Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-27
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 identity function generates Lebesgue measure

Example

Assuming the Axiom of Countable Choice, for F(x)=x the associated Lebesgue-Stieltjes measure is exactly Lebesgue measure. In particular, the interval formulas become the ordinary length formulas

λ((a,b])=b−a,λ((a,b))=b−a,λ([a,b])=b−a.

Facts & Assumptions

Given: The Axiom of Countable Choice and the identity function F(x)=x on R.

[L1]

Assuming Countable Choice, the Lebesgue-Stieltjes measure of the identity function is Lebesgue measure. (Lebesgue measure is the Lebesgue-Stieltjes measure of the identity function)

[L2]

The interval formulas for a Lebesgue-Stieltjes measure recover open, closed, and half-open interval values from the distribution function. (Interval formulas and atoms for a Lebesgue-Stieltjes measure)

Verification

technique · direct
1.1

By [L1], the measure attached to F(x)=x is λ.

L1
2.1

Applying [L2] with F(x)=x gives [step 1.1, L2] F(b)−F(a)=F(b−)−F(a)=F(b)−F(a−)=b−a, so every one of the four interval conventions has measure b−a.

step 1.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 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