| import Mathlib.Algebra.BigOperators.Ring.Finset |
| import Mathlib.Algebra.Order.BigOperators.Group.Finset |
| import Mathlib.Data.Fintype.Card |
| import Mathlib.Tactic.SplitIfs |
| import Lean.Elab.Tactic.Omega |
|
|
| namespace OAI |
|
|
| /-! Cardinal bounds for pairwise disjoint nonempty branch sets. -/ |
| namespace HadwigerMatching |
| open scoped BigOperators |
|
|
| variable {V : Type*} {t : ℕ} |
|
|
| /-- The number of one-vertex branches. -/ |
| def singletonBranchCount (B : Fin t → Finset V) : ℕ := |
| (Finset.univ.filter fun i => (B i).card = 1).card |
|
|
| /-- The number of two-vertex branches. -/ |
| def doubletonBranchCount (B : Fin t → Finset V) : ℕ := |
| (Finset.univ.filter fun i => (B i).card = 2).card |
|
|
| theorem sum_singleton_indicator (B : Fin t → Finset V) : |
| (∑ i, if (B i).card = 1 then 1 else 0) = singletonBranchCount B := by |
| simp [singletonBranchCount] |
|
|
| theorem sum_doubleton_indicator (B : Fin t → Finset V) : |
| (∑ i, if (B i).card = 2 then 1 else 0) = doubletonBranchCount B := by |
| simp [doubletonBranchCount] |
|
|
| /-- Disjoint branch sets occupy at most the ambient finite ground set. -/ |
| theorem branch_card_sum_le [Fintype V] [DecidableEq V] |
| (B : Fin t → Finset V) (hdisj : Pairwise (fun i j => Disjoint (B i) (B j))) : |
| (∑ i, (B i).card) ≤ Fintype.card V := by |
| have hd : (↑(Finset.univ : Finset (Fin t)) : Set (Fin t)).PairwiseDisjoint B := |
| fun i _ j _ hij => hdisj hij |
| rw [← Finset.card_biUnion hd] |
| exact Finset.card_le_univ _ |
|
|
| /-- Every nonempty branch costs at least three, except for the exact deficits |
| of two for a singleton and one for a doubleton. -/ |
| theorem branch_counting [Fintype V] [DecidableEq V] |
| (B : Fin t → Finset V) |
| (hne : ∀ i, (B i).Nonempty) |
| (hdisj : Pairwise (fun i j => Disjoint (B i) (B j))) : |
| 3 * t ≤ Fintype.card V + 2 * singletonBranchCount B + doubletonBranchCount B := by |
| have hpoint (i : Fin t) : |
| 3 ≤ (B i).card + 2 * (if (B i).card = 1 then 1 else 0) + |
| (if (B i).card = 2 then 1 else 0) := by |
| have hp : 0 < (B i).card := Finset.card_pos.mpr (hne i) |
| split_ifs <;> omega |
| have hsum := Finset.sum_le_sum (s := Finset.univ) (fun i _ => hpoint i) |
| have hsum' : 3 * t ≤ (∑ i, (B i).card) + |
| 2 * singletonBranchCount B + doubletonBranchCount B := by |
| simpa only [Finset.sum_add_distrib, ← Finset.mul_sum, |
| sum_singleton_indicator, sum_doubleton_indicator, Finset.sum_const, |
| Finset.card_univ, Fintype.card_fin, Nat.nsmul_eq_mul, mul_comm t 3] using hsum |
| exact hsum'.trans (Nat.add_le_add_right |
| (Nat.add_le_add_right (branch_card_sum_le B hdisj) _) _) |
|
|
| end HadwigerMatching |
|
|
| end OAI |
|
|
