ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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.

First exit time from an interval

Statement

If $X$ is an adapted real process and $a < b$ , then $τ = \inf {n \geq 0 : X_{n} \notin (a, b)}$ is a stopping time. It may equal infinity, and no integrability conclusion follows from the stopping-time property alone.

Facts & Assumptions

Given: The hypotheses, objects, and conventions in the Statement.

[F1]

First hitting time of an adapted process is a stopping time treats first hits of Borel sets.

Proof

1.1

The exit target $(- \infty, a] \cup [b, \infty)$ is Borel, so F1 applies. Explicitly, ${τ \leq n} = ⋃_{k = 0}^{n} ({X_{k} \leq a} \cup {X_{k} \geq b}) \in F_{n} .$

2.1

If a path remains in $(a, b)$ forever, the defining set of indices is empty and $τ = \infty$ . Thus the result asserts neither almost-sure finiteness nor integrability and leaves the cemetery convention relevant.

step 1.1∎

Depends on

First hitting time of an adapted process is a stopping time

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

van der Vaart, Martingales, Diffusions and Financial Mathematics, §2.3, pp. 9–11 (standard reference, not scraped)