ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck 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.

Intervals in Z are Folner sets

Example

For $(Z, +)$ and any finite subset $S \subseteq Z$ , the intervals $F_{n} = [- n, n] \cap Z$ are eventually $(S, ε)$ -Folner for every $ε > 0$ .

Facts & Assumptions

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

[L1]

$(S, ε)$ -Folner sets are defined by the symmetric-difference estimate (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

1.1

Let $M = \max_{s \in S} ∣ s ∣$ . For every $s \in S$ , the translate $s + F_{n}$ differs from $F_{n}$ only near the two ends of the interval, so $∣ (s + F_{n}) △ F_{n} ∣ \leq 2 M$ .

givenalgebra

2.1

Since $∣ F_{n} ∣ = 2 n + 1$ , the ratio $∣ (s + F_{n}) △ F_{n} ∣ / ∣ F_{n} ∣$ is at most $2 M / (2 n + 1)$ and therefore tends to $0$ uniformly in $s \in S$ . Hence $F_{n}$ is eventually $(S, ε)$ -Folner in the sense of [L1], illustrating the criterion [L2].

L1L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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