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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 | import OAI.Geometry.IsometricImmersion.Flows.FlowLowBounds
import OAI.Geometry.IsometricImmersion.Flows.FlowTransport
import OAI.Geometry.IsometricImmersion.Calculus.CoordinateJetBounds
namespace OAI
noncomputable section
open Set Filter Function
open scoped ContDiff Topology
namespace SmoothLocal.Flow
open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted
theorem coordPartial_time_of_pair_eq {f : Coord → ℝ} {F : ℝ × ℝ → ℝ}
(hf : ContDiffOn ℝ ∞ f capChartDomain)
(he : ∀ p ∈ capChartDomain, f p = F (p 0, p 1))
{p : Coord} (hp : p ∈ capChartDomain) :
coordPartial 0 f p = deriv (fun t => F (t, p 1)) (p 0) := by
have hd := ((hf.contDiffAt (capChartDomain_isOpen.mem_nhds hp)).differentiableAt (by simp)).hasFDerivAt
have hp' : boxPoint (p 0) (p 1) = p := boxPoint_coord_eta p
have hd' : HasFDerivAt f (fderiv ℝ f p) (boxPoint (p 0) (p 1)) := by rw [hp']; exact hd
have hline := hd'.comp_hasDerivAt (p 0) (boxPoint_hasDerivAt_t (p 0) (p 1))
have hevent : (fun t => F (t, p 1)) =ᶠ[𝓝 (p 0)] (fun t => f (boxPoint t (p 1))) := by
filter_upwards [isOpen_Ioo.mem_nhds hp.1] with t ht
have hm : boxPoint t (p 1) ∈ capChartDomain := by
simpa [capChartDomain, coordinateRectangle, boxPoint] using And.intro ht hp.2
simpa [boxPoint] using (he (boxPoint t (p 1)) hm).symm
exact (hline.congr_of_eventuallyEq hevent).deriv.symm
theorem coordPartial_initial_of_pair_eq {f : Coord → ℝ} {F : ℝ × ℝ → ℝ}
(hf : ContDiffOn ℝ ∞ f capChartDomain)
(he : ∀ p ∈ capChartDomain, f p = F (p 0, p 1))
{p : Coord} (hp : p ∈ capChartDomain) :
coordPartial 1 f p = deriv (fun s => F (p 0, s)) (p 1) := by
have hd := ((hf.contDiffAt (capChartDomain_isOpen.mem_nhds hp)).differentiableAt (by simp)).hasFDerivAt
have hp' : boxPoint (p 0) (p 1) = p := boxPoint_coord_eta p
have hd' : HasFDerivAt f (fderiv ℝ f p) (boxPoint (p 0) (p 1)) := by rw [hp']; exact hd
have hline := hd'.comp_hasDerivAt (p 1) (boxPoint_hasDerivAt_s (p 0) (p 1))
have hevent : (fun s => F (p 0, s)) =ᶠ[𝓝 (p 1)] (fun s => f (boxPoint (p 0) s)) := by
filter_upwards [isOpen_Ioo.mem_nhds hp.2] with s hs
have hm : boxPoint (p 0) s ∈ capChartDomain := by
simpa [capChartDomain, coordinateRectangle, boxPoint] using And.intro hp.1 hs
simpa [boxPoint] using (he (boxPoint (p 0) s) hm).symm
exact (hline.congr_of_eventuallyEq hevent).deriv.symm
section HeightWords
variable {Y : ℝ → ℝ → ℝ}
variable (hY : ContDiffOn ℝ ∞ (fun p : ℝ × ℝ => Y p.2 p.1) (pairRectangle 2 (-2) 2))
include hY
theorem capFlowHeight_partial_zero_eq {p : Coord} (hp : p ∈ capChartDomain) :
coordPartial 0 (capFlowHeight Y) p = timeFlowDerivative Y (p 0, p 1) :=
(capFlowHeight_hasDerivAt_time hY hp).deriv.symm
theorem capFlowHeight_partial_one_eq {p : Coord} (hp : p ∈ capChartDomain) :
coordPartial 1 (capFlowHeight Y) p = initialFlowDerivative Y (p 0, p 1) :=
(capFlowHeight_hasDerivAt_initial hY hp).deriv.symm
theorem capFlowHeight_word_00 {p : Coord} (hp : p ∈ capChartDomain) :
iteratedCoordPartial [0, 0] (capFlowHeight Y) p = timeSecondFlowDerivative Y (p 0, p 1) :=
coordPartial_time_of_pair_eq
(partial_contDiffOn (capFlowHeight_contDiffOn hY) capChartDomain_isOpen 0)
(fun _ hx => capFlowHeight_partial_zero_eq hY hx) hp
theorem capFlowHeight_word_10 {p : Coord} (hp : p ∈ capChartDomain) :
iteratedCoordPartial [1, 0] (capFlowHeight Y) p = mixedFlowDerivative Y (p 0, p 1) :=
coordPartial_initial_of_pair_eq
(partial_contDiffOn (capFlowHeight_contDiffOn hY) capChartDomain_isOpen 0)
(fun _ hx => capFlowHeight_partial_zero_eq hY hx) hp
theorem capFlowHeight_word_11 {p : Coord} (hp : p ∈ capChartDomain) :
iteratedCoordPartial [1, 1] (capFlowHeight Y) p = initialSecondFlowDerivative Y (p 0, p 1) :=
coordPartial_initial_of_pair_eq
(partial_contDiffOn (capFlowHeight_contDiffOn hY) capChartDomain_isOpen 1)
(fun _ hx => capFlowHeight_partial_one_eq hY hx) hp
theorem capFlowHeight_word_000 {p : Coord} (hp : p ∈ capChartDomain) :
iteratedCoordPartial [0, 0, 0] (capFlowHeight Y) p =
deriv (fun t => timeSecondFlowDerivative Y (t, p 1)) (p 0) :=
coordPartial_time_of_pair_eq
(LowQuotient.iterated_contDiffOn (capFlowHeight_contDiffOn hY) capChartDomain_isOpen [0, 0])
(fun _ hx => capFlowHeight_word_00 hY hx) hp
theorem capFlowHeight_word_100 {p : Coord} (hp : p ∈ capChartDomain) :
iteratedCoordPartial [1, 0, 0] (capFlowHeight Y) p =
deriv (fun s => timeSecondFlowDerivative Y (p 0, s)) (p 1) :=
coordPartial_initial_of_pair_eq
(LowQuotient.iterated_contDiffOn (capFlowHeight_contDiffOn hY) capChartDomain_isOpen [0, 0])
(fun _ hx => capFlowHeight_word_00 hY hx) hp
theorem capFlowHeight_word_110 {p : Coord} (hp : p ∈ capChartDomain) :
iteratedCoordPartial [1, 1, 0] (capFlowHeight Y) p =
deriv (fun s => mixedFlowDerivative Y (p 0, s)) (p 1) :=
coordPartial_initial_of_pair_eq
(LowQuotient.iterated_contDiffOn (capFlowHeight_contDiffOn hY) capChartDomain_isOpen [1, 0])
(fun _ hx => capFlowHeight_word_10 hY hx) hp
theorem capFlowHeight_word_111 {p : Coord} (hp : p ∈ capChartDomain) :
iteratedCoordPartial [1, 1, 1] (capFlowHeight Y) p =
deriv (fun s => initialSecondFlowDerivative Y (p 0, s)) (p 1) :=
coordPartial_initial_of_pair_eq
(LowQuotient.iterated_contDiffOn (capFlowHeight_contDiffOn hY) capChartDomain_isOpen [1, 1])
(fun _ hx => capFlowHeight_word_11 hY hx) hp
end HeightWords
theorem iterated_three_swap_outer {f : Coord → ℝ} {U : Set Coord}
(hf : ContDiffOn ℝ ∞ f U) (hU : IsOpen U) {p : Coord} (hp : p ∈ U) (i j k : Fin 2) :
iteratedCoordPartial [i, j, k] f p = iteratedCoordPartial [j, i, k] f p :=
coordPartial_comm (partial_contDiffOn hf hU k) hU hp i j
theorem iterated_three_swap_inner {f : Coord → ℝ} {U : Set Coord}
(hf : ContDiffOn ℝ ∞ f U) (hU : IsOpen U) {p : Coord} (hp : p ∈ U) (i j k : Fin 2) :
iteratedCoordPartial [i, j, k] f p = iteratedCoordPartial [i, k, j] f p := by
have he : coordPartial j (coordPartial k f) =ᶠ[𝓝 p] coordPartial k (coordPartial j f) := by
filter_upwards [hU.mem_nhds hp] with x hx
exact coordPartial_comm hf hU hx j k
exact coordPartial_eq_of_eventuallyEq he i
end SmoothLocal.Flow
end
end OAI
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