  We introduce a class of notions of forcing which we call $\Sigma$-Prikry, and
show that many of the known Prikry-type notions of forcing that centers around
singular cardinals of countable cofinality are $\Sigma$-Prikry. We show that
given a $\Sigma$-Prikry poset $\mathbb P$ and a name for a non-reflecting
stationary set $T$, there exists a corresponding $\Sigma$-Prikry poset that
projects to $\mathbb P$ and kills the stationarity of $T$. Then, in a sequel to
this paper, we develop an iteration scheme for $\Sigma$-Prikry posets. Putting
the two works together, we obtain a proof of the following.
  Theorem. If $\kappa$ is the limit of a countable increasing sequence of
supercompact cardinals, then there exists a cofinality-preserving forcing
extension in which $\kappa$ remains a strong limit, every finite collection of
stationary subsets of $\kappa^+$ reflects simultaneously, and
$2^\kappa=\kappa^{++}$.
