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8e4d533 | 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 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 | import Mathlib
import OAI.Geometry.Cannon.Energy.SquareMap
import OAI.Geometry.Cannon.Modulus.IndicesPreimageImage
namespace OAI
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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
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