| import OAI.Geometry.IsometricImmersion.Metrics.MetricQSpatialBundle | |
| import Mathlib.Analysis.Calculus.ContDiff.Bounds | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter Function | |
| open scoped ContDiff Topology BigOperators Matrix Matrix.Norms.Elementwise | |
| namespace SmoothLocal.HighEquation | |
| open SmoothLocal.Geometry SmoothLocal.Weighted SmoothLocal.ODE SmoothLocal.Hyperbolic | |
| theorem metricQSpatialBundle_mapsTo_domain {g : MetricField} {U : Set Coord} | |
| (hg : SmoothPositiveOn g U) (theta : ℝ) : | |
| MapsTo (metricQSpatialBundle g theta) (qFixedTimeDomain g U theta) universalMetricQDomain := by | |
| intro v hv | |
| exact ⟨metricDet_ne_zero hg hv.1, | |
| by simpa only [metricQSpatialBundle, Function.comp_apply, universalQDenominator_metricPBundle] using hv.2⟩ | |
| theorem actual_fixedTime_Q_jet_bound_from_universalTube | |
| {g : MetricField} {U : Set Coord} {theta : ℝ} {v : QSpatialState} | |
| (hg : SmoothPositiveOn g U) (hU : IsOpen U) | |
| (hv : v ∈ qFixedTimeDomain g U theta) | |
| {N : ℕ} {G W d c C : ℝ} (hG : 0 ≤ G) (hd : 0 < d) (hc : 0 < c) | |
| (hgB : HorizontalMetricTwoJetBound g (coordinatePoint (v 0) theta) N G) | |
| (hwB : ‖qFixedTimeState theta v‖ ≤ W) | |
| (hdet : d ≤ |(g (coordinatePoint (v 0) theta)).det|) | |
| (hden : c ≤ |stateQDenominator g (qFixedTimeState theta v)|) | |
| (hC : ∀ k ≤ N, ∀ a ∈ universalMetricQTube (max G W) d c, | |
| ‖iteratedFDeriv ℝ k universalMetricQ a‖ ≤ C) | |
| {n : ℕ} (hn : n ≤ N) : | |
| ‖iteratedFDeriv ℝ n (qFixedTimeQ g theta) v‖ ≤ n.factorial * C * (max 1 G)^n := by | |
| have hD := qFixedTimeDomain_isOpen hg hU theta | |
| have hsub : qFixedTimeDomain g U theta ⊆ qFixedTimeBaseDomain U theta := by | |
| intro w hw | |
| simpa only [statePoint_qFixedTimeState, qFixedTimeBaseDomain, Set.mem_ofPred_eq] using hw.1 | |
| have hf := (metricQSpatialBundle_contDiffOn hg hU theta).mono hsub | |
| have hmap := metricQSpatialBundle_mapsTo_domain hg theta | |
| have hmem : metricQSpatialBundle g theta v ∈ universalMetricQTube (max G W) d c := by | |
| refine ⟨⟨?_, ?_⟩, ?_⟩ | |
| · simpa only [Metric.mem_closedBall, dist_zero_right] using metricQSpatialBundle_norm_bound hG hgB hwB | |
| · simpa only [metricQSpatialBundle, Function.comp_apply, metricPBundle, statePoint_qFixedTimeState, | |
| Set.mem_preimage, Set.mem_Ici] using hdet | |
| · simpa only [Set.mem_preimage, Set.mem_ofPred_eq, metricQSpatialBundle, | |
| Function.comp_apply, universalQDenominator_metricPBundle] using hden | |
| have ho : ∀ i, i ≤ n → | |
| ‖iteratedFDerivWithin ℝ i universalMetricQ universalMetricQDomain (metricQSpatialBundle g theta v)‖ ≤ C := by | |
| intro i hi | |
| rw [iteratedFDerivWithin_of_isOpen i universalMetricQDomain_isOpen | |
| (universalMetricQTube_subset_domain (max G W) hd hc hmem)] | |
| exact hC i (hi.trans hn) _ hmem | |
| have hi : ∀ i, 1 ≤ i → i ≤ n → | |
| ‖iteratedFDerivWithin ℝ i (metricQSpatialBundle g theta) (qFixedTimeDomain g U theta) v‖ ≤ (max 1 G)^i := by | |
| intro i hi hin | |
| rw [iteratedFDerivWithin_of_isOpen i hD hv] | |
| exact (metricQSpatialBundle_positive_jet_bound hg hU (hsub hv) hG hgB hi (hin.trans hn)).trans | |
| (le_self_pow₀ (le_max_left _ _) (by omega)) | |
| have hcomp := norm_iteratedFDerivWithin_comp_le universalMetricQ_contDiffOn hf | |
| (WithTop.coe_le_coe.mpr le_top) universalMetricQDomain_isOpen.uniqueDiffOn hD.uniqueDiffOn | |
| hmap hv ho hi | |
| have heq : iteratedFDerivWithin ℝ n (qFixedTimeQ g theta) (qFixedTimeDomain g U theta) v = | |
| iteratedFDerivWithin ℝ n (universalMetricQ ∘ metricQSpatialBundle g theta) (qFixedTimeDomain g U theta) v := | |
| iteratedFDerivWithin_congr | |
| (fun w hw => sixVariableQ_eq_universalMetricQ hg hU hw.1) hv n | |
| rw [iteratedFDerivWithin_of_isOpen n hD hv] at heq | |
| rw [← heq] at hcomp | |
| exact hcomp | |
| theorem exists_local_fixedTime_Q_jet_bound (G W : ℝ) (hG : 0 ≤ G) | |
| {d c : ℝ} (hd : 0 < d) (hc : 0 < c) (N : ℕ) : | |
| ∃ C : ℝ, 0 ≤ C ∧ ∀ (g : MetricField) (U : Set Coord), | |
| SmoothPositiveOn g U → IsOpen U → ∀ (theta : ℝ) (v : QSpatialState), | |
| coordinatePoint (v 0) theta ∈ U → | |
| HorizontalMetricTwoJetBound g (coordinatePoint (v 0) theta) N G → | |
| ‖qFixedTimeState theta v‖ ≤ W → | |
| d ≤ |(g (coordinatePoint (v 0) theta)).det| → | |
| c ≤ |stateQDenominator g (qFixedTimeState theta v)| → | |
| ∀ n ≤ N, ‖iteratedFDeriv ℝ n (qFixedTimeQ g theta) v‖ ≤ C := by | |
| classical | |
| obtain ⟨C0, hC0, hCQ⟩ := universalMetricQTube_finite_bounds (max G W) hd hc N | |
| let C : ℝ := ∑ k : Fin (N + 1), (k.val.factorial : ℝ) * C0 * (max 1 G)^k.val | |
| have hC : 0 ≤ C := Finset.sum_nonneg (fun k _ => by positivity) | |
| refine ⟨C, hC, ?_⟩ | |
| intro g U hg hU theta v hp hgB hwB hdet hden n hn | |
| have hne : stateQDenominator g (qFixedTimeState theta v) ≠ 0 := by | |
| intro hz | |
| rw [hz, abs_zero] at hden | |
| linarith | |
| have hv : v ∈ qFixedTimeDomain g U theta := | |
| ⟨by simpa only [statePoint_qFixedTimeState] using hp, hne⟩ | |
| have hb := actual_fixedTime_Q_jet_bound_from_universalTube hg hU hv hG hd hc hgB hwB hdet hden hCQ hn | |
| let j : Fin (N + 1) := ⟨n, by omega⟩ | |
| have hsum : (j.val.factorial : ℝ) * C0 * (max 1 G)^j.val ≤ | |
| ∑ i : Fin (N+1), (i.val.factorial : ℝ) * C0 * (max 1 G)^i.val := | |
| Finset.single_le_sum | |
| (f := fun i : Fin (N+1) => (i.val.factorial : ℝ) * C0 * (max 1 G)^i.val) | |
| (fun i _ => by positivity) (Finset.mem_univ j) | |
| exact hb.trans hsum | |
| theorem qSolutionJet_norm_bound_of_coordinateBound | |
| {z : Coord → ℝ} {S : Set Coord} {R Z : ℝ} (hR : 0 ≤ R) | |
| (hzB : CoordinateBound z S 2 Z) | |
| (hcoords : ∀ p ∈ S, |p 0| ≤ R ∧ |p 1| ≤ R) {p : Coord} (hp : p ∈ S) : | |
| ‖qSolutionJet z p‖ ≤ max R Z := by | |
| apply (pi_norm_le_iff_of_nonneg (hR.trans (le_max_left _ _))).mpr | |
| intro i | |
| rw [Real.norm_eq_abs] | |
| fin_cases i | |
| · exact (hcoords p hp).1.trans (le_max_left _ _) | |
| · exact (hcoords p hp).2.trans (le_max_left _ _) | |
| · exact (hzB [0] (by norm_num) p hp).trans (le_max_right _ _) | |
| · exact (hzB [1] (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] (by norm_num) p hp).trans (le_max_right _ _) | |
| theorem exists_same_region_active_Q_coefficient_bound | |
| (G R Z : ℝ) (hG : 0 ≤ G) (hR : 0 ≤ R) | |
| {d c : ℝ} (hd : 0 < d) (hc : 0 < c) (m : ℕ) : | |
| ∃ C : ℝ, 0 ≤ C ∧ ∀ (g : MetricField) (z : Coord → ℝ) (U S : Set Coord), | |
| SmoothPositiveOn g U → IsOpen U → S ⊆ U → | |
| (∀ p ∈ S, |p 0| ≤ R ∧ |p 1| ≤ R) → | |
| (∀ p ∈ S, HorizontalMetricTwoJetBound g p (m + 3) G) → | |
| CoordinateBound z S 2 Z → | |
| (∀ p ∈ S, d ≤ |(g p).det|) → | |
| (∀ p ∈ S, c ≤ |covHessian g z p 0 0|) → | |
| ∀ w ∈ topResidualWords m, ∀ r, QActiveComponents w r → ∀ p ∈ S, | |
| |qCoordinateChainCoefficient g z w r p| ≤ C := by | |
| obtain ⟨C, hC, hQ⟩ := exists_local_fixedTime_Q_jet_bound G (max R Z) hG hd hc (m + 3) | |
| refine ⟨C, hC, ?_⟩ | |
| intro g z U S hg hU hSU hcoords hgB hzB hdet hden w hw r hactive 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 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 hzB hcoords hp) | |
| (by simpa only [hpoint] using hdet p hp) | |
| (by rw [he, stateQDenominator_qSolutionJet]; exact hden p hp) | |
| w.arity (q_residual_factor_orders hw r).2 | |
| exact (qCoordinateChainCoefficient_abs_le_fixedTime_derivative hg hU (hSU hp) hxx w r hactive).trans hbound | |
| end SmoothLocal.HighEquation | |
| end | |
| end OAI | |