openAI-math-dataset / lean /OAI /Geometry /IsometricImmersion /Darboux /QReferenceInitialTolerance.lean
| import OAI.Geometry.IsometricImmersion.Comparison.QFirstJetComparison | |
| import OAI.Geometry.IsometricImmersion.Taylor.TaylorCauchyData | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter | |
| open scoped ContDiff Topology Matrix Matrix.Norms.Elementwise | |
| namespace SmoothLocal.Pulse | |
| open SmoothLocal.Geometry SmoothLocal.HighEquation SmoothLocal.Taylor | |
| theorem exists_reference_XX_tolerance (B : ℝ) {d c : ℝ} (hd : 0 < d) (hc : 0 < c) : | |
| ∃ epsilon : ℝ, 0 < epsilon ∧ ∀ a b : QFirstInput, | |
| ‖a‖ ≤ B → d ≤ qFirstDet a → c ≤ |qFirstXX a| → | |
| ‖b - a‖ ≤ epsilon → c / 2 ≤ |qFirstXX b| := by | |
| obtain ⟨H, _, _, hH, _, _, hcompare⟩ := exists_uniform_qFirst_comparison B hd hc | |
| let D := 4 * max B 0 + 2 | |
| have hD : 0 < D := by dsimp [D]; positivity | |
| let epsilon := min 1 (min (d / (2 * D)) (c / (2 * (H + 1)))) | |
| have he : 0 < epsilon := by dsimp [epsilon]; positivity | |
| refine ⟨epsilon, he, ?_⟩ | |
| intro a b ha hdet hxx hdist | |
| have he1 : epsilon ≤ 1 := min_le_left _ _ | |
| have hed : epsilon ≤ d / (2 * D) := (min_le_right _ _).trans (min_le_left _ _) | |
| have hec : epsilon ≤ c / (2 * (H + 1)) := (min_le_right _ _).trans (min_le_right _ _) | |
| have hdist0 : 0 ≤ ‖b - a‖ := norm_nonneg _ | |
| have hdsmall : D * ‖b - a‖ ≤ d / 2 := by | |
| have hh := (le_div_iff₀ (show 0 < 2 * D by positivity)).mp (hdist.trans hed) | |
| nlinarith | |
| have hcsmall : H * ‖b - a‖ ≤ c / 2 := by | |
| have hh := (le_div_iff₀ (show 0 < 2 * (H + 1) by positivity)).mp (hdist.trans hec) | |
| nlinarith | |
| exact (hcompare a b ha hdet hxx (hdist.trans he1) hdsmall hcsmall).1 b | |
| (right_mem_segment ℝ a b) | |
| theorem exists_actual_reference_XX_tolerance (G W : ℝ) {d c : ℝ} (hd : 0 < d) (hc : 0 < c) : | |
| ∃ epsilon : ℝ, 0 < epsilon ∧ ∀ (g h : MetricField) (w : DarbouxState), | |
| ‖actualCurvatureFirstInput g (statePoint w)‖ ≤ G → ‖w‖ ≤ W → | |
| d ≤ (g (statePoint w)).det → c ≤ |stateQDenominator g w| → | |
| ‖actualCurvatureFirstInput h (statePoint w) - actualCurvatureFirstInput g (statePoint w)‖ ≤ epsilon → | |
| c / 2 ≤ |stateQDenominator h w| := by | |
| obtain ⟨epsilon, he, htol⟩ := exists_reference_XX_tolerance (max G W) hd hc | |
| refine ⟨epsilon, he, ?_⟩ | |
| intro g h w hg hw hdet hxx hdist | |
| have ha : ‖qFirstBundle g w‖ ≤ max G W := by | |
| change max ‖actualCurvatureFirstInput g (statePoint w)‖ ‖w‖ ≤ max G W | |
| exact max_le_max hg hw | |
| exact htol (qFirstBundle g w) (qFirstBundle h w) ha hdet hxx | |
| (by simpa only [qFirstBundle_distance] using hdist) | |
| theorem exists_reference_linearCauchy_XX_tolerance (G W : ℝ) {d c : ℝ} | |
| (hd : 0 < d) (hc : 0 < c) : | |
| ∃ epsilon : ℝ, 0 < epsilon ∧ ∀ (g h : MetricField) (z : Coord → ℝ) (U : Set Coord), | |
| ContDiffOn ℝ ∞ z U → IsOpen U → ∀ left right a : ℝ, | |
| (∀ x ∈ Ioo left right, (![x, a] : Coord) ∈ U) → | |
| ∀ x ∈ Ioo left right, | |
| ‖actualCurvatureFirstInput g ![x, a]‖ ≤ G → ‖qSolutionJet z ![x, a]‖ ≤ W → | |
| d ≤ (g ![x, a]).det → c ≤ |covHessian g z ![x, a] 0 0| → | |
| ‖actualCurvatureFirstInput h ![x, a] - actualCurvatureFirstInput g ![x, a]‖ ≤ epsilon → | |
| c / 2 ≤ |covHessian h (linearCauchy a (heightCauchyValue z a) (heightCauchyVelocity z a)) | |
| ![x, a] 0 0| := by | |
| obtain ⟨epsilon, he, htol⟩ := exists_actual_reference_XX_tolerance G W hd hc | |
| refine ⟨epsilon, he, ?_⟩ | |
| intro g h z U hz hU left right a hcut x hx hg hw hdet hxx hdist | |
| rw [linearCauchy_height_initial_hessian hz hU left right a hcut hx] | |
| have hb := htol g h (qSolutionJet z ![x, a]) | |
| (by simpa only [statePoint_qSolutionJet] using hg) hw | |
| (by simpa only [statePoint_qSolutionJet] using hdet) | |
| (by simpa only [stateQDenominator_qSolutionJet] using hxx) | |
| (by simpa only [statePoint_qSolutionJet] using hdist) | |
| simpa only [stateQDenominator_qSolutionJet] using hb | |
| end SmoothLocal.Pulse | |
| end | |
| end OAI | |