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.

1.1

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

given

2.1

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.

algebra∎

Depends on

Used by

A canonical random walk from product increments Example

Dependency tree · two levels

17 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)