| import Mathlib |
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| namespace OAI |
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| section |
| section |
| section |
| noncomputable section |
| namespace CannonClosedCover |
| open Set |
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| variable {X ι : Type*} [TopologicalSpace X] [Finite ι] |
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| theorem graph_preconnected (G : SimpleGraph ι) (C : ι → Set X) |
| (hclosed : ∀ i,IsClosed (C i)) |
| (hconn : IsPreconnected (⋃ i,C i)) (hne : ∀ i,(C i).Nonempty) |
| (hadj : ∀ i j,i≠j → (C i∩C j).Nonempty → G.Adj i j) : G.Preconnected := by |
| classical |
| intro a b |
| by_contra hb |
| let U : Set X := ⋃ i : {i : ι | G.Reachable a i},C i.val |
| let V : Set X := ⋃ i : {i : ι | ¬G.Reachable a i},C i.val |
| have hU : IsClosed U := isClosed_iUnion_of_finite fun i=>hclosed i.val |
| have hV : IsClosed V := isClosed_iUnion_of_finite fun i=>hclosed i.val |
| have hsub : (⋃ i,C i)⊆U∪V := by |
| intro x hx |
| obtain ⟨i,hi⟩ := mem_iUnion.mp hx |
| by_cases h : G.Reachable a i |
| · exact Or.inl (mem_iUnion.mpr ⟨⟨i,h⟩,hi⟩) |
| · exact Or.inr (mem_iUnion.mpr ⟨⟨i,h⟩,hi⟩) |
| obtain ⟨x,hx⟩ := hne a |
| obtain ⟨y,hy⟩ := hne b |
| obtain ⟨z,hz,hzU,hzV⟩ := isPreconnected_closed_iff.mp hconn U V hU hV hsub |
| ⟨x,mem_iUnion.mpr ⟨a,hx⟩,mem_iUnion.mpr ⟨⟨a,.rfl⟩,hx⟩⟩ |
| ⟨y,mem_iUnion.mpr ⟨b,hy⟩,mem_iUnion.mpr ⟨⟨b,hb⟩,hy⟩⟩ |
| obtain ⟨i,hi⟩ := mem_iUnion.mp hzU |
| obtain ⟨j,hj⟩ := mem_iUnion.mp hzV |
| have hneij : i.val≠j.val := fun h=>j.property (h ▸ i.property) |
| exact j.property (i.property.trans (hadj _ _ hneij ⟨z,hi,hj⟩).reachable) |
| end CannonClosedCover |
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| end |
| end |
| end |
| end |
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| end OAI |
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