lean/OAI/Geometry/Cannon/PlanarGraphs/ComponentRegion.lean

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

namespace OAI

section
section
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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noncomputable section
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

variable {X : Type*} [TopologicalSpace X]

lemma component_region_eq (V : Set X) (x : V) :
    region V ConnectedComponents.mk (ConnectedComponents.mk x)=connectedComponentIn V x.val := by
  rw [connectedComponentIn_eq_image x.property]
  ext y
  constructor
  · rintro ⟨hy,hye⟩
    exact ⟨⟨y,hy⟩,ConnectedComponents.coe_eq_coe'.mp hye,rfl⟩
  · rintro ⟨y,hy,rfl⟩
    exact ⟨y.property,ConnectedComponents.coe_eq_coe'.mpr hy⟩

lemma component_region_connected (V : Set X) (d : ConnectedComponents V) :
    IsConnected (region V ConnectedComponents.mk d) := by
  obtain ⟨x,rfl⟩ := ConnectedComponents.surjective_coe d
  rw [component_region_eq]
  exact isConnected_connectedComponentIn_iff.mpr x.property

lemma component_regions_disjoint (V : Set X) {d e : ConnectedComponents V} (hne : d≠e) :
    Disjoint (region V ConnectedComponents.mk d) (region V ConnectedComponents.mk e) := by
  apply disjoint_left.mpr
  rintro x ⟨hx,hd⟩ ⟨hx',he⟩
  exact hne (hd.symm.trans he)
end CannonFaceIncidence

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

end OAI