DefinitionDefinition: AI-adaptedProof: Not applicablejudge 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.

Subnet via an eventually cofinal index map

Definition

Let $x : D \to X$ be a net. A net $y : E \to X$ is a subnet of $x$ if $E$ is a directed preorder and there is a map $ϕ : E \to D$ such that $y_{e} = x_{ϕ (e)}$ for every $e \in E$ and

$for every d \in D there is e_{0} \in E such that e \geq e_{0} ⟹ ϕ (e) \geq d .$

The displayed condition says that $ϕ$ is eventually cofinal. No order-preservation condition is imposed on $ϕ$ .

Remarks

Stricter conventions require $ϕ$ to be order-preserving, or formulate subnets through a relation. They are not used here. Eventual cofinality is the property needed to carry eventual statements from a net to its subnet and to turn cluster points into convergent subnets.

Subnet via an eventually cofinal index map

Definition

Remarks

Depends on

Dependency tree · 2 levels

Sources