Chop the onion, carrot and celery together finely into a near paste. (A food processor makes short work of this, but a sharp knife and cutting board will do quite nicely as well.)
In a pot, preferably made of terracotta or enameled cast iron, sauté the onion, carrot and celery in abundant olive oil, along with a pinch of salt, over a gentle flame until tender.
Turn up the heat a bit and add the tripe strips and given them a good stir to coat them with the scented oil and aromatic vegetables. Let the tripe simmer over moderate heat for 5 minutes or so.
Add the tomatoes, which you will have either crushed with your hands or passed through a food mill, and stir again. Lower the heat, cover the pot partially, and let the tripe and tomatoes simmer until the tripe is tender and the sauce has nicely reduced. The total cooking time will depend on how fully the pre-cooked tripe is, as little as 30-45 minutes or as long as 2 hours.
If things are drying out, add a bit of water. Conversely, if the tripe is tender but the sauce a bit thin, uncover the pot and raise the heat so the sauce can reduce.
Serve your trippa alla fiorentina with a nice topping of freshly grated parmigiano-reggiano.