ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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 data-dependent threshold crossing

Example

Take two fair coin observations under the null $P_{0}$ , with the likelihood-ratio process $E_{0} = 1$ , $E_{1} = ℓ (X_{1})$ , $E_{2} = ℓ (X_{1}) ℓ (X_{2})$ and $E_{t} = E_{2}$ for $t \geq 2$ , where $ℓ (1) = 3 / 2$ and $ℓ (0) = 1 / 2$ . Set $τ = \inf {t \geq 0 : E_{t} \geq 2}$ . This is a data-dependent stopping rule. It crosses only on two heads, so $P_{0} (τ < \infty) = 1 / 4$ . Since $2 = 1 / α$ for $α = 1 / 2$ , the crossing probability is below Ville's bound $1 / 2$ .

Facts & Assumptions

Given: Two independent fair bits, their prefix filtration, the displayed likelihood-ratio process, and threshold $2$ . Assume AC for the cited test-supermartingale and optional-sampling conventions.

[F1]

Nonnegative test supermartingales: a test supermartingale is nonnegative, adapted, integrable, supermartingale under the null, and has initial expectation at most one.

[F2]

Ville's inequality for nonnegative test supermartingales: for $α \in (0, 1)$ , the probability that a nonnegative test supermartingale ever reaches $1 / α$ is at most $α$ .

[F3]

Discrete stopping time: a discrete-time map is a stopping time when each event ${τ \leq n}$ belongs to $F_{n}$ .

[A1]

The Axiom of Choice: AC is inherited by the conditional-expectation clauses used in F1 and F2.

Verification

1.1givenF1A1algebra

The one-step likelihood factor has null conditional mean $(3 / 2 + 1 / 2) / 2 = 1$ after either possible prefix. Thus $E_{0} = 1$ , $E_{1}$ and $E_{2}$ are nonnegative, adapted, integrable, and satisfy the supermartingale equality; the process is constant after time two. It is a test supermartingale under $P_{0}$ by F1.

2.1givenF2F3choosealgebra∎

For every $n \geq 0$ , ${τ \leq n} = ⋃_{s = 0}^{n} {E_{s} \geq 2}$ . This is empty for $n < 2$ and equals ${X_{1} = X_{2} = 1} \in F_{2} \subseteq F_{n}$ for $n \geq 2$ , so $τ$ is a stopping time by F3. The values $(E_{1}, E_{2})$ on $(X_{1}, X_{2}) = (1, 1), (1, 0), (0, 1), (0, 0)$ are respectively $(3 / 2, 9 / 4), (3 / 2, 3 / 4), (1 / 2, 3 / 4), (1 / 2, 1 / 4)$ . Thus the crossing event ${τ < \infty}$ is exactly ${X_{1} = X_{2} = 1}$ and has probability $1 / 4$ . Since $2 = 1 / α$ for $α = 1 / 2$ , F2 gives the general upper bound $1 / 2$ , and the exact probability is below it.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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.