openAI-math-dataset / lean /OAI /Geometry /Cannon /Uniformization /BoundaryUniformizationInputs.lean
| import Mathlib | |
| import OAI.Geometry.Cannon.Boundary.BoundaryModulusBounded | |
| import OAI.Geometry.Cannon.Energy.ApproxSelfSimilar | |
| namespace OAI | |
| section | |
| section | |
| section | |
| noncomputable section | |
| namespace CannonRelease | |
| open CannonGraph CannonVisual CannonGeometry CannonNet CannonCells CannonCharts Set Metric | |
| universe v | |
| variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G] | |
| variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ) | |
| theorem boundary_uniformization_inputs (φ : Boundary D ≃ₜ SphereTwo) : | |
| let _ := visualMetricSpace D hδ | |
| let a := Real.exp (2*visualExponent δ) | |
| let LE := 8*Real.exp (visualExponent δ*(4*(8*δ+6):ℕ)) | |
| ∀ d κ : ℝ, 0 < d → d < diam (univ : Set (Boundary D))/10 → | |
| d ≤ 1/(LE*3^2) → 2*a < κ → | |
| PathConnectedSpace (Boundary D) ∧ ApproxSelfSimilar (Boundary D) ∧ | |
| ∃ T : ℕ → Finset (Boundary D), ∃ N : ℕ → ℕ, ∃ B : ℝ, ∀ m, | |
| IsApproximation (fun i : T (N m) => i.val) | |
| (fun i : T (N m) => ball i.val (2*(a^(N m))⁻¹)) ((2:ℝ)^m)⁻¹ κ ∧ | |
| modulus (fun i : T (N m) => ball i.val (2*(a^(N m))⁻¹)) (macroFamily d) ≤ B := by | |
| classical | |
| let _ := visualMetricSpace D hδ | |
| dsimp only | |
| intro d κ hd hdia hdc hκ | |
| let a := Real.exp (2*visualExponent δ) | |
| have ha : 1 < a := Real.one_lt_exp_iff.mpr (mul_pos (by norm_num) (visualExponent_pos δ)) | |
| let _ : PathConnectedSpace SphereTwo := CannonCharts.sphere_pathConnected | |
| refine ⟨φ.symm.surjective.pathConnectedSpace φ.symm.continuous, | |
| boundary_approx_self_similar D hδ,?_⟩ | |
| obtain ⟨T,hT⟩ := exists_net_sequence (X:=Boundary D) ha | |
| have hscale (m : ℕ) : ∃ n : ℕ, | |
| ((2:ℝ)^m)⁻¹/a < (a^n)⁻¹ ∧ (a^n)⁻¹ ≤ ((2:ℝ)^m)⁻¹ := by | |
| apply exists_scale_below ha (by positivity) | |
| exact inv_le_one_of_one_le₀ (one_le_pow₀ (by norm_num)) | |
| choose N hNlow hNupp using hscale | |
| have hsep (n : ℕ) (i j : T n) (hij : i ≠ j) : (a^n)⁻¹ ≤ dist i.val j.val := | |
| (hT n).1 i.val i.property j.val j.property (fun hh => hij (Subtype.ext hh)) | |
| have hnet (n : ℕ) (x : Boundary D) : ∃ i : T n, dist i.val x ≤ (a^n)⁻¹ := by | |
| obtain ⟨y,hy,hh⟩ := (hT n).2 x | |
| exact ⟨⟨y,hy⟩,hh⟩ | |
| have hdia' : d < diam (univ : Set (Boundary D)) := by | |
| have := diam_nonneg (s:= (univ : Set (Boundary D))) | |
| linarith | |
| obtain ⟨B,hB⟩ := boundary_modulus_bounded D hδ φ d hd hdia' hdc | |
| (fun n => ↥(T n)) (fun n i => i.val) hsep hnet | |
| refine ⟨T,N,B,?_⟩ | |
| intro m | |
| exact ⟨balls_isApproximation (fun i : T (N m) => i.val) ha (by positivity) hκ | |
| (hNlow m) (hNupp m) (hsep (N m)) (hnet (N m)),hB (N m)⟩ | |
| end CannonRelease | |
| end | |
| end | |
| end | |
| end | |
| end OAI | |