You cannot prove a negative.
Sure you fucking can. Anything defined in such a way as to exclude other possible definitions can have the latter definitions be proven in the negative just as surely as the former definition can be in the positive.
3 != 4. A triangle is not a square. Red is not blue. Hydrogen is not helium. A dog is not a cat. If the coin landed heads-up, the coin did not land tails-up. If someone was in location A at time T, they could not have been in location B at time T committing crime C. You are not smart.
In your examples you are not actually proving a negative (that something didn't happen). You are proving that something is not possible or could not have happened.
Possible or not possible are easy by comparison. Proving a negative means, "take this thing that really could have possibly happened, and prove that it didn't happen". A shape cannot both be a triangle and a square. A pure color at a single wavelength cannot both be red and blue. You are drastically underestimating the scope of how difficult it is to prove a negative. "This couldn't have happened because it is impossible" is actually a positive claim and as such, can be proven.