| import Mathlib |
| import OAI.Geometry.Cannon.Energy.SquareMap |
| import OAI.Geometry.Cannon.Modulus.IndicesPreimageImage |
|
|
| namespace OAI |
|
|
| section |
| section |
| section |
| noncomputable section |
| section |
| section |
| section |
| section |
| section |
| open Set Topology |
| open Set Topology Filter Metric unitInterval |
| open scoped InnerProductSpace |
| open Set unitInterval |
| open Set Metric Topology |
| open Set Topology unitInterval |
|
|
| namespace CannonCharts |
|
|
| open CannonCells |
|
|
| theorem normalized_modulus {X ι : Type*} [Fintype ι] |
| (e : Plane → X) (C : ι → Set X) : |
| modulus (fun i => (e ∘ squareMap) ⁻¹' C i) CannonRectangle.leftRight = |
| modulus C (stripFamily e) := by |
| rw [modulus_preimage_image,stripFamily_eq_image] |
|
|
| end CannonCharts |
|
|
| namespace CannonApprox |
|
|
| open CannonCharts CannonRectangle CannonCells |
|
|
| theorem localized_reciprocal_weights {X : Type*} [MetricSpace X] [CompactSpace X] |
| (φ : X ≃ₜ CannonCharts.SphereTwo) (d : ℝ) (hd : 0 < d) |
| (ι : ℕ → Type*) [∀ n, Fintype (ι n)] |
| (C : (n : ℕ) → ι n → Set X) (hC : ∀ n x, ∃ i, x ∈ C n i) |
| (hM : ∀ n, 0 < modulus (C n) (macroFamily (X := X) d)) : |
| ∃ E : CannonGrid.Square → X, Topology.IsEmbedding E ∧ |
| ∃ A : ℝ, 0 < A ∧ ∃ N : ℕ → ℕ, StrictMono N ∧ |
| ∃ ρ : (n : ℕ) → ι (N n) → ℝ, ∀ n, |
| (∀ i, 0 ≤ ρ n i) ∧ Admissible (fun i => E ⁻¹' C (N n) i) (ρ n) ∧ |
| (∑ i, (ρ n i)^2) ≤ A / modulus (C (N n)) (macroFamily d) := by |
| obtain ⟨e,he,_,c,hc,N,hN,hlocal⟩ := rectangle_modulus_localization φ d hd ι C hC |
| let E : CannonGrid.Square → X := e ∘ squareMap |
| have hE : Topology.IsEmbedding E := he.isEmbedding.comp squareMap_embedding |
| have hweights (n : ℕ) := CannonRectangle.exists_dual_weights |
| (fun i => E ⁻¹' C (N n) i) (fun z => hC (N n) (E z)) |
| choose ρ hρ hadm heq using fun n => (hweights n).2 |
| refine ⟨E,hE,c⁻¹,inv_pos.mpr hc,N,hN,ρ,?_⟩ |
| intro n |
| refine ⟨hρ n,hadm n,?_⟩ |
| rw [heq] |
| have hloc : c*modulus (C (N n)) (macroFamily d) ≤ |
| modulus (fun i => E ⁻¹' C (N n) i) leftRight := by |
| simpa only [E,normalized_modulus] using hlocal n |
| calc |
| (modulus (fun i => E ⁻¹' C (N n) i) leftRight)⁻¹ ≤ |
| (c*modulus (C (N n)) (macroFamily d))⁻¹ := |
| inv_anti₀ (mul_pos hc (hM (N n))) hloc |
| _ = c⁻¹ / modulus (C (N n)) (macroFamily d) := by rw [mul_inv,div_eq_mul_inv] |
|
|
| theorem localized_chart_reciprocal_weights {X : Type*} [MetricSpace X] [CompactSpace X] |
| (φ : X ≃ₜ CannonCharts.SphereTwo) (d : ℝ) (hd : 0 < d) |
| (ι : ℕ → Type*) [∀ n, Fintype (ι n)] |
| (C : (n : ℕ) → ι n → Set X) (hC : ∀ n x, ∃ i, x ∈ C n i) |
| (hM : ∀ n, 0 < modulus (C n) (macroFamily (X := X) d)) : |
| ∃ e : Plane → X, Topology.IsOpenEmbedding e ∧ |
| ∃ A : ℝ, 0 < A ∧ ∃ N : ℕ → ℕ, StrictMono N ∧ |
| ∃ ρ : (n : ℕ) → ι (N n) → ℝ, ∀ n, |
| (∀ i, 0 ≤ ρ n i) ∧ |
| Admissible (fun i => (e ∘ squareMap) ⁻¹' C (N n) i) (ρ n) ∧ |
| (∑ i, (ρ n i)^2) ≤ A / modulus (C (N n)) (macroFamily d) := by |
| obtain ⟨e,he,_,c,hc,N,hN,hlocal⟩ := rectangle_modulus_localization φ d hd ι C hC |
| let E : CannonGrid.Square → X := e ∘ squareMap |
| have hweights (n : ℕ) := CannonRectangle.exists_dual_weights |
| (fun i => E ⁻¹' C (N n) i) (fun z => hC (N n) (E z)) |
| choose ρ hρ hadm heq using fun n => (hweights n).2 |
| refine ⟨e,he,c⁻¹,inv_pos.mpr hc,N,hN,ρ,?_⟩ |
| intro n |
| refine ⟨hρ n,hadm n,?_⟩ |
| rw [heq] |
| have hloc : c*modulus (C (N n)) (macroFamily d) ≤ |
| modulus (fun i => E ⁻¹' C (N n) i) leftRight := by |
| simpa only [E,normalized_modulus] using hlocal n |
| calc |
| (modulus (fun i => E ⁻¹' C (N n) i) leftRight)⁻¹ ≤ |
| (c*modulus (C (N n)) (macroFamily d))⁻¹ := |
| inv_anti₀ (mul_pos hc (hM (N n))) hloc |
| _ = c⁻¹ / modulus (C (N n)) (macroFamily d) := by rw [mul_inv,div_eq_mul_inv] |
|
|
| end CannonApprox |
|
|
| open Set Filter Topology |
|
|
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
|
|
| end OAI |
|
|
