File size: 2,977 Bytes
ea76aac | 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 | import Mathlib
import OAI.Computability.MinUncut.Estimates.Density
import OAI.Computability.MinUncut.Games.AffineLabels
namespace OAI
section
noncomputable section
open scoped BigOperators
namespace MinUncut.Composition
open BinaryFourier hiding F₂
open MinUncut.Inner
attribute [local instance] Classical.propDecidable BinaryFourier.dualFintype
variable {V W A B : Type*}
[AddCommGroup V] [Module F₂ V] [AddTorsor V A] [Fintype A]
[AddCommGroup W] [Module F₂ W] [AddTorsor W B] [Fintype B]
def labelList (f : Forms A → ℝ) (u : ℝ) : Finset A :=
Finset.univ.filter (fun a => u ≤ |coefficient f (labelFrequency a)|)
@[simp] lemma mem_labelList (f : Forms A → ℝ) (u : ℝ) (a : A) :
a ∈ labelList f u ↔ u ≤ |coefficient f (labelFrequency a)| := by
simp [labelList]
lemma labelList_card (f : Forms A → ℝ) {M u : ℝ} (hM : 0 ≤ M) (hu : 0<u)
(hf : ∀ a, |f a| ≤ M) : ((labelList f u).card : ℝ) ≤ M^2/u^2 := by
have hsub : (labelList f u).image labelFrequency ⊆ largeSpectrum f u := by
intro α hα
obtain ⟨a,ha,rfl⟩ := Finset.mem_image.mp hα
exact Finset.mem_filter.mpr ⟨Finset.mem_univ _, (mem_labelList f u a).mp ha⟩
have hcard := Finset.card_le_card hsub
rw [Finset.card_image_of_injective _ labelFrequency_injective] at hcard
exact (Nat.cast_le.mpr hcard).trans (largeSpectrum_card f hM hu hf)
lemma coefficient_sq_le_mean (f : Forms A → ℝ) (α : Module.Dual F₂ (Forms A)) :
coefficient f α ^2 ≤ 𝔼 a, f a ^2 := by
rw [← parseval]
exact Finset.single_le_sum (fun _ _ => sq_nonneg _) (Finset.mem_univ α)
theorem local_matching (π : A →ᵃ[F₂] B) (f : Forms A → ℝ) (g : Forms B → ℝ)
{u θ : ℝ} (hθ : 0<θ) (b : B)
(hb : b ∈ labelList g θ)
(hclose : (𝔼 z : Forms B, (g z-f (formPullback π z))^2) < (θ/4)^2)
(hrem : |coefficient (fun z => remainder f u (formPullback π z))
(labelFrequency b)| < θ/4) :
∃ a ∈ labelList f u, π a=b := by
have hc := coefficient_sq_le_mean (fun z => g z-f (formPullback π z)) (labelFrequency b)
rw [coefficient_sub] at hc
have hcabs : |coefficient g (labelFrequency b)-
coefficient (fun z => f (formPullback π z)) (labelFrequency b)| < θ/4 := by
have hs := hc.trans_lt hclose
nlinarith [sq_abs (coefficient g (labelFrequency b)-
coefficient (fun z => f (formPullback π z)) (labelFrequency b)),
abs_nonneg (coefficient g (labelFrequency b)-
coefficient (fun z => f (formPullback π z)) (labelFrequency b))]
obtain ⟨α,ha,hα,hodd⟩ := extract_matched_odd f g (formPullback π)
(AffineMap.const F₂ A 1) (AffineMap.const F₂ B 1) (formPullback_one π)
(labelFrequency b) hθ (labelFrequency_odd b) ((mem_labelList g θ b).mp hb) hcabs hrem
obtain ⟨a,rfl⟩ := oddFrequency_exists_label α hodd
refine ⟨a,(mem_labelList f u a).mpr ha,?_⟩
exact labelFrequency_injective hα
end MinUncut.Composition
end
end
end OAI
|