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.

Expectation under a measurable transformation

Example

Let $X$ take the values $- 1$ , $0$ , and $2$ with probabilities $1 / 4$ , $1 / 2$ , and $1 / 4$ , and let $g (x) = x^{2} + 1$ . Then $E [g (X)] = \int_{R} (x^{2} + 1) d P_{X} (x) = \frac{9}{4} .$

Facts & Assumptions

Given: The law $P_{X}$ with masses $P_{X} ({- 1}) = 1 / 4$ , $P_{X} ({0}) = 1 / 2$ , and $P_{X} ({2}) = 1 / 4$ , and the measurable map $g (x) = x^{2} + 1$ .

[L1]

Change of variables for expectation allows computation against the law instead of the original sample space (Change of variables for expectation).

Verification

technique · direct

1.1

By [L1], $E [g (X)] = \int_{R} g d P_{X} .$ Because the law is supported on ${- 1, 0, 2}$ , this is the weighted sum $g (- 1) \frac{1}{4} + g (0) \frac{1}{2} + g (2) \frac{1}{4} = \frac{2}{4} + \frac{1}{2} + \frac{5}{4} = \frac{9}{4} .$

L1algebra

2.1

The nonlinear expectation is therefore computed entirely from the law on the target space.

step 1.1∎

Depends on

Change of variables for expectation

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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

S. R. S. Varadhan, Probability Theory, Section 1.4 (standard reference, not scraped)