| 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 | |