import Mathlib
import OAI.Geometry.Cannon.Energy.Square

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 CannonChart

variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]

def liftPath (e : X → Y) (he : IsEmbedding e) {a b : X}
    (p : Path (e a) (e b)) (hp : ∀ t, p t ∈ range e) : Path a b where
  toFun := fun t => he.toHomeomorph.symm ⟨p t,hp t⟩
  continuous_toFun := he.toHomeomorph.symm.continuous.comp (p.continuous.subtype_mk hp)
  source' := by
    apply he.toHomeomorph.injective
    simp
  target' := by
    apply he.toHomeomorph.injective
    simp

lemma liftPath_apply (e : X → Y) (he : IsEmbedding e) {a b : X}
    (p : Path (e a) (e b)) (hp : ∀ t, p t ∈ range e) (t : unitInterval) :
    e (liftPath e he p hp t) = p t := by
  have hh := he.toHomeomorph.apply_symm_apply ⟨p t,hp t⟩
  exact congrArg Subtype.val hh

lemma lift_joinedIn (e : X → Y) (he : IsEmbedding e) {a b : X} {S : Set Y}
    (hS : S ⊆ range e) (hp : JoinedIn S (e a) (e b)) :
    JoinedIn (e ⁻¹' S) a b := by
  obtain ⟨p,hp⟩ := hp
  let hpr : ∀ t, p t ∈ range e := fun t => hS (hp t)
  refine ⟨liftPath e he p hpr,fun t => ?_⟩
  change e (liftPath e he p hpr t) ∈ S
  rw [liftPath_apply e he p hpr]
  exact hp t

end CannonChart

namespace CannonLevel

open CannonPlanar CannonRegularization

instance : PathConnectedSpace Square :=
  pathConnectedSpace_iff_connectedSpace.mpr inferInstance

theorem regularized_values {f : Square → ℝ} {A B : Set Square}
    (hA : A.Nonempty) (hB : B.Nonempty)
    (hf0 : ∀ z, 0 ≤ f z) (hf1 : ∀ z, f z ≤ 1)
    (hAf : ∀ z ∈ A, f z = 0) (hBf : ∀ z ∈ B, f z = 1) :
    (∀ z, drain (flood f A) B z ∈ Icc (0 : ℝ) 1) ∧
    (∀ z ∈ A, drain (flood f A) B z = 0) ∧
    (∀ z ∈ B, drain (flood f A) B z = 1) := by
  have hF0 : ∀ z, 0 ≤ flood f A z := fun z => (hf0 z).trans (le_flood hA hf1 z)
  have hF1 := flood_le_upper hA hf1
  have hH0 := drain_lower hB hF0
  have hHF := drain_le hB hF0
  refine ⟨fun z => ⟨hH0 z, (hHF z).trans (hF1 z)⟩, ?_, ?_⟩
  · intro z hz
    apply le_antisymm _ (hH0 z)
    simpa [flood_eq_on_band hA hf1 hz, hAf z hz] using hHF z
  · intro z hz
    rw [drain_eq_on_band hB hF0 hz]
    apply le_antisymm (hF1 z)
    simpa [hBf z hz] using le_flood hA hf1 z

theorem separating_level {f : Square → ℝ} {A B : Set Square}
    (hA : IsPathConnected A) (hB : IsPathConnected B)
    (hAedge : ∀ s, (s,0) ∈ A) (hBedge : ∀ s, (s,1) ∈ B)
    (hf : Continuous f) (hf0 : ∀ z, 0 ≤ f z) (hf1 : ∀ z, f z ≤ 1)
    (hAf : ∀ z ∈ A, f z = 0) (hBf : ∀ z ∈ B, f z = 1)
    {t : ℝ} (ht0 : 0 < t) (ht1 : t < 1) :
    let H := drain (flood f A) B
    ∃ C : Set Square, IsCompact C ∧ IsConnected C ∧
      (∀ z ∈ C, H z = t) ∧ (∃ z ∈ C, z.1 = 0) ∧ (∃ z ∈ C, z.1 = 1) ∧
      (∀ {x y : Square} (p : Path x y), H x < t → t < H y →
        ∃ s, p s ∈ C) := by
  let F := flood f A
  let H := drain F B
  have hF0 : ∀ z, 0 ≤ F z := fun z => (hf0 z).trans (le_flood hA.nonempty hf1 z)
  have hFc : Continuous F := continuous_flood hA.nonempty hf hf1
  have hHc : Continuous H := continuous_drain hB.nonempty hFc hF0
  obtain ⟨hHr,hHA,hHB⟩ := regularized_values hA.nonempty hB.nonempty hf0 hf1 hAf hBf
  obtain ⟨C,hCc,hCn,hCH,⟨a,ha,ha0⟩,⟨b,hb,hb1⟩⟩ := level_continuum hHc
    (c := t) (fun s => by rw [show H (s,0) = 0 from hHA _ (hAedge s)]; exact ht0.le)
    (fun s => by rw [show H (s,1) = 1 from hHB _ (hBedge s)]; exact ht1.le)
  refine ⟨C,hCc,hCn,hCH,⟨a,ha,ha0⟩,⟨b,hb,hb1⟩,?_⟩
  intro x y p hx hy
  obtain ⟨u,hu,lo,hlo⟩ := regularized_low_side hA.nonempty hB.nonempty hf hf0 hf1 hAf ht0 hx
  obtain ⟨v,hv,hi,hhi⟩ := drain_strict_side hB.nonempty hF0 hy
  obtain ⟨qa,hqa⟩ := hA.joinedIn u hu (0,0) (hAedge 0)
  obtain ⟨qb,hqb⟩ := hB.joinedIn v hv (0,1) (hBedge 0)
  let low : Path x (0,0) := lo.trans qa
  let high : Path y (0,1) := hi.trans qb
  have hlow : ∀ s, H (low s) < t := by
    intro s
    have hs : low s ∈ range lo ∪ range qa := by rw [← Path.trans_range]; exact mem_range_self s
    rcases hs with ⟨r,hr⟩ | ⟨r,hr⟩
    · rw [← hr]; exact hlo r
    · rw [← hr, show H (qa r) = 0 from hHA _ (hqa r)]; exact ht0
  have hhigh : ∀ s, t < H (high s) := by
    intro s
    have hs : high s ∈ range hi ∪ range qb := by rw [← Path.trans_range]; exact mem_range_self s
    rcases hs with ⟨r,hr⟩ | ⟨r,hr⟩
    · rw [← hr]; exact hhi r
    · rw [← hr, show H (qb r) = 1 from hHB _ (hqb r)]; exact ht1
  let q := high.symm.trans (p.symm.trans low)
  obtain ⟨s,hs⟩ := continuum_crosses_path hCn.isPreconnected ha hb ha0 hb1 q.continuous
    (by simp [q]) (by simp [q])
  have hqs : q s ∈ range high.symm ∪ (range p.symm ∪ range low) := by
    rw [← Path.trans_range, ← Path.trans_range]
    exact mem_range_self s
  rcases hqs with ⟨r,hr⟩ | ⟨r,hr⟩ | ⟨r,hr⟩
  · have ht := hhigh (unitInterval.symm r)
    have he := hCH _ hs
    change H (high.symm r) > t at ht
    rw [hr,he] at ht
    exact False.elim (lt_irrefl _ ht)
  · exact ⟨unitInterval.symm r, hr.symm ▸ hs⟩
  · have ht := hlow r
    rw [hr,hCH _ hs] at ht
    exact False.elim (lt_irrefl _ ht)

lemma slab_pathConnected {a b : ℝ} (ha : 0 ≤ a) (hab : a ≤ b) (hb : b ≤ 1) :
    IsPathConnected {z : Square | a ≤ (z.2 : ℝ) ∧ (z.2 : ℝ) ≤ b} := by
  let e : Square → ℝ × ℝ := Prod.map Subtype.val Subtype.val
  have he : Topology.IsInducing e := Topology.IsInducing.subtypeVal.prodMap
    Topology.IsInducing.subtypeVal
  rw [he.isPathConnected_iff]
  have himage : e '' {z : Square | a ≤ (z.2 : ℝ) ∧ (z.2 : ℝ) ≤ b} =
      Icc ((0 : ℝ),a) (1,b) := by
    ext z
    constructor
    · rintro ⟨w,hw,rfl⟩
      exact ⟨⟨w.1.property.1,hw.1⟩,⟨w.1.property.2,hw.2⟩⟩
    · intro hz
      refine ⟨(⟨z.1,hz.1.1,hz.2.1⟩,⟨z.2,ha.trans hz.1.2,hz.2.2.trans hb⟩), ?_, rfl⟩
      exact ⟨hz.1.2,hz.2.2⟩
  rw [himage]
  exact (convex_Icc _ _).isPathConnected ⟨(0,a),⟨le_rfl,⟨zero_le_one,hab⟩⟩⟩

def topBand : Set Square := {z | (z.2 : ℝ) ≤ 1 / 3}

def bottomBand : Set Square := {z | 2 / 3 ≤ (z.2 : ℝ)}

def centralRectangle : Set Square := {z | (z.2 : ℝ) ∈ Icc (1 / 3) (2 / 3)}

lemma topBand_pathConnected : IsPathConnected topBand := by
  convert slab_pathConnected (a := 0) (b := 1 / 3) (by norm_num) (by norm_num) (by norm_num) using 1
  ext z
  exact ⟨fun hz => ⟨z.2.property.1,hz⟩,fun hz => hz.2⟩

lemma bottomBand_pathConnected : IsPathConnected bottomBand := by
  convert slab_pathConnected (a := 2 / 3) (b := 1) (by norm_num) (by norm_num) (by norm_num) using 1
  ext z
  exact ⟨fun hz => ⟨hz,z.2.property.2⟩,fun hz => hz.1⟩

theorem rectangle_separating_level {f : Square → ℝ}
    (hf : Continuous f) (hf0 : ∀ z, 0 ≤ f z) (hf1 : ∀ z, f z ≤ 1)
    (hAf : ∀ z ∈ topBand, f z = 0) (hBf : ∀ z ∈ bottomBand, f z = 1)
    {t : ℝ} (ht0 : 0 < t) (ht1 : t < 1) :
    let H := drain (flood f topBand) bottomBand
    ∃ C : Set Square, IsCompact C ∧ IsConnected C ∧ C ⊆ centralRectangle ∧
      (∀ z ∈ C, H z = t) ∧ (∃ z ∈ C, z.1 = 0) ∧ (∃ z ∈ C, z.1 = 1) ∧
      (∀ {x y : Square} (p : Path x y), H x < t → t < H y → ∃ s, p s ∈ C) := by
  obtain ⟨C,hCc,hCn,hCH,hCl,hCr,hCs⟩ := separating_level topBand_pathConnected bottomBand_pathConnected
    (by intro s; change (0 : ℝ) ≤ 1 / 3; norm_num)
    (by intro s; change (2 : ℝ) / 3 ≤ 1; norm_num)
    hf hf0 hf1 hAf hBf ht0 ht1
  refine ⟨C,hCc,hCn,?_,hCH,hCl,hCr,hCs⟩
  obtain ⟨_,hHA,hHB⟩ := regularized_values topBand_pathConnected.nonempty
    bottomBand_pathConnected.nonempty hf0 hf1 hAf hBf
  intro z hz
  constructor
  · by_contra h
    have he := hHA z (show z ∈ topBand from (lt_of_not_ge h).le)
    rw [hCH z hz] at he
    exact (ne_of_gt ht0) he
  · by_contra h
    have he := hHB z (show z ∈ bottomBand from (lt_of_not_ge h).le)
    rw [hCH z hz] at he
    exact (ne_of_lt ht1) he

end CannonLevel

end
end
end
end
end
end
end
end
end

end OAI