ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-28

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.

Boxes in Z^n are Folner sets

Example

For $(Z^{d}, +)$ and any finite subset $S \subseteq Z^{d}$ , the boxes

$Q_{n} = [- n, n]^{d} \cap Z^{d}$

form a Folner family.

Facts & Assumptions

Given: A finite set $S \subseteq Z^{d}$ and a real $ε > 0$ .

[L1]

Folner sets are measured by relative symmetric-difference boundary (Folner sets and the Folner condition).

[L2]

Under the ultrafilter lemma, such Folner families witness amenability through the Folner criterion (Under the ultrafilter lemma, the Folner condition is equivalent to amenability).

Verification

technique · direct

1.1givenalgebra

Let $M = \max_{s \in S} ∥ s ∥_{\infty}$ . For each $s \in S$ , the translate $s + Q_{n}$ differs from $Q_{n}$ only inside the $M$ -thick boundary layers of the cube, so $∣ (s + Q_{n}) △ Q_{n} ∣ = O (n^{d - 1})$ .

2.1L1L2step 1.1∎

Because $∣ Q_{n} ∣ = (2 n + 1)^{d}$ , the ratio $∣ (s + Q_{n}) △ Q_{n} ∣ / ∣ Q_{n} ∣$ tends to $0$ as $n \to \infty$ uniformly in $s \in S$ . Hence the boxes are eventually $(S, ε)$ -Folner by [L1], illustrating [L2].

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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