CompleteGraphEdgeCount.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

2 declarations, 0 contain sorry, 0 declared as axioms

Imports (1)

Mathlib.Combinatorics.SimpleGraph.Finite

Imported by 1 file

FormalConjecturesForMathlib.lean

lemma SimpleGraph.card_edgeFinset_completeGraph The number of edges of the complete graph on a finite vertex type V is #V choose 2 line 32 source
Source documentation
The number of edges of the complete graph on a finite vertex type V is #V choose 2.
This is a thin completeGraph-flavoured restatement of Mathlib's SimpleGraph.card_edgeFinset_top_eq_card_choose_two, useful when a downstream definition is stated using the completeGraph V abbreviation rather than the ⊤ form. (Because completeGraph V reduces to ⊤, the two are definitionally equal.)

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, Eq, Nat, Finset.card, Sym2, SimpleGraph.edgeFinset, SimpleGraph.completeGraph, Nat.choose, Fintype.card, OfNat.ofNat
Proof uses: SimpleGraph.card_edgeFinset_top_eq_card_choose_two
4 instances, coercions and auxiliary constants left out.
```
lemma card_edgeFinset_completeGraph (V : Type*) [Fintype V] [DecidableEq V] :
    ((completeGraph V).edgeFinset).card = (Fintype.card V).choose 2 :=
…
```
lemma SimpleGraph.card_edgeFinset_completeGraph_fin Specialisation of card_edgeFinset_completeGraph to Fin n line 37 source
Source documentation
Specialisation of card_edgeFinset_completeGraph to Fin n.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Eq, Finset.card, Sym2, Fin, SimpleGraph.edgeFinset, SimpleGraph.completeGraph, Nat.choose, OfNat.ofNat
Proof uses: Eq.mpr, Compl.compl, Finset, Set.toFinset, Sym2.diagSet, Membership.mem, Set, id, congrFun', congrArg, Eq.trans, Top.top, SimpleGraph, SimpleGraph.edgeFinset_top, Set.toFinset_compl, Eq.mp, Fintype.card, congr, Fintype.card_fin, SimpleGraph.card_edgeFinset_completeGraph
24 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
lemma card_edgeFinset_completeGraph_fin (n : ℕ) :
    ((completeGraph (Fin n)).edgeFinset).card = n.choose 2 := by
…
```