axiom of Choice, of course, which is sometimes specifically mentioned.

• I thought about this but I think it would be worse. The assumptions are technical so I can't give them nice names like "axiom of choice". Then I'd be in the situation where someone would say "Assumption (H1)? Where the hell is that defined?"
– BBB
Mar 17 at 17:28
• It could be that the editor will accept the paper as is, of course. I don't actually find your original description of the paper so bad. OTOH, names that reveal the intention are very helpful both to the writer and the reader. Insight, again. But, no, "Assumption (H1)" is definitely worse. Mar 17 at 17:31

On one hand, if you can explain all the assumptions fairly directly, you might do so at the outset, and state the theorem there, too.

On another hand, if explanation of the equations and assumptions is very complicated, you might "recall" all those equations just before the statement and proof of your theorem, "for the convenience of the reader".

I do also generally like catch-phrases naming assumptions, both because they're at worst far more meaningful and explanatory than artifactual equation numbers, and therefore more memorable.