import Mathlib
import OAI.Geometry.Cannon.Energy.SquareMap
import OAI.Geometry.Cannon.Modulus.IndicesPreimageImage

namespace OAI

section
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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