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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 | 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
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