| 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 |
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|
