Alphabeta Math
ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10
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.

Conditioning on trivial and full sigma algebras

Example

Under the AC conditional-class convention, for integrable real X, E[X∣{∅,Ω}]=EX and E[X∣F]=X as classes.

Facts & Assumptions

Given: A probability space and real integrable X; AC is the conditional-class convention.

[F1]

A known variable conditions to itself, and a variable independent of the conditioning sigma-algebra conditions to its mean. (Conditioning a known variable and an independent variable)

[F2]

Finite weights summing to one define a probability space. (Finite probability spaces are exactly finite full-power-set probability spaces)

Verification

technique · direct
1.1

Every X is independent of the trivial sigma-algebra: for A empty both sides of the rectangle identity are zero, and for A=Omega both equal P(X∈B). Thus [F1] gives the first formula. Since X is F-measurable, the known-variable clause of [F1] gives the second.

F1
2.1

For example take two atoms a,b of masses 1/4,3/4 using [F2], and X(a)=0, X(b)=4. Then EX=0/4+12/4=3. Under trivial conditioning the version has values (3,3), with integral 3 on Omega; under full conditioning it has values (0,4), with integrals 0 on {a} and 3 on {b}. These verify the two formulas numerically.

step 1.1F2∎

Source notes

Durrett Examples 4.1.3–4.1.5, printed pp.207–208; van der Vaart Examples 1.4–1.5, printed p.2.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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