Chvatal.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

3 declarations, 1 contain sorry, 0 declared as axioms

Chvátal's Conjecture

References:

Imports (1)

FormalConjecturesUtil

def Chvatal.Decreasing A family F of sets is Decreasing if it is closed under taking subsets line 37 source
Source documentation
A family F of sets is Decreasing if it is closed under taking subsets.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset
Proof uses: LE.le, Membership.mem
5 instances, coercions and auxiliary constants left out.
```
def Decreasing (F : Finset (Finset α)) : Prop :=
…
```
def Chvatal.Intersecting A family F of sets is Intersecting if each pair of members has nonempty intersection line 41 source
Source documentation
A family F of sets is Intersecting if each pair of members has nonempty intersection.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: DecidableEq, Finset
Proof uses: Membership.mem, Ne, Inter.inter, EmptyCollection.emptyCollection
4 instances, coercions and auxiliary constants left out.
```
def Intersecting (F : Finset (Finset α)) : Prop :=
…
```
theorem Chvatal.exists_maximal_star Documented declaration: exists_maximal_star sorry line 50 source
Source documentation
If F is a decreasing family of sets of some finite type α, then there is some element x of α such that the family consisting of all members of F containing x is an intersecting subfamily of F with maximal cardinality.

depends on sorry what this means
Statement uses: Fintype, DecidableEq, Nonempty, Finset, Chvatal.Decreasing, Exists, LE.le, Chvatal.Intersecting, Nat, Finset.card, Finset.filter, Membership.mem
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
15 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem exists_maximal_star :
    ∀ F : Finset (Finset α), Decreasing F →
        ∃ x : α, ∀ G, G ⊆ F → Intersecting G → G.card ≤ { A ∈ F | x ∈ A }.card := by
…
```
Mentions: Chvatal.Decreasing, Chvatal.Intersecting