ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge 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.

A finite class shattering three points

Example

Let $X = {x_{1}, x_{2}, x_{3}}$ have the discrete sigma-algebra, and let $H$ be the class of all eight functions from $X$ to ${0, 1}$ . Then $VCdim (H) = 3$ .

Example references

Mohri et al., Foundations of Machine Learning, 2nd ed., Chapter 3

Statistical Learning Theory course notes, University of British Columbia, §7.2.2

Facts & Assumptions

Given: The finite measurable space $X$ and the class $H = {0, 1}^{X}$ .

[F1]

A class shatters an indexed finite set exactly when its trace family contains every subset of the index set; its VC dimension is the supremum of shattered cardinalities (Binary learning minimax excess risk, Shattering and the Vapnik–Chervonenkis dimension of a set family).

Verification

1.1givenF1construct

For each $A \subseteq [3]$ , the function $h_{A}$ defined by $h_{A} (x_{i}) = 1$ exactly when $i \in A$ belongs to $H$ ; hence the indexed trace family on ${x_{1}, x_{2}, x_{3}}$ is $P ([3])$ , so this set is shattered by [F1].

2.1givenF1algebra∎

The whole domain has only three points, so no subset of four distinct domain points exists. Thus no four-point set can be shattered, and the VC dimension is exactly three.

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

Mehryar Mohri, Afshin Rostamizadeh, and Ameet Talwalkar, Foundations of Machine Learning, 2nd ed., Chapter 3 §§3.3–3.4 (standard reference, not scraped)
Statistical Learning Theory course notes, University of British Columbia, Chapter 7 §§7.2.2–7.3 (standard reference, not scraped)