Whats an interesting name for the chat room?
Every site comes with its official chat room. This is created automatically and can be reached from the Site Switcher on the top left. The chat room is the third place of your community, after main and meta. It is like the lounge where everybody can get together to discuss, complain and have fun. Personalizing this space to reflect your community and what you are, is one of the small steps to take towards building an active and engaged community.
Try to get a few ideas up, discuss and vote on them. The name could be something esoteric in your field, or something that every person may understand. Whichever it may be, go ahead and start the discussion!
Really, all this proving stuff business isn't telling us anything we didn't already know. It's all just implications of the axioms. We are but explorers in the space of tautological statements; proof assistants are equipment.
Any on-topic discussion is talking about tautology.
Proof assistants almost inherently use libraries to collect, organize, and reuse definitions, constructions, and proofs. And while libraries are also a big part of most programming languages, "The Library" has a much more pencil-and-paper feel to me that is reminiscent of mathematics.
Suppose you are participating in a discussion related to some question, or a broader issue related to proof assistants, and you want to suggest taking it to chat. With a name like this you could say "Let's take this to The Reasoned Conversation."
For me that sets a good kind of tone, and reminds people that this site is about reasoning.