LemmaStatement: AI-adaptedProof: AI-adaptedprecheck 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.

If $F$ is closed under taking subsets then $F$ shatters every $F \in F$

Statement

Let $F \subseteq P ([n])$ be closed under taking subsets. Then every member $F \in F$ is shattered by $F$ .

Facts & Assumptions

Given: a downward-closed family $F \subseteq P ([n])$ and a set $F \in F$ .

[F1]

A set is shattered when every subset of it occurs as a trace (Shattering and the Vapnik–Chervonenkis dimension of a set family).

Proof

technique · direct

1.1F1given

If $A \subseteq F$ , then $A \in F$ by downward closure and $A \cap F = A$ . So every subset of $F$ is a trace of $F$ on $F$ .

2.1F1step 1.1∎

Hence $F ∣_{F} = P (F)$ , and $F$ is shattered by $F$ .

Remarks

This is the step that converts the structural output of shifting into the size bound of Sauer-Shelah.

Depends on

Used by

Sauer–Shelah: a family on [n] of VC dimension at most d has at most ∑ᵢ₌₀ᵈC(n, i) members Theorem

Dependency tree · two levels

16 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

L. Babai and P. Frankl, Linear Algebra Methods in Combinatorics, §7.5 (standard reference, not scraped)

If FF is closed under taking subsets then FF shatters every F∈FF∈F