I separated a pup today. I found it fairly simple using a (1) garden trowel; and (2) a rubber mallet. Since the garden trowel is smaller than a shovel and you do not want to damage your pot by trying to separate the pup, the trowel is small enough and the handle is long enough that you can pull it back out after you hammer it in. I don't know if this has ever been suggested before in a previous thread, but I'm just posting this for other people who want to separate a pup without removing the original plant from the pot.
Garden Trowel
Rubber Mallet