ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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.

A $0.01$ -regular pair restricted to two half-sized subsets is $0.02$ -regular

Statement

If $(X, Y)$ is $0.01$ -regular and $X^{'} \subseteq X$ , $Y^{'} \subseteq Y$ satisfy $∣ X^{'} ∣ \geq ∣ X ∣ / 2$ and $∣ Y^{'} ∣ \geq ∣ Y ∣ / 2$ , then $(X^{'}, Y^{'})$ is $0.02$ -regular and $∣ d (X^{'}, Y^{'}) - d (X, Y) ∣ \leq 0.01.$

Facts & Assumptions

Given: A pair and subsets satisfying the Statement.

[L1]

The slicing lemma gives new parameter $\max {ϵ / α, ϵ / β, 2 ϵ}$ and density shift at most $ϵ$ for restrictions of relative sizes at least $α, β$ (Slicing lemma: large subpairs remain regular and their density shifts by at most $ϵ$ ).

Verification

technique · direct

1.1givenL1algebra

Substitute $ϵ = 1 / 100$ and $α = β = 1 / 2$ in [L1]. Each of $ϵ / α$ , $ϵ / β$ , and $2 ϵ$ equals $1 / 50$ .

2.1step 1.1L1∎

The same application of [L1] retains the density-shift bound $1 / 100$ , proving both decimal-form assertions in the Statement.

Depends on

Slicing lemma: large subpairs remain regular and their density shifts by at most $\epsilon$

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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

Y. Zhao, Graph Theory and Additive Combinatorics, Exercise 2.1.4 (standard reference, not scraped)

A 0.010.01-regular pair restricted to two half-sized subsets is 0.020.02-regular