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caec230 | 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 | 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
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