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d4317d7 | 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.Falconer.Recursion.DescriptorMasks
namespace OAI
namespace PlanarFalconer
open Set MeasureTheory
theorem uniform_descriptor_good_transfer {sig : ℝ} (hsig : 0 < sig)
{J : ℕ} (hJ : 0 < J) :
∃ N₀ : ℕ, ∀ N ≥ N₀, ∀ (Q : DyadicSquare), Q.side = 1 →
∀ (ρ : Measure Plane) (m : ℝ → ℝ) (lam : ℝ) (i : ℕ)
(b : PolicyDescriptor), ValidDescriptor N b →
∀ P : CellPath, P.length = descriptorAnchor b → ∀ ω ω' : UnitCircle,
dist ω ω' ≤ sourceAngularRadius N sig (descriptorLength b) →
ω ∈ descriptorGoodSet Q ρ m N lam sig J i P b →
ω' ∈ descriptorGoodSet Q ρ m N lam sig J (i+1) P b := by
obtain ⟨N₁,h₁⟩ := uniform_parallel_width_transfer hsig hJ (C₀ := 1) zero_le_one
obtain ⟨N₂,h₂⟩ := uniform_projection_width_transfer hsig hJ (C₀ := 1) zero_le_one
refine ⟨max N₁ N₂,?_⟩
intro N hN Q hQ ρ m lam i b hb P hP ω ω' hd hg
obtain ⟨hN₁, hN₂⟩ := max_le_iff.mp hN
change dist (ω:Plane) (ω':Plane) ≤ _ at hd
cases b with
| parallel g p =>
dsimp only [descriptorGoodSet,parallelGoodSet,Set.mem_ofPred_eq] at hg ⊢
change (_ : ℝ) ≤ _ at hg ⊢
apply le_trans (Nat.cast_le.mpr (h₁ N hN₁
Q ρ g p hb.1 P i ω ω' (by simpa only [descriptorLength,one_mul] using hd))) hg
| projection p u =>
dsimp only [descriptorGoodSet,projectionGoodSet,Set.mem_ofPred_eq] at hg ⊢
change (_ : ℝ) ≤ _ at hg ⊢
apply le_trans (Nat.cast_le.mpr (h₂ N hN₂
Q hQ ρ p u hb.1 P hP i ω ω' (by simpa only [descriptorLength,one_mul] using hd))) hg
end PlanarFalconer
end OAI
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