Johnson.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, 0 contain sorry, 0 declared as axioms

Imports (2)

Imported by 1 file

FormalConjecturesForMathlib.lean

def SimpleGraph.johnson The Johnson graph $J(n, k)$ has as vertices the $k$-subsets of an $n$-set line 33 source
Source documentation
The Johnson graph $J(n, k)$ has as vertices the $k$-subsets of an $n$-set. Two vertices are adjacent if their intersection has size $k - 1$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, SimpleGraph, Subtype, Finset, Fin, Eq, Finset.card
Proof uses: SimpleGraph.mk, HAdd.hAdd, Inter.inter, OfNat.ofNat
8 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
def johnson (n k : ℕ) : SimpleGraph {s : Finset (Fin n) // #s = k} where
…
```
lemma SimpleGraph.johnson_adj_iff_ge line 40 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Subtype, Finset, Fin, Eq, Finset.card, Iff, SimpleGraph.Adj, SimpleGraph.johnson, And, Ne, LE.le, HAdd.hAdd, Inter.inter, OfNat.ofNat
Proof uses: Subtype.mk, Eq.mpr, Not, id, Eq.trans, congr, congrArg, SimpleGraph.johnson_adj, propext, le_antisymm_iff, congrFun', Subtype.mk.injEq, Iff.intro, False, Eq.mp, Finset.inter_self, LT.lt, Nat.add_one_le_iff, Eq.symm, Finset.card_lt_card_iff_of_subset and 5 more
27 instances, coercions and auxiliary constants left out.

Tactic trace: 16 steps
Open to load the checked before-and-after goal states.
lemma SimpleGraph.not_johnson_adj_iff_lt line 56 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Subtype, Finset, Fin, Eq, Finset.card, Iff, Not, SimpleGraph.Adj, SimpleGraph.johnson, Or, LT.lt, HAdd.hAdd, Inter.inter, OfNat.ofNat
Proof uses: Eq.mpr, id, congrFun', congrArg, Eq.trans, LE.le, And, implies_congr, Eq.refl, Iff.intro, Decidable.not_or_of_imp, Iff.mpr, Classical.or_iff_not_imp_left, False.elim, Decidable.of_not_not
27 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.