Alphabeta Math
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.

The trivial and discrete sigma-algebras are the two extremes

Example

For every set X, the trivial sigma-algebra is {∅,X} and the discrete sigma-algebra is P(X). Every sigma-algebra on X lies between them under inclusion. If X=∅, the two extremes coincide.

Facts & Assumptions

Given: A set X.

[L1]

A sigma-algebra contains the empty set and is closed under complements and countable unions (Sigma-algebras).

Verification

technique · direct
1.1L1algebra

The family {∅,X} satisfies the axioms in [L1]; when X=∅, it is the one-member family {∅}.

1.2L1algebra

The power set P(X) satisfies the axioms in [L1].

2.1step 1.1step 1.2L1∎

Every sigma-algebra contains ∅ and hence X, and every one is a subfamily of P(X). Thus the two verified families are the extremes, coinciding when X=∅.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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