import Mathlib
import OAI.Geometry.Cannon.PlanarGraphs.Region

namespace OAI

section
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open Set unitInterval Topology

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open Set Topology unitInterval

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open Set Metric Topology unitInterval

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open Set Metric Topology

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open Set unitInterval

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open Set Metric Topology

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open Set Metric Topology

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open Set Metric Topology

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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

namespace CannonFaceIncidence
open Set Topology Metric

variable {X D : Type*} [TopologicalSpace X] [TopologicalSpace D] [DiscreteTopology D]
theorem flat_chart_boundary_patch {V W : Set X} {π : V → D} (hπ : Continuous π)
    (H : (ℝ × ℝ) ≃ₜ X) (hW : IsOpen W) (h₀ : H (0,0)∈W)
    (hflat : ∀ z∈W, z∈V ↔ (H.symm z).2≠0)
    {d : D} (hd : d∈incident V π (H (0,0))) :
    ∃ U B : Set X, IsOpen U ∧ H (0,0)∈U ∧ U⊆W ∧
      IsConnected B ∧ B⊆W∩region V π d ∧ U∩Vᶜ⊆closure B := by
  obtain ⟨r,hr,hball⟩ := Metric.mem_nhds_iff.mp
    ((hW.preimage H.continuous).mem_nhds h₀)
  let C := openBox r
  let U := H '' C
  have hCU : C⊆H ⁻¹' W := by rw [show C=openBox r from rfl,openBox_eq_ball]; exact hball
  have hUW : U⊆W := by rintro _ ⟨p,hp,rfl⟩; exact hCU hp
  have hU : IsOpen U := H.isOpenMap C (by rw [show C=openBox r from rfl,openBox_eq_ball]; exact isOpen_ball)
  have h₀U : H (0,0)∈U := ⟨(0,0),by rw [show C=openBox r from rfl,openBox_eq_ball]; exact mem_ball_self hr,rfl⟩
  let S : Bool → Set X := fun b => H '' halfBox r b
  have hSc (b : Bool) : IsConnected (S b) :=
    (halfBox_connected hr b).image H H.continuous.continuousOn
  have hcover : U∩V=⋃ b, S b := by
    ext x
    constructor
    · rintro ⟨⟨p,hp,rfl⟩,hpV⟩
      have hp₂ : p.2≠0 := by simpa using (hflat (H p) (hCU hp)).mp hpV
      rcases lt_or_gt_of_ne hp₂ with hl|hr'
      · exact mem_iUnion.mpr ⟨false,⟨p,⟨hp.1,hp.2.1,hl⟩,rfl⟩⟩
      · exact mem_iUnion.mpr ⟨true,⟨p,⟨hp.1,hr',hp.2.2⟩,rfl⟩⟩
    · intro hx
      obtain ⟨b,p,hp,rfl⟩ := mem_iUnion.mp hx
      have hpC : p∈C := by
        cases b
        · exact ⟨hp.1,hp.2.1,hp.2.2.trans hr⟩
        · exact ⟨hp.1,(neg_neg_of_pos hr).trans hp.2.1,hp.2.2⟩
      refine ⟨⟨p,hpC,rfl⟩,(hflat (H p) (hCU hpC)).mpr ?_⟩
      simp only [H.symm_apply_apply]
      cases b
      · exact hp.2.2.ne
      · exact hp.2.1.ne'
  have hclose (b : Bool) : U∩Vᶜ⊆closure (S b) := by
    rintro x ⟨⟨p,hp,hpx⟩,hxV⟩
    subst x
    have hp₂ : p.2=0 := by
      by_contra h
      apply hxV
      apply (hflat (H p) (hCU hp)).mpr
      simpa using h
    rw [show S b=H '' halfBox r b from rfl,←H.image_closure]
    exact ⟨p,halfBox_closure hr hp hp₂ b,rfl⟩
  obtain ⟨labels,hsub,hlabels⟩ := sector_incidence hU inter_subset_left hπ S hSc hcover hclose
  have hnv : H (0,0)∉V := by
    intro h
    have hh := (hflat _ h₀).mp h
    simp at hh
  have hh := hlabels (H (0,0)) ⟨h₀U,hnv⟩
  obtain ⟨b,hb⟩ := hh ▸ hd
  refine ⟨U,S b,hU,h₀U,hUW,hSc b,?_,hclose b⟩
  intro z hz
  refine ⟨hUW ?_,hb ▸ hsub b hz⟩
  exact (hcover.symm ▸ mem_iUnion_of_mem b hz).1
end CannonFaceIncidence

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

end OAI