ExampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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 infinite fair-coin-toss space

Example

Assume countable choice and dependent choice. Take $E_{n} = {0, 1}$ with $μ_{n} ({0}) = μ_{n} ({1}) = 1 / 2$ . On ${0, 1}^{N}$ with its cylinder measure, $X_{n} (x) = x_{n}$ is the $n$ th fair toss; for distinct $n_{1}, \dots, n_{k}$ and $a_{j} \in {0, 1}$ ,

$P (X_{n_{1}} = a_{1}, \dots, X_{n_{k}} = a_{k}) = 2^{- k} .$

Verification

Given: Countable choice, dependent choice, and the fair Bernoulli laws.

1.1given

The countable product theorem supplies the measure with the displayed finite product marginals.

2.1algebra∎

Each specified coordinate atom has mass $1 / 2$ , so the finite rectangle has mass $\prod_{j = 1}^{k} (1 / 2) = 2^{- k}$ ; the coordinates are independent by the coordinate-independence corollary.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

23 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

Kajino, Probability Theory, Example 3.66 (standard reference, not scraped)