| import OAI.MeasureTheory.DyadicAvoidance.GridStabilityFin | |
| import OAI.MeasureTheory.DyadicAvoidance.WindowPlacement | |
| namespace OAI | |
| noncomputable section | |
| namespace Problem310 | |
| /-- Uniform stability for a finite sequence of windows. The gap is selected | |
| before any window lengths are fixed. -/ | |
| theorem exists_window_stability (K : ℕ) (p : ℝ) (hp : 0 < p) : | |
| ∃ g : ℕ, 2 ≤ g ∧ ∀ len : ℕ → ℕ, ∃ G : Set ℝ, | |
| MeasurableSet G ∧ (∀ x : ℝ, x + 1 ∈ G ↔ x ∈ G) ∧ | |
| MeasureTheory.volume (Gᶜ ∩ Set.Icc (0 : ℝ) 1) ≤ ENNReal.ofReal p ∧ | |
| ∀ x ∈ G, ∀ f e : ℕ, f < e → e < K → | |
| ∀ n ∈ WindowPlacement.window len g e, ∀ t ∈ Set.Icc (1 : ℝ) 2, | |
| GridSeparation.dyadicKey (WindowPlacement.finish len g f) | |
| (x + t * ((2 : ℝ)⁻¹) ^ n) = | |
| GridSeparation.dyadicKey (WindowPlacement.finish len g f) x := by | |
| obtain ⟨g, hg, hstable⟩ := exists_uniform_grid_stability (ι := Fin K) p hp | |
| refine ⟨g, hg, fun len ↦ ?_⟩ | |
| obtain ⟨G, hGm, hGp, hGbudget, hGkey⟩ := | |
| hstable (fun i : Fin K ↦ WindowPlacement.finish len g i.val) | |
| refine ⟨G, hGm, hGp, hGbudget, ?_⟩ | |
| intro x hx f e hfe heK n hn t ht | |
| have hfK : f < K := hfe.trans heK | |
| apply hGkey x hx ⟨f, hfK⟩ n ?_ t ht | |
| have hgap := WindowPlacement.gap_of_lt len g hfe | |
| have hstart := (WindowPlacement.mem_window.mp hn).1 | |
| dsimp | |
| omega | |
| end Problem310 | |
| end | |
| end OAI | |