Abstract Corrected Iterations

Abstract: We consider $(&lt;\lambda)$-support iterations of a version of $(&lt;\lambda)$-strategically complete $\lambda<sup>+$-c.c. definable forcing notions along partial orders. We show that such iterations can be corrected to yield an analog of a result by Judah and Shelah for finite support iterations of Suslin ccc forcing, namely that if $(\mathbb{P}<em>{\alpha}, \underset{\sim}{\mathbb Q</em>{\beta}} : \alpha \leq \delta, \beta &lt;\delta)$ is a FS iteration of Suslin ccc forcing and $U\subseteq \delta$ is sufficiently closed, then letting $\mathbb{P}<em>U$ be the iteration along $U$, we have $\mathbb{P}_U \lessdot \mathbb{P}</em>{\delta}$.