Alphabeta Math
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→X be a net. A net y:E→X is a subnet of x if E is a directed preorder and there is a map ϕ:E→D such that ye=xϕ(e) for every e∈E and

for every d∈D there is e0∈E such that e≥e0⟹ϕ(e)≥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.

Depends on

Used by

Dependency tree · two levels

4 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