There are certain generalizations of the notion of a proper forcing to uncountable cardinals in the context of forcing iterations. For example, the ones introduced by Eisworth, and by Roslanowski and Shelah. They usually considered such a notion under some cardinal arithmetic assumptions and in the context of iterated forcings.

I am simply looking for a reference for the following definition:

• Definition: Suppose $\mathcal S$ is a set of uncountable elementary submodels in some large $H_\theta$, and $\mathbb P$ is a forcing notion belonging to every model in $\mathcal S$. As in the definition of a proper forcing, let us say that $\mathbb P$ is $\mathcal S$-proper if, for every $M\in\mathcal S$ and every $p\in\mathbb P\cap M$, there is an $(M,\mathbb P)$-generic condition $q\leq p$.

I considered the above definition and the following theorem as folklore, but I was asked by a referee to provide a reference for them.

• Theorem: Suppose $\kappa$ is a regular cardinal, and $\mathcal S\subseteq\mathcal P_\kappa(H_\theta)$ is stationary. If $\mathbb P$ is $\mathcal S$-proper, then $\mathbb P$ preserves $\kappa$.

I think calling it $\kappa$-properness can be confusing for several reasons, including the fact that the set $\mathcal S$ is just stationary.

Question 1: Does the above notion have a name in the literature? If not, is my terminology convenient? What about "$\mathbb P$ is proper for $\mathcal S$"?

Question 2: What is the most appropriate work to cite for the above theorem?