ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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.

The uniform random variable on [0,1]

Example

Let $Ω = [0, 1]$ with its Borel sets and uniform probability measure, and let $X (ω) = ω$ . Then $F_{X} (x) = {\begin{cases} 0, & x < 0, \\ x, & 0 \leq x \leq 1, \\ 1, & x \geq 1, \end{cases}$ and $E [X] = \frac{1}{2}, E [X^{2}] = \frac{1}{3}, \int_{0}^{\infty} P (X > t) d t = \frac{1}{2} .$

Facts & Assumptions

Given: The identity random variable $X (ω) = ω$ on $[0, 1]$ with uniform probability.

[L1]

The cumulative distribution function is $F_{X} (x) = P (X \leq x)$ (Cumulative distribution function of a real random variable).

[L2]

Expectation is integration against the probability measure, and for a nonnegative random variable the layer-cake formula computes it from the tail probabilities (Expectation of a nonnegative or integrable random variable, Layer-cake formulas for random variables).

Verification

technique · direct

1.1L1given

By direct interval computation, $P (X \leq x) = 0 (x < 0), P (X \leq x) = x (0 \leq x \leq 1), P (X \leq x) = 1 (x \geq 1),$ which is exactly the displayed $F_{X}$ .

1.2L2algebra

Direct integration gives $E [X] = \int_{0}^{1} x d x = \frac{1}{2}, E [X^{2}] = \int_{0}^{1} x^{2} d x = \frac{1}{3} .$

2.1step 1.2L2algebra

The tail is $P (X > t) = 1 - t$ for $0 \leq t \leq 1$ and $0$ for $t > 1$ , so $\int_{0}^{\infty} P (X > t) d t = \int_{0}^{1} (1 - t) d t = \frac{1}{2} = E [X],$ in agreement with [L2].

3.1step 1.1step 1.2step 2.1∎

Steps 1.1, 1.2, and 2.1 verify the CDF, moments, and tail integral.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 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

Jean-Francois Le Gall, Integration, Probabilities and Stochastic Processes, Section 8.1.5 (standard reference, not scraped)