| import OAI.Geometry.IsometricImmersion.Calculus.HorizontalJetCalculus | |
| import OAI.Geometry.IsometricImmersion.Darboux.QPrincipal | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter Function | |
| open scoped ContDiff Topology Matrix BigOperators | |
| namespace SmoothLocal.HighEquation | |
| open SmoothLocal.Geometry SmoothLocal.Weighted SmoothLocal.ODE SmoothLocal.Hyperbolic | |
| def qSolutionJetCurve (z : Coord → ℝ) (y : ℝ) : ℝ → DarbouxState := | |
| fun x => qSolutionJet z (coordinatePoint x y) | |
| theorem qSolutionJet_contDiffOn {z : Coord → ℝ} {U : Set Coord} | |
| (hU : IsOpen U) (hz : ContDiffOn ℝ ∞ z U) : | |
| ContDiffOn ℝ ∞ (qSolutionJet z) U := by | |
| apply contDiffOn_pi.mpr | |
| intro i | |
| fin_cases i | |
| · exact contDiffOn_apply ℝ ℝ 0 U | |
| · exact contDiffOn_apply ℝ ℝ 1 U | |
| · exact partial_contDiffOn hz hU 0 | |
| · exact partial_contDiffOn hz hU 1 | |
| · exact partial_contDiffOn (partial_contDiffOn hz hU 1) hU 0 | |
| · exact partial_contDiffOn (partial_contDiffOn hz hU 0) hU 0 | |
| theorem qSolutionJetCurve_contDiffAt {z : Coord → ℝ} {U : Set Coord} {x y : ℝ} | |
| (hU : IsOpen U) (hz : ContDiffOn ℝ ∞ z U) (hp : coordinatePoint x y ∈ U) : | |
| ContDiffAt ℝ ∞ (qSolutionJetCurve z y) x := | |
| ((qSolutionJet_contDiffOn hU hz).contDiffAt (hU.mem_nhds hp)).comp x | |
| (horizontalPoint_contDiff y).contDiffAt | |
| theorem sixVariableQ_contDiffAt_solutionJet | |
| {g : MetricField} {z : Coord → ℝ} {U : Set Coord} {p : Coord} | |
| (hg : SmoothPositiveOn g U) (hU : IsOpen U) (hp : p ∈ U) | |
| (hxx : covHessian g z p 0 0 ≠ 0) : | |
| ContDiffAt ℝ ∞ (sixVariableQ g) (qSolutionJet z p) := | |
| (sixVariableQ_contDiffOn hg hU).contDiffAt | |
| ((darbouxQStateDomain_isOpen hg hU).mem_nhds (qSolutionJet_mem_domain hp hxx)) | |
| theorem qSolutionJetCurve_iteratedDeriv_positive | |
| {z : Coord → ℝ} {U : Set Coord} {x y : ℝ} | |
| (hU : IsOpen U) (hz : ContDiffOn ℝ ∞ z U) (hp : coordinatePoint x y ∈ U) | |
| (n : ℕ) (hn : 0 < n) : | |
| iteratedDeriv n (qSolutionJetCurve z y) x = | |
| ![if n = 1 then 1 else 0, 0, | |
| spatialJet (spatialFirstJet z 0) n (coordinatePoint x y), | |
| spatialJet (spatialFirstJet z 1) n (coordinatePoint x y), | |
| spatialJet (spatialFirstJet z 1) (n + 1) (coordinatePoint x y), | |
| spatialJet (spatialFirstJet z 0) (n + 1) (coordinatePoint x y)] := by | |
| have hc := qSolutionJetCurve_contDiffAt hU hz hp | |
| ext i | |
| rw [iteratedDeriv_state_eval hc n i] | |
| fin_cases i | |
| · simpa [qSolutionJetCurve, qSolutionJet, coordinatePoint, hn.ne'] using | |
| (iteratedDeriv_fun_id (n := n) (x := x)) | |
| · simp [qSolutionJetCurve, qSolutionJet, coordinatePoint, iteratedDeriv_const, hn.ne'] | |
| · change iteratedDeriv n (fun t => coordPartial 0 z (coordinatePoint t y)) x = _ | |
| rw [horizontalJet_slice hU (partial_contDiffOn hz hU 0) y n x hp, | |
| horizontalJet_eq_spatialJet] | |
| rfl | |
| · change iteratedDeriv n (fun t => coordPartial 1 z (coordinatePoint t y)) x = _ | |
| rw [horizontalJet_slice hU (partial_contDiffOn hz hU 1) y n x hp, | |
| horizontalJet_eq_spatialJet] | |
| rfl | |
| · change iteratedDeriv n (fun t => coordPartial 0 (coordPartial 1 z) (coordinatePoint t y)) x = _ | |
| rw [horizontalJet_slice hU (partial_contDiffOn (partial_contDiffOn hz hU 1) hU 0) y n x hp, | |
| horizontalJet_coordPartial hU (partial_contDiffOn hz hU 1) 0 n _ hp, | |
| horizontalJet_eq_spatialJet] | |
| rfl | |
| · change iteratedDeriv n (fun t => coordPartial 0 (coordPartial 0 z) (coordinatePoint t y)) x = _ | |
| rw [horizontalJet_slice hU (partial_contDiffOn (partial_contDiffOn hz hU 0) hU 0) y n x hp, | |
| horizontalJet_coordPartial hU (partial_contDiffOn hz hU 0) 0 n _ hp, | |
| horizontalJet_eq_spatialJet] | |
| rfl | |
| theorem qSolutionJetCurve_iteratedDeriv | |
| {z : Coord → ℝ} {U : Set Coord} {x y : ℝ} | |
| (hU : IsOpen U) (hz : ContDiffOn ℝ ∞ z U) (hp : coordinatePoint x y ∈ U) | |
| (n : ℕ) (hn : 2 ≤ n) : | |
| iteratedDeriv n (qSolutionJetCurve z y) x = | |
| ![0, 0, coordPartial 0 (horizontalJet z n) (coordinatePoint x y), | |
| coordPartial 1 (horizontalJet z n) (coordinatePoint x y), | |
| coordPartial 0 (coordPartial 1 (horizontalJet z n)) (coordinatePoint x y), | |
| coordPartial 0 (coordPartial 0 (horizontalJet z n)) (coordinatePoint x y)] := by | |
| have hc := qSolutionJetCurve_contDiffAt hU hz hp | |
| ext i | |
| rw [iteratedDeriv_state_eval hc n i] | |
| fin_cases i | |
| · simp [qSolutionJetCurve, qSolutionJet, coordinatePoint, iteratedDeriv_fun_id, | |
| show n ≠ 0 by omega, show n ≠ 1 by omega] | |
| · simp [qSolutionJetCurve, qSolutionJet, coordinatePoint, iteratedDeriv_const, | |
| show n ≠ 0 by omega] | |
| · change iteratedDeriv n (fun t => coordPartial 0 z (coordinatePoint t y)) x = _ | |
| rw [horizontalJet_slice hU (partial_contDiffOn hz hU 0) y n x hp, | |
| horizontalJet_coordPartial hU hz 0 n _ hp] | |
| rfl | |
| · change iteratedDeriv n (fun t => coordPartial 1 z (coordinatePoint t y)) x = _ | |
| rw [horizontalJet_slice hU (partial_contDiffOn hz hU 1) y n x hp, | |
| horizontalJet_coordPartial hU hz 1 n _ hp] | |
| rfl | |
| · change iteratedDeriv n (fun t => coordPartial 0 (coordPartial 1 z) (coordinatePoint t y)) x = _ | |
| rw [horizontalJet_slice hU (partial_contDiffOn (partial_contDiffOn hz hU 1) hU 0) y n x hp, | |
| horizontalJet_two_coordPartials hU hz hp n 0 1] | |
| rfl | |
| · change iteratedDeriv n (fun t => coordPartial 0 (coordPartial 0 z) (coordinatePoint t y)) x = _ | |
| rw [horizontalJet_slice hU (partial_contDiffOn (partial_contDiffOn hz hU 0) hU 0) y n x hp, | |
| horizontalJet_two_coordPartials hU hz hp n 0 0] | |
| rfl | |
| end SmoothLocal.HighEquation | |
| end | |
| end OAI | |