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.

Empty pools, singleton classes, and zero query budgets

Example

In a finite pool-based instance with finite nonempty classifier class $H$ , exact target $h^{*} \in H$ , ordered pool $U = (u_{1}, \dots, u_{m})$ , and exact oracle, run the ascending-scan procedure with any query budget $b \geq 0$ . If $m = 0$ , it returns $V = H$ with no pool records and no queries. If $H = {h_{0}}$ and $h^{*} = h_{0}$ , it predicts every pool entry without querying, including when $b = 0$ . For a general $H$ with $b = 0$ , it predicts each unanimous entry before the first disagreement, then marks that index and the remaining pool suffix as abstentions; it makes no queries.

Example references

Finite pool-based realizable active-learning instance: fixes the finite-pool, exact-target model used in the boundary cases.

Finite-pool version-space query procedure: specifies the ascending scan, unanimous-prediction priority, and zero-budget stopping behavior.

Facts & Assumptions

Given: The three boundary cases and the exact-target finite-pool model stated in the Example.

[F1]

Finite pool-based realizable active-learning instance: $h^{*} \in H$ and every exact oracle response equals $h^{*}$ at the queried point.

[F2]

Finite-pool version-space query procedure: the procedure checks unanimity before the query-budget condition and marks an unresolved suffix as abstentions when it reaches a disagreement with no budget remaining.

Verification

1.1givenF2algebra

If $m = 0$ , the initialized scan index is $i = 1 > m$ . The loop has no iteration, so the procedure returns the initialized $V = H$ and empty query log.

1.2givenF1F2construct

If $H = {h_{0}}$ , then initially $V = {h_{0}}$ ; after any query-free scan step it remains that singleton. Thus every $u_{i}$ is unanimous with label $h_{0} (u_{i}) = h^{*} (u_{i})$ by [F1]. Since [F2] tests unanimity before the budget condition, the procedure predicts each index and never queries, even when $b = 0$ .

2.1givenF2algebra∎

If $b = 0$ for a general class, the query count remains $q = 0$ until the first disagreement. At every preceding unanimous index, [F2] records a prediction and advances without a query. At the first disagreement, $q = b$ , so [F2] marks that index and the remaining suffix as abstentions and returns; if no disagreement occurs, the scan reaches its end. In either case no oracle query is made.

Depends on

Used by

Nothing in the library uses this result yet.

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