The event $A_0$ is mutually independent (under $\mu$) of all sign choices of the events $A_i$ for $i \in S$: for every assignment of each $B_i$ to either $A_i$ or $A_i^c$, $\mu(A_0 \cap \bigcap_{i \in S} B_i) = \mu(A_0)\,\mu(\bigcap_{i \in S} B_i)$.
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A directed graph $G$ is a dependency digraph for the family $(A_i)_{i \in \iota}$ if, for every $i$ and every finite set $S$ of indices distinct from $i$ and not adjacent to $i$ in $G$, the event $A_i$ is mutually independent of the events $\{A_j\}_{j \in S}$.
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The out-neighborhood of $i$ in the digraph $G$, viewed as a finite set.
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Inductive bound built from hcond: given a uniform conditional estimate up to size $n$, the probability of avoiding all events in $S_1 \cup S_2$ is at least $\bigl(\prod_{j \in S_1}(1 - x_j)\bigr) \cdot \mu(\bigcap_{j \in S_2} A_j^c)$ whenever $S_1, S_2$ are disjoint and $|S_1 \cup S_2| \le n$.
The induction underlying the general Lovász Local Lemma: under the dependency-digraph hypothesis hG and the bound $\mu(A_i) \le x_i \prod_{j \in N(i)}(1 - x_j)$, for every finite $S$ not containing $i$ we have $\mu(A_i \cap \bigcap_{j \in S} A_j^c) \le x_i \, \mu(\bigcap_{j \in S} A_j^c)$.
General Lovász Local Lemma (Theorem 6.1.9): given events $A_i$ with dependency digraph $G$ and weights $x_i \in [0,1)$ such that $\mu(A_i) \le x_i \prod_{j \in N(i)}(1 - x_j)$ for every $i$, one has $\mu(\bigcap_i \overline{A_i}) \ge \prod_i (1 - x_i)$, so in particular it is positive.
Weierstrass-style inequality: for $f_i \in [0,1]$ on a finite set $s$, $1 - \sum_{i \in s} f_i \le \prod_{i \in s} (1 - f_i)$.
Corollary 6.1.10 of LLL: if every $p_i < 1/2$ and for every $i$ the sum $\sum_{j \in N(i)} p_j \le 1/4$, then $\mu(\bigcap_i \overline{A_i}) > 0$.
Symmetric Lovász Local Lemma (Theorem 6.1.7): if each event satisfies $\mu(A_i) \le p$, the dependency digraph has maximum out-degree at most $d \ge 1$, and $e p (d+1) \le 1$, then $\mu(\bigcap_i \overline{A_i}) > 0$.