ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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.

F-sigma and G-delta subsets of the real line are Borel

Example

Every $F_{σ}$ subset and every $G_{δ}$ subset of $R$ is Borel.

Facts & Assumptions

Given: A subset $A \subseteq R$ .

[L1]

An $F_{σ}$ set is a countable union of closed subsets of $R$ , and a $G_{δ}$ set is a countable intersection of open subsets ( $F_{σ}$ and $G_{δ}$ subsets of $R$ ).

[L2]

The Borel sigma-algebra is generated by the open sets (The Borel sigma-algebra of a topological space).

[L3]

Sigma-algebras are closed under countable intersections (Sigma-algebras are closed under countable intersections, differences, symmetric differences, and set limits).

Verification

technique · direct

1.1L1L2

If $A$ is $F_{σ}$ , [L1] writes it as a countable union of closed sets. Closed sets are complements of the open generators in [L2], so they and their countable union are Borel.

2.1L1L2L3∎

If $A$ is $G_{δ}$ , [L1] writes it as a countable intersection of open, hence Borel, sets; [L3] makes the intersection Borel.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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

T. Tao, An Introduction to Measure Theory, Definition 1.4.16 (standard reference, not scraped)