LemmaStatement: Literature-sourcedProof: 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.

Nested-event probabilities factor into conditional levels

Statement

For the nested event levels of Nested rare-event levels and conditional laws, $P (A_{L}) = \prod_{ℓ = 1}^{L} p_{ℓ} .$

Facts & Assumptions

Given: A nested sequence $A_{0} = Ω \supseteq A_{1} \supseteq \dots \supseteq A_{L}$ with all prefix probabilities positive.

[F1]

Nested rare-event levels and conditional laws: $p_{ℓ} = P (A_{ℓ}) / P (A_{ℓ - 1})$ and $P (A_{0}) = 1$ .

Proof

1.1F1givenalgebra

The definition gives $P (A_{ℓ}) = p_{ℓ} P (A_{ℓ - 1})$ for every $1 \leq ℓ \leq L$ .

2.1F1step 1.1algebra∎

Applying these identities successively from $ℓ = 1$ to $L$ yields $P (A_{L}) = P (A_{0}) \prod_{ℓ = 1}^{L} p_{ℓ} = \prod_{ℓ = 1}^{L} p_{ℓ}$ , since $P (A_{0}) = 1$ .

Depends on

Nested rare-event levels and conditional laws

Used by

Dependency tree · two levels

2 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

Gino Biondini, An Introduction to Rare Event Simulation and Importance Sampling, Handbook of Statistics 33, Chapter 2, §8 (standard reference, not scraped)