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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 | import OAI.Geometry.IsometricImmersion.Caps.CapReflectionGeometry
namespace OAI
noncomputable section
open Set Function Filter
open scoped ContDiff Topology Interval
namespace SmoothLocal.Flow.Reflection
open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted SmoothLocal.HighEquation
theorem reflectPoint_coordinatePoint (t s : ℝ) :
reflectPoint (coordinatePoint t s) = coordinatePoint t (-s) := by
ext i
fin_cases i <;> simp [coordinatePoint]
theorem reflectPoint_mem_capChartDomain (p : Coord) :
reflectPoint p ∈ capChartDomain ↔ p ∈ capChartDomain := by
change (reflectPoint p 0 ∈ Ioo (-2 : ℝ) 2 ∧ reflectPoint p 1 ∈ Ioo (-2 : ℝ) 2) ↔ _
simp only [reflectPoint_zero,reflectPoint_one,mem_Ioo]
change ((-2 < p 0 ∧ p 0 < 2) ∧ -2 < -p 1 ∧ -p 1 < 2) ↔
((-2 < p 0 ∧ p 0 < 2) ∧ -2 < p 1 ∧ p 1 < 2)
constructor <;> rintro ⟨ht,hs⟩ <;> refine ⟨ht,?_,?_⟩ <;> linarith [hs.1,hs.2]
theorem coordPartial_reflectedScalar {f : Coord → ℝ} {p : Coord}
(hf : DifferentiableAt ℝ f (reflectPoint p)) (i : Fin 2) :
coordPartial i (reflectedScalar f) p =
if i=0 then coordPartial 0 f (reflectPoint p) else -coordPartial 1 f (reflectPoint p) := by
have hfa : DifferentiableAt ℝ f (affineCoordinates 0 reflectionMatrix p) := by
rwa [←reflectPoint_eq_affine]
have hh := coordPartial_affine_comp 0 reflectionMatrix p hfa i
rw [←reflectPoint_eq_affine] at hh
fin_cases i <;>
simpa [reflectedScalar,reflectionMatrix,Fin.sum_univ_two] using hh
theorem verticalJet_reflectedScalar {f : Coord → ℝ} {U : Set Coord}
(hf : ContDiffOn ℝ ∞ f U) (hU : IsOpen U) (n : ℕ)
{p : Coord} (hp : reflectPoint p ∈ U) :
verticalJet (reflectedScalar f) n p = (-1 : ℝ)^n * verticalJet f n (reflectPoint p) := by
induction n generalizing p with
| zero => simp [verticalJet,reflectedScalar]
| succ n ih =>
have he : verticalJet (reflectedScalar f) n =ᶠ[𝓝 p]
(fun q => (-1 : ℝ)^n * verticalJet f n (reflectPoint q)) := by
filter_upwards [(reflectedDomain_isOpen hU).mem_nhds hp] with q hq
exact ih hq
have hfn := verticalJet_contDiffOn hU hf n
have hdr : DifferentiableAt ℝ (reflectedScalar (verticalJet f n)) p :=
((reflectedScalar_contDiffOn hfn).contDiffAt
((reflectedDomain_isOpen hU).mem_nhds hp)).differentiableAt (by simp)
have hd : DifferentiableAt ℝ (verticalJet f n) (reflectPoint p) :=
(hfn.contDiffAt (hU.mem_nhds hp)).differentiableAt (by simp)
change coordPartial 1 (verticalJet (reflectedScalar f) n) p = _
rw [coordPartial_eq_of_eventuallyEq he 1]
change coordPartial 1 (fun q => (-1 : ℝ)^n * reflectedScalar (verticalJet f n) q) p = _
rw [AffineCalculus.coordPartial_const_mul_at hdr,
coordPartial_reflectedScalar hd 1]
simp only [show (1 : Fin 2) ≠ 0 by decide,ite_false,verticalJet,pow_succ]
ring
theorem negOnePow_square (n : ℕ) : ((-1 : ℝ)^n)^2 = 1 := by
rw [←pow_mul,Nat.mul_comm,pow_mul]
norm_num
def conjugateFlow (Y : ℝ → ℝ → ℝ) (s t : ℝ) : ℝ := -Y (-s) t
theorem conjugateFlow_chart (Y : ℝ → ℝ → ℝ) (p : Coord) :
capChart (conjugateFlow Y) p = reflectPoint (capChart Y (reflectPoint p)) := by
simp only [capChart,capFlowHeight,conjugateFlow,reflectPoint_zero,reflectPoint_one,
reflectPoint_coordinatePoint]
theorem conjugateFlow_start {Y : ℝ → ℝ → ℝ}
(h : ∀ s ∈ Icc (-2 : ℝ) 2, Y s 0 = s) :
∀ s ∈ Icc (-2 : ℝ) 2, conjugateFlow Y s 0 = s := by
intro s hs
have hn : -s ∈ Icc (-2 : ℝ) 2 := by constructor <;> linarith [hs.1,hs.2]
simp [conjugateFlow,h (-s) hn]
theorem conjugateFlow_chart_contDiffOn {Y : ℝ → ℝ → ℝ}
(hY : ContDiffOn ℝ ∞ (capChart Y) capChartDomain) :
ContDiffOn ℝ ∞ (capChart (conjugateFlow Y)) capChartDomain := by
have hh := reflectPoint_contDiff.comp_contDiffOn
(hY.comp reflectPoint_contDiff.contDiffOn
(fun p hp => (reflectPoint_mem_capChartDomain p).mpr hp))
exact hh.congr (fun p _ => conjugateFlow_chart Y p)
theorem conjugateFlow_displacement {Y : ℝ → ℝ → ℝ}
(hY : ∀ p ∈ capChartDomain, |capFlowHeight Y p-p 1| ≤ (1 : ℝ)/50) :
∀ p ∈ capChartDomain, |capFlowHeight (conjugateFlow Y) p-p 1| ≤ (1 : ℝ)/50 := by
intro p hp
have hh := hY (reflectPoint p) ((reflectPoint_mem_capChartDomain p).mpr hp)
change |-Y (-p 1) (p 0)-p 1| ≤ _
have he : -Y (-p 1) (p 0)-p 1 = -(Y (-p 1) (p 0)-(-p 1)) := by ring
rw [he,abs_neg]
simpa only [capFlowHeight,reflectPoint_zero,reflectPoint_one] using hh
theorem conjugateFlow_ode
{g : MetricField} {z : Coord → ℝ} {U : Set Coord} {Y : ℝ → ℝ → ℝ}
(hg : SmoothPositiveOn g U) (hz : ContDiffOn ℝ ∞ z U) (hU : IsOpen U)
(hmap : MapsTo (capChart Y) capChartDomain (reflectPoint ⁻¹' U))
(hode : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2,
HasDerivWithinAt (Y s)
(-hessianQuotient (reflectedMetric g) (reflectedScalar z) (coordinatePoint t (Y s t)))
(Icc (-2 : ℝ) 2) t) :
∀ s ∈ Ioo (-2 : ℝ) 2, ∀ t ∈ Ioo (-2 : ℝ) 2,
HasDerivWithinAt (conjugateFlow Y s)
(-hessianQuotient g z (coordinatePoint t (conjugateFlow Y s t)))
(Ioo (-2 : ℝ) 2) t := by
intro s hs t ht
have hsneg : -s ∈ Icc (-2 : ℝ) 2 := by constructor <;> linarith [hs.1,hs.2]
have hp : coordinatePoint t (-s) ∈ capChartDomain := by
change (coordinatePoint t (-s) 0 ∈ Ioo (-2 : ℝ) 2 ∧
coordinatePoint t (-s) 1 ∈ Ioo (-2 : ℝ) 2)
simp only [coordinatePoint,Pi.add_apply,Pi.smul_apply,Pi.single_apply]
norm_num
exact ⟨ht,by constructor <;> linarith [hs.1,hs.2]⟩
have hmem : reflectPoint (coordinatePoint t (Y (-s) t)) ∈ U := by
simpa [capChart,capFlowHeight,coordinatePoint] using hmap hp
have hh := ((hode (-s) hsneg t ⟨ht.1.le,ht.2.le⟩).neg).mono Ioo_subset_Icc_self
simp only [neg_neg,reflectedMetric_quotient hg hz hU hmem,
reflectPoint_coordinatePoint] at hh
exact hh
theorem rectangleIntegral_reflectedScalar (left right bottom top : ℝ) (f : Coord → ℝ) :
rectangleIntegral left right bottom top (reflectedScalar f) =
rectangleIntegral left right (-top) (-bottom) f := by
unfold rectangleIntegral
have he (t s : ℝ) : reflectedScalar f (boxPoint t s) = f (boxPoint t (-s)) := by
exact congrArg f (reflectPoint_coordinatePoint t s)
simp_rw [he]
exact intervalIntegral.integral_comp_neg (fun s : ℝ => ∫ t in left..right, f (boxPoint t s))
theorem reflectedScalar_gradient_square {u : Coord → ℝ} {p : Coord}
(hu : DifferentiableAt ℝ u (reflectPoint p)) :
(coordPartial 0 (reflectedScalar u) p)^2 + (coordPartial 1 (reflectedScalar u) p)^2 =
(coordPartial 0 u (reflectPoint p))^2 + (coordPartial 1 u (reflectPoint p))^2 := by
simp only [coordPartial_reflectedScalar hu,ite_true,
show (1 : Fin 2) ≠ 0 by decide,ite_false,neg_sq]
theorem rectangleGradient_reflectedScalar {u : Coord → ℝ} {U : Set Coord}
(hu : ContDiffOn ℝ ∞ u U) (hU : IsOpen U)
{left right bottom top : ℝ} (hlr : left ≤ right) (hbt : bottom ≤ top)
(hmap : MapsTo reflectPoint (closedRectangle left right bottom top) U) :
rectangleIntegral left right bottom top (fun p =>
(coordPartial 0 (reflectedScalar u) p)^2 + (coordPartial 1 (reflectedScalar u) p)^2) =
rectangleIntegral left right (-top) (-bottom) (fun p =>
(coordPartial 0 u p)^2 + (coordPartial 1 u p)^2) := by
calc
_ = rectangleIntegral left right bottom top
(reflectedScalar (fun p => (coordPartial 0 u p)^2+(coordPartial 1 u p)^2)) := by
apply rectangleIntegral_congr hlr hbt
intro p hp
exact reflectedScalar_gradient_square
((hu.contDiffAt (hU.mem_nhds (hmap hp))).differentiableAt (by simp))
_ = _ := rectangleIntegral_reflectedScalar _ _ _ _ _
end SmoothLocal.Flow.Reflection
end
end OAI
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