So come again: who/what decide (i.e. judge), is cheating A who claims his last transaction to B is N+1, or B who claims the last transaction he received from A is N?
There
isn't an N+1 unless
both parties agree that there is an N+1. Otherwise N is the current state, regardless of what anyone "
claims". In this example, B can withdraw their funds from N without a penalty. There's no cheating involved, because no one is trying to spend from an old state. N is the current state.
However, it's worth pointing out that B does have a financial incentive to agree to N+1, because it sounds like A wants to send them more BTC. It's a strange example you're using. I assume in your example you are positioning A as a malicious actor attempting to trick B to steal their coins? If so, A won't have much luck with that.
N+1 can only become the current state if A and B both lock N+1 with a new key and have also sent each other the old keys for N to revoke payments from that state. Once payments have been revoked from N, if either party then attempts to spend from N,
that's where a penalty would come into play.
I think the problem you might be having is that you associate the word "
transaction" with a one-sided push payment like it works with regular on-chain transactions. Once you've sent it, it belongs to someone else. In Lightning, however, a transaction involves the other party effectively approving the transaction. If you send a payment, it only belongs to the other person once they've accepted the latest state and revoked payments from the old state.
You keep each other honest.
It's rediculous to require to be trustfull in trustless environment, isn't it?
But it's not "
trust" keeping people honest, it's "
consequence". Play by the rules or risk losing funds.