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8e4d533 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 | import Mathlib
namespace OAI
section
section
section
noncomputable section
namespace CannonClosedCover
open Set
variable {X ι : Type*} [TopologicalSpace X] [Finite ι]
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
end
end
end
end
end OAI
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