ExampleConstruction: AI-generatedVerification: 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.

Binary zero-one loss and risk

Example

Let $X = {u, v}$ , $Y = A = {0, 1}$ with their discrete sigma-algebras, and let the joint law be $P (u, 0) = \frac{1}{8}, P (u, 1) = \frac{3}{8}, P (v, 0) = \frac{3}{8}, P (v, 1) = \frac{1}{8} .$ Use zero-one loss $ℓ (a, y) = 1_{{a \neq y}}$ . The predictor $f_{10}$ with $f_{10} (u) = 1$ , $f_{10} (v) = 0$ has population risk $1 / 4$ ; the predictor $f_{01}$ with the opposite labels has risk $3 / 4$ . The two constant predictors each have risk $1 / 2$ . Thus $f_{10}$ is the unique population-risk minimizer among the four predictors.

Facts & Assumptions

Given: The finite measurable spaces, probability law, loss and predictors stated in the example.

[F1]

A supervised problem specifies the measurable instance/label/action spaces, joint law, predictors and loss. Supervised prediction problems

[F2]

Population risk is the integral of the nonnegative loss under the joint law. Population and empirical risk

Verification

1.1givenF1F2algebra

The table entries sum to one, and each of $u$ and $v$ has marginal probability $1 / 2$ . The loss of $f_{10}$ is one exactly at $(u, 0)$ and $(v, 1)$ , so its risk is $1 / 8 + 1 / 8 = 1 / 4$ .

2.1givenF1F2step 1.1algebra∎

The loss of $f_{01}$ is one at $(u, 1)$ and $(v, 0)$ , so its risk is $3 / 8 + 3 / 8 = 3 / 4$ . The constant-one predictor errs on label-zero mass $1 / 2$ , and the constant-zero predictor errs on label-one mass $1 / 2$ . These are all four functions from $X$ to $Y$ , so $f_{10}$ uniquely minimizes risk.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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

Shai Shalev-Shwartz and Shai Ben-David, Understanding Machine Learning: From Theory to Algorithms, Chapter 3 §3.2.2 (standard reference, not scraped)
Mehryar Mohri, Afshin Rostamizadeh, and Ameet Talwalkar, Foundations of Machine Learning, 2nd ed., Chapter 2 §2.1 (standard reference, not scraped)