How to Write a Proof in a paper

Reviewers are not required to read supplementary materials, but many do. Therefore, making your proof easy to read is important.

General Guidelines:

1. Placement:
   • Conference Papers: Given the page limit, it’s usually acceptable to include the proof within the main text if it’s not too long. For longer proofs, consider placing them in supplementary materials if allowed. Note that some conference does not limit the length of supplementary materials. So, some authors submit the statements in the main text (the paper), and provide the proofs in supplementary materials.
   • Journal Papers: Proofs are often provided in the appendix and referenced in the main text.
2. Structure:
   • Highlight Key Ideas: Start by outlining the main ideas of the proof.
   • Break into Subsections: Divide the proof into logical subsections if possible.
   • Clear Notation: Clearly state any necessary notations. You can also provide a table of notations if needed.
   • Equation Numbering: Numbering equations can be helpful for reviewers to provide feedback, although it’s not mandatory for all equations.
3. Clarity:
   • Ensure there is a clear connection between different parts of the proof, similar to writing a story.
   • When using lemma(s), explicitly state if the proof is for the lemma or the main statement.

Example:

Theorem 1: Let and be real numbers. If , then .

The proof of the above theorem is based on the following lemma:

Lemma 1: If and both and are positive, then .

Proof of Lemma 1:
Assume . Since both and are positive, we can write:

Squaring both sides, we get:

Since , it follows that .

Proof of Theorem 1:

We consider two cases:

• Both and are positive.
• and are not both positive.

Case 1: If and are positive, the result follows directly from the lemma.

Case 2: If either or (or both) are non-positive, without loss of generality, assume . Since , we have . Hence:

In both cases, , completing the proof.

This example demonstrates a structured approach to writing a proof, including an outline, a preliminary result, and the proof of the main statement. Breaking down the proof into manageable sections and clearly stating the logical flow can help reviewers understand and follow the argument.

More examples in published papers:

An example of a not too long proof being placed in the main text can be seen in page 4-5 of this Combining datasets to increase sample size paper, a 5-page long proof being broken down into lemma, subsection can be seen in pages 12-17 in this EPEM algorithm paper.