GraphConjecture20.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

Written on the Wall II - Conjecture 20

Reference: E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc

Imports (1)

FormalConjecturesUtil

theorem WrittenOnTheWallII.GraphConjecture20.conjecture20 Documented declaration: conjecture20 sorry line 43 source
Source documentation
WOWII Conjecture 20
For a simple connected graph G, b(G) ≥ n(G) / ⌊deg_avg(G)⌋, where b(G) is the size of a largest induced bipartite subgraph, n(G) is the number of vertices, and deg_avg(G) = (∑ v, deg(v)) / n(G) is the average degree.

L4DEL explanation
WOWII Conjecture 20 For a simple connected graph G, b(G) ≥ n(G) / ⌊deg_avg(G)⌋, where b(G) is the size of a largest induced bipartite subgraph, n(G) is the number of vertices, and deg_avg(G) = (∑ v, deg(v)) / n(G) is the average degree.
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Fintype, DecidableEq, Nontrivial, SimpleGraph, DecidableRel, SimpleGraph.Adj, SimpleGraph.Connected, Real, HDiv.hDiv, Finset.sum, Finset.univ, SimpleGraph.degree, Membership.mem, Set, SimpleGraph.neighborSet, Fintype.card, LE.le, Int.floor, SimpleGraph.b
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
31 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture20 (G : SimpleGraph α) [DecidableRel G.Adj] (h : G.Connected) :
    let deg_avg : ℝ := (∑ v : α, (G.degree v : ℝ)) / (Fintype.card α : ℝ)
…
```
example WrittenOnTheWallII.GraphConjecture20.(example at line 52) line 52 source
Source documentation
The number of vertices of K₃ is 3.
```
example : Fintype.card (Fin 3) = 3 := by decide
…
```
example WrittenOnTheWallII.GraphConjecture20.(example at line 56) line 56 source
Source documentation
The average degree of K₃ is 2 (every vertex has degree 2 in the complete graph on 3 vertices).
```
example : averageDegree (⊤ : SimpleGraph (Fin 3)) = 2 := by
…
```