I got the game in the mail early today and noticed the super trainer thing in the very early game.
While you are exploring on the top screen you can have the super trainer on the bottom. You gets to choose between different punching bags for your Pokemon and will punch it once every 10 seconds, but it will also punch the screen if you tap the bottom screen. Punching will improve the stats for the pokemon depending on the punching bag.
The first thing I thought of was the cookie clicker web game. And that mindless tapping for training your Pokemon during downtime is a smart idea.