"Beyond a shadow of doubt" is an idiom that means "with absolute certainty," or "without any possibility of error or mistake." It means that something is so clear and definite that it cannot be questioned or disputed.

The idiom "pay lip service" means to give the appearance of doing something but not actually doing it, or to only say good things about something or someone but not truly support them. It's like giving a fake hug or nodding in agreement when you don't actually mean it.

"Bell the cat" is an idiomatic expression that means to reveal a secret or a surprise, often for someone else's benefit. It can also mean to outsmart someone or to take action against them.

The idiom "once in a blue moon" means that something happens very rarely or almost never. It refers to a situation or event that is so unlikely to occur that it is considered a rare occurrence.

The idiom "green with envy" means to feel intense jealousy or resentment towards someone who has something that one desires, such as wealth, success, power, or a desirable relationship.