| import Mathlib |
| import OAI.Geometry.Cannon.Flows.BoundaryRecordFlow |
| import OAI.Geometry.Cannon.Flows.CenterNeighborhood |
|
|
| namespace OAI |
|
|
| section |
| section |
| section |
| noncomputable section |
| namespace CannonRelease |
| open CannonGraph CannonVisual CannonMagnification CannonPacking CannonApprox CannonCells CannonCharts |
| open CannonNode CannonAnnular Set Metric Filter Topology |
|
|
| universe v |
| variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G] |
| variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ) |
|
|
| theorem boundary_modulus_bounded (φ : Boundary D ≃ₜ SphereTwo) : |
| let _ := visualMetricSpace D hδ |
| let a := Real.exp (2*visualExponent δ) |
| let LE := 8*Real.exp (visualExponent δ*(4*(8*δ+6):ℕ)) |
| ∀ d : ℝ, 0 < d → d < diam (univ : Set (Boundary D)) → d ≤ 1/(LE*3^2) → |
| ∀ (ι : ℕ → Type*) [∀ n, Fintype (ι n)] (p : ∀ n, ι n → Boundary D), |
| (∀ n i j, i ≠ j → (a^n)⁻¹ ≤ dist (p n i) (p n j)) → |
| (∀ n x, ∃ i, dist (p n i) x ≤ (a^n)⁻¹) → |
| ∃ B : ℝ, ∀ n, modulus (fun i => ball (p n i) (2*(a^n)⁻¹)) (macroFamily d) ≤ B := by |
| classical |
| let _ := visualMetricSpace D hδ |
| let _ : MeasurableSpace (Boundary D) := borel (Boundary D) |
| dsimp only |
| intro d hd hdia hdc ι hι p hpsep hpnet |
| by_contra! hunbounded |
| obtain ⟨Q,hQ,CP,hCP,hpack,L,hL,hpaths,Ns,hNs,R,⟨F⟩⟩ := |
| boundary_record_flow D hδ φ d hd hdia hdc ι p hpsep hpnet hunbounded |
| let a := Real.exp (2*visualExponent δ) |
| have ha : 1 < a := Real.one_lt_exp_iff.mpr (mul_pos (by norm_num) (visualExponent_pos δ)) |
| have hrlim : Tendsto (fun n => (a^(Ns n))⁻¹) atTop (𝓝 0) := by |
| simpa only [inv_pow,Function.comp_def] using |
| (tendsto_pow_atTop_nhds_zero_of_lt_one (inv_pos.mpr (zero_lt_one.trans ha)).le |
| (inv_lt_one_of_one_lt₀ ha)).comp hNs.tendsto_atTop |
| exact boundary_no_record_flow D hδ (fun n => ι (Ns n)) (fun n => p (Ns n)) |
| (fun n => (a^(Ns n))⁻¹) (fun _ => inv_pos.mpr (pow_pos (zero_lt_one.trans ha) _)) |
| hrlim (fun n => hpnet (Ns n)) R Q (lt_of_lt_of_le (by norm_num : 0 < 2) hQ) |
| CP hCP.le hpack L hL hpaths F |
|
|
| end CannonRelease |
|
|
| end |
| end |
| end |
| end |
|
|
| end OAI |
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|
