File size: 2,433 Bytes
e63eb7f | 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 65 66 67 68 69 70 71 72 | import Mathlib.Data.Fintype.Powerset
import Mathlib.Data.Finset.Union
import Mathlib.Basic.Real.Basic
import Mathlib.Algebra.Order.BigOperators.Group.Finset
namespace OAI
/-!
# Finite families, containment covers, and exceptional sets
All families are ordinary finite sets: repeated generators carry no extra cost.
In contrast, the members of a union tuple may repeat. The exceptional family
uses containment in a union, without any monotonicity assumption on the input.
-/
namespace TalagrandDiscreteConvexity
open scoped BigOperators
abbrev Family (α : Type*) := Finset (Finset α)
variable {α : Type*} [Fintype α] [DecidableEq α]
noncomputable section
/-- The mass of a subset under independent Bernoulli coordinates. -/
def bernoulliMass (p : ℝ) (s : Finset α) : ℝ :=
p ^ s.card * (1 - p) ^ (Finset.univ \ s).card
/-- The Bernoulli measure of an arbitrary finite family. -/
def familyMeasure (p : ℝ) (D : Family α) : ℝ :=
∑ s ∈ D, bernoulliMass p s
/-- Cost at the specified density, counting each generator once. -/
def cost (p : ℝ) (G : Family α) : ℝ := ∑ I ∈ G, p ^ I.card
/-- A cover supplies a contained generator for every target set. -/
def Covers (G A : Family α) : Prop := ∀ S ∈ A, ∃ I ∈ G, I ⊆ S
/-- Smallness is witnessed by an actual containment cover of cost at most one half. -/
def Small (p : ℝ) (A : Family α) : Prop :=
∃ G : Family α, Covers G A ∧ cost p G ≤ 1 / 2
/-- A union of exactly `k` entries; repetitions are allowed. -/
def unionTuple {k : ℕ} (f : Fin k → Finset α) : Finset α :=
Finset.univ.biUnion f
/-- All unions of `k` members of a family. -/
def unionFamily (k : ℕ) (D : Family α) : Family α := by
classical
exact Finset.univ.filter (fun S =>
∃ f : Fin k → Finset α, (∀ j, f j ∈ D) ∧ unionTuple f = S)
/-- Sets contained in no union of `k` members of `D`. -/
def exceptional (k : ℕ) (D : Family α) : Family α := by
classical
exact Finset.univ.filter (fun S =>
∀ f : Fin k → Finset α, (∀ j, f j ∈ D) → ¬ S ⊆ unionTuple f)
/-- The subset represented by a Boolean coordinate vector. -/
def cubeSet (x : α → Bool) : Finset α :=
Finset.univ.filter (fun i => x i = true)
/-- The indicator of a family on the Boolean cube. -/
def familyIndicator (D : Family α) (x : α → Bool) : ℝ :=
if cubeSet x ∈ D then 1 else 0
end
end TalagrandDiscreteConvexity
end OAI
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