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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 | import Mathlib
import OAI.Geometry.Cannon.PlaneTopology.BoundaryControlledEmbeddedArcs
import OAI.Geometry.Cannon.PlaneTopology.SeparatedArcColoring
namespace OAI
section
section
section
noncomputable section
open Set unitInterval Topology
noncomputable section
noncomputable section
namespace CannonRelease
open Set Metric CannonGraph CannonPacking 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_embedded_graph (φ : Boundary D ≃ₜ SphereTwo) :
let _ := visualMetricSpace D hδ
∃ L r₀ : ℝ, 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) →
∃ n : ℕ, ∃ F : Fin n → Arc (Boundary D),
Pairwise (fun e f => Compatible (F e) (F f)) ∧
(∀ x∈T, ∀ y∈T, x≠y → dist x y<4*r →
Vertex F x ∧ Vertex F y ∧ ∃ b : Arc (Boundary D), b 0=x ∧ b 1=y ∧
range b ⊆ (⋃ j, range (F j)) ∧ diam (range b)≤3*L*r) ∧
(∀ j, diam (range (F j))≤L*r) ∧
(∀ x, intersectionCount F (ball x (2*L*r))≤K) := by
let _ := visualMetricSpace D hδ
dsimp only
obtain ⟨L,s₀,hL,hs₀,harc⟩ := boundary_controlled_embedded_arcs D hδ φ
obtain ⟨Q,C,_,hC,hp⟩ := relative_polynomial_packing D hδ
have hLpos : 0<L := zero_lt_one.trans_le hL
obtain ⟨K,hK⟩ := net_arc_graph Q C (2*L) hp hC.le (by linarith)
refine ⟨2*L,min (s₀/2) (1/(14*L)),by linarith,
lt_min (half_pos hs₀) (by positivity),K,?_⟩
intro r hr hrs T hsep
have hrs₀ : 2*r ≤ s₀ := by have := hrs.trans (min_le_left ..); linarith
have hscale : 7*(2*L)*r≤1 := by
have h := hrs.trans (min_le_right ..)
have hh := (le_div_iff₀ (by positivity : 0<14*L)).mp h
nlinarith
apply hK r hr hscale T hsep
intro x hx y hy hxy hnear
obtain ⟨a,h₀,h₁,_,hd⟩ := harc (2*r) (by positivity) hrs₀ x y hxy (by linarith)
exact ⟨a,h₀,h₁,by nlinarith [hd]⟩
end CannonRelease
end
end
end
end
end
end
end OAI
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