what kind of information does someone need to prove they have not left their country after being deported and 10 year bar has lapsed

You simply need to show you have not returned to the United States. There are a variety of records that could be used to show you complied with the ban. You could submit copies of leases, utility bills, bank statements, income tax returns, employment records, school records, etc. You essentially need to provide documentation showing you were in your native country during the past ten years and not the United States.