Anapanasati is usually described as "mindfulness of breathing" but I've seen a member of this site describing it as "mindfulness with breathing".
Sati or mindfulness (as seen in the parable of the bowl filled to the brim with oil, of SN 47.20) is keeping the mind on the task, and not allowing the mind to become distracted.
What is the simplest version of anapanasati?
I found a very simple instruction set in SN 54.3 (quoted below). There may be other suttas too. But I find this description from SN 54.3 insufficient and requires elaboration.
While breathing, what is it that one is mindful of? Is one mindful of the breath, or mindful of something else?
The sutta states: "They practice like this: ‘I’ll breathe in observing letting go.’ They practice like this: ‘I’ll breathe out observing letting go.’"
What does "breathe in observing letting go" and "breathe out observing letting go" mean? Letting go of what? Do you let go, or do you observe the letting go? How do you observe the letting go (rather than letting go)?
What is the simplest instruction for the practice of anapanasati? Especially for people who are not well-versed in the teachings and maybe even non-Buddhist? Such people may not have the patience for lengthy and complicated instructions.
From SN 54.3:
“Mendicants, when mindfulness of breathing is developed and cultivated it’s very fruitful and beneficial. And how is mindfulness of breathing developed and cultivated to be very fruitful and beneficial?
It’s when a mendicant has gone to a wilderness, or to the root of a tree, or to an empty hut. They sit down cross-legged, with their body straight, and establish mindfulness right there.
Just mindful, they breathe in. Mindful, they breathe out. …
They practice like this: ‘I’ll breathe in observing letting go.’ They practice like this: ‘I’ll breathe out observing letting go.’
Mindfulness of breathing, when developed and cultivated in this way, is very fruitful and beneficial.”