lean/OAI/Geometry/Cannon/Modulus/ModulusEnergy.lean

import Mathlib
import OAI.Geometry.Cannon.Modulus.Cost

namespace OAI

section
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noncomputable 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 CannonCells

open CannonDuality Set Filter

variable {X ι J : Type*} [Fintype ι] [Fintype J]

lemma modulus_le_energy (C : ι → Set X) (F : Set (Set X)) (w : Weights ι)
    (hw : Admissible (familyVectors C F : Set (Weights ι)) w) :
    modulus C F ≤ ‖w‖^2 := by
  apply csInf_le
  · refine ⟨0,?_⟩
    rintro a ⟨v,_,rfl⟩
    exact sq_nonneg _
  · exact ⟨w,hw,rfl⟩

theorem modulus_le_card_mul_sum (C : ι → Set X)
    (hC : ∀ x, ∃ i, x ∈ C i) (F : Set (Set X)) (H : J → Set (Set X))
    (hH : ∀ j, (H j).Nonempty) (hHn : ∀ j A, A ∈ H j → A.Nonempty)
    (hsub : ∀ A ∈ F, ∃ j, ∃ B ∈ H j, B ⊆ A) :
    modulus C F ≤ Fintype.card J * ∑ j, modulus C (H j) := by
  classical
  choose w hw hwe using fun j => (modulus_minimum C hC (hH j) (hHn j)).2
  have hwp (j : J) : Nonnegative (w j) := (hw j).1
  let W : Weights ι := ∑ j, w j
  have hW : Admissible (familyVectors C F : Set (Weights ι)) W := by
    rw [admissible_iff]
    have hpoint (i : ι) : W i = ∑ j, w j i := by simp [W]
    have hnon : Nonnegative W := by
      intro i
      rw [hpoint]
      exact Finset.sum_nonneg (fun j _ => hwp j i)
    refine ⟨hnon,?_⟩
    intro A hA
    obtain ⟨j,B,hB,hBA⟩ := hsub A hA
    have hwj := (admissible_iff C (H j) (w j)).mp (hw j)
    calc
      1 ≤ ∑ i ∈ indices C B, w j i := hwj.2 B hB
      _ ≤ ∑ i ∈ indices C A, w j i := Finset.sum_le_sum_of_subset_of_nonneg
        (indices_mono C hBA) (fun i _ _ => hwp j i)
      _ ≤ ∑ i ∈ indices C A, W i := by
        apply Finset.sum_le_sum
        intro i _
        rw [hpoint]
        exact Finset.single_le_sum (fun k _ => hwp k i) (Finset.mem_univ j)
  calc
    modulus C F ≤ ‖W‖^2 := modulus_le_energy C F W hW
    _ ≤ (∑ j, ‖w j‖)^2 := by
      apply pow_le_pow_left₀ (norm_nonneg W)
      exact norm_sum_le _ _
    _ ≤ (Fintype.card J : ℝ)*∑ j, ‖w j‖^2 := by
      simpa using (sq_sum_le_card_mul_sum_sq (s := Finset.univ) (f := fun j => ‖w j‖))
    _ = _ := by simp only [hwe]

theorem exists_modulus_localization [Nonempty J] (C : ι → Set X)
    (hC : ∀ x, ∃ i, x ∈ C i) (F : Set (Set X)) (H : J → Set (Set X))
    (hH : ∀ j, (H j).Nonempty) (hHn : ∀ j A, A ∈ H j → A.Nonempty)
    (hsub : ∀ A ∈ F, ∃ j, ∃ B ∈ H j, B ⊆ A) :
    ∃ j, modulus C F ≤ (Fintype.card J : ℝ)^2 * modulus C (H j) := by
  obtain ⟨j,_,hj⟩ := Finset.univ.exists_max_image (fun j => modulus C (H j)) Finset.univ_nonempty
  refine ⟨j,(modulus_le_card_mul_sum C hC F H hH hHn hsub).trans ?_⟩
  calc
    (Fintype.card J : ℝ)*∑ k, modulus C (H k) ≤
        (Fintype.card J : ℝ)*∑ _k : J, modulus C (H j) := by
      apply mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun k hk => hj k hk))
      exact Nat.cast_nonneg _
    _ = _ := by simp [pow_two,mul_assoc]

theorem fixed_localization_subsequence (M : ℕ → ℝ) (m : J → ℕ → ℝ) (C : ℝ)
    (h : ∀ n, ∃ j, M n ≤ C*m j n) :
    ∃ j, ∃ N : ℕ → ℕ, StrictMono N ∧ ∀ n, M (N n) ≤ C*m j (N n) := by
  have hf : ∃ᶠ n in atTop, ∃ j, M n ≤ C*m j n :=
    (Eventually.of_forall h).frequently
  obtain ⟨j,hj⟩ := frequently_exists.mp hf
  obtain ⟨N,hN,hNj⟩ := extraction_of_frequently_atTop hj
  exact ⟨j,N,hN,hNj⟩

end CannonCells

end
end
end
end
end
end
end
end
end

end OAI