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.

A canonical random walk from product increments

Example

On the fair-coin space set $ξ_{n} = 2 X_{n} - 1 \in {- 1, 1}$ and $S_{0} = 0$ , $S_{n} = \sum_{r = 0}^{n - 1} ξ_{r}$ for $n \geq 1$ . The process $(S_{n})_{n \geq 0}$ is the canonical simple random walk built from independent increments.

Verification

Given: The fair-coin coordinate process.

1.1

The coin-toss example gives independent fair $X_{n}$ , hence independent signs $ξ_{n}$ with $P (ξ_{n} = 1) = P (ξ_{n} = - 1) = 1 / 2$ .

given

2.1

For any finite list of distinct increment times, their joint law is the uniform law on the corresponding sign vectors. The process coordinates are the partial sums of these increments, not the increments themselves.

step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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

Durrett, Probability: Theory and Examples, Section 2.1.4 (standard reference, not scraped)