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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 | import Mathlib.Basic.Real.Basic
import Mathlib.Data.Fintype.Basic
import Mathlib.Algebra.BigOperators.Ring.Finset
namespace OAI
/-!
# Finite weighted truncation
The heavy set uses a strict cutoff. Weights equal to the cutoff are already
saturated when the cutoff increases, as required by the maximal truncation
argument.
-/
namespace TalagrandThreshold.Selector
open scoped BigOperators
variable {α : Type*} [Fintype α] [DecidableEq α]
/-- The points whose weight is strictly above the cutoff. -/
noncomputable def heavySet (w : α → ℝ) (u : ℝ) : Finset α := by
classical
exact Finset.univ.filter (fun x => u < w x)
/-- The excess mass after every coordinate is truncated at the same cutoff. -/
noncomputable def truncatedExcess (w : α → ℝ) (A : Finset α) (d u : ℝ) : ℝ :=
∑ x, min (w x) u * ((if x ∈ A then 1 else 0) - (1 - d))
/-- An admissible cutoff includes all heavy points and meets the truncated
weighted-excess test. -/
def Admissible (w : α → ℝ) (A : Finset α) (m d u : ℝ) : Prop :=
0 ≤ u ∧ u ≤ 1 ∧ (∀ x, u < w x → x ∈ A) ∧
m * u ≤ truncatedExcess w A d u
omit [DecidableEq α] in
@[simp] theorem mem_heavySet {w : α → ℝ} {u : ℝ} {x : α} :
x ∈ heavySet w u ↔ u < w x := by
classical
simp [heavySet]
theorem Admissible.heavy_subset {w : α → ℝ} {A : Finset α} {m d u : ℝ}
(hu : Admissible w A m d u) : heavySet w u ⊆ A := by
intro x hx
exact hu.2.2.1 x (mem_heavySet.mp hx)
theorem Admissible.outside_le {w : α → ℝ} {A : Finset α} {m d u : ℝ}
(hu : Admissible w A m d u) {x : α} (hx : x ∉ A) : w x ≤ u := by
exact le_of_not_gt (fun h => hx (hu.2.2.1 x h))
end TalagrandThreshold.Selector
end OAI
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