openAI-math-dataset / lean /OAI /Geometry /NodalSets /SmoothLimit /SphereIterationPositiveLimit.lean
| import OAI.Geometry.NodalSets.SmoothLimit.IterationGeometricTails | |
| import OAI.Geometry.NodalSets.SmoothLimit.SphereCoefficientSummablePositiveLimit | |
| import OAI.Geometry.NodalSets.SmoothLimit.SphereIterationSummability | |
| import OAI.Geometry.NodalSets.SmoothLimit.SphereSummableIterationSequence | |
| namespace OAI | |
| namespace Yau.Target | |
| open Manifold Yau.Geometry Yau.Analysis Filter | |
| open scoped ContDiff Topology | |
| noncomputable section | |
| theorem sphere_iteration_positive_limit : | |
| ∃ (r delta : ℝ) (s : (k : ℕ) → SphereNodalStage r delta k), | |
| (∀ k, k+1 ≤ (s k).frequency) ∧ | |
| ∃ (b : SphereEnergyData) (_ : ContMDiff (𝓡 4) 𝓘(ℝ,ℝ) ∞ b.density), | |
| ∀ k, sphereCoefficientDistance (s k).charts 8 | |
| (s k).energy.tensor (s k).energy.density b.tensor b.density < (s k).radius := by | |
| obtain ⟨P,hP⟩ := finite_sphere_coefficient_atlas | |
| obtain ⟨eta,heta,Hlimit⟩ := sphere_coefficient_summable_positive_limit P hP | |
| roundSphereEnergyData roundSphereEnergyData_density_smooth | |
| obtain ⟨r,delta,hr,hdelta,s,hfirst,hfreq,hgap,hsupp,hdiag,hzero,hpast⟩ := | |
| sphere_summable_iteration_sequence P eta heta | |
| let d : ℕ → SphereEnergyData := fun n ↦ Nat.casesOn n roundSphereEnergyData (fun k ↦ (s k).energy) | |
| have hd : ∀ n, ContMDiff (𝓡 4) 𝓘(ℝ,ℝ) ∞ (d n).density := by | |
| rintro (_|n) | |
| · exact roundSphereEnergyData_density_smooth | |
| · exact (s n).density_smooth | |
| have hds (n : ℕ) (p : Base) := intrinsic_coefficient_chart_smooth | |
| (d n).tensor (d n).smooth (d n).symm (d n).pos (d n).density (hd n) p | |
| let D (Q : Finset Base) (J n : ℕ) : ℝ := sphereCoefficientDistance Q J | |
| (d n).tensor (d n).density (d (n+1)).tensor (d (n+1)).density | |
| have hnonneg (Q : Finset Base) (J n : ℕ) : 0 ≤ D Q J n := | |
| sphereCoefficientDistance_nonneg Q J (d n).tensor (d (n+1)).tensor | |
| (d n).density (d (n+1)).density (fun p _ ↦ hds n p) (fun p _ ↦ hds (n+1) p) | |
| have hsum (J : ℕ) : Summable (D P J) := by | |
| apply (summable_nat_add_iff 1).mp | |
| exact sphere_coefficient_diagonal_summable P (fun k ↦ (s k).energy) | |
| (fun k ↦ (s k).density_smooth) hdiag J | |
| have hzero_le (n : ℕ) : D P 0 (n+1) ≤ eta*(1/2:ℝ)^(n+3) := (hzero n).le | |
| have htotal : (∑' n, D P 0 n) < eta := by | |
| have ht : (∑' n, D P 0 (n+1)) ≤ eta/4 := by | |
| rw [← tsum_scaled_geometric_three eta] | |
| exact ((summable_nat_add_iff 1).mpr (hsum 0)).tsum_le_tsum hzero_le | |
| (hasSum_scaled_geometric_shift eta 3).summable | |
| rw [(hsum 0).tsum_eq_zero_add] | |
| have hf : D P 0 0 < eta/4 := hfirst | |
| linarith | |
| obtain ⟨b,hb,Htail⟩ := Hlimit d hd rfl hsum htotal | |
| refine ⟨r,delta,s,hfreq,b,hb,?_⟩ | |
| intro i | |
| have hpast_le (j : ℕ) : D (s i).charts 8 (j+(i+1)) ≤ (s i).radius*(1/2:ℝ)^(j+2) := by | |
| have h := hpast (i+j) i (Nat.le_add_right i j) | |
| simpa only [D,d,show j+(i+1) = (i+j)+1 by omega,Nat.casesOn, | |
| Nat.add_sub_cancel_left] using h.le | |
| have hsumtail : Summable (fun j ↦ D (s i).charts 8 (j+(i+1))) := | |
| Summable.of_nonneg_of_le (fun j ↦ hnonneg _ _ _) hpast_le | |
| (hasSum_scaled_geometric_shift (s i).radius 2).summable | |
| have hsumall : Summable (D (s i).charts 8) := (summable_nat_add_iff (i+1)).mp hsumtail | |
| have ht := (Htail (s i).charts 8 hsumall).2 (i+1) | |
| have htsum : (∑' j, D (s i).charts 8 ((i+1)+j)) ≤ (s i).radius/2 := by | |
| rw [← tsum_scaled_geometric_two (s i).radius] | |
| have hs' : Summable (fun j ↦ D (s i).charts 8 ((i+1)+j)) := by | |
| simpa only [Nat.add_comm] using hsumtail | |
| exact hs'.tsum_le_tsum | |
| (fun j ↦ by simpa only [Nat.add_comm] using hpast_le j) | |
| (hasSum_scaled_geometric_shift (s i).radius 2).summable | |
| exact ht.trans_lt (htsum.trans_lt (half_lt_self (s i).radius_pos)) | |
| end | |
| end Yau.Target | |
| end OAI | |