openAI-math-dataset / lean /OAI /Geometry /IsometricImmersion /Estimates /BoundedClassHyperbolicity.lean
| import OAI.Geometry.IsometricImmersion.Coordinates.ActualShearHyperbolicity | |
| import OAI.Geometry.IsometricImmersion.Immersions.BoundedHeightClasses | |
| namespace OAI | |
| noncomputable section | |
| open Set | |
| open scoped ContDiff Matrix | |
| namespace SmoothLocal.Perturbation | |
| open SmoothLocal.Geometry SmoothLocal.Pulse SmoothLocal.HighEquation | |
| theorem boundedClassSpeed_sq {kappa : ℝ} (hkappa : 0 < kappa) | |
| {M : ℕ} (hM : 0 < M) : | |
| (boundedClassSpeed kappa M) ^ 2 = kappa / (2 * (M : ℝ) ^ 3) := by | |
| have hMp : (0 : ℝ) < (M : ℝ) := Nat.cast_pos.mpr hM | |
| have he : ((M : ℝ) ^ (-(3 : ℝ) / 2)) ^ (2 : ℕ) = ((M : ℝ) ^ (3 : ℕ))⁻¹ := by | |
| rw [← Real.rpow_mul_natCast hMp.le] | |
| norm_num [Real.rpow_neg, Real.rpow_natCast] | |
| unfold boundedClassSpeed | |
| rw [mul_pow, Real.sq_sqrt (by positivity), he] | |
| ring | |
| theorem boundedClassSpeed_sq_energy {kappa : ℝ} (hkappa : 0 < kappa) | |
| {M : ℕ} (hM : 0 < M) : | |
| (boundedClassSpeed kappa M) ^ 2 = | |
| kappa * (1 / (M : ℝ)) / (2 * (M : ℝ) ^ 2) := by | |
| rw [boundedClassSpeed_sq hkappa hM] | |
| field_simp | |
| theorem bounded_class_shear_estimates {g : MetricField} {z : Coord → ℝ} | |
| {M : ℕ} (h : BoundedAdmissibleHeight g M z) {kappa q0 : ℝ} | |
| (hkappa : 0 < kappa) {p : Coord} | |
| (hp : inverseShearCoordinates q0 p ∈ modelSquare) | |
| (hK : gaussianCurvature g (inverseShearCoordinates q0 p) ≤ -kappa / 2) | |
| (hq : |hessianQuotient g z (inverseShearCoordinates q0 p) - q0| ≤ | |
| 1 / (10 * boundedClassWidth kappa M)) : | |
| (boundedClassSpeed kappa M) ^ 2 / 2 ≤ -shearTimeFactor g z q0 p ∧ | |
| (boundedClassSpeed kappa M) ^ 2 / (2 * (M : ℝ)) ≤ | |
| |covHessian (metricInShearCoordinates g q0) (heightInShearCoordinates z q0) p 0 0| ∧ | |
| 0 < qFirstCoefficient (metricInShearCoordinates g q0) 5 | |
| (qSolutionJet (heightInShearCoordinates z q0) p) ∧ | |
| |qFirstCoefficient (metricInShearCoordinates g q0) 4 | |
| (qSolutionJet (heightInShearCoordinates z q0) p)| + | |
| Real.sqrt (qFirstCoefficient (metricInShearCoordinates g q0) 5 | |
| (qSolutionJet (heightInShearCoordinates z q0) p)) < boundedClassWidth kappa M / 4 := by | |
| obtain ⟨hM, hadm, hjet, hHb, hEb⟩ := h | |
| obtain ⟨U, hU, hSU, hg, hz, hD, hE, hyy, hsmall⟩ := hadm | |
| have hMp : (0 : ℝ) < (M : ℝ) := Nat.cast_pos.mpr hM | |
| have hv := boundedClassSpeed_pos hkappa hM | |
| obtain ⟨hd, hxx, hS, hchar⟩ := actual_shear_hyperbolicity hg hz hU (hSU hp) | |
| hkappa (one_div_pos.mpr hMp) hv (boundedClassSpeed_sq_energy hkappa hM).le | |
| (hyy _ hp) (hHb _ hp).2 (hEb _ hp) hK hq (hD _ hp) | |
| have hdneg : shearTimeFactor g z q0 p < 0 := by | |
| have hv2 : 0 < (boundedClassSpeed kappa M) ^ 2 / 2 := half_pos (sq_pos_of_pos hv) | |
| linarith | |
| have hfac : covHessian (metricInShearCoordinates g q0) (heightInShearCoordinates z q0) p 0 0 = | |
| shearTimeFactor g z q0 p * covHessian g z (inverseShearCoordinates q0 p) 1 1 := | |
| sheared_Hxx_factor hg hz hU q0 (hSU hp) (hyy _ hp) (hD _ hp) | |
| refine ⟨hd, ?_, hS, hchar⟩ | |
| calc | |
| _ = ((boundedClassSpeed kappa M) ^ 2 / 2) * (1 / (M : ℝ)) := by ring | |
| _ ≤ (-shearTimeFactor g z q0 p) * |covHessian g z (inverseShearCoordinates q0 p) 1 1| := | |
| mul_le_mul hd (hHb _ hp).1 (one_div_nonneg.mpr hMp.le) (neg_pos.mpr hdneg).le | |
| _ = _ := by rw [hfac, abs_mul, abs_of_neg hdneg] | |
| theorem boundedClassShearHxxFloor_pos {kappa : ℝ} (hkappa : 0 < kappa) | |
| {M : ℕ} (hM : 0 < M) : | |
| 0 < (boundedClassSpeed kappa M) ^ 2 / (2 * (M : ℝ)) := | |
| div_pos (sq_pos_of_pos (boundedClassSpeed_pos hkappa hM)) | |
| (mul_pos (by norm_num) (Nat.cast_pos.mpr hM)) | |
| end SmoothLocal.Perturbation | |
| end | |
| end OAI | |