openAI-math-dataset / lean /OAI /Combinatorics /ExpectationThreshold /SelectorTruncationDefinitions.lean
| 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 | |