Alphabeta Math
ExampleConstruction: AI-adaptedVerification: 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.

The Borel sigma-algebra of the Cantor set is the trace of the real Borel sigma-algebra

Example

For the Cantor middle-thirds set C⊆R with its subspace topology,

B(C)={B∩C:B∈B(R)}.

Facts & Assumptions

Given: The Cantor middle-thirds set C with the subspace topology inherited from R.

[L1]

The Cantor middle-thirds set is C=⋂n∈NCn⊆[0,1] (The Cantor middle-thirds set as the intersection of the sets Cn obtained by removing open middle thirds).

[L2]

The Borel sigma-algebra of a subspace is the trace of the ambient Borel sigma-algebra (The Borel sigma-algebra of a subspace is the trace of the ambient Borel sigma-algebra).

Verification

technique · direct
1.1L1

The description in [L1] makes C a specified subset of R, equipped here with the subspace topology.

2.1step 1.1L2

The subspace theorem [L2] applies to this inclusion C⊆R.

3.1step 2.1L2∎

Therefore [L2] gives B(C)={B∩C:B∈B(R)}, as claimed.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 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