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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 | import OAI.Geometry.IsometricImmersion.Caps.CapDensityBounds
namespace OAI
noncomputable section
open Set Filter Function MeasureTheory
open scoped ContDiff Topology Interval
namespace SmoothLocal.Flow
open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted
def coordinateGradientSquare (f : Coord → ℝ) (p : Coord) : ℝ :=
(coordPartial 0 f p)^2+(coordPartial 1 f p)^2
theorem coordinateGradientSquare_nonneg (f : Coord → ℝ) (p : Coord) :
0 ≤ coordinateGradientSquare f p := add_nonneg (sq_nonneg _) (sq_nonneg _)
theorem coordinateGradientSquare_contDiffOn {f : Coord → ℝ} {V : Set Coord}
(hf : ContDiffOn ℝ ∞ f V) (hV : IsOpen V) :
ContDiffOn ℝ ∞ (coordinateGradientSquare f) V :=
((partial_contDiffOn hf hV 0).pow 2).add ((partial_contDiffOn hf hV 1).pow 2)
theorem triangular_gradient_forward_scalar {a b H x y : ℝ}
(ha : |a| ≤ 1) (hb : 0 ≤ b) (hbH : b ≤ H) :
(x+a*y)^2+(b*y)^2 ≤ (2+H^2)*(x^2+y^2) := by
have hH : 0 ≤ H := hb.trans hbH
have ha2 : a^2 ≤ 1 := (sq_le_one_iff_abs_le_one a).2 ha
have hb2 : b^2 ≤ H^2 := (sq_le_sq₀ hb hH).2 hbH
have hay := mul_le_mul_of_nonneg_right ha2 (sq_nonneg y)
have hby := mul_le_mul_of_nonneg_right hb2 (sq_nonneg y)
have hsum : (x+a*y)^2 ≤ 2*(x^2+(a*y)^2) := add_sq_le
have hx := mul_nonneg (sq_nonneg H) (sq_nonneg x)
nlinarith only [hay,hby,hsum,hx]
theorem triangular_gradient_inverse_scalar {a b H x y : ℝ}
(ha : |a| ≤ 1) (hH : 0 < H) (hb : 1/H ≤ b) :
x^2+y^2 ≤ (2+3*H^2)*((x+a*y)^2+(b*y)^2) := by
have ha2 : a^2 ≤ 1 := (sq_le_one_iff_abs_le_one a).2 ha
have hHb : 1 ≤ b*H := (div_le_iff₀ hH).1 hb
have hHb2 : 1 ≤ (b*H)^2 := by
simpa only [one_pow] using (sq_le_sq₀ (show (0:ℝ) ≤ 1 by norm_num) (by linarith : 0 ≤ b*H)).2 hHb
have hy := mul_le_mul_of_nonneg_right hHb2 (sq_nonneg y)
have hay := mul_le_mul_of_nonneg_right ha2 (sq_nonneg y)
have hx : ((x+a*y)+(-a*y))^2 ≤ 2*((x+a*y)^2+(-a*y)^2) := add_sq_le
have hgap : 0 ≤ 3*H^2*(x+a*y)^2+2*(b*y)^2 := by positivity
nlinarith only [hy,hay,hx,hgap]
theorem capPullback_gradient_pointwise
{Y : ℝ → ℝ → ℝ} {q f : Coord → ℝ} {M : ℝ}
(hYs : ContDiffOn ℝ ∞ (fun p : ℝ × ℝ => Y p.2 p.1) (pairRectangle 2 (-2) 2))
(hode : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2,
HasDerivWithinAt (Y s) (-q (coordinatePoint t (Y s t))) (Icc (-2 : ℝ) 2) t)
{p : Coord} (hp : p ∈ capChartDomain)
(hf : DifferentiableAt ℝ f (capChart Y p))
(hq : |q (capChart Y p)| ≤ 1)
(hJ : Real.exp (-2*M) ≤ coordPartial 1 (capFlowHeight Y) p ∧
coordPartial 1 (capFlowHeight Y) p ≤ Real.exp (2*M)) :
coordinateGradientSquare (capPullback Y f) p ≤
(2+(Real.exp (2*M))^2)*capPullback Y (coordinateGradientSquare f) p ∧
capPullback Y (coordinateGradientSquare f) p ≤
(2+3*(Real.exp (2*M))^2)*coordinateGradientSquare (capPullback Y f) p := by
have hJpos : 0 < coordPartial 1 (capFlowHeight Y) p := (Real.exp_pos _).trans_le hJ.1
have hJinv : 1/Real.exp (2*M) ≤ coordPartial 1 (capFlowHeight Y) p := by
simpa only [show -2*M = -(2*M) by ring,Real.exp_neg,one_div] using hJ.1
have htime := capPullback_partial_time hYs hode hp hf
have hinit := capPullback_partial_initial hYs hp hf
change (coordPartial 0 (capPullback Y f) p)^2+(coordPartial 1 (capPullback Y f) p)^2 ≤
(2+(Real.exp (2*M))^2)*((coordPartial 0 f (capChart Y p))^2+(coordPartial 1 f (capChart Y p))^2) ∧
(coordPartial 0 f (capChart Y p))^2+(coordPartial 1 f (capChart Y p))^2 ≤
(2+3*(Real.exp (2*M))^2)*((coordPartial 0 (capPullback Y f) p)^2+(coordPartial 1 (capPullback Y f) p)^2)
rw [htime,hinit]
simp only [coordinateDrift,capPullback]
have ha : |-q (capChart Y p)| ≤ 1 := by simpa only [abs_neg] using hq
constructor
· simpa only [neg_mul,sub_eq_add_neg,mul_comm] using
triangular_gradient_forward_scalar (x := coordPartial 0 f (capChart Y p))
(y := coordPartial 1 f (capChart Y p)) ha hJpos.le hJ.2
· simpa only [neg_mul,sub_eq_add_neg,mul_comm] using
triangular_gradient_inverse_scalar (x := coordPartial 0 f (capChart Y p))
(y := coordPartial 1 f (capChart Y p)) ha (Real.exp_pos _) hJinv
theorem cap_gradient_integral_bounds
{Y : ℝ → ℝ → ℝ} {q f : Coord → ℝ} {V : Set Coord}
{e : OpenPartialHomeomorph (ℝ × ℝ) (ℝ × ℝ)}
(hYs : ContDiffOn ℝ ∞ (fun p : ℝ × ℝ => Y p.2 p.1) (pairRectangle 2 (-2) 2))
(hvar : ∀ s ∈ Ioo (-2 : ℝ) 2, ∀ t ∈ Ioo (-2 : ℝ) 2, 0 < deriv (fun r => Y r t) s)
(hode : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2,
HasDerivWithinAt (Y s) (-q (coordinatePoint t (Y s t))) (Icc (-2 : ℝ) 2) t)
(hsource : e.source = pairRectangle 2 (-2) 2)
(heq : (e : (ℝ × ℝ) → (ℝ × ℝ)) = triangularFlow Y)
(hf : ContDiffOn ℝ ∞ f V) (hV : IsOpen V)
(hmap : MapsTo (capChart Y) capChartDomain V)
{tl tr sb st M : ℝ} (ht : tl ≤ tr) (hs : sb ≤ st)
(hbox : closedRectangle tl tr sb st ⊆ capChartDomain)
(hq : ∀ p ∈ closedRectangle tl tr sb st, |q (capChart Y p)| ≤ 1)
(hJ : ∀ p ∈ closedRectangle tl tr sb st,
Real.exp (-2*M) ≤ coordPartial 1 (capFlowHeight Y) p ∧
coordPartial 1 (capFlowHeight Y) p ≤ Real.exp (2*M)) :
rectangleIntegral tl tr sb st (coordinateGradientSquare (capPullback Y f)) ≤
((2+(Real.exp (2*M))^2)*Real.exp (2*M))*
(∫ p in capChart Y '' closedRectangle tl tr sb st, coordinateGradientSquare f p) ∧
(∫ p in capChart Y '' closedRectangle tl tr sb st, coordinateGradientSquare f p) ≤
(Real.exp (2*M)*(2+3*(Real.exp (2*M))^2))*
rectangleIntegral tl tr sb st (coordinateGradientSquare (capPullback Y f)) := by
have hGp := coordinateGradientSquare_contDiffOn hf hV
have hGc : ContinuousOn (coordinateGradientSquare f) (capChart Y '' closedRectangle tl tr sb st) :=
hGp.continuousOn.mono (by rintro _ ⟨p,hp,rfl⟩; exact hmap (hbox hp))
have hP : ContDiffOn ℝ ∞ (capPullback Y f) capChartDomain := capPullback_contDiffOn hYs hf hmap
have hGPc : ContinuousOn (coordinateGradientSquare (capPullback Y f)) (closedRectangle tl tr sb st) :=
(coordinateGradientSquare_contDiffOn hP capChartDomain_isOpen).continuousOn.mono hbox
have hPGc : ContinuousOn (capPullback Y (coordinateGradientSquare f)) (closedRectangle tl tr sb st) :=
(capPullback_contDiffOn hYs hGp hmap).continuousOn.mono hbox
have hpoint (p : Coord) (hp : p ∈ closedRectangle tl tr sb st) :=
capPullback_gradient_pointwise hYs hode (hbox hp)
((hf.contDiffAt (hV.mem_nhds (hmap (hbox hp)))).differentiableAt (by simp)) (hq p hp) (hJ p hp)
have hzero := fun p (_hp : p ∈ capChart Y '' closedRectangle tl tr sb st) =>
coordinateGradientSquare_nonneg f p
have hforward := rectangleIntegral_mono ht hs hGPc (continuousOn_const.mul hPGc)
(fun p hp => (hpoint p hp).1)
simp only [Pi.mul_def, rectangleIntegral_const_mul] at hforward
have hinverse := rectangleIntegral_mono ht hs hPGc (continuousOn_const.mul hGPc)
(fun p hp => (hpoint p hp).2)
simp only [Pi.mul_def, rectangleIntegral_const_mul] at hinverse
have hlower := cap_pullback_density_integral_le_image hYs hvar hsource heq ht hs hbox hJ hGc hzero
have hupper := (cap_rectangle_density_integral_bounds hYs hvar hsource heq ht hs hbox hJ hGc hzero).2
constructor
· calc
_ ≤ (2+(Real.exp (2*M))^2)*rectangleIntegral tl tr sb st
(capPullback Y (coordinateGradientSquare f)) := hforward
_ ≤ (2+(Real.exp (2*M))^2)*(Real.exp (2*M)*
(∫ p in capChart Y '' closedRectangle tl tr sb st, coordinateGradientSquare f p)) :=
mul_le_mul_of_nonneg_left hlower (by positivity)
_ = _ := by ring
· calc
_ ≤ Real.exp (2*M)*rectangleIntegral tl tr sb st
(capPullback Y (coordinateGradientSquare f)) := hupper
_ ≤ Real.exp (2*M)*((2+3*(Real.exp (2*M))^2)*
rectangleIntegral tl tr sb st (coordinateGradientSquare (capPullback Y f))) :=
mul_le_mul_of_nonneg_left hinverse (Real.exp_pos _).le
_ = _ := by ring
end SmoothLocal.Flow
end
end OAI
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