import Mathlib
import OAI.Geometry.Cannon.Energy.PathCosts
import OAI.Geometry.Cannon.ScalarLimits.LocalOscillation

namespace OAI

section
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noncomputable section
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open Set Topology
open Set Metric Topology
open Set Topology Filter Metric unitInterval
open scoped InnerProductSpace
open Set unitInterval

namespace CannonApprox

open CannonGrid CannonRegularization CannonLevel CannonCells CannonBands

variable {ι : Type*} [Fintype ι]

local instance : PathConnectedSpace (Strip height (1 / 3) (2 / 3)) :=
  strip_pathConnected _ _ (by norm_num) (by norm_num) (by norm_num)

def rawFunction (C : ι → Set CannonGrid.Square) (ρ : ι → ℝ) : CannonGrid.Square → ℝ :=
  rawExtended height (1/3) (2/3) C ρ

def Admissible (C : ι → Set CannonGrid.Square) (ρ : ι → ℝ) : Prop :=
  ∀ u : Strip height (1/3) (2/3), height u.val = 1/3 →
    ∀ v : Strip height (1/3) (2/3), height v.val = 2/3 →
      ∀ p : Path u v, 1 ≤ cost (fun i => Subtype.val ⁻¹' C i) ρ (range p)

theorem rawFunction_properties (C : ι → Set CannonGrid.Square) (ρ : ι → ℝ)
    (hρ : ∀ i, 0 ≤ ρ i) (hcross : Admissible C ρ) :
    (∀ z, rawFunction C ρ z ∈ Icc 0 1) ∧
    (∀ z ∈ topBand, rawFunction C ρ z = 0) ∧
    (∀ z ∈ bottomBand, rawFunction C ρ z = 1) ∧
    (∀ u v, ∀ p : Path u v, |rawFunction C ρ u-rawFunction C ρ v| ≤ cost C ρ (range p)) := by
  exact rawExtended_properties height (by unfold height; fun_prop) (1/3) (2/3) (by norm_num) C ρ hρ
    (strip_top_nonempty _ _ (by norm_num) (by norm_num) (by norm_num)) hcross

def initial (C : ι → Set CannonGrid.Square) (ρ : ι → ℝ) (n : ℕ) (hn : 0 < n) :
    CannonGrid.Square → ℝ := interpolate (3*n) (by omega) (rawFunction C ρ)

def scalar (C : ι → Set CannonGrid.Square) (ρ : ι → ℝ) (n : ℕ) (hn : 0 < n) :
    CannonGrid.Square → ℝ := drain (flood (initial C ρ n hn) topBand) bottomBand

lemma initial_properties (C : ι → Set CannonGrid.Square) (ρ : ι → ℝ)
    (hρ : ∀ i, 0 ≤ ρ i) (hcross : Admissible C ρ) (n : ℕ) (hn : 0 < n) :
    Continuous (initial C ρ n hn) ∧ (∀ z, initial C ρ n hn z ∈ Icc 0 1) ∧
    (∀ z ∈ topBand, initial C ρ n hn z = 0) ∧
    (∀ z ∈ bottomBand, initial C ρ n hn z = 1) := by
  obtain ⟨hrange,htop,hbottom,_⟩ := rawFunction_properties C ρ hρ hcross
  have hbands := interpolate_preserves_bands n hn (rawFunction C ρ) htop hbottom
  exact ⟨continuous_interpolate _ _ _, interpolate_bounds _ _ _ hrange,hbands⟩

theorem scalar_properties (C : ι → Set CannonGrid.Square) (ρ : ι → ℝ)
    (hρ : ∀ i, 0 ≤ ρ i) (hcross : Admissible C ρ) (n : ℕ) (hn : 0 < n) :
    Continuous (scalar C ρ n hn) ∧ (∀ z, scalar C ρ n hn z ∈ Icc 0 1) ∧
    (∀ z ∈ topBand, scalar C ρ n hn z = 0) ∧
    (∀ z ∈ bottomBand, scalar C ρ n hn z = 1) := by
  obtain ⟨hf,hfI,hT,hB⟩ := initial_properties C ρ hρ hcross n hn
  have hF := continuous_flood topBand_pathConnected.nonempty hf (fun z => (hfI z).2)
  have hFk : ∀ z, 0 ≤ flood (initial C ρ n hn) topBand z := fun z =>
    (hfI z).1.trans (le_flood topBand_pathConnected.nonempty (fun w => (hfI w).2) z)
  exact ⟨continuous_drain bottomBand_pathConnected.nonempty hF hFk,
    regularized_values topBand_pathConnected.nonempty bottomBand_pathConnected.nonempty
      (fun z => (hfI z).1) (fun z => (hfI z).2) hT hB⟩

theorem scalar_level (C : ι → Set CannonGrid.Square) (ρ : ι → ℝ)
    (hρ : ∀ i, 0 ≤ ρ i) (hcross : Admissible C ρ) (n : ℕ) (hn : 0 < n)
    {t : ℝ} (ht : t ∈ Ioo (0:ℝ) 1) :
    ∃ A : Set CannonGrid.Square, IsCompact A ∧ IsConnected A ∧ A ⊆ centralRectangle ∧
      (∀ z ∈ A, scalar C ρ n hn z = t) ∧ (∃ z ∈ A, z.1 = 0) ∧ (∃ z ∈ A, z.1 = 1) ∧
      (∀ {x y : CannonGrid.Square} (p : Path x y), scalar C ρ n hn x < t →
        t < scalar C ρ n hn y → ∃ s, p s ∈ A) := by
  obtain ⟨hf,hfI,hT,hB⟩ := initial_properties C ρ hρ hcross n hn
  exact rectangle_separating_level hf (fun z => (hfI z).1) (fun z => (hfI z).2) hT hB ht.1 ht.2

theorem exists_mesh {X : Type*} [MetricSpace X] (e : CannonGrid.Square → X)
    (he : Continuous e) {r : ℝ} (hr : 0 < r) :
    ∃ n : ℕ, ∃ hn : 0 < n, ∀ z : CannonGrid.Square, ∀ i : Index (3*n),
      weight (3*n) z i ≠ 0 → dist (e z) (e (vertex (3*n) (by omega) i)) < r :=
  exists_metric_mesh e he hr

theorem scalar_local_oscillation {X : Type*} [MetricSpace X]
    (e : CannonGrid.Square → X) (he : Topology.IsEmbedding e)
    (L J r : ℝ) (hL : 1 ≤ L) (hJ : 0 < J) (hr : 0 < r)
    (hpaths : ∀ (o u v : X) (s : ℝ), 0 < s → dist o u ≤ s → dist o v ≤ s →
      JoinedIn {z | dist o z < L*s} u v)
    (x : X) (hchart : Metric.ball x (L*(L*J+1)*r) ⊆ range e)
    (q : ι → X) (ρ : ι → ℝ) (hρ : ∀ i, 0 ≤ ρ i)
    (hcross : Admissible (fun i => e ⁻¹' Metric.ball (q i) (2*r)) ρ)
    (n : ℕ) (hn : 0 < n)
    (hmesh : ∀ z : CannonGrid.Square, ∀ i : Index (3*n), weight (3*n) z i ≠ 0 →
      dist (e z) (e (vertex (3*n) (by omega) i)) < r)
    (a b : CannonGrid.Square) (ha : dist x (e a) ≤ J*r) (hb : dist x (e b) ≤ J*r) :
    let C := fun i => e ⁻¹' Metric.ball (q i) (2*r)
    |scalar C ρ n hn a-scalar C ρ n hn b| ≤
      ∑ i ∈ Finset.univ.filter (fun i => dist (q i) x ≤ (L*(L*J+1)+3)*r), ρ i := by
  let C := fun i => e ⁻¹' Metric.ball (q i) (2*r)
  obtain ⟨hrawI,_,_,hraw⟩ := rawFunction_properties C ρ hρ hcross
  exact CannonScalarLocal.local_oscillation e he L J r hL hJ hr hpaths x hchart q ρ hρ
    (rawFunction C ρ) hrawI hraw (3*n) (by omega) hmesh topBand bottomBand
    topBand_pathConnected.nonempty bottomBand_pathConnected.nonempty a b ha hb

end CannonApprox

namespace CannonRectangle

open CannonGrid CannonCells CannonBands

abbrev Middle := Strip height (1/3) (2/3)

def normalize (z : Middle) : CannonGrid.Square :=
  (z.val.1, ⟨3*(z.val.2:ℝ)-1, by
    have hz := z.property
    change (z.val.2:ℝ) ∈ Icc (1/3:ℝ) (2/3) at hz
    constructor <;> linarith [hz.1,hz.2]⟩)

lemma continuous_normalize : Continuous normalize := by
  unfold normalize
  fun_prop

lemma normalize_injective : Function.Injective normalize := by
  intro x y h
  apply Subtype.ext
  apply Prod.ext
  · simpa only [normalize] using congrArg Prod.fst h
  · apply Subtype.ext
    have hh := congrArg (fun z : CannonGrid.Square => (z.2:ℝ)) h
    change 3*(x.val.2:ℝ)-1 = 3*(y.val.2:ℝ)-1 at hh
    linarith

def leftRight : Set (Set CannonGrid.Square) :=
  {A | ∃ u v : Middle, u.val.1 = 0 ∧ v.val.1 = 1 ∧
    ∃ p : Path u v, A = Subtype.val '' range p}

def topBottom : Set (Set CannonGrid.Square) :=
  {A | ∃ u v : Middle, height u.val = 1/3 ∧ height v.val = 2/3 ∧
    ∃ p : Path u v, A = Subtype.val '' range p}

lemma leftRight_nonempty : leftRight.Nonempty := by
  let u : Middle := ⟨(0,⟨1/2,by norm_num⟩),by norm_num [height]⟩
  let v : Middle := ⟨(1,⟨1/2,by norm_num⟩),by norm_num [height]⟩
  let : PathConnectedSpace Middle := strip_pathConnected _ _ (by norm_num) (by norm_num) (by norm_num)
  exact ⟨Subtype.val '' range (PathConnectedSpace.somePath u v),
    u,v,rfl,rfl,PathConnectedSpace.somePath u v,rfl⟩

lemma leftRight_members_nonempty (A : Set CannonGrid.Square) (hA : A ∈ leftRight) :
    A.Nonempty := by
  obtain ⟨u,v,_,_,p,rfl⟩ := hA
  exact ⟨(p 0).val,⟨p 0,⟨0,rfl⟩,rfl⟩⟩

lemma crossings_intersect (A : Set CannonGrid.Square) (hA : A ∈ leftRight)
    (B : Set CannonGrid.Square) (hB : B ∈ topBottom) : (A ∩ B).Nonempty := by
  obtain ⟨u,v,hu,hv,p,rfl⟩ := hA
  obtain ⟨w,z,hw,hz,q,rfl⟩ := hB
  have hp0 : (normalize (p 0)).1 = 0 := by simpa [normalize] using hu
  have hp1 : (normalize (p 1)).1 = 1 := by simpa [normalize] using hv
  have hq0 : (normalize (q.symm 0)).2 = 1 := by
    apply Subtype.ext
    change 3*((q.symm 0).val.2:ℝ)-1 = 1
    simpa only [Path.source] using show 3*(z.val.2:ℝ)-1 = 1 by
      change (z.val.2:ℝ) = 2/3 at hz
      linarith
  have hq1 : (normalize (q.symm 1)).2 = 0 := by
    apply Subtype.ext
    change 3*((q.symm 1).val.2:ℝ)-1 = 0
    simpa only [Path.target] using show 3*(w.val.2:ℝ)-1 = 0 by
      change (w.val.2:ℝ) = 1/3 at hw
      linarith
  obtain ⟨s,t,hst⟩ := CannonPlanar.paths_cross
    (continuous_normalize.comp p.continuous) (continuous_normalize.comp q.symm.continuous)
    hp0 hp1 hq0 hq1
  have he := normalize_injective hst
  refine ⟨(p s).val,⟨p s,⟨s,rfl⟩,rfl⟩,?_⟩
  refine ⟨q.symm t,?_,congrArg Subtype.val he.symm⟩
  exact (q.symm_range ▸ mem_range_self t)

theorem exists_dual_weights {ι : Type*} [Fintype ι]
    (C : ι → Set CannonGrid.Square) (hcover : ∀ x, ∃ i, x ∈ C i) :
    0 < modulus C leftRight ∧ ∃ ρ : ι → ℝ, (∀ i, 0 ≤ ρ i) ∧
      CannonApprox.Admissible C ρ ∧ (∑ i, (ρ i)^2) = (modulus C leftRight)⁻¹ := by
  obtain ⟨hpos,_⟩ := modulus_minimum C hcover leftRight_nonempty leftRight_members_nonempty
  obtain ⟨ρ,hρ,he⟩ := transverse_weights C hcover leftRight_nonempty
    leftRight_members_nonempty crossings_intersect
  obtain ⟨hn,hadm⟩ := (admissible_iff C topBottom ρ).mp hρ
  refine ⟨hpos,ρ,hn,?_,?_⟩
  · intro u hu v hv p
    rw [cost_preimage_image]
    exact hadm _ ⟨u,v,hu,hv,p,rfl⟩
  · simpa only [EuclideanSpace.real_norm_sq_eq] using he

end CannonRectangle

end
end
end
end
end
end
end
end
end

end OAI