  We show that the following two theories are equiconsistent:
  (T) ZFC, CH and "There is a dense ideal on the first uncountable cardinal
such that if j is the generic embedding associated with it then its restriction
on ordinals is independent of the generic object is". (S) ZF, ADR and "Theta is
a regular cardinal."
  The main result of this paper is that T implies that the minimal model of S
exists. Woodin, in unpublished work, showed that the consistency of S implies
the consistency of T. We will also give a proof of this result, which, together
with our main theorem, establishes the equiconsistency of T and S. Our main
result partially resolves a well-known conjecture of Woodin, and completely
solves one of the main Core Model Induction problems dating back to 90s.
