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e320e82 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 | 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
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