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

namespace CannonFaceIncidence
variable {X D : Type*} [TopologicalSpace X] [TopologicalSpace D] [DiscreteTopology D]

lemma closure_region_inside {V : Set X} (hV : IsOpen V) {π : V → D}
    (hπ : Continuous π) {d : D} {x : X} (hx : x∈closure (region V π d))
    (hxV : x∈V) : x∈region V π d := by
  let d₀ := π ⟨x,hxV⟩
  have hR := region_isOpen hV hπ d₀
  have hxR : x∈region V π d₀ := ⟨hxV,rfl⟩
  obtain ⟨y,⟨hyV,hyd₀⟩,⟨hyV',hyd⟩⟩ :=
    (mem_closure_iff_nhds.mp hx) _ (hR.mem_nhds hxR)
  exact ⟨hxV,hyd₀.symm.trans hyd⟩

theorem region_boundary_hits {K T : Set X} (hK : IsClosed K) (hT : IsClosed T)
    (hTK : T⊆K) (hconn : IsPreconnected Tᶜ) (hne : (K\T).Nonempty)
    {π : (Kᶜ : Set X) → D} (hπ : Continuous π) (d : D)
    (hd : (region Kᶜ π d).Nonempty) :
    ∃ x∈K, x∉T ∧ d∈incident Kᶜ π x := by
  by_contra hn
  have hbound : closure (region Kᶜ π d)⊆region Kᶜ π d∪T := by
    intro x hx
    by_cases hxK : x∈K
    · right
      by_contra hxT
      exact hn ⟨x,hxK,hxT,hx⟩
    · exact Or.inl (closure_region_inside hK.isOpen_compl hπ hx hxK)
  have hclosed : IsClosed (region Kᶜ π d∪T) := by
    apply isClosed_of_closure_subset
    rw [closure_union,hT.closure_eq]
    exact union_subset hbound subset_union_right
  let _ : PreconnectedSpace (Tᶜ : Set X) := isPreconnected_iff_preconnectedSpace.mp hconn
  let C : Set (Tᶜ : Set X) := Subtype.val ⁻¹' region Kᶜ π d
  have hC : IsClopen C := by
    constructor
    · have he : C=Subtype.val ⁻¹' (region Kᶜ π d∪T) := by
        ext x
        simp only [C,mem_preimage,mem_union]
        exact or_iff_left x.property |>.symm
      rw [he]
      exact hclosed.preimage continuous_subtype_val
    · exact (region_isOpen hK.isOpen_compl hπ d).preimage continuous_subtype_val
  obtain ⟨x,hx⟩ := hd
  have hxT : x∉T := fun hxT => (region_subset Kᶜ π d hx) (hTK hxT)
  have hCe : C=univ := hC.eq_univ ⟨⟨x,hxT⟩,hx⟩
  obtain ⟨y,hyK,hyT⟩ := hne
  have hyC : (⟨y,hyT⟩ : (Tᶜ : Set X))∈C := hCe.symm ▸ mem_univ _
  exact (region_subset Kᶜ π d hyC) hyK
end CannonFaceIncidence

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

end OAI