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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 | import Mathlib
import OAI.Geometry.Cannon.PlaneTopology.SquareSide
import OAI.Geometry.Cannon.Boundary.ExpansionGlobalUpper
import OAI.Geometry.Cannon.PlaneTopology.Lift
namespace OAI
section
section
section
noncomputable section
open Set unitInterval Topology
noncomputable section
noncomputable section
noncomputable section
open Set Topology unitInterval
namespace CannonRelease
open Set Metric Topology unitInterval CannonGraph CannonEmbeddedArc CannonEmbeddedArc.Arc
variable {G : Type*} [Group G] [TopologicalSpace G] [DiscreteTopology G]
variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ)
theorem boundary_uniform_complement_graph (φ : Boundary D ≃ₜ SphereTwo) :
let _ := visualMetricSpace D hδ
∃ A L r₀ : ℝ, 1≤A ∧ 1≤L ∧ 0<r₀ ∧ ∃ K : ℕ,
∀ r : ℝ, 0<r → r≤r₀ → ∀ T : Finset (Boundary D),
(∀ x∈T, ∀ y∈T, x≠y → r≤dist x y) →
(∀ x : Boundary D, ∃ y∈T, dist x y≤r) →
∃ n : ℕ, ∃ F : Fin n → Arc (Boundary D),
Pairwise (fun e f => Compatible (F e) (F f)) ∧
(∀ j, diam (range (F j))≤A*r) ∧
(∀ x, intersectionCount F (ball x (2*A*r))≤K) ∧
(∀ S : Set (Boundary D), IsPreconnected S →
S⊆((⋃ j, range (F j))∪(T:Set (Boundary D)))ᶜ →
∀ x∈S, ∀ y∈S, dist x y<L*r) := by
let _ := visualMetricSpace D hδ
dsimp only
obtain ⟨A,B,rG,hA,hB,hrG,K,hgraph⟩ := boundary_uniform_loop_graph D hδ φ
obtain ⟨C,η,hC,hη,hexpand⟩ := boundary_uniform_expanding_homeomorphs D hδ
obtain ⟨L,hL,hseparate⟩ := CannonWinding.loop_approximation_separates φ hC hB hη hexpand
have hLp : 0<L := zero_lt_one.trans_le hL
refine ⟨A,L,min rG (1/L),hA,hL,lt_min hrG (one_div_pos.mpr hLp),K,?_⟩
intro r hr hrs T hsep hcover
obtain ⟨n,F,hFc,hFd,hFt,happrox⟩ := hgraph r hr (hrs.trans (min_le_left ..)) T hsep hcover
refine ⟨n,F,hFc,hFd,hFt,?_⟩
apply hseparate r hr (hrs.trans (min_le_right ..))
intro f hf
obtain ⟨H,hH₀,hH₁,hHc,hHd⟩ := happrox f hf
let g : C(I,Boundary D) := ⟨fun t => H (1,t),by fun_prop⟩
refine ⟨g,hHc 1,?_,?_⟩
· rintro _ ⟨t,rfl⟩
exact hH₁ t
· intro t
change dist (H (1,t)) (f t)<B*r
rw [←hH₀ t]
exact hHd 1 0 t
end CannonRelease
end
end
end
end
end
end
end
end OAI
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