Let's say program $P$ with given input $i$ is found to halt (or doesn’t halt) by a Turing machine. Is it true that the same program $P$ with input $F(i)$ also halts (or not, respectively), where $F$ is a computable function?
In general, is it true that the behavior of program $G(P)$ with the input $F(i)$ identical to the behavior of P with input i? (Both $G$ and $F$ are computable.)