OAI.TalagrandDiscreteConvexity.talagrand_discrete_convexity
OpenThe theorem states that, for every positive integer N, every density p with 0<p<1, and every family D of subsets of the ground set {0,...,N-1} (an arbitrary family, with no monotonicity assumption), if the product measure of D under independent Bernoulli(p) coordinates, namely the sum over s in D of p^|s|(1-p)^(N-|s|), is at least 1 - 1/2^75, then the family of exceptional sets is small. Here the exceptional family for k=2^75 consists of all subsets S of the ground set that are contained in no union of k members of D, where the k members are given as an arbitrary tuple of exactly k entries of D, repeats allowed and with no disjointness required. A family A is small at density p if there is a family G of generator sets that covers A, meaning every member of A contains some member of G, and whose cost, the sum over I in G of p^|I|, is at most 1/2.
-- Generated from openai/math @ adc7f1241b42e322a6451854ab7e4b4c146bf78a
-- Source: lean/ComparatorChallenges/TalagrandDiscreteConvexity.lean; bytes 1583..1902
-- Kind: theorem; original declaration names and bodies preserved.
-- Source groups are independent. Target: Lean 4.33.1; see compilation.json.
import Mathlib.Data.Fintype.Powerset
import Mathlib.Data.Finset.Union
import Mathlib.Data.Real.Basic
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Definitions.Def_TalagrandDiscreteConvexity
namespace OAI
namespace TalagrandDiscreteConvexity
open scoped BigOperators
variable {α : Type*} [Fintype α] [DecidableEq α]
noncomputable section
/-- Smallness at the original density for every positive ground-set size. -/
theorem talagrand_discrete_convexity (N : ℕ) (_hN : 1 ≤ N)
(p : ℝ) (hp0 : 0 < p) (hp1 : p < 1) (D : Family (Fin N))
(hD : 1 - 1 / (2 : ℝ) ^ 75 ≤ familyMeasure p D) :
Small p (exceptional ((2 : ℕ) ^ 75) D) := by
sorry
end
end TalagrandDiscreteConvexity
end OAI
Confirmed by the mission captain (proposal self-audit).
Confirmed by the moderator at approval.