Nisan comments on EDT solves 5 and 10 with conditional oracles

[Nisan]

I’m having trouble following this step of the proof of Theorem 4: “Obviously, the first conditional probability is 1”. Since the COD isn’t necessarily reflective, couldn’t the conditional be anything?

[jessicata]

By definition $U^O\left(\right)={FiveTen}^O(\lceil COED{T}^O(\lceil U\rceil )\rceil )=COED{T}^O(\lceil U\rceil )$, regardless of $O$. (The subscript $Q_i^j$ to $P$ only affects the distribution of $O$)

EDIT: clarified notation in the post