As Wolfgang pointed out, if the search space is limited and searchable, proving a negative is quite possible. You can prove there is no tiger in the living room without much difficulty. Proving there is no tiger in the whole country is a lot harder, but in principle possible, because it would at least in theory be poossoble to conduct a thorough search. Proving that there never was a tiger in the country is impossible, because in principle you cannot search the past.

In legal situations the presumption of innocence until proved guilty applies. Until recently it was recognised that to require people to prove their innocence of a charge was a clear indication of the exitence of an essentially tyrannical and opppressive regime.

Myself, I'd say that that is still the case, and I very much regret the lurch in that direction by a number of regimes which, however imperfect, used to draw the line at that kind of thing.