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59d1c80 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 | import OAI.Geometry.IsometricImmersion.Calculus.MovingFirstJetBase
import OAI.Geometry.IsometricImmersion.Calculus.VerticalSliceJets
namespace OAI
noncomputable section
open Set Filter Function
open scoped ContDiff Topology Matrix
namespace SmoothLocal.HighEquation
open SmoothLocal.Geometry SmoothLocal.Weighted SmoothLocal.ODE SmoothLocal.Hyperbolic
def horizontalJet {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]
(f : Coord → V) : ℕ → Coord → V
| 0 => f
| n + 1 => coordPartial 0 (horizontalJet f n)
theorem horizontalJet_eq_spatialJet (f : Coord → ℝ) (n : ℕ) :
horizontalJet f n = spatialJet f n := by
induction n with
| zero => rfl
| succ n hn => simp only [horizontalJet, spatialJet, hn]
theorem horizontalJet_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 ∞ (horizontalJet f n) U := by
intro n
induction n with
| zero => exact hf
| succ _ hn => exact normed_coordPartial_contDiffOn hn hU 0
theorem horizontalPoint_contDiff (y : ℝ) :
ContDiff ℝ ∞ (fun x : ℝ => coordinatePoint x y) := by
unfold coordinatePoint
fun_prop
theorem horizontalJet_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 : coordinatePoint x y ∈ U) :
HasDerivAt (fun t => horizontalJet f n (coordinatePoint t y))
(horizontalJet f (n + 1) (coordinatePoint x y)) x := by
have hd : DifferentiableAt ℝ (horizontalJet f n) (coordinatePoint x y) :=
((horizontalJet_contDiffOn hU hf n).contDiffAt (hU.mem_nhds hp)).differentiableAt (by simp)
exact hd.hasFDerivAt.comp_hasDerivAt x (point_hasDerivAt_t x y)
theorem horizontalJet_slice
{V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]
{f : Coord → V} {U : Set Coord} (hU : IsOpen U) (hf : ContDiffOn ℝ ∞ f U)
(y : ℝ) : ∀ n x, coordinatePoint x y ∈ U →
iteratedDeriv n (fun t => f (coordinatePoint t y)) x =
horizontalJet f n (coordinatePoint x y) := by
intro n
induction n with
| zero => intro _ _; rfl
| succ n hn =>
intro x hx
rw [iteratedDeriv_succ]
have heq : iteratedDeriv n (fun t => f (coordinatePoint t y)) =ᶠ[𝓝 x]
(fun t => horizontalJet f n (coordinatePoint t y)) := by
filter_upwards [(hU.preimage (horizontalPoint_contDiff y).continuous).mem_nhds hx] with t ht
exact hn t ht
rw [heq.deriv_eq]
exact (horizontalJet_slice_hasDerivAt hU hf n hx).deriv
theorem horizontalJet_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 →
horizontalJet (coordPartial i f) n p = coordPartial i (horizontalJet f n) p := by
intro n
induction n with
| zero => intro _ _; rfl
| succ n hn =>
intro p hp
have heq : horizontalJet (coordPartial i f) n =ᶠ[𝓝 p] coordPartial i (horizontalJet f n) := by
filter_upwards [hU.mem_nhds hp] with q hq
exact hn q hq
have hd : coordPartial 0 (horizontalJet (coordPartial i f) n) p =
coordPartial 0 (coordPartial i (horizontalJet f n)) p :=
congrArg (fun L : Coord →L[ℝ] V => L (Pi.single 0 1)) heq.fderiv_eq
change coordPartial 0 (horizontalJet (coordPartial i f) n) p =
coordPartial i (coordPartial 0 (horizontalJet f n)) p
rw [hd]
exact coordPartial_comm (horizontalJet_contDiffOn hU hf n) hU hp 0 i
theorem horizontalJet_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) :
horizontalJet (coordPartial i (coordPartial j f)) n p =
coordPartial i (coordPartial j (horizontalJet f n)) p := by
rw [horizontalJet_coordPartial hU (normed_coordPartial_contDiffOn hf hU j) i n p hp]
have heq : horizontalJet (coordPartial j f) n =ᶠ[𝓝 p] coordPartial j (horizontalJet f n) := by
filter_upwards [hU.mem_nhds hp] with q hq
exact horizontalJet_coordPartial hU hf j n q hq
exact congrArg (fun L : Coord →L[ℝ] V => L (Pi.single i 1)) heq.fderiv_eq
end SmoothLocal.HighEquation
end
end OAI
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