openAI-math-dataset / lean /OAI /Geometry /IsometricImmersion /Obstructions /BoundedClassTopology.lean
| import OAI.Geometry.IsometricImmersion.Comparison.BoundedClassApproximation | |
| import Mathlib.Topology.GDelta.Basic | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter | |
| open scoped ContDiff Topology Matrix Matrix.Norms.Elementwise | |
| namespace SmoothLocal.Perturbation | |
| open SmoothLocal.Geometry SmoothLocal.Pulse SmoothLocal.Flow | |
| theorem boundedHeightClass_nonempty_parameters | |
| {g0 : MetricField} {kappa : ℝ} {M : ℕ} {q0 : ℚ} | |
| (hclass : (boundedHeightClass g0 kappa M q0).Nonempty) : | |
| 0 < M ∧ |(q0 : ℝ)| < 1/20 := by | |
| obtain ⟨eta,z,hz,hcenter⟩ := hclass | |
| exact ⟨hz.1,boundedHeightClass_label_abs_lt ⟨eta,z,hz,hcenter⟩⟩ | |
| theorem boundedHeightClass_closure_parameters | |
| {g0 : MetricField} {kappa : ℝ} {M : ℕ} {q0 : ℚ} | |
| {eta : metricPatchSet g0 kappa} | |
| (heta : eta ∈ closure (boundedHeightClass g0 kappa M q0)) : | |
| 0 < M ∧ |(q0 : ℝ)| < 1/20 := | |
| boundedHeightClass_nonempty_parameters (Set.Nonempty.of_closure ⟨eta,heta⟩) | |
| theorem bounded_class_open_closure_pulse_family_at_radius | |
| {g0 : MetricField} {V : Set Coord} {kappa : ℝ} | |
| (hg0 : SmoothPositiveOn g0 V) (hV : IsOpen V) (hSV : modelSquare ⊆ V) | |
| (M : ℕ) (q0 : ℚ) | |
| {O : Set (metricPatchSet g0 kappa)} (hO : IsOpen O) | |
| {etaStar : metricPatchSet g0 kappa} (hStar : etaStar ∈ O) | |
| (hOclosure : O ⊆ closure (boundedHeightClass g0 kappa M q0)) | |
| {r : ℝ} (hr : 0 < r) (hLr : boundedClassWidth kappa M*r ≤ 1/20) : | |
| ∃ N : ℕ, 10 < N ∧ ∀ delta : ℝ, 0 < delta → | |
| ∀ᶠ tau : ℕ in atTop, 0 < tau ∧ | |
| ∃ (eta : metricPatchSet g0 kappa) (z : Coord → ℝ), | |
| BoundedAdmissibleHeight (perturbedMetric g0 eta.val) M z ∧ | |
| |hessianQuotient (perturbedMetric g0 eta.val) z 0-(q0 : ℝ)| ≤ | |
| 1/(100*boundedClassWidth kappa M) ∧ | |
| (∀ i j : Fin 2, ∀ k ≤ tau, ∀ p ∈ modelSquare, | |
| ‖iteratedFDeriv ℝ k (fun x => perturbedMetric g0 eta.val x i j- | |
| testMetric (perturbedMetric g0 etaStar.val) (q0 : ℝ) | |
| (boundedClassWidth kappa M*r/16) N delta (tau : ℝ) x i j) p‖ ≤ | |
| metricApproximationAccuracy tau) := by | |
| have hparams := boundedHeightClass_closure_parameters (hOclosure hStar) | |
| let a := boundedClassWidth kappa M*r/16 | |
| have ha : 0 < a := div_pos (mul_pos (boundedClassWidth_pos kappa M) hr) (by norm_num) | |
| have hax : a ≤ 1/10 := by | |
| dsimp only [a] | |
| linarith | |
| have hq : |(q0 : ℝ)| *a < 1/10 := by | |
| have hm := mul_le_mul hparams.2.le hax ha.le (by norm_num : (0 : ℝ) ≤ 1/20) | |
| norm_num at hm | |
| linarith | |
| obtain ⟨N,hN,hfamily⟩ := exists_N_before_delta_eventually_bounded_height_approximation | |
| hg0 hV hSV M q0 hO hStar hOclosure a ha hax hq | |
| refine ⟨N,hN,?_⟩ | |
| intro delta hdelt | |
| filter_upwards [hfamily delta hdelt] with tau htau | |
| obtain ⟨eta,z,hclass,hcenter,_,_,hC⟩ := htau.2 | |
| exact ⟨htau.1,eta,z,hclass,hcenter.le,fun i j k hk p _ => hC i j k hk p⟩ | |
| end SmoothLocal.Perturbation | |
| end | |
| end OAI | |