| import OAI.Geometry.IsometricImmersion.Taylor.ActualDarbouxFaa |
| import OAI.Geometry.IsometricImmersion.Calculus.NormedCoordinateCalculus |
| import Mathlib.Analysis.Calculus.Deriv.Prod |
| import Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter Function |
| open scoped ContDiff Topology BigOperators Matrix |
|
|
| namespace SmoothLocal.HighEquation |
| open SmoothLocal.Geometry |
|
|
| def verticalJet {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] |
| (f : Coord → V) : ℕ → Coord → V |
| | 0 => f |
| | n + 1 => coordPartial 1 (verticalJet f n) |
|
|
| theorem verticalJet_contDiffOn |
| {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] |
| {f : Coord → V} {U : Set Coord} (hU : IsOpen U) (hf : ContDiffOn ℝ ∞ f U) : |
| ∀ n, ContDiffOn ℝ ∞ (verticalJet f n) U := by |
| intro n |
| induction n with |
| | zero => exact hf |
| | succ n hn => exact normed_coordPartial_contDiffOn hn hU 1 |
|
|
| theorem verticalPoint_hasDerivAt (x y : ℝ) : |
| HasDerivAt (fun t : ℝ => (![x, t] : Coord)) (Pi.single 1 (1 : ℝ)) y := by |
| apply hasDerivAt_pi.mpr |
| intro i |
| fin_cases i |
| · simpa using hasDerivAt_const y x |
| · convert hasDerivAt_id y using 1 <;> rfl |
|
|
| theorem verticalJet_slice_hasDerivAt |
| {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] |
| {f : Coord → V} {U : Set Coord} {x y : ℝ} |
| (hU : IsOpen U) (hf : ContDiffOn ℝ ∞ f U) (n : ℕ) |
| (hp : (![x, y] : Coord) ∈ U) : |
| HasDerivAt (fun t => verticalJet f n ![x, t]) (verticalJet f (n + 1) ![x, y]) y := by |
| have hd : DifferentiableAt ℝ (verticalJet f n) ![x, y] := |
| (((verticalJet_contDiffOn hU hf n).contDiffAt (hU.mem_nhds hp)).differentiableAt (by simp)) |
| exact hd.hasFDerivAt.comp_hasDerivAt y (verticalPoint_hasDerivAt x y) |
|
|
| theorem verticalJet_slice |
| {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] |
| {f : Coord → V} {U : Set Coord} (hU : IsOpen U) (hf : ContDiffOn ℝ ∞ f U) |
| (x : ℝ) : ∀ n y, (![x, y] : Coord) ∈ U → |
| iteratedDeriv n (fun t => f ![x, t]) y = verticalJet f n ![x, y] := by |
| intro n |
| induction n with |
| | zero => intro y hy; rfl |
| | succ n hn => |
| intro y hy |
| rw [iteratedDeriv_succ] |
| have heq : iteratedDeriv n (fun t => f ![x, t]) =ᶠ[𝓝 y] |
| (fun t => verticalJet f n ![x, t]) := by |
| filter_upwards [(hU.preimage (verticalPoint_contDiff x).continuous).mem_nhds hy] with t ht |
| exact hn t ht |
| rw [heq.deriv_eq] |
| exact (verticalJet_slice_hasDerivAt hU hf n hy).deriv |
|
|
| theorem verticalJet_coordPartial |
| {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] |
| {f : Coord → V} {U : Set Coord} (hU : IsOpen U) (hf : ContDiffOn ℝ ∞ f U) |
| (i : Fin 2) : ∀ n p, p ∈ U → |
| verticalJet (coordPartial i f) n p = coordPartial i (verticalJet f n) p := by |
| intro n |
| induction n with |
| | zero => intro p hp; rfl |
| | succ n hn => |
| intro p hp |
| have heq : verticalJet (coordPartial i f) n =ᶠ[𝓝 p] coordPartial i (verticalJet f n) := by |
| filter_upwards [hU.mem_nhds hp] with q hq |
| exact hn q hq |
| have hd : coordPartial 1 (verticalJet (coordPartial i f) n) p = |
| coordPartial 1 (coordPartial i (verticalJet f n)) p := by |
| exact congrArg (fun L : Coord →L[ℝ] V => L (Pi.single 1 1)) heq.fderiv_eq |
| change coordPartial 1 (verticalJet (coordPartial i f) n) p = |
| coordPartial i (coordPartial 1 (verticalJet f n)) p |
| rw [hd] |
| exact coordPartial_comm (verticalJet_contDiffOn hU hf n) hU hp 1 i |
|
|
| theorem verticalJet_two_coordPartials |
| {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] |
| {f : Coord → V} {U : Set Coord} {p : Coord} |
| (hU : IsOpen U) (hf : ContDiffOn ℝ ∞ f U) (hp : p ∈ U) (n : ℕ) (i j : Fin 2) : |
| verticalJet (coordPartial i (coordPartial j f)) n p = |
| coordPartial i (coordPartial j (verticalJet f n)) p := by |
| rw [verticalJet_coordPartial hU (normed_coordPartial_contDiffOn hf hU j) i n p hp] |
| have heq : verticalJet (coordPartial j f) n =ᶠ[𝓝 p] coordPartial j (verticalJet f n) := by |
| filter_upwards [hU.mem_nhds hp] with q hq |
| exact verticalJet_coordPartial hU hf j n q hq |
| exact congrArg (fun L : Coord →L[ℝ] V => L (Pi.single i 1)) heq.fderiv_eq |
|
|
| theorem iteratedDeriv_state_eval {f : ℝ → DarbouxState} {y : ℝ} |
| (hf : ContDiffAt ℝ ∞ f y) (n : ℕ) (i : Fin 6) : |
| (iteratedDeriv n f y) i = iteratedDeriv n (fun t => f t i) y := by |
| let L : DarbouxState →L[ℝ] ℝ := ContinuousLinearMap.proj i |
| symm |
| change iteratedFDeriv ℝ n (L ∘ f) y (fun _ => 1) = L (iteratedDeriv n f y) |
| rw [L.iteratedFDeriv_comp_left hf (WithTop.coe_le_coe.mpr le_top)] |
| rfl |
|
|
| theorem solutionJetCurve_iteratedDeriv |
| {z : Coord → ℝ} {U : Set Coord} {x y : ℝ} |
| (hU : IsOpen U) (hz : ContDiffOn ℝ ∞ z U) (hp : (![x, y] : Coord) ∈ U) |
| (n : ℕ) (hn : 2 ≤ n) : |
| iteratedDeriv n (solutionJetCurve z x) y = |
| ![0, 0, coordPartial 0 (verticalJet z n) ![x, y], |
| coordPartial 1 (verticalJet z n) ![x, y], |
| coordPartial 0 (coordPartial 1 (verticalJet z n)) ![x, y], |
| coordPartial 1 (coordPartial 1 (verticalJet z n)) ![x, y]] := by |
| have hc := solutionJetCurve_contDiffAt hU hz hp |
| ext i |
| rw [iteratedDeriv_state_eval hc n i] |
| fin_cases i |
| · change iteratedDeriv n (fun _ : ℝ => x) y = 0 |
| simp [iteratedDeriv_const, show n ≠ 0 by omega] |
| · change iteratedDeriv n id y = 0 |
| simp [iteratedDeriv_id, show n ≠ 0 by omega, show n ≠ 1 by omega] |
| · change iteratedDeriv n (fun t => coordPartial 0 z ![x, t]) y = _ |
| rw [verticalJet_slice hU (normed_coordPartial_contDiffOn hz hU 0) x n y hp, |
| verticalJet_coordPartial hU hz 0 n ![x, y] hp] |
| rfl |
| · change iteratedDeriv n (fun t => coordPartial 1 z ![x, t]) y = _ |
| rw [verticalJet_slice hU (normed_coordPartial_contDiffOn hz hU 1) x n y hp, |
| verticalJet_coordPartial hU hz 1 n ![x, y] hp] |
| rfl |
| · change iteratedDeriv n (fun t => coordPartial 0 (coordPartial 1 z) ![x, t]) y = _ |
| rw [verticalJet_slice hU (normed_coordPartial_contDiffOn (normed_coordPartial_contDiffOn hz hU 1) hU 0) x n y hp, |
| verticalJet_two_coordPartials hU hz hp n 0 1] |
| rfl |
| · change iteratedDeriv n (fun t => coordPartial 1 (coordPartial 1 z) ![x, t]) y = _ |
| rw [verticalJet_slice hU (normed_coordPartial_contDiffOn (normed_coordPartial_contDiffOn hz hU 1) hU 1) x n y hp, |
| verticalJet_two_coordPartials hU hz hp n 1 1] |
| rfl |
|
|
| theorem actualDarboux_faaExpansion_coord |
| {g : MetricField} {z : Coord → ℝ} {U : Set Coord} {x y : ℝ} |
| (hg : SmoothPositiveOn g U) (hU : IsOpen U) (hz : ContDiffOn ℝ ∞ z U) |
| (hD : ∀ p ∈ U, (covHessian g z p).det = gaussianCurvature g p * heightEnergy g z p) |
| (hyy : ∀ p ∈ U, covHessian g z p 1 1 ≠ 0) |
| (hp : (![x, y] : Coord) ∈ U) (ell : ℕ) : |
| coordPartial 0 (coordPartial 0 (verticalJet z ell)) ![x, y] = |
| ∑ c : OrderedFinpartition ell, |
| iteratedFDeriv ℝ c.length (sixVariableP g) (solutionJetCurve z x y) |
| (fun j => iteratedDeriv (c.partSize j) (solutionJetCurve z x) y) := by |
| have hh := actualDarboux_faaExpansion hg hU hz hD hyy hp ell |
| rw [verticalJet_slice hU (normed_coordPartial_contDiffOn (normed_coordPartial_contDiffOn hz hU 0) hU 0) x ell y hp, |
| verticalJet_two_coordPartials hU hz hp ell 0 0] at hh |
| exact hh |
|
|
| end SmoothLocal.HighEquation |
|
|
| end |
|
|
| end OAI |
|
|
