  We discuss the mechanism of truncations driven by the imposition of
constraints. We show how the consistency of such truncations is controlled, and
give general theorems that establish conditions for the correct uplifting of
solutions. We show in some particular examples how one can get correct
upliftings from 7d supergravities to 10d type IIB supergravity, even in cases
when the truncation is not initially consistent by its own.
