Given a channel that exists between Alice and Bob, where Alice wants to transfer 0.0001 BTC to Bob, Alice makes an HTLC payment to Bob contingent on the secret for the payment hash within a timelock of say 100 blocks.
Alice and Bob will each have a copy of the commitment transaction.
Suppose Alice decides to close the channel at the 50th block height by broadcasting her copy of commitment transaction which has two outputs:
HTLC success and
HTLC timeout. She bears the transaction fee of the commitment transaction. As a result, Bob can immediately spend the
HTLC success transaction by using the preimage of the payment hash and his signature. In order to guarantee his
HTLC success transaction is prioritized, Bob increases the transaction fee of the
HTLC success transaction (using Replace-by-Fee).
So does Bob have to bear the cost of the increased transaction fee in order to spend his output?