TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-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.

Finite-candidate power and sample-size search is correct

Statement

For the finite-candidate algorithm Finite-candidate power and sample-size search, suppose the input list is ordered by nondecreasing total size and its evaluator returns the exact feasibility truth value for every candidate. The algorithm terminates after at most $m$ evaluations. If it returns a plan, that plan is feasible and its total size is least among feasible plans in the supplied list; if it returns $none$ , no plan in that list is feasible. When several feasible plans have the same least size, the algorithm returns the first such plan in the supplied tie order.

Facts & Assumptions

Given: A finite list of $m \geq 0$ candidate plans ordered by nondecreasing positive-integer size, an exact evaluator that terminates for each plan, and the algorithm of Finite-candidate power and sample-size search.

[F1]

Feasibility is the exact set of test, precision, allocation, and resource conditions declared by the planning specification. Power and sample-size planning specifications

[F2]

The algorithm evaluates each listed candidate once, returns at the first true result, and returns $none$ after a finite list is exhausted. Finite-candidate power and sample-size search

Proof

1.1givenF2algebra

Each evaluator call terminates by the hypothesis. The loop makes at most $m$ calls; for $m = 0$ it returns immediately. Thus the algorithm terminates after finitely many calls.

1.2givenF1F2algebra

If the algorithm returns $d_{j}$ , it does so only after the evaluator returns true, which by [F1] means $d_{j}$ is feasible. Each earlier $d_{k}$ , $k < j$ , was evaluated false; since the list is ordered by size, no feasible listed plan has smaller total size than $d_{j}$ . The first passing index also gives the declared tie rule.

2.1givenF1F2∎

If the algorithm returns $none$ , it evaluated every listed plan and each exact result was false, so no candidate in that list is feasible.

Depends on

Used by

Nothing in the library uses this result yet.

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Sources

Gary W. Oehlert, A First Course in Design and Analysis of Experiments, Chapter 7, §§7.1 and 7.3–7.4, printed pp. 149–150, 153–159 (standard reference, not scraped)