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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 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 | 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
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