If the lease is UP and they have moved, as in any tenant (even good ones), there is no need to evict once a tenant moves out. It sounds like they never had more than a month to month. So if they left after the month was over (or if 7/31/2016 is the last day of the month and they have already moved) and they are no longer tenants, it would appear that you have nothing to evict. They have not renewed, nor have they "held over."
Make sure you followed the letter of the law with your 3 day notice. That notice would typically say you have 3 days to pay or move out/vacate or else then I evict. In your case, I think you are saying they have vacated... I'd wait til after the full time (after Tuesday? since first day doesn't count and weekends don't count)... to make sure. If they don't pay the rent and have vacated, you have no need to evict.
Now, the question becomes, how to you prove, if necessary, that the tenant vacated or abandoned; this article can help: http://www.evict.com/?page=legnew1006
And read the statute on abandonment. How long has the tenant NOT been there? How do you know?
I would video tape the entire place, inside and out, just in case you later need it.