Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 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.

Closed left rays form a pi-system generating the Borel sigma-algebra on the real line

Example

The family P:={(−∞,a]:a∈R} is a pi-system and σR(P)=B(R).

Facts & Assumptions

Given: The family P of closed left rays.

[L1]

A pi-system is a nonempty family closed under binary intersections (Pi-systems).

[L2]

The open right rays (a,∞) generate B(R) (Seven generating families for the Borel sigma-algebra on the real line).

Verification

technique · direct
1.1

The family is nonempty, and (−∞,a]∩(−∞,b]=(−∞,min⁡{a,b}], so it is a pi-system by [L1].

L1algebra
2.1

Complementation exchanges (−∞,a] with (a,∞). A generated sigma-algebra is complement-closed, so [L2] implies that the closed left rays generate B(R).

step 1.1L2algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 36 results over 10 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources