ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

Tails, limits, and diagonal intersection

Example

Let $κ$ be regular uncountable. Each strict tail $C_{ξ} = {α < κ : ξ < α}$ and the set of nonzero limit ordinals are club. Nevertheless $⋂_{ξ < κ} C_{ξ} = \emptyset$ , while $△_{ξ < κ} C_{ξ} = κ$ . A superset of a club need not be closed.

Facts & Assumptions

[F1]

Closed unbounded subsets of ordinals: Closed means containing every nonzero limit point below the ambient ordinal.

[F2]

The diagonal intersection of clubs is club: Diagonal membership at alpha tests exactly the indices below alpha.

Verification

Given: The objects and hypotheses in the statement.

1.1

Tails are unbounded; a nonzero limit point of a tail lies above its cutoff and hence in the tail. Nonzero limits are unbounded since $β + ω < κ$ for $β < κ$ ; a nonzero limit point of limit ordinals is itself a limit.

1.2

No alpha lies in $C_{α}$ , so the full intersection is empty. But for every $ξ < α$ we have $α \in C_{ξ}$ , including the vacuous test at zero. Thus the diagonal intersection is all of kappa, consistently with its club theorem.

2.1

The set $[ω + 1, κ) \cup {1, 2, 3, \dots}$ contains a club tail, but omits its nonzero limit point omega, and is therefore not closed.

F1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

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

Vasey, Examples 14.2 and 14.10, pp.79–81 (standard reference, not scraped)