/-
Copyright 2026 The Formal Conjectures Authors.
Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at
https://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
-/
module
public import Mathlib.Combinatorics.SimpleGraph.Finite@[expose] public sectionnamespace SimpleGraph
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.)
lemma card_edgeFinset_completeGraph (V : Type*) [Fintype V] [DecidableEq V] :
((completeGraph V).edgeFinset).card = (Fintype.card V).choose 2 :=
card_edgeFinset_top_eq_card_choose_two
Specialisation of card_edgeFinset_completeGraph to Fin n.
lemma card_edgeFinset_completeGraph_fin (n : ℕ) :
((completeGraph (Fin n)).edgeFinset).card = n.choose 2 := n:ℕ⊢ (completeGraph (Fin n)).edgeFinset.card = n.choose 2
All goals completed! 🐙end SimpleGraph