import Mathlib
import OAI.Geometry.Cannon.Energy.RawFunction
import OAI.Geometry.Cannon.Energy.StripCrossing
import OAI.Geometry.Cannon.Modulus.ModulusEnergy

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 CannonCharts

open CannonCells

theorem rectangle_modulus_localization {X : Type*} [MetricSpace X] [CompactSpace X]
    (φ : X ≃ₜ 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) :
    ∃ e : Plane → X, IsOpenEmbedding e ∧ diam (e '' box 8) < d ∧
      ∃ c : ℝ, 0 < c ∧ ∃ N : ℕ → ℕ, StrictMono N ∧ ∀ n,
        c * modulus (C (N n)) (macroFamily (X := X) d) ≤
          modulus (C (N n)) (stripFamily e) := by
  classical
  obtain ⟨J,e,hJ,he,hdiam,hsub⟩ := finite_strip_localization φ d hd
  let : Nonempty J := hJ.coe_sort
  let H := fun j : J × Fin 4 => stripFamily (e j)
  let K : ℝ := (Fintype.card (J × Fin 4) : ℝ)^2
  have hK : 0 < K := pow_pos (by exact_mod_cast Fintype.card_pos) _
  have hlocal (n : ℕ) : ∃ j : J × Fin 4,
      modulus (C n) (macroFamily (X := X) d) ≤ K*modulus (C n) (H j) := by
    exact exists_modulus_localization (C n) (hC n) (macroFamily d) H
      (fun j => stripFamily_nonempty (e j))
      (fun _ _ h => stripFamily_member_nonempty h) hsub
  obtain ⟨j,N,hN,hNj⟩ := fixed_localization_subsequence
    (fun n => modulus (C n) (macroFamily (X := X) d))
    (fun j n => modulus (C n) (H j)) K hlocal
  refine ⟨e j,he j,hdiam j,K⁻¹,inv_pos.mpr hK,N,hN,?_⟩
  intro n
  calc
    K⁻¹ * modulus (C (N n)) (macroFamily d) ≤
        K⁻¹ * (K * modulus (C (N n)) (H j)) :=
      mul_le_mul_of_nonneg_left (hNj n) (inv_nonneg.mpr hK.le)
    _ = modulus (C (N n)) (stripFamily (e j)) := by rw [← mul_assoc,inv_mul_cancel₀ hK.ne',one_mul]

end CannonCharts

namespace CannonCharts

def squareMap (z : CannonGrid.Square) : Plane := (1+(z.1:ℝ),12*(z.2:ℝ)-6)

lemma continuous_squareMap : Continuous squareMap := by
  unfold squareMap
  fun_prop

lemma squareMap_injective : Function.Injective squareMap := by
  intro z w h
  have hx := congrArg Prod.fst h
  have hy := congrArg Prod.snd h
  apply Prod.ext
  · apply Subtype.ext
    dsimp [squareMap] at hx
    linarith
  · apply Subtype.ext
    dsimp [squareMap] at hy
    linarith

lemma squareMap_embedding : IsEmbedding squareMap :=
  (continuous_squareMap.isClosedEmbedding squareMap_injective).isEmbedding

lemma squareMap_in_box (z : CannonGrid.Square) :
    squareMap z ∈ Metric.closedBall 0 8 := by
  simp only [Metric.mem_closedBall,dist_zero_right,Prod.norm_def,Real.norm_eq_abs,
    max_le_iff,abs_le,squareMap]
  constructor <;> constructor <;> nlinarith [z.1.property.1,z.1.property.2,z.2.property.1,z.2.property.2]

theorem lift_strip_crossing (q : I → Plane) (hq : StripCrossing q) :
    ∃ u v : CannonRectangle.Middle, u.val.1 = 0 ∧ v.val.1 = 1 ∧
      ∃ p : Path u v, ∀ t, squareMap (p t).val = q t := by
  let m : I → CannonRectangle.Middle := fun t =>
    ⟨(⟨(q t).1-1,by constructor <;> linarith [(hq.2.2.2 t).1.1,(hq.2.2.2 t).1.2]⟩,
      ⟨((q t).2+6)/12,by
        have hy := abs_le.mp (hq.2.2.2 t).2
        constructor <;> linarith [hy.1,hy.2]⟩),by
      have hy := abs_le.mp (hq.2.2.2 t).2
      change ((q t).2+6)/12 ∈ Icc (1/3:ℝ) (2/3)
      constructor <;> linarith [hy.1,hy.2]⟩
  have hqc : Continuous q := hq.1
  have hm : Continuous m := by dsimp [m]; fun_prop
  let p : Path (m 0) (m 1) := ⟨⟨m,hm⟩,rfl,rfl⟩
  refine ⟨m 0,m 1,?_,?_,p,?_⟩
  · apply Subtype.ext
    change (q 0).1-1 = 0
    rw [hq.2.1]; norm_num
  · apply Subtype.ext
    change (q 1).1-1 = 1
    rw [hq.2.2.1]; norm_num
  · intro t
    apply Prod.ext <;> dsimp [squareMap,p,m] <;> ring

theorem stripFamily_eq_image {X : Type*} (e : Plane → X) :
    stripFamily e = {A | ∃ B ∈ CannonRectangle.leftRight, A = (e ∘ squareMap) '' B} := by
  ext A
  constructor
  · rintro ⟨q,hq,rfl⟩
    obtain ⟨u,v,hu,hv,p,hp⟩ := lift_strip_crossing q hq
    refine ⟨Subtype.val '' range p,⟨u,v,hu,hv,p,rfl⟩,?_⟩
    ext x
    constructor
    · rintro ⟨t,rfl⟩
      exact ⟨(p t).val,⟨p t,⟨t,rfl⟩,rfl⟩,by dsimp; rw [hp]⟩
    · rintro ⟨z,⟨m,⟨t,rfl⟩,rfl⟩,rfl⟩
      exact ⟨t,by dsimp; rw [hp]⟩
  · rintro ⟨B,⟨u,v,hu,hv,p,rfl⟩,rfl⟩
    let q : I → Plane := fun t => squareMap (p t).val
    have hq : Continuous q := continuous_squareMap.comp (continuous_subtype_val.comp p.continuous)
    refine ⟨q,⟨hq,?_,?_,?_⟩,?_⟩
    · change 1+((p 0).val.1:ℝ) = 1
      rw [p.source,hu]
      norm_num
    · change 1+((p 1).val.1:ℝ) = 2
      rw [p.target,hv]
      norm_num
    · intro t
      have hy := (p t).property
      change ((p t).val.2:ℝ) ∈ Icc (1/3:ℝ) (2/3) at hy
      change 1+((p t).val.1:ℝ) ∈ Icc 1 2 ∧ |12*((p t).val.2:ℝ)-6| ≤ 2
      constructor
      · constructor <;> nlinarith [(p t).val.1.property.1,(p t).val.1.property.2]
      · rw [abs_le]
        constructor <;> linarith [hy.1,hy.2]
    · ext x
      constructor
      · rintro ⟨_,⟨_,⟨t,rfl⟩,rfl⟩,rfl⟩
        exact ⟨t,rfl⟩
      · rintro ⟨t,rfl⟩
        exact ⟨(p t).val,⟨p t,⟨t,rfl⟩,rfl⟩,rfl⟩

end CannonCharts

end
end
end
end
end
end
end
end
end

end OAI