lean/OAI/Geometry/Cannon/Modulus/PathCellIndices.lean

import Mathlib
import OAI.Geometry.Cannon.Boundary.PolynomialPacking
import OAI.Geometry.Cannon.Modulus.Clip

namespace OAI

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open Set MeasureTheory
open scoped BigOperators
open Set unitInterval

namespace CannonAnnular

open Metric

noncomputable def pathCellIndices {Y β : Type*} [PseudoMetricSpace Y] [Fintype β]
    (ξ : β → Y) (s : ℝ) {u v : Y} (p : Path u v) : Finset β := by
  classical
  exact Finset.univ.filter (fun j => ∃ t, dist (p t) (ξ j) < 2*s)

def MacroscopicAdmissible {Y β : Type*} [PseudoMetricSpace Y] [Fintype β]
    (ξ : β → Y) (s d₀ : ℝ) (ω : β → ℝ) : Prop :=
  (∀ j, 0 ≤ ω j) ∧ ∀ (u v : Y) (p : Path u v),
    d₀ ≤ diam (Set.range p) → 1 ≤ ∑ j ∈ pathCellIndices ξ s p, ω j

lemma exists_outer_point {X : Type*} [PseudoMetricSpace X] (o : X) (R : ℝ)
    (hR : 0 ≤ R) {K : Set X} (hdiam : 4*R < diam K) :
    ∃ y ∈ K, 2*R ≤ dist y o := by
  by_contra hn
  push Not at hn
  have hh : diam K ≤ 4*R := by
    apply diam_le_of_forall_dist_le (by positivity)
    intro x hx y hy
    have hh := dist_triangle x o y
    rw [dist_comm o y] at hh
    linarith [hn x hx, hn y hy]
  exact (not_lt_of_ge hh) hdiam

theorem annular_interval_cover {X Y α β : Type*} [MetricSpace X] [MetricSpace Y]
    [LocallyPathConnectedSpace X] [Fintype α] [Fintype β]
    (q : α → X) (ξ : β → Y) (h : X → Y) (hh : Continuous h)
    (f : X → ℝ) (L U : α → ℝ) (o : X) (R r C d₀ a b : ℝ)
    (hR : 0 < R) (hr : 0 < r) (hrR : r ≤ R/2) (hC : 0 ≤ C)
    (hnet : ∀ y, R ≤ dist y o → dist y o ≤ 2*R → ∃ i, dist (q i) y ≤ r)
    (hzoom : ∀ u v, dist u o ≤ 3*R → dist v o ≤ 3*R →
      dist (h u) (h v) ≤ C/R * dist u v)
    (hlarge : ∀ u v, dist u o ≤ 2*R → dist v o ≤ 2*R →
      R ≤ dist u v → d₀ ≤ dist (h u) (h v))
    (hrange : ∀ i z, dist z (q i) ≤ 2*r → f z ∈ Icc (L i) (U i))
    (ω : β → ℝ) (hω : MacroscopicAdmissible ξ (r/R) d₀ ω)
    (Clevel : ℝ → Set X)
    (hlevel : ∀ t ∈ Ioo a b, IsPreconnected (Clevel t) ∧
      4*R < diam (Clevel t) ∧ (∃ x ∈ Clevel t, dist x o ≤ R/2) ∧
      ∀ z ∈ Clevel t, f z = t) :
    ∀ t ∈ Ioo a b, 1 ≤ ∑ i,
      (Icc (L i) (U i)).indicator
        (fun _ : ℝ => ∑ j ∈ Finset.univ.filter
          (fun j => dist (ξ j) (h (q i)) ≤ (C+2)*(r/R)), ω j) t := by
  classical
  intro t ht
  obtain ⟨hc, hd, ⟨x, hx, hxr⟩, hf⟩ := hlevel t ht
  obtain ⟨y, hy, hyr⟩ := exists_outer_point o R hR.le hd
  let V : Set X := ⋃ z ∈ Clevel t, ball z (r/4)
  have hV : IsOpen V := isOpen_iUnion (fun z => isOpen_iUnion (fun _ => isOpen_ball))
  have hCV : Clevel t ⊆ V := by
    intro z hz
    exact mem_iUnion.mpr ⟨z, mem_iUnion.mpr ⟨hz, by simp [hr]⟩⟩
  obtain ⟨u, v, p, hpV, hu, hv, huv, hp⟩ := annular_path_near_continuum
    o R (2*R) (by linarith) (Clevel t) V hc hV hCV hx hy (by linarith) hyr
  let p' : Path (h u) (h v) := p.map hh
  have hmacro : d₀ ≤ diam (Set.range p') := by
    have hsep := hlarge u v (by linarith) hv.le (by linarith)
    apply hsep.trans
    exact dist_le_diam_of_mem (isCompact_range p'.continuous).isBounded
      ⟨0, p'.source⟩ ⟨1, p'.target⟩
  have hadm := hω.2 (h u) (h v) p' hmacro
  apply interval_cover_of_charging _ ω L U hω.1 t (pathCellIndices ξ (r/R) p') hadm
  intro j hj
  obtain ⟨s, hs⟩ := (Finset.mem_filter.mp hj).2
  change dist (h (p s)) (ξ j) < 2*(r/R) at hs
  have hsV := hpV (Set.mem_range_self s)
  obtain ⟨z, hz⟩ := mem_iUnion.mp hsV
  obtain ⟨hzC, hzs⟩ := mem_iUnion.mp hz
  have hnear : dist z (p s) ≤ r/4 := by
    rw [mem_ball, dist_comm] at hzs
    exact hzs.le
  obtain ⟨i, _, _, hit, hij⟩ := annular_net_charge q h f L U o (p s) z (ξ j)
    R r C t hR hr.le hrR hC (hp s) (hnet (p s) (hp s).1 (hp s).2)
    hzoom (by simpa only [dist_comm] using hs.le) hnear (hf z hzC) hrange
  exact ⟨i, hit, Finset.mem_filter.mpr ⟨Finset.mem_univ _, hij⟩⟩

end CannonAnnular

namespace CannonAnnular

open Metric

theorem annular_comparison {X Y α β γ : Type*} [MetricSpace X] [MetricSpace Y]
    [LocallyPathConnectedSpace X] [Fintype α] [Fintype β] [Fintype γ]
    (q : α → X) (ξ : β → Y) (p : γ → X) (h : X → Y) (hh : Continuous h)
    (f : X → ℝ) (L U : α → ℝ) (o : X) (R r C d d₀ a b : ℝ)
    (hR : 0 < R) (hr : 0 < r) (hrR : r ≤ R/2) (hC : 0 ≤ C)
    (hd : d ≤ R/4) (hab : a ≤ b) (hLU : ∀ i, L i ≤ U i)
    (hq : ∀ i, R/2 ≤ dist (q i) o ∧ dist (q i) o ≤ 3*R)
    (hnet : ∀ y, R ≤ dist y o → dist y o ≤ 2*R → ∃ i, dist (q i) y ≤ r)
    (hzoom : ∀ u v, dist u o ≤ 3*R → dist v o ≤ 3*R →
      dist (h u) (h v) ≤ C/R * dist u v)
    (hlarge : ∀ u v, dist u o ≤ 2*R → dist v o ≤ 2*R →
      R ≤ dist u v → d₀ ≤ dist (h u) (h v))
    (hrange : ∀ i z, dist z (q i) ≤ 2*r → f z ∈ Icc (L i) (U i))
    (ω : β → ℝ) (hω : MacroscopicAdmissible ξ (r/R) d₀ ω)
    (ρ : γ → ℝ) (hρ : ∀ j, 0 ≤ ρ j)
    (hlocal : ∀ i, U i - L i ≤ ∑ j ∈ Finset.univ.filter
      (fun j => dist (p j) (q i) ≤ d), ρ j)
    (Clevel : ℝ → Set X)
    (hlevel : ∀ t ∈ Ioo a b, IsPreconnected (Clevel t) ∧
      4*R < diam (Clevel t) ∧ (∃ x ∈ Clevel t, dist x o ≤ R/2) ∧
      ∀ z ∈ Clevel t, f z = t)
    (kr kc lr lc : ℕ)
    (hTrow : ∀ i, (Finset.univ.filter
      (fun j => dist (ξ j) (h (q i)) ≤ (C+2)*(r/R))).card ≤ kr)
    (hTcol : ∀ j, (Finset.univ.filter
      (fun i => dist (ξ j) (h (q i)) ≤ (C+2)*(r/R))).card ≤ kc)
    (hSrow : ∀ i, (Finset.univ.filter
      (fun j => dist (p j) (q i) ≤ d)).card ≤ lr)
    (hScol : ∀ j, (Finset.univ.filter
      (fun i => dist (p j) (q i) ≤ d)).card ≤ lc) :
    (b-a)^2 ≤ ((kr:ℝ)*kc*lr*lc) * (∑ j, (ω j)^2) *
      (∑ j ∈ Finset.univ.filter
        (fun j => R/4 ≤ dist (p j) o ∧ dist (p j) o ≤ 4*R), (ρ j)^2) := by
  classical
  let T (i : α) := Finset.univ.filter
    (fun j => dist (ξ j) (h (q i)) ≤ (C+2)*(r/R))
  let S (i : α) := Finset.univ.filter (fun j => dist (p j) (q i) ≤ d)
  let A := Finset.univ.filter
    (fun j => R/4 ≤ dist (p j) o ∧ dist (p j) o ≤ 4*R)
  let w (j : γ) := if j ∈ A then ρ j else 0
  have hn : ∀ j, 0 ≤ w j := by
    intro j
    dsimp [w]
    split_ifs
    · exact hρ j
    · exact le_rfl
  have hloc : ∀ i, U i - L i ≤ ∑ j ∈ S i, w j := by
    intro i
    apply (hlocal i).trans_eq
    apply Finset.sum_congr rfl
    intro j hj
    have hja : j ∈ A := Finset.mem_filter.mpr ⟨Finset.mem_univ _,
      charged_center_in_annulus o (q i) (p j) R d hR.le (hq i)
        (Finset.mem_filter.mp hj).2 hd⟩
    simp [w, hja]
  have hcover := annular_interval_cover q ξ h hh f L U o R r C d₀ a b
    hR hr hrR hC hnet hzoom hlarge hrange ω hω Clevel hlevel
  have henergy := annular_energy_transfer T S ω w L U a b hab hω.1 hn hLU
    hloc kr kc lr lc hTrow (by simpa [T] using hTcol)
    hSrow (by simpa [S] using hScol) hcover
  have hwE : (∑ j, (w j)^2) = ∑ j ∈ A, (ρ j)^2 := by
    simp [w]
  rw [hwE] at henergy
  exact henergy

end CannonAnnular

open Set Metric

namespace CannonAnnular

open CannonPacking Metric

theorem exists_packed_annular_constant {X : Type*} [MetricSpace X]
    [LocallyPathConnectedSpace X] (Q : ℕ) (CP C c A : ℝ)
    (hpack : PolynomialPacking (X := X) Q CP) (hCP : 0 ≤ CP)
    (hC : 0 ≤ C) (hc : 0 < c) (hA : 0 < A) :
    ∃ E : ℝ, 0 ≤ E ∧ ∀ {α β γ : Type*} [Fintype α] [Fintype β] [Fintype γ]
      (q : α → X) (ξ : β → X) (p : γ → X) (h : X → X), Continuous h →
      ∀ (f : X → ℝ) (L U : α → ℝ) (o : X) (R r d₀ a b : ℝ),
      0 < R → 0 < r → r ≤ R/2 → A*r ≤ R/4 →
      max (C+2) (max c 1)*(r/R) ≤ 1 → max A 1*r ≤ 1 →
      a ≤ b → (∀ i, L i ≤ U i) →
      (∀ i, R/2 ≤ dist (q i) o ∧ dist (q i) o ≤ 3*R) →
      (∀ i j, i ≠ j → r ≤ dist (q i) (q j)) →
      (∀ i j, i ≠ j → r/R ≤ dist (ξ i) (ξ j)) →
      (∀ i j, i ≠ j → r ≤ dist (p i) (p j)) →
      (∀ y, R ≤ dist y o → dist y o ≤ 2*R → ∃ i, dist (q i) y ≤ r) →
      (∀ u v, dist u o ≤ 3*R → dist v o ≤ 3*R →
        dist (h u) (h v) ≤ C/R * dist u v) →
      (∀ u v, dist u o ≤ 3*R → dist v o ≤ 3*R →
        c/R * dist u v ≤ dist (h u) (h v)) →
      (∀ u v, dist u o ≤ 2*R → dist v o ≤ 2*R →
        R ≤ dist u v → d₀ ≤ dist (h u) (h v)) →
      (∀ i z, dist z (q i) ≤ 2*r → f z ∈ Icc (L i) (U i)) →
      ∀ (ω : β → ℝ), MacroscopicAdmissible ξ (r/R) d₀ ω →
      ∀ (ρ : γ → ℝ), (∀ j, 0 ≤ ρ j) →
      (∀ i, U i-L i ≤ ∑ j ∈ Finset.univ.filter
        (fun j => dist (p j) (q i) ≤ A*r), ρ j) →
      ∀ (Clevel : ℝ → Set X),
      (∀ t ∈ Ioo a b, IsPreconnected (Clevel t) ∧
        4*R < diam (Clevel t) ∧ (∃ x ∈ Clevel t, dist x o ≤ R/2) ∧
        ∀ z ∈ Clevel t, f z = t) →
      (b-a)^2 ≤ E*(∑ j, (ω j)^2)*
        (∑ j ∈ Finset.univ.filter
          (fun j => R/4 ≤ dist (p j) o ∧ dist (p j) o ≤ 4*R), (ρ j)^2) := by
  classical
  have hC2 : 0 < C+2 := by linarith
  obtain ⟨kr, hkr⟩ := exists_overlap_bound Q CP (C+2) 1 hpack hCP hC2 zero_lt_one
  obtain ⟨kc, hkc⟩ := exists_overlap_bound Q CP (C+2) c hpack hCP hC2 hc
  obtain ⟨lr, hlr⟩ := exists_overlap_bound Q CP A 1 hpack hCP hA zero_lt_one
  obtain ⟨lc, hlc⟩ := exists_overlap_bound Q CP A 1 hpack hCP hA zero_lt_one
  refine ⟨(kr:ℝ)*kc*lr*lc, by positivity, ?_⟩
  intro α β γ _ _ _ q ξ p h hh f L U o R r d₀ a b hR hr hrR hAr
    hscaleT hscaleS hab hLU hq hqsep hξsep hpsep hnet hzoom hlow hlarge
    hrange ω hω ρ hρ hlocal Clevel hlevel
  have hrdiv : 0 < r/R := div_pos hr hR
  have hscale1 : max (C+2) 1*(r/R) ≤ 1 :=
    (mul_le_mul_of_nonneg_right (max_le_max le_rfl (le_max_right ..)) hrdiv.le).trans hscaleT
  have hscalec : max (C+2) c*(r/R) ≤ 1 :=
    (mul_le_mul_of_nonneg_right (max_le_max le_rfl (le_max_left ..)) hrdiv.le).trans hscaleT
  have hrow (i : α) := hkr (ι := β) ξ Finset.univ (h (q i)) (r/R) hrdiv hscale1
    (by simpa using hξsep)
  have himsep (i j : α) (hne : i ≠ j) : c*(r/R) ≤ dist (h (q i)) (h (q j)) := by
    calc
      c*(r/R) = c/R*r := by ring
      _ ≤ c/R*dist (q i) (q j) :=
        mul_le_mul_of_nonneg_left (hqsep i j hne) (div_nonneg hc.le hR.le)
      _ ≤ dist (h (q i)) (h (q j)) := hlow _ _ (hq i).2 (hq j).2
  have hcol (j : β) := hkc (ι := α) (fun i => h (q i)) Finset.univ (ξ j) (r/R)
    hrdiv hscalec (by simpa using himsep)
  have hsrow (i : α) := hlr (ι := γ) p Finset.univ (q i) r hr hscaleS
    (by simpa using hpsep)
  have hscol (j : γ) := hlc (ι := α) q Finset.univ (p j) r hr hscaleS
    (by simpa using hqsep)
  exact annular_comparison q ξ p h hh f L U o R r C (A*r) d₀ a b
    hR hr hrR hC hAr hab hLU hq hnet hzoom hlarge hrange ω hω ρ hρ hlocal
    Clevel hlevel kr kc lr lc
    (by simpa only [dist_comm] using hrow) hcol
    (by simpa only [one_mul, dist_comm] using hsrow) hscol

end CannonAnnular

end

namespace CannonCapacity

open Set Metric Finset CannonAnnular

attribute [local instance] Classical.propDecidable

lemma level_radial_hit {X : Type*} [MetricSpace X] (x : X) (K : Set X)
    (hK : IsPreconnected K) {b : ℝ} (hb : 0 ≤ b)
    (hinner : ∃ y ∈ K, dist y x ≤ b) (hdiam : 2*b < diam K) :
    ∃ w ∈ K, dist w x = b := by
  obtain ⟨y,hy,hyb⟩ := hinner
  obtain ⟨z,hz,hzb⟩ := exists_outer_point x (b/2) (by linarith) (by linarith : 4*(b/2) < diam K)
  exact hK.intermediate_value hy hz (continuous_id.dist continuous_const).continuousOn
    ⟨hyb,by dsimp; linarith⟩

theorem unmarked_level_detection {X : Type*} [MetricSpace X]
    [LocallyPathConnectedSpace X] {ι β : Type*} [Fintype ι] [Fintype β]
    (q : ι → X) (ξ : β → X) (h : X → X) (hh : Continuous h)
    (x w z : X) (A Ω K : Set X) (f : X → ℝ) (lo hi σ : ι → ℝ)
    (large : ι → Prop) (r s C d₀ b t : ℝ)
    (hr : 0 < r) (hs : 0 < s) (hrs : r ≤ s) (hC : 0 ≤ C)
    (hK : IsPreconnected K) (hwK : w ∈ K) (hdiam : 4*s < diam K)
    (hwb : dist w x = b) (hwz : dist w z ≤ s)
    (hregion : ∀ v, dist v w ≤ s → v ∈ A)
    (hdomain : ∀ v, dist v w ≤ s+r/4 → v ∈ Ω)
    (hlevel : ∀ v ∈ K, v ∈ Ω → f v = t)
    (hnet : ∀ v ∈ A, ∃ i, dist (q i) v ≤ r)
    (hupper : ∀ v v', dist v z ≤ 3*s → dist v' z ≤ 3*s →
      dist (h v) (h v') ≤ C/s*dist v v')
    (hlower : ∀ v v', dist v z ≤ 2*s → dist v' z ≤ 2*s →
      s ≤ dist v v' → d₀ ≤ dist (h v) (h v'))
    (hrange : ∀ i v, v ∈ Ω → dist v (q i) ≤ 2*r → f v ∈ Icc (lo i) (hi i))
    (ω : β → ℝ) (hω : MacroscopicAdmissible ξ (r/s) d₀ ω)
    (hσ : ∀ i, 0 ≤ σ i)
    (hpay : ∀ i, dist (q i) z ≤ 3*s →
      (∑ j ∈ univ.filter (fun j => dist (ξ j) (h (q i)) ≤ (C+2)*(r/s)), ω j) ≤ σ i)
    (hunmarked : ∀ i, large i → |dist (q i) x-b| ≤ 5*s → t ∈ Icc (lo i) (hi i) → False) :
    1 ≤ ∑ i, (Icc (lo i) (hi i)).indicator
      (fun _ : ℝ => if |dist (q i) x-b| ≤ 5*s ∧ ¬ large i then σ i else 0) t := by
  classical
  let V : Set X := ⋃ v ∈ K, ball v (r/4)
  have hV : IsOpen V := isOpen_iUnion (fun v => isOpen_iUnion (fun _ => isOpen_ball))
  have hKV : K ⊆ V := by
    intro v hv
    exact mem_iUnion.mpr ⟨v,mem_iUnion.mpr ⟨hv,by simp [hr]⟩⟩
  obtain ⟨v,hv,hvs⟩ := exists_outer_point w s hs.le hdiam
  obtain ⟨u,v',π,hπV,hu,hv',huv,hπ⟩ := annular_path_near_continuum w 0 s hs K V
    hK hV hKV hwK hv (by simp) (by linarith)
  have hzπ : ∀ a, dist (π a) z ≤ 2*s := by
    intro a
    exact (dist_triangle (π a) w z).trans (by linarith [(hπ a).2])
  let π' := π.map hh
  have hmacro : d₀ ≤ diam (range π') := by
    have h₁ : dist u z ≤ 2*s := by simpa only [π.source] using hzπ 0
    have h₂ : dist v' z ≤ 2*s := by simpa only [π.target] using hzπ 1
    exact (hlower u v' h₁ h₂ (by linarith)).trans
      (dist_le_diam_of_mem (isCompact_range π'.continuous).isBounded
        ⟨0,π'.source⟩ ⟨1,π'.target⟩)
  have hadm := hω.2 (h u) (h v') π' hmacro
  let eligible := univ.filter (fun i => |dist (q i) x-b| ≤ 5*s ∧
    ¬ large i ∧ t ∈ Icc (lo i) (hi i) ∧ dist (q i) z ≤ 3*s)
  let rel i := univ.filter (fun j => dist (ξ j) (h (q i)) ≤ (C+2)*(r/s))
  have hcharge : ∀ j ∈ pathCellIndices ξ (r/s) π', ∃ i ∈ eligible, j ∈ rel i := by
    intro j hj
    obtain ⟨a,ha⟩ := (mem_filter.mp hj).2
    obtain ⟨i,hqi⟩ := hnet (π a) (hregion _ (hπ a).2)
    have hiw : dist (q i) w ≤ r+s := (dist_triangle (q i) (π a) w).trans (by linarith [(hπ a).2])
    have hiz : dist (q i) z ≤ 3*s := (dist_triangle (q i) w z).trans (by linarith)
    obtain ⟨v',hv'⟩ := mem_iUnion.mp (hπV (mem_range_self a))
    obtain ⟨hv'K,hv'π⟩ := mem_iUnion.mp hv'
    have hv'π : dist v' (π a) < r/4 := by simpa only [mem_ball,dist_comm] using hv'π
    have hv'Ω : v' ∈ Ω := hdomain _ ((dist_triangle v' (π a) w).trans (by linarith [(hπ a).2]))
    have htv : t ∈ Icc (lo i) (hi i) := by
      rw [← hlevel v' hv'K hv'Ω]
      apply hrange i v' hv'Ω
      have hd := dist_triangle v' (π a) (q i)
      rw [dist_comm (π a) (q i)] at hd
      linarith
    have hib : |dist (q i) x-b| ≤ 5*s := by
      rw [← hwb]
      exact (abs_dist_sub_le (q i) w x).trans (by linarith)
    refine ⟨i,mem_filter.mpr ⟨mem_univ _,hib,fun hl => hunmarked i hl hib htv,htv,hiz⟩,?_⟩
    apply mem_filter.mpr
    refine ⟨mem_univ _,?_⟩
    have ht : dist (h (π a)) (h (q i)) ≤ C/s*r := by
      apply (hupper _ _ (by linarith [hzπ a]) hiz).trans
      exact mul_le_mul_of_nonneg_left (by simpa only [dist_comm] using hqi) (div_nonneg hC hs.le)
    have ha' : dist (ξ j) (h (π a)) ≤ 2*(r/s) := by
      simpa only [Path.map_coe,π',dist_comm,Function.comp_apply] using ha.le
    calc
      _ ≤ dist (ξ j) (h (π a)) + dist (h (π a)) (h (q i)) := dist_triangle _ _ _
      _ ≤ 2*(r/s)+C/s*r := add_le_add ha' ht
      _ = _ := by ring
  have hchargeSum := finite_charge_cover eligible (pathCellIndices ξ (r/s) π') rel ω hω.1 hcharge
  have hbound : (∑ i ∈ eligible, ∑ j ∈ rel i, ω j) ≤ ∑ i ∈ eligible, σ i := by
    apply sum_le_sum
    intro i hi
    exact hpay i (mem_filter.mp hi).2.2.2.2
  apply hadm.trans (hchargeSum.trans (hbound.trans _))
  rw [sum_filter]
  apply sum_le_sum
  intro i _
  by_cases he : |dist (q i) x-b| ≤ 5*s ∧ ¬ large i ∧ t ∈ Icc (lo i) (hi i) ∧ dist (q i) z ≤ 3*s
  · have hnear : |dist (q i) x-b| ≤ 5*s ∧ ¬ large i := ⟨he.1,he.2.1⟩
    simp only [ite_eq_left he,indicator_of_mem he.2.2.1,ite_eq_left hnear]
    exact le_rfl
  · rw [ite_eq_right he]
    exact Set.indicator_nonneg (fun _ _ => by split_ifs; exact hσ i; exact le_rfl) _

end CannonCapacity

end

section

namespace CannonCapacity

open Set Metric Finset CannonAnnular CannonPacking

attribute [local instance] Classical.propDecidable

lemma sparse_transfer_energy {ι β : Type*} [Fintype ι] [Fintype β]
    (f : β → ι → ℝ) (near : β → ι → Prop) (N : ℕ) (E : ℝ)
    (hsupport : ∀ z i, ¬ near z i → f z i = 0)
    (hmult : ∀ i, (univ.filter (fun z => near z i)).card ≤ N)
    (he : ∀ z, ∑ i, (f z i)^2 ≤ E) :
    (∑ i, (∑ z, f z i)^2) ≤ (N : ℝ) * Fintype.card β * E := by
  classical
  have hrow (i : ι) : (∑ z, f z i)^2 ≤ (N : ℝ)*∑ z, (f z i)^2 := by
    let S := univ.filter (fun z => near z i)
    have heq : ∑ z, f z i = ∑ z ∈ S, f z i := by
      symm
      apply sum_filter_of_ne
      intro z _ hz
      by_contra hn
      exact hz (hsupport z i hn)
    rw [heq]
    have hCS := sum_mul_sq_le_sq_mul_sq S (fun _ => (1:ℝ)) (fun z => f z i)
    simp only [one_mul,one_pow,sum_const,nsmul_eq_mul,mul_one] at hCS
    calc
      _ ≤ (S.card : ℝ)*(∑ z ∈ S, (f z i)^2) := hCS
      _ ≤ (N : ℝ)*(∑ z, (f z i)^2) := by
        apply mul_le_mul (by exact_mod_cast hmult i)
          (sum_le_sum_of_subset_of_nonneg (filter_subset _ _) (fun z _ _ => sq_nonneg _))
          (sum_nonneg (fun _ _ => sq_nonneg _)) (by positivity)
  calc
    _ ≤ ∑ i, (N : ℝ)*∑ z, (f z i)^2 := sum_le_sum (fun i _ => hrow i)
    _ = (N : ℝ)*∑ z, ∑ i, (f z i)^2 := by rw [← mul_sum,sum_comm]
    _ ≤ (N : ℝ)*∑ _z : β, E := mul_le_mul_of_nonneg_left (sum_le_sum (fun z _ => he z)) (by positivity)
    _ = _ := by simp; ring

theorem exists_local_transfer_constant {X : Type*} [MetricSpace X]
    (Q : ℕ) (CP A c : ℝ) (hpack : PolynomialPacking (X := X) Q CP)
    (hCP : 0 ≤ CP) (hA : 0 < A) (hc : 0 < c) :
    ∃ E : ℝ, 0 ≤ E ∧ ∀ {ι β : Type*} [Fintype ι] [Fintype β]
      (q : ι → X) (ξ : β → X) (h : X → X) (z : X) (r s : ℝ),
      0 < r → 0 < s → max A (max c 1)*(r/s) ≤ 1 →
      (∀ i j, i ≠ j → r ≤ dist (q i) (q j)) →
      (∀ i j, i ≠ j → r/s ≤ dist (ξ i) (ξ j)) →
      (∀ v v', dist v z ≤ 3*s → dist v' z ≤ 3*s →
        c/s*dist v v' ≤ dist (h v) (h v')) →
      ∀ ω : β → ℝ,
      (∑ i, (if dist (q i) z ≤ 3*s then
        ∑ j ∈ univ.filter (fun j => dist (ξ j) (h (q i)) ≤ A*(r/s)), ω j else 0)^2) ≤
        E*∑ j, (ω j)^2 := by
  classical
  obtain ⟨kr,hkr⟩ := exists_overlap_bound Q CP A 1 hpack hCP hA zero_lt_one
  obtain ⟨kc,hkc⟩ := exists_overlap_bound Q CP A c hpack hCP hA hc
  refine ⟨(kr:ℝ)*kc,by positivity,?_⟩
  intro ι β _ _ q ξ h z r s hr hs hscale hq hξ hlow ω
  let rel i := if dist (q i) z ≤ 3*s then
    univ.filter (fun j => dist (ξ j) (h (q i)) ≤ A*(r/s)) else ∅
  let S := univ.filter (fun i => dist (q i) z ≤ 3*s)
  have hratio : 0 < r/s := div_pos hr hs
  have hscale₁ : max A 1*(r/s) ≤ 1 :=
    (mul_le_mul_of_nonneg_right (max_le_max le_rfl (le_max_right ..)) hratio.le).trans hscale
  have hscalec : max A c*(r/s) ≤ 1 :=
    (mul_le_mul_of_nonneg_right (max_le_max le_rfl (le_max_left ..)) hratio.le).trans hscale
  have hrow (i : ι) : (rel i).card ≤ kr := by
    dsimp only [rel]
    split_ifs
    · have hh := hkr ξ univ (h (q i)) (r/s) hratio hscale₁ (by simpa using hξ)
      simpa only [dist_comm] using hh
    · simp
  have hcol (j : β) : (univ.filter (fun i => j ∈ rel i)).card ≤ kc := by
    have him : ∀ i ∈ S, ∀ j ∈ S, i ≠ j → c*(r/s) ≤ dist (h (q i)) (h (q j)) := by
      intro i hi j hj hij
      calc
        _ = c/s*r := by ring
        _ ≤ c/s*dist (q i) (q j) := mul_le_mul_of_nonneg_left (hq i j hij) (div_nonneg hc.le hs.le)
        _ ≤ _ := hlow _ _ (mem_filter.mp hi).2 (mem_filter.mp hj).2
    have hh := hkc (fun i => h (q i)) S (ξ j) (r/s) hratio hscalec him
    convert hh using 1
    congr 1
    ext i
    by_cases hi : dist (q i) z ≤ 3*s <;> simp [S,rel,hi]
  have he := overlap_energy rel ω kr kc hrow hcol
  convert he using 1
  congr 1
  funext i
  simp only [rel]
  split_ifs <;> simp

end CannonCapacity

end

namespace CannonCapacity

open Set Metric Finset CannonAnnular

attribute [local instance] Classical.propDecidable

theorem node_separated_capacity {X : Type*} [MetricSpace X]
    [LocallyPathConnectedSpace X]
    {ι β ζ δ : Type*} [Fintype ι] [Fintype β] [Fintype ζ] [Fintype δ]
    (q : ι → X) (bin : β → ℝ) (aux : ζ → X) (ξ : δ → X)
    (h : ζ → X ≃ₜ X) (x : X) (Ω : Set X) (f : X → ℝ)
    (K : ℝ → Set X) (lo hi t : ι → ℝ)
    (a b A B₀ R r s C c d₀ H L Q E M BinMul : ℝ) (N Naux : ℕ)
    (hab : a < b) (hA : 1 ≤ A) (hAB : A+2 ≤ B₀)
    (hR : 0 < R) (hr : 0 < r) (hs : 0 < s) (hrs : r ≤ s) (hsR : s ≤ R/4)
    (hC : 0 ≤ C) (hc : 0 < c) (hdc : d₀ ≤ c)
    (hbin : ∀ j, (A+1)*R ≤ bin j ∧ bin j ≤ (B₀-1)*R)
    (hnet : ∀ v, (A-1)*R ≤ dist v x → dist v x ≤ (B₀+1)*R →
      ∃ i, dist (q i) v ≤ r)
    (haux : ∀ v, (A-1)*R ≤ dist v x → dist v x ≤ (B₀+1)*R →
      ∃ z, dist v (aux z) ≤ s)
    (hΩ : ∀ v, dist v x ≤ (B₀+2)*R → v ∈ Ω)
    (hK : ∀ v ∈ Ioo a b, IsPreconnected (K v))
    (hinner : ∀ v ∈ Ioo a b, ∃ y ∈ K v, dist y x ≤ A*R)
    (hdiam : ∀ v ∈ Ioo a b, 2*B₀*R < diam (K v))
    (hlevel : ∀ v ∈ Ioo a b, ∀ y ∈ K v, y ∈ Ω → f y = v)
    (hlohi : ∀ i, lo i ≤ hi i) (ht : ∀ i, 0 ≤ t i)
    (hlen : ∀ i, clip a b (hi i)-clip a b (lo i) = (b-a)*t i)
    (hrange : ∀ i v, v ∈ Ω → dist v (q i) ≤ 2*r → f v ∈ Icc (lo i) (hi i))
    (hupper : ∀ z v v', dist v (aux z) ≤ 3*s → dist v' (aux z) ≤ 3*s →
      dist (h z v) (h z v') ≤ C/s*dist v v')
    (hlower : ∀ z v v', dist v (aux z) ≤ 3*s → dist v' (aux z) ≤ 3*s →
      c/s*dist v v' ≤ dist (h z v) (h z v'))
    (ω : δ → ℝ) (hω : MacroscopicAdmissible ξ (r/s) d₀ ω)
    (hωmass : ∑ j, (ω j)^2 ≤ M)
    (hlocal : ∀ z, (∑ i, (if dist (q i) (aux z) ≤ 3*s then
      ∑ j ∈ univ.filter (fun j => dist (ξ j) (h z (q i)) ≤ (C+2)*(r/s)), ω j else 0)^2)
      ≤ E*∑ j, (ω j)^2)
    (hmultaux : ∀ i, (univ.filter (fun z => dist (q i) (aux z) ≤ 3*s)).card ≤ Naux)
    (hL : 0 < L) (hQ : 0 < Q) (hE : 0 ≤ E) (hM : 0 ≤ M)
    (hBinMul : 0 < BinMul) (hHr : 0 ≤ H*r) (hN : 0 < N)
    (hdegree : ∀ i, (univ.filter (fun j => dist (q i) (q j) ≤ H*r)).card ≤ N)
    (hmult : ∀ i, ((univ.filter (fun j => |dist (q i) x-bin j| ≤ 5*s)).card : ℝ) ≤ BinMul)
    (hlarge : 2*BinMul*N ≤ L*Fintype.card β)
    (hsmall : 4*BinMul^2*((Naux:ℝ)*Fintype.card ζ*(E*M))*N < Q*(Fintype.card β : ℝ)^2) :
    ∃ S : Finset ι, (∀ i ∈ S, ∀ j ∈ S, i ≠ j → H*r < dist (q i) (q j)) ∧
      1 ≤ ∑ i ∈ S, min (L*t i) (Q*(t i)^2) := by
  classical
  let σz : ζ → ι → ℝ := fun z i => if dist (q i) (aux z) ≤ 3*s then
    ∑ j ∈ univ.filter (fun j => dist (ξ j) (h z (q i)) ≤ (C+2)*(r/s)), ω j else 0
  let σ : ι → ℝ := fun i => ∑ z, σz z i
  have hσz : ∀ z i, 0 ≤ σz z i := by
    intro z i
    dsimp only [σz]
    split_ifs
    · exact sum_nonneg (fun j _ => hω.1 j)
    · exact le_rfl
  have hσ : ∀ i, 0 ≤ σ i := fun i => sum_nonneg (fun z _ => hσz z i)
  have he := sparse_transfer_energy σz (fun z i => dist (q i) (aux z) ≤ 3*s) Naux (E*M)
    (by intro z i hn; simp only [σz,ite_eq_right hn]) hmultaux
    (fun z => (hlocal z).trans (mul_le_mul_of_nonneg_left hωmass hE))
  apply separated_capacity_of_detection q lo hi t σ
    (fun j i => |dist (q i) x-bin j| ≤ 5*s) a b BinMul L Q
    ((Naux:ℝ)*Fintype.card ζ*(E*M)) H r N hab hlohi ht hlen hσ hL hQ hBinMul
    (by positivity) hHr hN hdegree hmult he hlarge hsmall
  intro j v hv hunmarked
  have hbnon : 0 ≤ bin j := by nlinarith only [(hbin j).1,hA,hR]
  obtain ⟨w,hwK,hwb⟩ := level_radial_hit x (K v) (hK v hv) hbnon (by
    obtain ⟨y,hy,hyA⟩ := hinner v hv
    exact ⟨y,hy,by nlinarith only [(hbin j).1,hyA,hR]⟩) (by nlinarith only [hdiam v hv,(hbin j).2,hR])
  have hwann : (A-1)*R ≤ dist w x ∧ dist w x ≤ (B₀+1)*R := by
    rw [hwb]
    constructor <;> nlinarith only [(hbin j).1,(hbin j).2,hR]
  obtain ⟨z,hwz⟩ := haux w hwann.1 hwann.2
  apply unmarked_level_detection q ξ (h z) (h z).continuous x w (aux z)
    {v | (A-1)*R ≤ dist v x ∧ dist v x ≤ (B₀+1)*R} Ω (K v) f lo hi σ
    (fun i => L ≤ Q*t i) r s C d₀ (bin j) v hr hs hrs hC (hK v hv) hwK
    (by nlinarith only [hdiam v hv,hAB,hA,hR,hsR]) hwb hwz
    (by
      intro y hy
      have hyw := dist_triangle y w x
      have hwy := dist_triangle w y x
      rw [dist_comm w y] at hwy
      constructor <;> nlinarith only [(hbin j).1,(hbin j).2,hy,hyw,hwy,hwb,hsR,hR])
    (by
      intro y hy
      apply hΩ
      have hh := dist_triangle y w x
      nlinarith only [(hbin j).2,hh,hy,hwb,hsR,hrs,hR])
    (hlevel v hv) (by intro y hy; exact hnet y hy.1 hy.2)
    (hupper z)
    (by
      intro y y' hy hy' hyy'
      have hh := hlower z y y' (by linarith only [hy,hs]) (by linarith only [hy',hs])
      have hsne : s ≠ 0 := ne_of_gt hs
      have hcs : c/s*s = c := by field_simp
      have hm := mul_le_mul_of_nonneg_left hyy' (div_nonneg hc.le hs.le)
      rw [hcs] at hm
      exact hdc.trans (hm.trans hh))
    hrange ω hω hσ (by
      intro i hi
      have hh : σz z i ≤ σ i := single_le_sum (fun z _ => hσz z i) (mem_univ z)
      simpa only [σz,ite_eq_left hi] using hh)
    hunmarked

end CannonCapacity

end

section

namespace CannonCapacity

open Set Metric Finset CannonAnnular CannonPacking

attribute [local instance] Classical.propDecidable

end CannonCapacity

end

end
end
end
end
end
end
end
end
end
end
end
end
end

end OAI