DefinitionDefinition: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)judge pass (z-ai/glm-5.2)audited 2026-07-31

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.

A net is eventually or frequently in a subset of its codomain

Definition

Let $x:D\to X$ be a net (Directed preorders and nets) and let $S\subseteq X$ .

$x$ is eventually in $S$ if some $d_0\in D$ satisfies $x_d\in S$ for every $d\ge d_0$ .
$x$ is frequently in $S$ if, for every $d_0\in D$ , there is $d\ge d_0$ with $x_d\in S$ .

The net is frequently in $S$ exactly when it is not eventually in $X\setminus S$ : negating the first displayed existential-universal condition gives the second one.

Depends on

Directed preorders and nets

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 3 results over 3 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

WVU Math 581 Topology I (standard reference, not scraped)
Net (mathematics) (Wikipedia) (standard reference, not scraped)