The overhead of bounds checking varies a lot. In the common case it's negligible (few percents), but in some cases, depending on what you build, it can go up to even 20%. And if it prevents autovectorization it can cost even more.

There are techniques to minimize the perf loss, though (safely), and of course you can use unsafe code. If you do it smartly, in the vast majority of cases bound checks do not matter (in fact, even in C++ there is a push for a hardened standard library that does bound checks, and e.g. Google uses that).

Rust will never include full proofs, but it might include ranged integers which can minimize bound checks even more.