Theorem 7.7 — $K_t \subseteq P_{i_1,\dots,i_t}(K)$ for any $t$ distinct 0-1 indices

This is Theorem 7.7 of Balas's Disjunctive Programming (Sherali–Adams [112]). Let $K = \{x : \tilde A x \ge \tilde b\}$ be the LP relaxation of a mixed 0-1 program with 0-1 index set $N'$, the system containing the bound rows $x \ge 0$ and $x_j \le 1$ ($j \in N'$), and let $K_t$ be the level-$t$ Sherali–Adams relaxation. Then for every sequence $i_1,\dots,i_t$ of distinct elements of $N'$,

where $P_{i_1,\dots,i_t}(K) = P_{i_t}(\cdots P_{i_1}(K)\cdots)$ and $P_j(S) = \mathrm{conv}(S \cap \{x_j \in \{0,1\}\})$. The book states the case $i_k = k$; $K_t$ is symmetric in the indices of $N'$.

Formalization Note. The retired version allowed systems without the bound rows; with no rows $K_t = \mathbb R^n$. Throughout Chapter 7 the book works with $K = \{x \in \mathbb R^n : Ax \ge b,\ x \ge 0,\ x_j \le 1,\ j \in N'\} = \{x : \tilde A x \ge \tilde b\}$: the bound constraints are rows of the system that every construction multiplies. This is now the explicit hypothesis HasBoundRows A b N' (the system contains the rows $x_k \ge 0$ for every $k$ and $-x_j \ge -1$ for every $j \in N'$); it is satisfiable with nonempty $K$ (e.g. the unit box) and excludes the disproof's instance with no rows. The right-hand side is now the iterated operator $P_{i_1,\dots,i_t}(K)$ of the book (the retired version wrote it as $\mathrm{conv}(K \cap \{x_S \in \{0,1\}\})$, which equals it only by Theorem 7.2).