When I started writing proofs, I focused on correctness only. If each logical step was valid, I assumed the proof was finished. Over time I noticed that this standard is too low for real communication.
A proof is not only a chain of implications. It is also a guided path through structure. If a reader cannot see why the path is natural, the argument might be true but still poorly designed.
My current rule is simple: write for future collaborators, not for exam graders. That means naming key moves, exposing the global strategy early, and avoiding hidden leaps in notation.
I now revise proofs the way I revise essays. First pass: make it correct. Second pass: make it legible. Third pass: make it teachable.
Anjou Zhao 