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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 | import Mathlib
import OAI.Geometry.Cannon.PlanarGraphs.Region
namespace OAI
section
section
section
noncomputable section
open Set unitInterval Topology
noncomputable section
noncomputable section
noncomputable section
open Set Topology unitInterval
noncomputable section
open Set Metric Topology unitInterval
noncomputable section
open Set Metric Topology
noncomputable section
noncomputable section
open Set unitInterval
noncomputable section
noncomputable section
open Set Metric Topology
noncomputable section
open Set Metric Topology
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open Set Metric Topology
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open Set Topology
noncomputable section
open Set Topology
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open Set Topology
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open Set Topology unitInterval
noncomputable section
namespace CannonFaceIncidence
open Set Topology
variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
noncomputable def homeomorphComponents (e : X ≃ₜ Y) : ConnectedComponents X ≃ ConnectedComponents Y :=
Equiv.ofBijective e.continuous.connectedComponentsMap
(e.isQuotientMap.isCoinducing.connectedComponentsMap_bijective (fun y => by
have he : e ⁻¹' {y}={e.symm y} := by
ext x
exact ⟨fun h => by simpa using congrArg e.symm h, fun h => by
change x=e.symm y at h
rw [h]
exact e.apply_symm_apply y⟩
rw [he]
exact isConnected_singleton))
noncomputable def componentsDeletionEquiv (V : Set X) (S : Set X)
[DiscreteTopology (ConnectedComponents V)] :
ConnectedComponents (V\S : Set X) ≃
Σ d : ConnectedComponents V, ConnectedComponents (region V ConnectedComponents.mk d\S : Set X) := by
let U (d : ConnectedComponents V) : Set (V\S : Set X) :=
{x | x.val∈region V ConnectedComponents.mk d}
let π : (V\S : Set X) → ConnectedComponents V := fun x => ConnectedComponents.mk ⟨x.val,x.property.1⟩
have hπ : Continuous π := ConnectedComponents.continuous_coe.comp (continuous_subtype_val.subtype_mk _)
have hU (d : ConnectedComponents V) : U d=π ⁻¹' {d} := by
ext x
constructor
· exact fun h => h.2
· exact fun h => ⟨x.property.1,h⟩
have hcl (d : ConnectedComponents V) : IsClopen (U d) := by
rw [hU]
exact (isClopen_discrete {d}).preimage hπ
have hdis : Pairwise (Function.onFun Disjoint U) := by
intro d e hde
apply Set.disjoint_left.mpr
intro x hd he
exact hde (hd.2.symm.trans he.2)
have hcover : ⋃ d,U d=univ := by
apply eq_univ_of_forall
intro x
exact mem_iUnion.mpr ⟨π x,x.property.1,rfl⟩
let E (d : ConnectedComponents V) : U d ≃ₜ (region V ConnectedComponents.mk d\S : Set X) :=
{ toFun := fun z => ⟨z.val.val,z.property,z.val.property.2⟩
invFun := fun z => ⟨⟨z.val,(region_subset _ _ d z.property.1),z.property.2⟩,z.property.1⟩
left_inv := fun _ => rfl
right_inv := fun _ => rfl
continuous_toFun := (continuous_subtype_val.comp continuous_subtype_val).subtype_mk _
continuous_invFun := (continuous_subtype_val.subtype_mk _).subtype_mk _ }
exact (ConnectedComponents.equivOfIsClopen hcl hdis hcover).trans
(Equiv.sigmaCongrRight fun d => homeomorphComponents (E d))
lemma region_disjoint_components (V : Set X) {d e : ConnectedComponents V} (hne : d≠e) :
Disjoint (region V ConnectedComponents.mk d) (region V ConnectedComponents.mk e) := by
apply Set.disjoint_left.mpr
intro x hx hy
exact hne (hx.2.symm.trans hy.2)
lemma connected_components_card_one {U : Set X} (h : IsConnected U) :
Nat.card (ConnectedComponents U)=1 := by
let _ : ConnectedSpace U := isConnected_iff_connectedSpace.mp h
exact Nat.card_eq_one_iff_unique.mpr ⟨inferInstance,inferInstance⟩
end CannonFaceIncidence
end
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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