| import OAI.Geometry.IsometricImmersion.Metrics.LocalMetricQActive |
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| namespace OAI |
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| noncomputable section |
| open Set Filter Function |
| open scoped ContDiff Topology BigOperators Matrix |
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| namespace SmoothLocal.HighEquation |
| open SmoothLocal.Geometry SmoothLocal.Weighted SmoothLocal.ODE SmoothLocal.Hyperbolic |
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| theorem qSpatialProjection_norm_le (w : DarbouxState) : ‖qSpatialProjection w‖ ≤ ‖w‖ := by |
| apply (pi_norm_le_iff_of_nonneg (norm_nonneg w)).mpr |
| intro i |
| fin_cases i |
| · exact norm_le_pi_norm w 0 |
| · exact norm_le_pi_norm w 2 |
| · exact norm_le_pi_norm w 3 |
| · exact norm_le_pi_norm w 4 |
| · exact norm_le_pi_norm w 5 |
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| theorem qSpatialCLM_projection_of_time_zero {w : DarbouxState} (hw : w 1 = 0) : |
| qSpatialCLM (qSpatialProjection w) = w := by |
| have he := qFixedTimeState_projection w |
| simpa only [qFixedTimeState, hw, Pi.single_zero, add_zero] using he |
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| theorem coordPartial_qSolutionJet_xi_norm_bound |
| {z : Coord → ℝ} {U S : Set Coord} {Z : ℝ} |
| (hU : IsOpen U) (hz : ContDiffOn ℝ ∞ z U) (hSU : S ⊆ U) |
| (hzB : CoordinateBound z S 3 Z) {p : Coord} (hp : p ∈ S) : |
| ‖coordPartial 0 (qSolutionJet z) p‖ ≤ max 1 Z := by |
| have he := coordinate_qSolutionJet_horizontal_positive hU hz (hSU hp) 1 (by norm_num) |
| change coordPartial 0 (qSolutionJet z) p = _ at he |
| rw [he] |
| apply (pi_norm_le_iff_of_nonneg (zero_le_one.trans (le_max_left _ _))).mpr |
| intro i |
| rw [Real.norm_eq_abs] |
| fin_cases i |
| · simp |
| · simp |
| · exact (hzB [0, 0] (by norm_num) p hp).trans (le_max_right _ _) |
| · exact (hzB [0, 1] (by norm_num) p hp).trans (le_max_right _ _) |
| · exact (hzB [0, 0, 1] (by norm_num) p hp).trans (le_max_right _ _) |
| · exact (hzB [0, 0, 0] (by norm_num) p hp).trans (le_max_right _ _) |
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| theorem heightQCoefficient_xi_eq_fixedTime_second |
| {g : MetricField} {z : Coord → ℝ} {U : Set Coord} {p : Coord} |
| (hg : SmoothPositiveOn g U) (hU : IsOpen U) (hz : ContDiffOn ℝ ∞ z U) |
| (hp : p ∈ U) (hxx : covHessian g z p 0 0 ≠ 0) (i : Fin 6) (hi : i ≠ 1) : |
| coordPartial 0 (heightQCoefficient g z i) p = |
| iteratedFDeriv ℝ 2 (qFixedTimeQ g (p 1)) (qSpatialProjection (qSolutionJet z p)) |
| ![qSpatialProjection (coordPartial 0 (qSolutionJet z) p), Pi.single (qSpatialIndex i) 1] := by |
| have he : qFixedTimeState (p 1) (qSpatialProjection (qSolutionJet z p)) = qSolutionJet z p := |
| qFixedTimeState_projection (qSolutionJet z p) |
| have hv : qSpatialProjection (qSolutionJet z p) ∈ qFixedTimeDomain g U (p 1) := by |
| change qFixedTimeState (p 1) (qSpatialProjection (qSolutionJet z p)) ∈ darbouxQStateDomain g U |
| rw [he] |
| exact qSolutionJet_mem_domain hp hxx |
| have htime : coordPartial 0 (qSolutionJet z) p 1 = 0 := by |
| have hh := congrArg (fun v : DarbouxState => v 1) |
| (coordinate_qSolutionJet_horizontal_positive hU hz hp 1 (by norm_num)) |
| exact hh |
| rw [qFixedTimeQ_iteratedFDeriv hg hU hv 2, ContinuousMultilinearMap.compContinuousLinearMap_apply, he] |
| have hvec : (fun j : Fin 2 => qSpatialCLM |
| (![qSpatialProjection (coordPartial 0 (qSolutionJet z) p), Pi.single (qSpatialIndex i) 1] j)) = |
| ![coordPartial 0 (qSolutionJet z) p, Pi.single i 1] := by |
| funext j |
| fin_cases j |
| · exact qSpatialCLM_projection_of_time_zero htime |
| · change qSpatialCLM (Pi.single (qSpatialIndex i) 1) = Pi.single i 1 |
| rw [qSpatialCLM_single, qSpatialAxis_index hi] |
| rw [hvec] |
| exact (second_fderiv_comp_constant_direction_xi |
| (sixVariableQ_contDiffAt_solutionJet hg hU hp hxx) |
| (((qSolutionJet_contDiffOn hU hz).contDiffAt (hU.mem_nhds hp)).differentiableAt (by simp)) |
| (Pi.single i 1)).symm |
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| theorem heightQCoefficient_xi_bound_of_fixedTime_second |
| {g : MetricField} {z : Coord → ℝ} {U S : Set Coord} {Z C : ℝ} |
| (hg : SmoothPositiveOn g U) (hU : IsOpen U) (hz : ContDiffOn ℝ ∞ z U) |
| (hSU : S ⊆ U) (hzB : CoordinateBound z S 3 Z) (hC : 0 ≤ C) |
| {p : Coord} (hp : p ∈ S) (hxx : covHessian g z p 0 0 ≠ 0) |
| (hQ : ‖iteratedFDeriv ℝ 2 (qFixedTimeQ g (p 1)) (qSpatialProjection (qSolutionJet z p))‖ ≤ C) |
| (i : Fin 6) (hi : i ≠ 1) : |
| |coordPartial 0 (heightQCoefficient g z i) p| ≤ C * max 1 Z := by |
| rw [heightQCoefficient_xi_eq_fixedTime_second hg hU hz (hSU hp) hxx i hi, ← Real.norm_eq_abs] |
| have hd := (qSpatialProjection_norm_le (coordPartial 0 (qSolutionJet z) p)).trans |
| (coordPartial_qSolutionJet_xi_norm_bound hU hz hSU hzB hp) |
| have hb := (iteratedFDeriv ℝ 2 (qFixedTimeQ g (p 1)) |
| (qSpatialProjection (qSolutionJet z p))).le_opNorm |
| ![qSpatialProjection (coordPartial 0 (qSolutionJet z) p), Pi.single (qSpatialIndex i) (1 : ℝ)] |
| simp only [Fin.prod_univ_two, Matrix.cons_val_zero, Matrix.cons_val_one, Pi.norm_single, norm_one, mul_one] at hb |
| exact hb.trans (mul_le_mul hQ hd (norm_nonneg _) hC) |
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| theorem exists_same_region_Q_first_xi_bound |
| (G R Z : ℝ) (hG : 0 ≤ G) (hR : 0 ≤ R) |
| {d c : ℝ} (hd : 0 < d) (hc : 0 < c) : |
| ∃ Cfirst : ℝ, 0 ≤ Cfirst ∧ ∀ (g : MetricField) (z : Coord → ℝ) (U S : Set Coord), |
| SmoothPositiveOn g U → IsOpen U → ContDiffOn ℝ ∞ z U → S ⊆ U → |
| (∀ p ∈ S, |p 0| ≤ R ∧ |p 1| ≤ R) → |
| (∀ p ∈ S, HorizontalMetricTwoJetBound g p 2 G) → |
| CoordinateBound z S 3 Z → |
| (∀ p ∈ S, d ≤ |(g p).det|) → |
| (∀ p ∈ S, c ≤ |covHessian g z p 0 0|) → |
| ∀ i : Fin 6, i ≠ 1 → ∀ p ∈ S, |coordPartial 0 (heightQCoefficient g z i) p| ≤ Cfirst := by |
| obtain ⟨C, hC, hQ⟩ := exists_local_fixedTime_Q_jet_bound G (max R Z) hG hd hc 2 |
| refine ⟨C * max 1 Z, mul_nonneg hC (zero_le_one.trans (le_max_left _ _)), ?_⟩ |
| intro g z U S hg hU hz hSU hcoords hgB hzB hdet hden i hi p hp |
| have hxx : covHessian g z p 0 0 ≠ 0 := by |
| intro he |
| have hh := hden p hp |
| rw [he, abs_zero] at hh |
| linarith |
| have he : qFixedTimeState (p 1) (qSpatialProjection (qSolutionJet z p)) = qSolutionJet z p := |
| qFixedTimeState_projection (qSolutionJet z p) |
| have hpoint : coordinatePoint (qSpatialProjection (qSolutionJet z p) 0) (p 1) = p := point_eta p |
| have hz2 : CoordinateBound z S 2 Z := fun ds hds p hp => hzB ds (by omega) p hp |
| have hbound := hQ g U hg hU (p 1) (qSpatialProjection (qSolutionJet z p)) |
| (by simpa only [hpoint] using hSU hp) |
| (by simpa only [hpoint] using hgB p hp) |
| (by rw [he]; exact qSolutionJet_norm_bound_of_coordinateBound hR hz2 hcoords hp) |
| (by simpa only [hpoint] using hdet p hp) |
| (by rw [he, stateQDenominator_qSolutionJet]; exact hden p hp) 2 le_rfl |
| exact heightQCoefficient_xi_bound_of_fixedTime_second hg hU hz hSU hzB hC hp hxx hbound i hi |
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| end SmoothLocal.HighEquation |
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| end |
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| end OAI |
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