ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge 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 pilot with no rare-event hit returns failure

Example

Let $P$ be Bernoulli on ${0, 1}$ with $P ({1}) = 1 / 10$ , set $A = {1}$ , and use the one-member proposal family $q = P$ . A pilot of size $m \geq 1$ misses $A$ with positive probability even though the target probability is positive.

Facts & Assumptions

Given: $Y_{1}, \dots, Y_{m}$ are IID Bernoulli $(1 / 10)$ , the target event is $A = {1}$ , and the pilot rule in Independent-pilot cross-entropy proposal fitting and production is used.

[F1]

Independent-pilot cross-entropy proposal fitting and production: If no pilot sample lies in $A$ , the algorithm returns the status pilot-no-hit.

[F2]

Independent cross-entropy production is conditionally unbiased: Conditional unbiasedness applies only after a successful pilot and admissible production proposal.

Verification

technique · direct

1.1givenalgebra

By the IID assumption, $P (Y_{1} \notin A, \dots, Y_{m} \notin A) = \prod_{j = 1}^{m} P (Y_{j} = 0) = (9 / 10)^{m} > 0.$ Also $P (A) = 1 / 10 > 0$ .

2.1F1F2step 1.1given∎

On this positive-probability pilot event, [F1] returns pilot-no-hit, so no fitted proposal or production estimate is returned. The positive target probability therefore does not imply pilot success; [F2] makes no production-unbiasedness assertion on this failure branch.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

Patrick Laub, Algorithms for Rare Event Estimation, Short Course, Lecture 2, Cross-Entropy Method (standard reference, not scraped)
Gino Biondini, An Introduction to Rare Event Simulation and Importance Sampling, Handbook of Statistics 33, Chapter 2, §5 (standard reference, not scraped)