I've leased my last 7 cars, and never once paid any attention to the actual price of the car when going into the dealer. Sure, I know what the price should be beforehand, and I take that, factor in the residual value at leases end along with my money factor, and see exactly what my payment should be. I know what my monthly payment should be. What do I care how the dealer got there, as long as he gets there? Whether he fudges the numbers on my trade or lowers the price of the new car, means absolutely nothing to me. As long as you know what you should be paying per month, that is absolutely all that matters.
Putting money down on a lease can be a waste, but the same can be said for buying a car. If you buy a car and trade it in before the end of the term (effectively leasing, but at a higher monthly payment) and you put money down on the car, you're never going to get that back either.
However, if you lease your car all the way to the end of the term, putting a bit of money down isn't a waste. Let's say you put $1,000 down. That will lower your payment by about $25 per month. On a 36 month lease, that comes out to $900. Factor in taxes, you're probably spot on.