I'm sure this is a dumb question, but, I don't fully understand how the refunds work.

So, we have a refund for the TX_0 transaction, but don't announce it. If Carol announces a transaction  that spends TX_0, and then announces a refund for TX_1, how does the refund work to prevent cheating? Isn't TX_0 spent at that point?

I think I am missing something re: the time-out redemptions. Do these refund transactions somehow supersede other transactions, even if the other transaction is announced before the refund?