| import OAI.Geometry.IsometricImmersion.Taylor.TaylorSeparatedProduct |
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| namespace OAI |
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| noncomputable section |
| open Set Filter Function |
| open scoped ContDiff Topology Matrix |
|
|
| namespace SmoothLocal.Taylor |
| open SmoothLocal.Geometry SmoothLocal.HighEquation |
|
|
| theorem secondCoordPartial_add {P R : Coord → ℝ} {V : Set Coord} |
| (hV : IsOpen V) (hP : ContDiffOn ℝ ∞ P V) (hR : ContDiffOn ℝ ∞ R V) |
| {p : Coord} (hp : p ∈ V) (i j : Fin 2) : |
| coordPartial i (coordPartial j (fun q => P q + R q)) p = |
| coordPartial i (coordPartial j P) p + coordPartial i (coordPartial j R) p := by |
| have he : coordPartial j (fun q => P q + R q) =ᶠ[𝓝 p] |
| (fun q => coordPartial j P q + coordPartial j R q) := by |
| filter_upwards [hV.mem_nhds hp] with q hq |
| exact HessianCalculus.coordPartial_add_at |
| ((hP.contDiffAt (hV.mem_nhds hq)).differentiableAt (by simp)) |
| ((hR.contDiffAt (hV.mem_nhds hq)).differentiableAt (by simp)) j |
| rw [coordPartial_eq_of_eventuallyEq he i] |
| exact HessianCalculus.coordPartial_add_at |
| (((partial_contDiffOn hP hV j).contDiffAt (hV.mem_nhds hp)).differentiableAt (by simp)) |
| (((partial_contDiffOn hR hV j).contDiffAt (hV.mem_nhds hp)).differentiableAt (by simp)) i |
|
|
| def correctionState (C : ℝ → ℝ) (a : ℝ) (n : ℕ) (p : Coord) : DarbouxState := |
| ![0, 0, |
| deriv C (p 0) * normalizedTimePower a (n + 2) (p 1), |
| C (p 0) * normalizedTimePower a (n + 1) (p 1), |
| deriv C (p 0) * normalizedTimePower a (n + 1) (p 1), |
| deriv (deriv C) (p 0) * normalizedTimePower a (n + 2) (p 1)] |
|
|
| theorem correctionState_slice_contDiff (C : ℝ → ℝ) (a : ℝ) (n : ℕ) (x : ℝ) : |
| ContDiff ℝ ∞ (fun t => correctionState C a n ![x, t]) := by |
| apply contDiff_pi.mpr |
| intro i |
| fin_cases i |
| · exact contDiff_const |
| · exact contDiff_const |
| · change ContDiff ℝ ∞ (fun t => deriv C x * normalizedTimePower a (n + 2) t) |
| exact contDiff_const.mul (normalizedTimePower_contDiff a (n + 2)) |
| · change ContDiff ℝ ∞ (fun t => C x * normalizedTimePower a (n + 1) t) |
| exact contDiff_const.mul (normalizedTimePower_contDiff a (n + 1)) |
| · change ContDiff ℝ ∞ (fun t => deriv C x * normalizedTimePower a (n + 1) t) |
| exact contDiff_const.mul (normalizedTimePower_contDiff a (n + 1)) |
| · change ContDiff ℝ ∞ (fun t => deriv (deriv C) x * normalizedTimePower a (n + 2) t) |
| exact contDiff_const.mul (normalizedTimePower_contDiff a (n + 2)) |
|
|
| theorem correctionState_time_jets_zero (C : ℝ → ℝ) (a : ℝ) (n : ℕ) (x : ℝ) |
| (k : ℕ) (hk : k ≤ n) : |
| iteratedDeriv k (fun t => correctionState C a n ![x, t]) a = 0 := by |
| ext i |
| rw [iteratedDeriv_state_eval (correctionState_slice_contDiff C a n x).contDiffAt k i] |
| fin_cases i |
| · exact iteratedDeriv_fun_const_zero |
| · exact iteratedDeriv_fun_const_zero |
| · change iteratedDeriv k (fun t => deriv C x * normalizedTimePower a (n + 2) t) a = 0 |
| rw [coefficient_timePower_jet, ite_eq_right (by omega)] |
| · change iteratedDeriv k (fun t => C x * normalizedTimePower a (n + 1) t) a = 0 |
| rw [coefficient_timePower_jet, ite_eq_right (by omega)] |
| · change iteratedDeriv k (fun t => deriv C x * normalizedTimePower a (n + 1) t) a = 0 |
| rw [coefficient_timePower_jet, ite_eq_right (by omega)] |
| · change iteratedDeriv k (fun t => deriv (deriv C) x * normalizedTimePower a (n + 2) t) a = 0 |
| rw [coefficient_timePower_jet, ite_eq_right (by omega)] |
|
|
| theorem qSolutionJet_add_taylorCorrection {P : Coord → ℝ} {C : ℝ → ℝ} {I : Set ℝ} |
| (hI : IsOpen I) (hP : ContDiffOn ℝ ∞ P (spatialStrip I)) |
| (hC : ContDiffOn ℝ ∞ C I) (a : ℝ) (n : ℕ) |
| {p : Coord} (hp : p ∈ spatialStrip I) : |
| qSolutionJet (fun q => P q + taylorCorrection C a n q) p = |
| qSolutionJet P p + correctionState C a n p := by |
| have hV := spatialStrip_isOpen hI |
| have hR := taylorCorrection_contDiffOn hC a n |
| have hdP := (hP.contDiffAt (hV.mem_nhds hp)).differentiableAt (by simp) |
| have hdR := (hR.contDiffAt (hV.mem_nhds hp)).differentiableAt (by simp) |
| ext i |
| fin_cases i |
| · exact (add_zero (p 0)).symm |
| · exact (add_zero (p 1)).symm |
| · change coordPartial 0 (fun q => P q + taylorCorrection C a n q) p = _ |
| rw [HessianCalculus.coordPartial_add_at hdP hdR 0, |
| taylorCorrection_partial_x hC hI a n hp] |
| rfl |
| · change coordPartial 1 (fun q => P q + taylorCorrection C a n q) p = _ |
| rw [HessianCalculus.coordPartial_add_at hdP hdR 1, |
| taylorCorrection_partial_t hC hI a n hp] |
| rfl |
| · change coordPartial 0 (coordPartial 1 (fun q => P q + taylorCorrection C a n q)) p = _ |
| rw [secondCoordPartial_add hV hP hR hp 0 1, |
| taylorCorrection_partial_xt hC hI a n hp] |
| rfl |
| · change coordPartial 0 (coordPartial 0 (fun q => P q + taylorCorrection C a n q)) p = _ |
| rw [secondCoordPartial_add hV hP hR hp 0 0, |
| taylorCorrection_partial_xx hC hI a n hp] |
| rfl |
|
|
| theorem qSolutionJet_time_slice_contDiffAt {P : Coord → ℝ} {I : Set ℝ} |
| (hI : IsOpen I) (hP : ContDiffOn ℝ ∞ P (spatialStrip I)) |
| {x : ℝ} (hx : x ∈ I) (a : ℝ) : |
| ContDiffAt ℝ ∞ (fun t => qSolutionJet P ![x, t]) a := |
| ((qSolutionJet_contDiffOn (spatialStrip_isOpen hI) hP).contDiffAt |
| ((spatialStrip_isOpen hI).mem_nhds hx)).comp a (verticalPoint_contDiff x).contDiffAt |
|
|
| theorem qSolutionJet_taylorCorrection_time_jets |
| {P : Coord → ℝ} {C : ℝ → ℝ} {I : Set ℝ} |
| (hI : IsOpen I) (hP : ContDiffOn ℝ ∞ P (spatialStrip I)) |
| (hC : ContDiffOn ℝ ∞ C I) (a : ℝ) (n : ℕ) {x : ℝ} (hx : x ∈ I) : |
| ∀ k ≤ n, |
| iteratedDeriv k (fun t => qSolutionJet |
| (fun q => P q + taylorCorrection C a n q) ![x, t]) a = |
| iteratedDeriv k (fun t => qSolutionJet P ![x, t]) a := by |
| have he : (fun t => qSolutionJet (fun q => P q + taylorCorrection C a n q) ![x, t]) = |
| (fun t => qSolutionJet P ![x, t] + correctionState C a n ![x, t]) := by |
| funext t |
| exact qSolutionJet_add_taylorCorrection hI hP hC a n hx |
| rw [he] |
| exact iteratedDeriv_add_eq_of_zero_jets |
| (qSolutionJet_time_slice_contDiffAt hI hP hx a) |
| (correctionState_slice_contDiff C a n x).contDiffAt |
| (fun k hk => correctionState_time_jets_zero C a n x k hk) |
|
|
| theorem sixVariableQ_taylorCorrection_time_jets |
| {g : MetricField} {U : Set Coord} {P : Coord → ℝ} {C : ℝ → ℝ} {I : Set ℝ} |
| (hg : SmoothPositiveOn g U) (hU : IsOpen U) |
| (hI : IsOpen I) (hP : ContDiffOn ℝ ∞ P (spatialStrip I)) |
| (hC : ContDiffOn ℝ ∞ C I) (a : ℝ) (n : ℕ) {x : ℝ} (hx : x ∈ I) |
| (hp : (![x, a] : Coord) ∈ U) (hxx : covHessian g P ![x, a] 0 0 ≠ 0) : |
| ∀ k ≤ n, |
| iteratedDeriv k (fun t => sixVariableQ g (qSolutionJet |
| (fun q => P q + taylorCorrection C a n q) ![x, t])) a = |
| iteratedDeriv k (fun t => sixVariableQ g (qSolutionJet P ![x, t])) a := by |
| intro k hk |
| have hnew := hP.add (taylorCorrection_contDiffOn hC a n) |
| have hj := qSolutionJet_taylorCorrection_time_jets hI hP hC a n hx |
| have hvalue : qSolutionJet (fun q => P q + taylorCorrection C a n q) ![x, a] = |
| qSolutionJet P ![x, a] := by simpa only [iteratedDeriv_zero] using hj 0 (by omega) |
| have hQ := sixVariableQ_contDiffAt_solutionJet hg hU hp hxx |
| rw [← hvalue] at hQ |
| exact iteratedDeriv_comp_eq_of_jets (Q := sixVariableQ g) |
| (qSolutionJet_time_slice_contDiffAt hI hnew hx a) |
| (qSolutionJet_time_slice_contDiffAt hI hP hx a) hQ |
| (fun j hjk => hj j (hjk.trans hk)) |
|
|
| end SmoothLocal.Taylor |
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| end |
|
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| end OAI |
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