Random oracles and independence

Question

I'm reading an unpublished paper in which the author makes the following conclusions several times:

Assumptions: finite probability space, $H$ is a random oracle, $X$ and $Y$ are two (not necessarily independent) random variables, $X \neq Y$ for every possible outcome.

Conclusion: $H(X)$ and $H(Y)$ are independent and uniformly distributed random variables.

"Proof" (not actually stated as such): obvious, $P(H(x) = a|D) = P(H(x) = a)$ for every event $D$.

Is this a valid conclusion? Are there any counterexamples which don't explicitly involve events $D$ similar to $H(x) \neq a$?

Answer 1

If $H(X)$ and $H(Y)$ were not evaluated as a part of selecting $X$ and $Y$, then yes; the assumption of a random Oracle is that $H(Z)$ is an independent and uniformly distributed variable for every new (not previously submitted to the Oracle) value $Z$.

If this isn't the case, then this need not hold. One obvious counterexample is:

$X := 1; Y := 2 \textit{ if H(1) < H(2)}\\ X := 2; Y :=1 \textit{ otherwise}$