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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 | import Mathlib
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
namespace CannonGraphCore
variable {X : Type*} [MetricSpace X] [PreconnectedSpace X]
theorem net_joined {r : ℝ} (hr : 0<r) (T Q : Set X) (hTQ : T⊆Q)
(hcover : ∀ x : X, ∃ t∈T, dist x t≤r)
(hjoin : ∀ x∈T, ∀ y∈T, dist x y<4*r → JoinedIn Q x y)
{a : X} (ha : a∈T) : ∀ b∈T, JoinedIn Q a b := by
let U := ⋃ (t : X) (_ : t∈T) (_ : JoinedIn Q a t), ball t (2*r)
let V := ⋃ (t : X) (_ : t∈T) (_ : ¬JoinedIn Q a t), ball t (2*r)
have hU : IsOpen U := isOpen_iUnion fun _ => isOpen_iUnion fun _ => isOpen_iUnion fun _ => isOpen_ball
have hV : IsOpen V := isOpen_iUnion fun _ => isOpen_iUnion fun _ => isOpen_iUnion fun _ => isOpen_ball
have hdisj : Disjoint U V := by
apply Set.disjoint_left.mpr
intro z hzU hzV
simp only [U,mem_iUnion] at hzU
obtain ⟨t,ht,hjt,hzt⟩ := hzU
simp only [V,mem_iUnion] at hzV
obtain ⟨u,hu,hju,hzu⟩ := hzV
have hdt : dist z t<2*r := hzt
have hdu : dist z u<2*r := hzu
have htu : dist t u<4*r := by
have hh := dist_triangle t z u
rw [dist_comm t z] at hh
linarith
exact hju (hjt.trans (hjoin t ht u hu htu))
have hcover' : (univ : Set X)⊆U∪V := by
intro x _
obtain ⟨t,ht,hxt⟩ := hcover x
have hxt' : x∈ball t (2*r) := hxt.trans_lt (by linarith)
by_cases hj : JoinedIn Q a t
· exact Or.inl (mem_iUnion.mpr ⟨t,mem_iUnion.mpr ⟨ht,mem_iUnion.mpr ⟨hj,hxt'⟩⟩⟩)
· exact Or.inr (mem_iUnion.mpr ⟨t,mem_iUnion.mpr ⟨ht,mem_iUnion.mpr ⟨hj,hxt'⟩⟩⟩)
have haU : a∈U := mem_iUnion.mpr ⟨a,mem_iUnion.mpr ⟨ha,
mem_iUnion.mpr ⟨JoinedIn.refl (hTQ ha),mem_ball_self (by positivity)⟩⟩⟩
have hall := isPreconnected_univ.subset_left_of_subset_union hU hV hdisj hcover' ⟨a,mem_univ a,haU⟩
intro b hb
have hbU := hall (mem_univ b)
simp only [U,mem_iUnion] at hbU
obtain ⟨t,ht,hjt,hbt⟩ := hbU
apply hjt.trans (hjoin t ht b hb ?_)
have hbt' : dist b t<2*r := hbt
rw [dist_comm]
linarith
end CannonGraphCore
namespace CannonGraphCore
variable {X ι : Type*} [TopologicalSpace X]
theorem component_whole_edges (E : ι → C(I,X)) (T : Set X) (a : X)
(_ha : a∈T)
(hT : ∀ t∈T, JoinedIn ((⋃ i, range (E i))∪T) a t) :
let Q := (⋃ i, range (E i))∪T
let J := {i : ι // E i 0∈pathComponentIn Q a}
pathComponentIn Q a = (⋃ j : J, range (E j.val))∪T := by
dsimp only
let Q := (⋃ i, range (E i))∪T
let _ : PathConnectedSpace I := isPathConnected_iff_pathConnectedSpace.mp
((convex_Icc (0:ℝ) 1).isPathConnected ⟨0,by norm_num⟩)
have hed (i : ι) : range (E i)⊆Q := by
intro x hx
exact Or.inl (mem_iUnion.mpr ⟨i,hx⟩)
have hc (i : ι) : IsPathConnected (range (E i)) := isPathConnected_range (E i).continuous
apply Set.Subset.antisymm
· intro x hx
rcases pathComponentIn_subset hx with h|h
· obtain ⟨i,hi⟩ := mem_iUnion.mp h
have h₀ : E i 0∈pathComponentIn Q a := hx.trans ((hc i).joinedIn x hi _ (mem_range_self 0) |>.mono (hed i))
exact Or.inl (mem_iUnion.mpr ⟨⟨i,h₀⟩,hi⟩)
· exact Or.inr h
· intro x hx
rcases hx with h|h
· obtain ⟨i,hi⟩ := mem_iUnion.mp h
exact i.property.trans ((hc i.val).joinedIn _ (mem_range_self 0) _ hi |>.mono (hed i.val))
· exact hT x h
theorem path_range_in_component {Q : Set X} {a x y : X} (hax : JoinedIn Q a x)
(p : Path x y) (hp : range p⊆Q) : range p⊆pathComponentIn Q a := by
let _ : PathConnectedSpace I := isPathConnected_iff_pathConnectedSpace.mp
((convex_Icc (0:ℝ) 1).isPathConnected ⟨0,by norm_num⟩)
intro z hz
have hc : IsPathConnected (range p) := isPathConnected_range p.continuous
have hx : x∈range p := ⟨0,p.source⟩
exact hax.trans ((hc.joinedIn x hx z hz).mono hp)
end CannonGraphCore
namespace CannonGraphCore
variable {X ι : Type*} [TopologicalSpace X]
theorem component_retains_edges (E : ι → C(I,X)) {Q : Set X} {a x : X}
(hEQ : (⋃ i, range (E i))⊆Q) (hx : x∈pathComponentIn Q a)
(hxE : x∈⋃ i, range (E i)) :
x∈⋃ j : {i : ι // E i 0∈pathComponentIn Q a}, range (E j.val) := by
let _ : PathConnectedSpace I := isPathConnected_iff_pathConnectedSpace.mp
((convex_Icc (0:ℝ) 1).isPathConnected ⟨0,by norm_num⟩)
obtain ⟨i,hi⟩ := mem_iUnion.mp hxE
have h₀ : E i 0∈pathComponentIn Q a := hx.trans
(((isPathConnected_range (E i).continuous).joinedIn x hi _ (mem_range_self 0)).mono
(fun z hz => hEQ (mem_iUnion.mpr ⟨i,hz⟩)))
exact mem_iUnion.mpr ⟨⟨i,h₀⟩,hi⟩
end CannonGraphCore
end
end
end
end
end
end
end
end
end OAI
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