Ultimately you have to decide what you can/can't live with. Personally I can't live with someone I don't trust.
Me either. So while all of the above is true -- you can't control, focus on yourself, all of that good stuff -- I'm with Jay in that I would tell the A that I knew. I wouldn't do it as an accusation, I wouldn't do it as an opening to a discussion, I would probably just say, "I found a receipt for a Bacardi in your laundry, and I want you to know that when I believe you are lying to me, it makes me feel like I can't trust you." And walk away.