openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /GraphGeometry /EllipseSeparation.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.EllipseGrid | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.EllipseAngles | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.EllipseLinear | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open Set Function | |
| namespace CriticalPercolation.EuclideanEllipse | |
| theorem separated_neighboring_ellipses (W U zeta : ℝ) | |
| (hW : 0<W) (hWU : W≤U) (hzeta : 0<zeta) | |
| (chooseCenter : (ℂ → ℂ) → Circle) | |
| (hboundary : ∀ phi : ℝ, | |
| |((Circle.exp phi : ℂ)⁻¹*(chooseCenter (normalize U W (Circle.exp phi)) : ℂ)).re|<1/8) : | |
| ∃ p∈parameterDisk, ∃ q∈parameterDisk, | |
| (∀ z : ℂ, ‖diskNormalize W U p (diskExpand W U q z)-z‖≤zeta*‖z‖) ∧ | |
| (1 : ℝ)/4≤|EllipseLinear.det | |
| (chooseCenter (diskNormalize W U p) : ℂ).re (chooseCenter (diskNormalize W U p) : ℂ).im | |
| (chooseCenter (diskNormalize W U q) : ℂ).re (chooseCenter (diskNormalize W U q) : ℂ).im| := by | |
| obtain ⟨N,K,hN,hK,hangle,hcomp⟩ := exists_grid_comparison W U zeta hW hWU hzeta | |
| have hNr : (N : ℝ)≠0 := (Nat.cast_pos.mpr hN).ne' | |
| have hKr : (K : ℝ)≠0 := (Nat.cast_pos.mpr hK).ne' | |
| let phi : ℕ → ℝ := fun k => (k : ℝ)*Real.pi/K | |
| let centers : ℕ → ℕ → Circle := fun i k => chooseCenter (diskNormalize W U (gridPoint N K i k)) | |
| let F : ℕ → ℕ → Real.Angle := fun i k => lineAngle (centers i k) | |
| have hp0 : phi 0=0 := by simp [phi] | |
| have hpK : phi K=Real.pi := by dsimp [phi]; field_simp | |
| have hn : ∀ k, diskNormalize W U (gridPoint N K N k)=normalize U W (Circle.exp (phi k)) := by | |
| intro k | |
| simp only [diskNormalize,gridPoint,div_self hNr,diskRadius,mul_one,add_sub_cancel,phi] | |
| have hc : ∀ k≤K, F 0 k=F 0 0 := by | |
| intro k _hk | |
| dsimp [F,centers,gridPoint] | |
| simp only [Nat.cast_zero,zero_div,zero_mul] | |
| rw [diskNormalize_center W U ((k : ℝ)*Real.pi/K)] | |
| have hs : ∀ i≤N, F i K=F i 0 := by | |
| intro i _hi | |
| have he : gridPoint N K i K=((i : ℝ)/N,Real.pi) := by | |
| simp only [gridPoint] | |
| congr 1 | |
| have he0 : gridPoint N K i 0=((i : ℝ)/N,0) := by simp [gridPoint] | |
| dsimp [F,centers] | |
| rw [he,he0,diskNormalize_seam] | |
| let err : ℕ → ℝ := fun k => boundaryError (phi k) (centers N k) | |
| let b : ℕ → ℝ := fun k => 2*phi k+Real.pi+err k | |
| have herr : ∀ k≤K, |err k|<Real.pi/8 := by | |
| intro k _hk | |
| apply boundary_error_lt | |
| simpa only [centers,hn] using hboundary (phi k) | |
| have hlift : ∀ k≤K, F N k=(b k : Real.Angle) := by | |
| intro k _hk | |
| exact boundary_lift (phi k) (centers N k) | |
| have hmesh : ∀ k<K, |phi (k+1)-phi k|<Real.pi/32 := by | |
| intro k _hk | |
| have he := gridPoint_angular_step N K k hK | |
| exact he.trans_lt hangle | |
| have hstep := DiskWinding.boundary_mesh_lifts phi err K hmesh herr | |
| have hturn : b K-b 0=2*Real.pi := by | |
| have hcseam : centers N K=centers N 0 := by | |
| dsimp [centers] | |
| rw [hn,hn,hpK,hp0,circle_exp_pi] | |
| congr 1 | |
| funext z | |
| exact normalize_neg U W (Circle.exp 0) z | |
| have heseam : err K=err 0 := by | |
| dsimp [err] | |
| rw [hpK,hp0,hcseam,boundary_error_seam] | |
| dsimp [b] | |
| rw [hpK,hp0,heseam] | |
| ring | |
| have hex := DiskWinding.exists_large_edge F N K b hc hs hlift hstep hturn | |
| have edge : ∀ i≤N, ∀ k≤K, ∀ a≤1, ∀ b≤1, i+a≤N → k+b≤K → | |
| Real.pi/4≤|DiskWinding.increment (F i k) (F (i+a) (k+b))| → | |
| ∃ p∈parameterDisk, ∃ q∈parameterDisk, | |
| (∀ z : ℂ, ‖diskNormalize W U p (diskExpand W U q z)-z‖≤zeta*‖z‖) ∧ | |
| (1 : ℝ)/4≤|EllipseLinear.det | |
| (chooseCenter (diskNormalize W U p) : ℂ).re (chooseCenter (diskNormalize W U p) : ℂ).im | |
| (chooseCenter (diskNormalize W U q) : ℂ).re (chooseCenter (diskNormalize W U q) : ℂ).im| := by | |
| intro i hi k hk a ha b hb hia hkb hsep | |
| refine ⟨gridPoint N K i k,gridPoint_mem hN hK hi hk, | |
| gridPoint N K (i+a) (k+b),gridPoint_mem hN hK hia hkb,hcomp i hi k hk a ha b hb hia hkb,?_⟩ | |
| have hd := EllipseLinear.det_lower_of_separation (centers i k : ℂ).arg (centers (i+a) (k+b) : ℂ).arg hsep | |
| simpa only [circle_arg_re,circle_arg_im] using hd | |
| rcases hex with ⟨i,hi,k,hk,hik⟩ | ⟨i,hi,k,hk,hik⟩ | ⟨i,hi,k,hk,hik⟩ | |
| · exact edge i hi k (by omega) 0 (by omega) 1 (by omega) (by omega) (by omega) (by simpa using hik) | |
| · exact edge i (by omega) k hk 1 (by omega) 0 (by omega) (by omega) (by omega) (by simpa using hik) | |
| · exact edge i (by omega) k (by omega) 1 (by omega) 1 (by omega) (by omega) (by omega) hik | |
| end CriticalPercolation.EuclideanEllipse | |
| end | |
| end | |
| end | |
| end OAI | |