import Mathlib
import OAI.Geometry.Cannon.PlaneTopology.RegionEdges
import OAI.Geometry.Cannon.PlaneTopology.VertexLines
import OAI.Geometry.Cannon.PlaneTopology.OtherComponentDeleteArc

namespace OAI

section
section
section
noncomputable section
open Set unitInterval Topology

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noncomputable section
noncomputable section
open Set Topology unitInterval

noncomputable section
open Set Metric Topology unitInterval

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open Set Metric Topology

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noncomputable section
open Set unitInterval

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noncomputable section
open Set Metric Topology

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open Set Metric Topology

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open Set Metric Topology

noncomputable section
open Set Topology

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open Set Topology

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open Set Topology

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open Set Topology unitInterval

noncomputable section
namespace CannonPlanarEuler
open Set Topology unitInterval CannonEmbeddedArc CannonEmbeddedArc.Arc CannonFaceIncidence

variable {ι : Type*} [DecidableEq ι]
def support (F : ι → Arc ℂ) (S : Finset ι) : Set ℂ := ⋃ i∈S,range (F i)
def vertices (F : ι → Arc ℂ) (S : Finset ι) : Finset ℂ := by
  classical
  exact S.biUnion fun i => {F i 0,F i 1}
def faces (F : ι → Arc ℂ) (S : Finset ι) : ℕ :=
  Nat.card (ConnectedComponents ((support F S)ᶜ : Set ℂ))
lemma support_insert (F : ι → Arc ℂ) (S : Finset ι) (i : ι) :
    support F (insert i S)=support F S∪range (F i) := by
  simp only [support,Finset.mem_insert,iUnion_iUnion_eq_or_left,union_comm]
lemma vertices_insert (F : ι → Arc ℂ) (S : Finset ι) (i : ι) :
    vertices F (insert i S)=insert (F i 0) (insert (F i 1) (vertices F S)) := by
  classical
  ext x
  simp only [vertices,Finset.biUnion_insert,Finset.mem_union,Finset.mem_insert,Finset.mem_singleton]
  tauto
omit [DecidableEq ι] in
lemma support_closed (F : ι → Arc ℂ) (S : Finset ι) : IsClosed (support F S) :=
  isClosed_biUnion_finset fun i _ => (F i).compact_range.isClosed
omit [DecidableEq ι] in
lemma mem_vertices (F : ι → Arc ℂ) (S : Finset ι) (x : ℂ) :
    x∈vertices F S ↔ ∃ i∈S,F i 0=x ∨ F i 1=x := by
  classical
  simp [vertices,eq_comm]
omit [DecidableEq ι] in
lemma vertices_subset (F : ι → Arc ℂ) (S : Finset ι) : (vertices F S:Set ℂ)⊆support F S := by
  intro x hx
  obtain ⟨i,hi,hx⟩ := (mem_vertices F S x).mp hx
  exact mem_iUnion.mpr ⟨i,mem_iUnion.mpr ⟨hi,hx.elim (fun h => ⟨0,h⟩) (fun h => ⟨1,h⟩)⟩⟩
omit [DecidableEq ι] in
lemma support_subtype (F : ι → Arc ℂ) (S : Finset ι) :
    support F S=⋃i:S,range (F i.val) := by ext x; simp [support]
omit [DecidableEq ι] in
lemma support_tame (F : ι → Arc ℂ) (S : Finset ι) (Q : Finset (ℂ × ℂ))
    (hQ : ∀ i,range (F i)⊆⋃q∈Q,segment ℝ q.1 q.2) :
    ∃ P : Finset (ℂ × ℂ),support F S=⋃p∈P,segment ℝ p.1 p.2 := by
  rw [support_subtype]
  exact finite_graph_exact_segments (fun i:S => F i.val) Q (fun i => hQ i.val)
omit [DecidableEq ι] in
lemma faces_finite (F : ι → Arc ℂ) (S : Finset ι) (hS : S.Nonempty)
    (hF : Pairwise (fun i j => Compatible (F i) (F j)))
    (Q : Finset (ℂ × ℂ)) (hQ : ∀ i,range (F i)⊆⋃q∈Q,segment ℝ q.1 q.2) :
    Finite (ConnectedComponents ((support F S)ᶜ : Set ℂ)) := by
  let _ : Nonempty S := hS.to_subtype
  rw [support_subtype]
  exact CannonEmbeddedArc.Arc.faces_finite (fun i:S => F i.val)
    (fun _ _ h => hF (fun he => h (Subtype.ext he))) Q (fun i => hQ i.val)
omit [DecidableEq ι] in
lemma edge_interior_avoids (F : ι → Arc ℂ) (S : Finset ι)
    (hF : Pairwise (fun i j => Compatible (F i) (F j))) {i : ι} (hi : i∉S) :
    F i '' Ioo (0:I) 1⊆(support F S)ᶜ := by
  rintro x ⟨t,ht,rfl⟩ hx
  simp only [support,mem_iUnion] at hx
  obtain ⟨j,hj,s,hs⟩ := hx
  have hij : i≠j := fun h => hi (h ▸ hj)
  rcases (hF hij).1 t s hs.symm |>.1 with he|he
  · exact ht.1.ne' he
  · exact ht.2.ne he
omit [DecidableEq ι] in
lemma edge_end_mem_support (F : ι → Arc ℂ) (S : Finset ι)
    (hF : Pairwise (fun i j => Compatible (F i) (F j))) {i : ι} (hi : i∉S) (t : I) :
    F i t∈support F S ↔ F i t∈vertices F S := by
  constructor
  · intro h
    simp only [support,mem_iUnion] at h
    obtain ⟨j,hj,s,hs⟩ := h
    refine (mem_vertices F S (F i t)).mpr ⟨j,hj,?_⟩
    rcases (hF (show i≠j from fun h => hi (h ▸ hj))).1 t s hs.symm |>.2 with he|he
    · exact Or.inl (he ▸ hs)
    · exact Or.inr (he ▸ hs)
  · exact fun h => vertices_subset F S h

omit [DecidableEq ι] in
lemma edge_face (F : ι → Arc ℂ) (S : Finset ι)
    (hF : Pairwise (fun i j => Compatible (F i) (F j))) {i : ι} (hi : i∉S) :
    ∃ d : ConnectedComponents ((support F S)ᶜ : Set ℂ),
      F i '' Ioo (0:I) 1⊆region _ ConnectedComponents.mk d ∧
      ∀ t:I, F i t∉support F S → F i t∈region _ ConnectedComponents.mk d := by
  let _ : LocallyConnectedSpace ((support F S)ᶜ : Set ℂ) :=
    (support_closed F S).isOpen_compl.locallyConnectedSpace
  let _ : DiscreteTopology (ConnectedComponents ((support F S)ᶜ : Set ℂ)) := inferInstance
  have hc : IsConnected (F i '' Ioo (0:I) 1) :=
    (isConnected_Ioo (show (0:I)<1 from zero_lt_one)).image _ (F i).continuous.continuousOn
  obtain ⟨d,hd⟩ := label_on_connected ConnectedComponents.continuous_coe hc (edge_interior_avoids F S hF hi)
  have hcl : range (F i)⊆closure (region _ ConnectedComponents.mk d) :=
    range_in_closed_of_interior (F i) isClosed_closure (fun t ht₀ ht₁ => subset_closure (hd ⟨t,⟨ht₀,ht₁⟩,rfl⟩))
  refine ⟨d,hd,fun t ht => ?_⟩
  exact closure_region_inside (support_closed F S).isOpen_compl ConnectedComponents.continuous_coe
    (hcl (mem_range_self t)) ht

lemma single_arc_faces (a : Arc ℂ) (Q : Finset (ℂ × ℂ))
    (hQ : range a⊆⋃q∈Q,segment ℝ q.1 q.2) :
    Nat.card (ConnectedComponents ((range a)ᶜ : Set ℂ))=1 := by
  have hconn : IsPreconnected ({a 1}ᶜ : Set ℂ) := by
    convert CannonPlanarNeighborhood.open_preconnected_remove_point isOpen_univ isPreconnected_univ (a 1) using 1
    ext x
    simp only [mem_compl_iff,mem_singleton_iff,mem_sdiff,mem_univ,true_and]
  have ha₀ : a 0∈({a 1}ᶜ : Set ℂ) := fun h => zero_ne_one (a.injective h)
  have hUa : a '' Ioo (0:I) 1⊆({a 1}ᶜ : Set ℂ) := by
    rintro x ⟨t,ht,rfl⟩ h
    exact ht.2.ne (a.injective h)
  have hc := tame_slit_connected a Q hQ isClosed_singleton.isOpen_compl hconn ha₀ hUa (by simp)
  have heq : ({a 1}ᶜ : Set ℂ)\range a=(range a)ᶜ := by
    ext z
    constructor
    · exact fun h => h.2
    · exact fun h => ⟨fun he => h (he ▸ mem_range_self 1),h⟩
  rw [heq] at hc
  exact connected_components_card_one hc

lemma attached_edge_euler (F : ι → Arc ℂ) (S : Finset ι) (hS : S.Nonempty)
    (hconn : IsPreconnected (support F S))
    (hF : Pairwise (fun i j => Compatible (F i) (F j)))
    (Q : Finset (ℂ × ℂ)) (hQ : ∀ i,range (F i)⊆⋃q∈Q,segment ℝ q.1 q.2)
    {i : ι} (hi : i∉S) (hattach : (support F S∩range (F i)).Nonempty) :
    faces F (insert i S)+(vertices F (insert i S)).card=
      faces F S+(vertices F S).card+1 := by
  classical
  let _ := faces_finite F S hS hF Q hQ
  obtain ⟨d,hd,hdend⟩ := edge_face F S hF hi
  have hdist : F i 0≠F i 1 := fun h => zero_ne_one ((F i).injective h)
  have hatt : F i 0∈support F S ∨ F i 1∈support F S := by
    obtain ⟨x,hx,t,rfl⟩ := hattach
    by_cases ht₀ : t=0
    · exact Or.inl (ht₀ ▸ hx)
    by_cases ht₁ : t=1
    · exact Or.inr (ht₁ ▸ hx)
    exact (edge_interior_avoids F S hF hi ⟨t,⟨lt_of_le_of_ne (nonneg t) (Ne.symm ht₀),
      lt_of_le_of_ne (le_one t) ht₁⟩,rfl⟩ hx).elim
  by_cases h₀ : F i 0∈support F S
  · have hv₀ := (edge_end_mem_support F S hF hi 0).mp h₀
    by_cases h₁ : F i 1∈support F S
    · have hv₁ := (edge_end_mem_support F S hF hi 1).mp h₁
      obtain ⟨P,hP⟩ := support_tame F S Q hQ
      have hf : faces F (insert i S)=faces F S+1 := by
        unfold faces
        rw [support_insert]
        have aux : ∀ d : ConnectedComponents ((support F S)ᶜ : Set ℂ),
            F i '' Ioo (0:I) 1⊆region _ ConnectedComponents.mk d →
            F i 0∈support F S → F i 1∈support F S →
            Nat.card (ConnectedComponents ((support F S∪range (F i))ᶜ : Set ℂ))=
              Nat.card (ConnectedComponents ((support F S)ᶜ : Set ℂ))+1 := by
          rw [hP]
          let _ : Finite (ConnectedComponents ((⋃p∈P,segment ℝ p.1 p.2)ᶜ : Set ℂ)) := by
            rw [←hP]
            exact faces_finite F S hS hF Q hQ
          intro e he he₀ he₁
          exact crosscut_face_count P Q (hP ▸ hconn) (F i) (hQ i) e he₀ he₁ he
        exact aux d hd h₀ h₁
      rw [hf,vertices_insert,Finset.insert_eq_of_mem hv₁,Finset.insert_eq_of_mem hv₀]
      omega
    · have hv₁ : F i 1∉vertices F S := fun h => h₁ (vertices_subset F S h)
      have hf : faces F (insert i S)=faces F S := by
        unfold faces
        rw [support_insert,←reverse_range (F i)]
        apply leaf_face_count (support F S) (support_closed F S) (F i).reverse Q
          (by simpa only [reverse_range] using hQ i) d
        · simpa only [reverse_zero] using hdend 1 h₁
        · simpa only [reverse_one] using h₀
        · rintro x ⟨t,ht,rfl⟩
          apply hd
          refine ⟨unitInterval.symm t,⟨?_,?_⟩,rfl⟩
          · change (0:ℝ)<1-(t:ℝ)
            have hh : (t:ℝ)<1 := ht.2
            linarith
          · change 1-(t:ℝ)<(1:ℝ)
            have hh : (0:ℝ)<t := ht.1
            linarith
      rw [hf,vertices_insert,Finset.insert_eq_of_mem (Finset.mem_insert_of_mem hv₀),Finset.card_insert_of_notMem hv₁]
      omega
  · have h₁ := hatt.resolve_left h₀
    have hv₀ : F i 0∉vertices F S := fun h => h₀ (vertices_subset F S h)
    have hv₁ := (edge_end_mem_support F S hF hi 1).mp h₁
    have hf : faces F (insert i S)=faces F S := by
      unfold faces
      rw [support_insert]
      exact leaf_face_count _ (support_closed F S) (F i) Q (hQ i) d (hdend 0 h₀) h₁ hd
    rw [hf,vertices_insert,Finset.insert_eq_of_mem hv₁,Finset.card_insert_of_notMem hv₀]
    omega
omit [DecidableEq ι] in
lemma mem_support (F : ι → Arc ℂ) (S : Finset ι) (x : ℂ) :
    x∈support F S ↔ ∃ i∈S, x∈range (F i) := by simp only [support,mem_iUnion,exists_prop]
omit [DecidableEq ι] in
lemma support_mono (F : ι → Arc ℂ) {S T : Finset ι} (h : S⊆T) : support F S⊆support F T := by
  intro x hx
  obtain ⟨i,hi,hx⟩ := (mem_support F S x).mp hx
  exact (mem_support F T x).mpr ⟨i,h hi,hx⟩
omit [DecidableEq ι] in
lemma support_nonempty (F : ι → Arc ℂ) (S : Finset ι) (hS : S.Nonempty) : (support F S).Nonempty := by
  obtain ⟨i,hi⟩ := hS
  exact ⟨F i 0,(mem_support ..).mpr ⟨i,hi,mem_range_self 0⟩⟩

lemma exists_attached_edge [Fintype ι] (F : ι → Arc ℂ) (S : Finset ι)
    (hconn : IsPreconnected (support F Finset.univ)) (hS : S.Nonempty) (hSne : S≠Finset.univ) :
    ∃ i∉S, (support F S∩range (F i)).Nonempty := by
  classical
  have hT : (Finset.univ\S).Nonempty := Finset.sdiff_nonempty.mpr (by simpa only [Finset.univ_subset_iff] using hSne)
  have hcover : support F Finset.univ⊆support F S∪support F (Finset.univ\S) := by
    intro x hx
    obtain ⟨i,_,hi⟩ := (mem_support F _ x).mp hx
    by_cases h : i∈S
    · exact Or.inl ((mem_support F S x).mpr ⟨i,h,hi⟩)
    · exact Or.inr ((mem_support F _ x).mpr ⟨i,by simp [h],hi⟩)
  obtain ⟨u,hu⟩ := support_nonempty F S hS
  obtain ⟨v,hv⟩ := support_nonempty F (Finset.univ\S) hT
  obtain ⟨x,_,hxS,hxT⟩ := isPreconnected_closed_iff.mp hconn
    (support F S) (support F (Finset.univ\S)) (support_closed F S) (support_closed F _)
    hcover ⟨u,support_mono F (Finset.subset_univ S) hu,hu⟩
      ⟨v,support_mono F (Finset.subset_univ _) hv,hv⟩
  obtain ⟨i,hi,hix⟩ := (mem_support F _ x).mp hxT
  exact ⟨i,(Finset.mem_sdiff.mp hi).2,x,hxS,hix⟩

theorem planar_euler [Fintype ι] [Nonempty ι] (F : ι → Arc ℂ)
    (hF : Pairwise (fun i j => Compatible (F i) (F j)))
    (Q : Finset (ℂ × ℂ)) (hQ : ∀ i,range (F i)⊆⋃q∈Q,segment ℝ q.1 q.2)
    (hconn : IsPreconnected (support F Finset.univ)) :
    faces F Finset.univ+(vertices F Finset.univ).card=Fintype.card ι+2 := by
  classical
  let good : Finset ι → Prop := fun S => S.Nonempty ∧ IsPreconnected (support F S) ∧
    faces F S+(vertices F S).card=S.card+2
  let A : Finset (Finset ι) := Finset.univ.powerset.filter good
  have hA (S : Finset ι) : S∈A ↔ good S := by simp [A]
  obtain ⟨i⟩ := ‹Nonempty ι›
  have hsupp : support F {i}=range (F i) := by
    simp only [support,Finset.mem_singleton,iUnion_iUnion_eq_left]
  have hv : (vertices F {i}).card=2 := by
    have hn : F i 0≠F i 1 := fun h => zero_ne_one ((F i).injective h)
    simp [vertices,hn]
  have hf : faces F {i}=1 := by
    unfold faces
    rw [hsupp]
    exact single_arc_faces (F i) Q (hQ i)
  have hg : good {i} := ⟨Finset.singleton_nonempty _,hsupp ▸ isPreconnected_range (F i).continuous,by simp [hf,hv]⟩
  obtain ⟨S,hSA,hmax⟩ := A.exists_max_image Finset.card ⟨{i},(hA {i}).mpr hg⟩
  obtain ⟨hSne,hSC,hSE⟩ := (hA S).mp hSA
  have hSuniv : S=Finset.univ := by
    by_contra hn
    obtain ⟨j,hj,hjS⟩ := exists_attached_edge F S hconn hSne hn
    have hnew : good (insert j S) := by
      refine ⟨Finset.insert_nonempty ..,?_,?_⟩
      · rw [support_insert]
        exact hSC.union' hjS (isPreconnected_range (F j).continuous)
      · rw [attached_edge_euler F S hSne hSC hF Q hQ hj hjS,hSE,Finset.card_insert_of_notMem hj]
    have hm := hmax (insert j S) ((hA _).mpr hnew)
    rw [Finset.card_insert_of_notMem hj] at hm
    omega
  subst S
  simpa only [Finset.card_univ] using hSE
end CannonPlanarEuler

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

end OAI