File size: 2,366 Bytes
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 65 66 67 68 69 70 71 72 73 | import Mathlib.Data.Fintype.Powerset
import Mathlib.Data.Finset.Union
import Mathlib.Basic.Real.Basic
import Mathlib.Algebra.Order.Archimedean.Real.Basic
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.BigOperators.Group.Finset.Piecewise
namespace OAI
/-!
# Containment covers and expectation thresholds
Both costs use budget one half. Fractional coefficients are indexed by every
subset of the ground set, without a support-size bound. The empty subset is
included, and ordinary natural powers give `p^0 = 1`, including at zero.
The thresholds are real suprema of feasible densities in the closed unit interval.
-/
namespace TalagrandThreshold
open scoped BigOperators
abbrev Family (α : Type*) := Finset (Finset α)
variable {α : Type*} [Fintype α] [DecidableEq α]
noncomputable section
def cost (p : ℝ) (G : Family α) : ℝ := ∑ S ∈ G, p ^ S.card
def Covers (G F : Family α) : Prop := ∀ H ∈ F, ∃ S ∈ G, S ⊆ H
def Small (p : ℝ) (F : Family α) : Prop :=
∃ G : Family α, Covers G F ∧ cost p G ≤ 1 / 2
def fracCost (p : ℝ) (g : Finset α → ℝ) : ℝ :=
∑ S : Finset α, g S * p ^ S.card
def FractionalCover (F : Family α) (g : Finset α → ℝ) : Prop :=
(∀ S, 0 ≤ g S ∧ g S ≤ 1) ∧
∀ H ∈ F, 1 ≤ ∑ S ∈ H.powerset, g S
def FractionalSmall (p : ℝ) (F : Family α) : Prop :=
∃ g : Finset α → ℝ, FractionalCover F g ∧ fracCost p g ≤ 1 / 2
def Increasing (F : Family α) : Prop :=
∀ H ∈ F, ∀ K : Finset α, H ⊆ K → K ∈ F
def integralFeasible (F : Family α) : Set ℝ :=
{p | 0 ≤ p ∧ p ≤ 1 ∧ Small p F}
def fractionalFeasible (F : Family α) : Set ℝ :=
{p | 0 ≤ p ∧ p ≤ 1 ∧ FractionalSmall p F}
/-- The integral expectation threshold. -/
def q (F : Family α) : ℝ := sSup (integralFeasible F)
/-- The fractional expectation threshold. -/
def qf (F : Family α) : ℝ := sSup (fractionalFeasible F)
omit [Fintype α] [DecidableEq α] in
@[simp] theorem mem_integralFeasible {F : Family α} {p : ℝ} :
p ∈ integralFeasible F ↔ 0 ≤ p ∧ p ≤ 1 ∧ Small p F := Iff.rfl
omit [DecidableEq α] in
@[simp] theorem mem_fractionalFeasible {F : Family α} {p : ℝ} :
p ∈ fractionalFeasible F ↔ 0 ≤ p ∧ p ≤ 1 ∧ FractionalSmall p F := Iff.rfl
end
end TalagrandThreshold
end OAI
|