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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 | import OAI.Geometry.IsometricImmersion.Darboux.QActualSegmentTransfer
import OAI.Geometry.IsometricImmersion.Energy.MultiplierIntegral
import Mathlib.Analysis.Calculus.MeanValue
namespace OAI
noncomputable section
open Set Filter
open scoped ContDiff Topology
namespace SmoothLocal.HighEquation
open SmoothLocal.Geometry SmoothLocal.Weighted
theorem horizontal_partial_eq_of_slice_eq {f f0 : Coord → ℝ} {U : Set Coord}
(hf : ContDiffOn ℝ ∞ f U) (hf0 : ContDiffOn ℝ ∞ f0 U) (hU : IsOpen U)
{left right a : ℝ}
(hline : ∀ x ∈ Ioo left right, boxPoint x a ∈ U)
(heq : ∀ x ∈ Ioo left right, f0 (boxPoint x a) = f (boxPoint x a))
{x : ℝ} (hx : x ∈ Ioo left right) :
coordPartial 0 f0 (boxPoint x a) = coordPartial 0 f (boxPoint x a) := by
have hd : HasDerivAt (fun t => f (boxPoint t a)) (coordPartial 0 f (boxPoint x a)) x :=
((hf.contDiffAt (hU.mem_nhds (hline x hx))).differentiableAt (by simp)).hasFDerivAt.comp_hasDerivAt
x (boxPoint_hasDerivAt_t x a)
have hd0 : HasDerivAt (fun t => f0 (boxPoint t a)) (coordPartial 0 f0 (boxPoint x a)) x :=
((hf0.contDiffAt (hU.mem_nhds (hline x hx))).differentiableAt (by simp)).hasFDerivAt.comp_hasDerivAt
x (boxPoint_hasDerivAt_t x a)
have hev : (fun t => f0 (boxPoint t a)) =ᶠ[𝓝 x] (fun t => f (boxPoint t a)) := by
filter_upwards [isOpen_Ioo.mem_nhds hx] with t ht
exact heq t ht
rw [← hd0.deriv, ← hd.deriv]
exact hev.deriv_eq
theorem qSolutionJet_eq_of_cauchy_data {z z0 : Coord → ℝ} {U : Set Coord}
(hz : ContDiffOn ℝ ∞ z U) (hz0 : ContDiffOn ℝ ∞ z0 U) (hU : IsOpen U)
{left right a : ℝ}
(hline : ∀ x ∈ Ioo left right, boxPoint x a ∈ U)
(hvalue : ∀ x ∈ Ioo left right, z0 (boxPoint x a) = z (boxPoint x a))
(hvelocity : ∀ x ∈ Ioo left right,
coordPartial 1 z0 (boxPoint x a) = coordPartial 1 z (boxPoint x a))
{x : ℝ} (hx : x ∈ Ioo left right) :
qSolutionJet z0 (boxPoint x a) = qSolutionJet z (boxPoint x a) := by
have hxfirst : ∀ x ∈ Ioo left right,
coordPartial 0 z0 (boxPoint x a) = coordPartial 0 z (boxPoint x a) :=
fun x hx => horizontal_partial_eq_of_slice_eq hz hz0 hU hline hvalue hx
have hxx := horizontal_partial_eq_of_slice_eq (partial_contDiffOn hz hU 0)
(partial_contDiffOn hz0 hU 0) hU hline hxfirst hx
have hxy := horizontal_partial_eq_of_slice_eq (partial_contDiffOn hz hU 1)
(partial_contDiffOn hz0 hU 1) hU hline hvelocity hx
ext i
fin_cases i
· rfl
· rfl
· exact hxfirst x hx
· exact hvelocity x hx
· exact hxy
· exact hxx
theorem coordinate_word_vertical_variation {f : Coord → ℝ} {U : Set Coord}
(hf : ContDiffOn ℝ ∞ f U) (hU : IsOpen U)
{left right a b x t : ℝ}
(hbox : closedRectangle left right a b ⊆ U)
(hx : x ∈ Icc left right) (hab : a ≤ b) (ht : t ∈ Icc a b)
{M : ℝ} (hB : CoordinateBound f (closedRectangle left right a b) 3 M)
(ds : List (Fin 2)) (hds : ds.length ≤ 2) :
|iteratedCoordPartial ds f (boxPoint x t) -
iteratedCoordPartial ds f (boxPoint x a)| ≤ M * |t - a| := by
have hd (s : ℝ) (hs : s ∈ Icc a b) :
HasDerivAt (fun y => iteratedCoordPartial ds f (boxPoint x y))
(coordPartial 1 (iteratedCoordPartial ds f) (boxPoint x s)) s :=
(LowQuotient.iterated_differentiableAt hf hU (hbox (boxPoint_mem hx hs)) ds).hasFDerivAt.comp_hasDerivAt
s (boxPoint_hasDerivAt_s x s)
have hb (s : ℝ) (hs : s ∈ Icc a b) :
‖deriv (fun y => iteratedCoordPartial ds f (boxPoint x y)) s‖ ≤ M := by
rw [(hd s hs).deriv, Real.norm_eq_abs]
exact hB (1 :: ds) (Nat.succ_le_succ hds) _ (boxPoint_mem hx hs)
simpa only [Real.norm_eq_abs] using
(convex_Icc a b).norm_image_sub_le_of_norm_deriv_le
(fun s hs => (hd s hs).differentiableAt) hb (show a ∈ Icc a b from ⟨le_rfl, hab⟩) ht
theorem coordinate_word_difference_on_strip {z z0 : Coord → ℝ} {U : Set Coord}
(hz : ContDiffOn ℝ ∞ z U) (hz0 : ContDiffOn ℝ ∞ z0 U) (hU : IsOpen U)
{left right a b x t : ℝ}
(hbox : closedRectangle left right a b ⊆ U)
(hx : x ∈ Icc left right) (hab : a ≤ b) (ht : t ∈ Icc a b)
{M M0 : ℝ} (hB : CoordinateBound z (closedRectangle left right a b) 3 M)
(hB0 : CoordinateBound z0 (closedRectangle left right a b) 3 M0)
(ds : List (Fin 2)) (hds : ds.length ≤ 2)
(hinitial : iteratedCoordPartial ds z0 (boxPoint x a) =
iteratedCoordPartial ds z (boxPoint x a)) :
|iteratedCoordPartial ds z0 (boxPoint x t) - iteratedCoordPartial ds z (boxPoint x t)| ≤
(M + M0) * |t - a| := by
have h0 := coordinate_word_vertical_variation hz0 hU hbox hx hab ht hB0 ds hds
have h1 := coordinate_word_vertical_variation hz hU hbox hx hab ht hB ds hds
calc
_ = |(iteratedCoordPartial ds z0 (boxPoint x t) - iteratedCoordPartial ds z0 (boxPoint x a)) -
(iteratedCoordPartial ds z (boxPoint x t) - iteratedCoordPartial ds z (boxPoint x a))| := by
rw [hinitial]
congr 1
ring
_ ≤ |iteratedCoordPartial ds z0 (boxPoint x t) - iteratedCoordPartial ds z0 (boxPoint x a)| +
|iteratedCoordPartial ds z (boxPoint x t) - iteratedCoordPartial ds z (boxPoint x a)| := abs_sub _ _
_ ≤ M0 * |t - a| + M * |t - a| := add_le_add h0 h1
_ = (M + M0) * |t - a| := by ring
theorem qSolutionJet_distance_on_same_strip {z z0 : Coord → ℝ} {U : Set Coord}
(hz : ContDiffOn ℝ ∞ z U) (hz0 : ContDiffOn ℝ ∞ z0 U) (hU : IsOpen U)
{left right a b x t : ℝ}
(hbox : closedRectangle left right a b ⊆ U)
(hx : x ∈ Ioo left right) (hab : a ≤ b) (ht : t ∈ Icc a b)
{M M0 : ℝ} (hM : 0 ≤ M) (hM0 : 0 ≤ M0)
(hB : CoordinateBound z (closedRectangle left right a b) 3 M)
(hB0 : CoordinateBound z0 (closedRectangle left right a b) 3 M0)
(hvalue : ∀ x ∈ Ioo left right, z0 (boxPoint x a) = z (boxPoint x a))
(hvelocity : ∀ x ∈ Ioo left right,
coordPartial 1 z0 (boxPoint x a) = coordPartial 1 z (boxPoint x a)) :
‖qSolutionJet z0 (boxPoint x t) - qSolutionJet z (boxPoint x t)‖ ≤
(M + M0) * (b - a) := by
have hx' : x ∈ Icc left right := ⟨hx.1.le, hx.2.le⟩
have hinitial := qSolutionJet_eq_of_cauchy_data hz hz0 hU
(fun y hy => hbox (boxPoint_mem ⟨hy.1.le, hy.2.le⟩ ⟨le_rfl, hab⟩)) hvalue hvelocity hx
have hword (ds : List (Fin 2)) (hds : ds.length ≤ 2)
(he : iteratedCoordPartial ds z0 (boxPoint x a) = iteratedCoordPartial ds z (boxPoint x a)) :
|iteratedCoordPartial ds z0 (boxPoint x t) - iteratedCoordPartial ds z (boxPoint x t)| ≤
(M + M0) * (b - a) := by
apply (coordinate_word_difference_on_strip hz hz0 hU hbox hx' hab ht hB hB0 ds hds he).trans
apply mul_le_mul_of_nonneg_left _ (add_nonneg hM hM0)
rw [abs_of_nonneg (sub_nonneg.mpr ht.1)]
exact sub_le_sub_right ht.2 a
apply state_distance_le_of_coordinate_error (mul_nonneg (add_nonneg hM hM0) (sub_nonneg.mpr hab))
apply qSolutionJet_coordinate_error z z0 (boxPoint x t)
(mul_nonneg (add_nonneg hM hM0) (sub_nonneg.mpr hab))
· intro i
fin_cases i
· exact hword [0] (by norm_num) (congrFun hinitial 2)
· exact hword [1] (by norm_num) (congrFun hinitial 3)
· exact hword [0, 1] (by norm_num) (congrFun hinitial 4)
· exact hword [0, 0] (by norm_num) (congrFun hinitial 5)
end SmoothLocal.HighEquation
end
end OAI
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