DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Dependency digraphs for a finite family of bad events

Definition

Let $(A_{i})_{i \in I}$ be a finite family of events. A loopless digraph $D$ on $I$ is a dependency digraph for the family when, for every $i \in I$ and every set $S \subseteq I ∖ ({i} \cup N_{D}^{+} (i))$ , $P ⁣ (A_{i} \cap ⋂_{j \in S} A_{j}^{c}) = P (A_{i}) P ⁣ (⋂_{j \in S} A_{j}^{c}) .$ Thus $A_{i}$ is independent of every conjunction of complements indexed by its non-out-neighbours. An undirected dependency graph is the special case in which every edge is replaced by both directed arcs.

Depends on

Used by

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. Matousek and J. Vondrak, The Probabilistic Method, Section 5.1 (standard reference, not scraped)