LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-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.

Generalized inverse sublevel identity for a CDF

Statement

Assume Countable Choice. Let $F (x) = P (X \leq x)$ be the CDF of a real random variable. For $u \in (0, 1)$ define $Q (u) = \inf {y \in R : F (y) \geq u} .$ The set in this infimum is nonempty and bounded below, so $Q (u) \in R$ , and for every $x \in R$ , $Q (u) \leq x ⟺ u \leq F (x) .$

Facts & Assumptions

Given: Countable Choice, a real random variable $X$ , its CDF $F$ , and $u \in (0, 1)$ .

[F1]

A CDF is nondecreasing and right-continuous, with limits $0$ and $1$ at the two infinities. Probability laws correspond to distribution functions

[F2]

Every nonempty subset of $R$ bounded below has a finite infimum, and for every $ε > 0$ it has an element below its infimum plus $ε$ . Complete ordered field (least-upper-bound property)

[A1]

Countable Choice is assumed for the supplied CDF theorem; the local infimum argument makes only finitely many individual choices and uses no stronger choice principle. The Axiom of Countable Choice ( ${A C}_{ω}$ )

Proof

Given: As in the Statement. Put $S_{u} = {y : F (y) \geq u}$ .

1.1givenF1F2A1choose

By the upper-tail limit in [F1], some real $b$ has $F (b) > u$ , so $S_{u}$ is nonempty. By the lower-tail limit, some real $a$ has $F (a) < u$ ; monotonicity then gives $F (y) < u$ for all $y \leq a$ , so $S_{u}$ is bounded below. Fact [F2] gives a finite infimum $Q (u)$ .

2.1givenstep 1.1

If $u \leq F (x)$ , then $x \in S_{u}$ , hence $\inf S_{u} \leq x$ .

2.2givenF1step 1.1

Conversely, suppose $Q (u) \leq x$ . For every $ε > 0$ , the definition of infimum gives some $y \in S_{u}$ with $y < Q (u) + ε \leq x + ε$ . Monotonicity gives $F (x + ε) \geq F (y) \geq u$ . Right continuity at $x$ as $ε ↓ 0$ yields $F (x) \geq u$ .

3.1step 1.1step 2.1step 2.2∎

Steps 2.1 and 2.2 prove the equivalence, and step 1.1 proves the required well-definedness.

Depends on

Used by

Dependency tree · two levels

17 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

Luc Devroye, Non-Uniform Random Variate Generation, Chapter II §2.1, The Inversion Principle (standard reference, not scraped)
D. Pollard, A User's Guide to Measure Theoretic Probability, generalized inverse discussion (standard reference, not scraped)