ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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 two-valued random variable attains equality in Markov's inequality

Example

Let $a > 0$ and $p \in [0, 1]$ . On a two-outcome finite probability space, give an event $E$ probability $p$ and define $X = a$ on $E$ and $X = 0$ off $E$ . Then equality holds in Markov's inequality at threshold $a$ .

Facts & Assumptions

Given: Parameters $a > 0$ , $p \in [0, 1]$ , and the construction in the Example.

[L1]

Zero outcome weights are permitted in a finite probability space (Finite probability spaces, outcome weights, events, and event probabilities).

[L2]

Real random variables and expectation are finite functions and weighted sums (Real random variables on finite probability spaces and their finite distributions, Expectation of a real random variable on a finite probability space).

[L3]

Markov gives $P (X \geq a) \leq E [X] / a$ for nonnegative $X$ and $a > 0$ (Markov's inequality on a finite probability space).

Verification

technique · direct

1.1

Give the two outcomes weights $p$ and $1 - p$ ; these are nonnegative and sum to $1$ , including at $p = 0, 1$ .

2.1

The variable is nonnegative, $E [X] = a p$ , and ${X \geq a} = E$ has probability $p$ .

step 1.1L2algebra

3.1

Hence $P (X \geq a) = p = a p / a = E [X] / a$ , so [L3] is sharp.

step 2.1L3algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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

H. Pishro-Nik, Introduction to Probability, Statistics, and Random Processes, Section 6.2.2 (standard reference, not scraped)