How it works:
1. In the beginning the door is locked, cat is outside
2. Cat walks up to the gate, RF tag gets read within about 4 inches
3. Door unlocks and a light comes on.
4. Timer ensures that door stays unlocked long enough to give cat a chance to respond
5. Cat pushes open the flap and enters
6. Flap falls back, activates Hall effect sensor, and the door locks
7. Going from the inside out, an infrared proximity sensor detects the presence of an animal if within about 10 inches and unlocks the door.
Most animals will quickly learn to push the flap in response to the click (lock) and the light. Make sure that the edges of the door and the doorway are padded with soft material for when the tail gets caught! A few painful experiences may be enough to make the cat never want to go in there again.
The electronics are shown without enclosure, but it is advised to put an enclosure around the electronic parts once you're done. It is not further discussed in this Instructable however.