What do you drive? F(ront)WD, RWD, or AWD?
Treadlife warranties are riddled with exceptions and generally worthless. Read the fine print of your warranty. First, it probably requires documentation that you rotate your tires at least every 10,000 miles. (And you should rotate your tires every time you get an oil change). you'll need the receipts to prove this.
Second, you generally have to be down to 2/32s of tread in order for the warranty to kick in. Are you down to that?
Third, the wear has to be even across all 4 tires. If it isn't, Goodyear won't honor the warranty - they'll blame it on bad alignment. If it's only your front tires, that's a problem.
Fourth, the warranty is prorated (meaning you've already burned a third of your warranty amount), and can only be applied against another set of Goodyear tires, usually at MSRP.
In summary, you'll probably be better off just buying a new set at a cheaper price. Or, you might consider just rotating your tires and trying to squeeze another 5k of life out of them.