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.

Holm step-down versus one-step Bonferroni

Example

For a fixed four-hypothesis family, take the observed, already ordered p-value vector $(1 / 250, 3 / 200, 3 / 100, 1 / 5)$ and $α = 1 / 20$ . Holm's rank cutoffs are $1 / 80$ , $1 / 60$ , $1 / 40$ , and $1 / 20$ . The first two comparisons pass and the third fails, so Holm rejects $H_{01}$ and $H_{02}$ and retains $H_{03}$ and $H_{04}$ . The one-step Bonferroni comparison uses $α / 4 = 1 / 80$ for every hypothesis; it rejects only $H_{01}$ .

Facts & Assumptions

Given: A family $(H_{01}, H_{02}, H_{03}, H_{04})$ fixed before observing the p-values, the ordered vector $p = (1 / 250, 3 / 200, 3 / 100, 1 / 5)$ , and $α = 1 / 20$ . In this example, one-step Bonferroni means rejecting index $i$ exactly when $p_{i} \leq α / 4$ . Determine both rejection sets and state the limit of this numerical illustration.

[F1]

Holm step-down procedure: Holm rejects an initial ordered prefix, comparing rank $j$ with $α / (m - j + 1)$ and stopping at the first failed comparison.

[F2]

Holm step-down controls strong FWER: if every p-value is valid for its fixed null, then the Holm procedure has FWER at most $α$ for every parameter, without an independence assumption.

Verification

technique · direct arithmetic

1.1givenF1algebra

At rank $1$ , $1 / 250 < 1 / 80$ . At rank $2$ , $3 / 200 \leq 1 / 60$ because $180 \leq 200$ . At rank $3$ , $3 / 100 > 1 / 40$ because $120 > 100$ . By [F1] the procedure stops at rank $3$ and rejects exactly indices ${1, 2}$ .

2.1givenstep 1.1algebra

The one-step Bonferroni cutoff is $1 / 80$ . The first p-value is below it; the second is above it because $3 / 200 > 1 / 80$ is equivalent to $240 > 200$ , and the third and fourth are larger still. Thus Bonferroni rejects exactly ${1}$ , so the two observed rejection sets differ.

3.1F2step 1.1step 2.1∎

This computation determines decisions for one observed vector only. It does not establish that the p-value statistics are valid under their nulls or calculate an error probability; those distributional hypotheses are needed for the guarantee in [F2].

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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