The textbook proofs of Commoner's theorem characterizing liveness in free-choice Petri nets are given in contexts of technical notions and claims that make the proofs look a bit long. The aim of this note is to give a concise self-contained proof.

翻译：教科书上关于Commoner定理（刻画自由选择Petri网活性）的证明，往往置于技术概念及声明的上下文中，这使得证明略显冗长。本文旨在给出一个简洁的自包含证明。