Datasets:

roseking
/

openAI-math-dataset

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

	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 (2Ar))≤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