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We have a Padding Oracle if there is a different response from the server gives us an indication of the correctness of the pad (say if this needs proving). We can establish this by playing a game where we send badly padded cipher-text and random strings to the server, finally submitting some at random and seeing if we can get a non-negligible Advantage is ...

Prove? Why does the attacker need to "prove" it? For example, the attacker can check whether there is an oracle by looking at the code and seeing whether such an attack is possible. Or, the attacker can guess that such an attack might be possible and then try the attack. If the attack succeeds, the attacker knows the system is vulnerable. There might ...