PS, Everyone is making good points. Your H is in replay because it is easier. The lying is his survival defense mechanism, one which he is not proud of underneath. We all feel better when we are operating from the best part of ourselves, and he has lowered himself once more. And when you react to his actions, he picks up that you can be brought down too.
It is important to show him that he does not get to you and that you are moving forward and living life the way it was meant to be. This is healthy for you and your boys.