ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26

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.

The $n$ hyperplanes $x_{i} = 1$ cover ${0, 1}^{n}$ except the origin, so the Alon–Füredi bound is tight

Example

For each $i < n$ , let

$H_{i} : = {x \in R^{n} : x_{i} = 1} .$

Facts & Assumptions

Given: the hyperplanes $H_{0}, \dots, H_{n - 1}$ .

[L1]

Covering ${0, 1}^{n} ∖ {0}$ by hyperplanes missing the origin needs at least $n$ hyperplanes (Covering ${0, 1}^{n}$ minus the origin by affine hyperplanes avoiding the origin needs at least $n$ of them).

Verification

technique · direct

1.1

Every nonzero vertex of the cube has some coordinate equal to $1$ , so it lies on at least one of the hyperplanes $H_{i}$ . The origin has no coordinate equal to $1$ , so it lies on none of them.

given

2.1

Thus the $n$ hyperplanes $H_{0}, \dots, H_{n - 1}$ cover exactly the nonzero cube vertices. By [L1], no smaller family can do so.

L1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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

N. Alon, Combinatorial Nullstellensatz, Theorem 6.3 (standard reference, not scraped)

The nn hyperplanes xi=1xi​=1 cover {0,1}n{0,1}n except the origin, so the Alon–Füredi bound is tight