openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Couplings /ProductCircleConditionalEvents.lean
| import OAI.Analysis.IrrationalRotation.Couplings.ProductConditionalEvents | |
| import OAI.Analysis.IrrationalRotation.Product.ProductCircleConditioning | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory | |
| namespace Invariant233.OrderedProductCoupling | |
| theorem finite_threshold_attenuation_circle (P : Parameters) (I : Finset ℕ) | |
| (u w : ℕ → ℝ) | |
| (hwidth : ∀ i ∈ I, w i = P.a i ∨ w i = P.b i) | |
| (hthreshold : ∀ i ∈ I, w i ≤ u i ∧ u i ≤ 1 - w i) | |
| (m ρ L : ℝ) (hρ : 0 ≤ ρ) (hL : L ≤ m * ρ) | |
| (hI : m / 20 ≤ (I.card : ℝ)) | |
| (hrho : ∀ i ∈ I, ρ ≤ CouplingMarginalBounds.rho (P.κ i) (P.m i) (w i)) | |
| (R : Sample → ℝ) (hR : ∀ᵐ ω ∂law P, R ω ≤ 1) | |
| (hcross : ∀ᵐ ω ∂law P, ∀ i ∈ I, | |
| (leftEndpoint ω i < u i ∧ u i ≤ rightEndpoint ω i) → R ω ≤ Real.exp (-L)) : | |
| ∃ H₁ H₂ : Set Sample, MeasurableSet H₁ ∧ MeasurableSet H₂ ∧ | |
| (∀ᵐ ω ∂law P, ((law P)[H₁.indicator (fun _ => (1 : ℝ)) | | |
| MeasurableSpace.comap circleLeftEndpoint inferInstance]) ω ≤ Real.exp (-L / 400)) ∧ | |
| (∀ᵐ ω ∂law P, ((law P)[H₂.indicator (fun _ => (1 : ℝ)) | | |
| MeasurableSpace.comap circleRightEndpoint inferInstance]) ω ≤ Real.exp (-L / 400)) ∧ | |
| (∀ᵐ ω ∂law P, R ω ≤ Real.exp (-L) + H₁.indicator (fun _ => (1 : ℝ)) ω + | |
| H₂.indicator (fun _ => (1 : ℝ)) ω) := by | |
| obtain ⟨H₁, H₂, hH₁, hH₂, hleft, hright, hratio⟩ := | |
| finite_threshold_attenuation P I u w hwidth hthreshold m ρ L hρ hL hI hrho | |
| R hR hcross | |
| refine ⟨H₁, H₂, hH₁, hH₂, ?_, ?_, hratio⟩ | |
| · filter_upwards [hleft, left_condExp_eq_circle P (H₁.indicator (fun _ => (1 : ℝ)))] | |
| with ω hω heq | |
| rw [← heq] | |
| exact hω | |
| · filter_upwards [hright, right_condExp_eq_circle P (H₂.indicator (fun _ => (1 : ℝ)))] | |
| with ω hω heq | |
| rw [← heq] | |
| exact hω | |
| end Invariant233.OrderedProductCoupling | |
| end | |
| end OAI | |