i lost my airpods but i don’t have any proof of purchase because they were a gift. what are my options?

Warranty does not cover lost or stolen devices.

If you lost the Airpods and the case, you will have to buy a brand new set.

If you lost a single Airpod or both, it makes financial sense to buy replacements.

You need the serial number on the case, go here:

AirPods and AirPods Case Replacement - Official Apple Support