Alphabeta Math
TheoremStatement: AI-adaptedProof: AI-generatedprecheck passjudge 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.

The tail-filter and derived-net constructions preserve convergence and cluster points in both directions

Statement

Passing from a net to its tail filter, or from a filter to its derived net, preserves and reflects convergence and cluster points.

Facts & Assumptions

Given: A net, a filter, and a point of the relevant topological space.

[L1]

A net and its tail filter have the same limits and cluster points (A net and its tail filter have the same limits and cluster points).

[L2]

A filter and its derived net have the same limits and cluster points (A filter and its canonical derived net have the same limits and cluster points).

Proof

technique · direct
1.1

Apply [L1] to the given net.

L1
1.2

Apply [L2] to the given filter.

L2
2.1

These are precisely the two asserted correspondences.

step 1.1step 1.2∎

Depends on

Used by

Dependency tree · two levels

6 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