| import OAI.Geometry.IsometricImmersion.Caps.CapDensityBounds |
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| namespace OAI |
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| noncomputable section |
| open Set Filter Function MeasureTheory |
| open scoped ContDiff Topology Interval |
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| namespace SmoothLocal.Flow |
| open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted |
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| def coordinateGradientSquare (f : Coord → ℝ) (p : Coord) : ℝ := |
| (coordPartial 0 f p)^2+(coordPartial 1 f p)^2 |
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| theorem coordinateGradientSquare_nonneg (f : Coord → ℝ) (p : Coord) : |
| 0 ≤ coordinateGradientSquare f p := add_nonneg (sq_nonneg _) (sq_nonneg _) |
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| 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) |
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| 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] |
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| 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] |
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| 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 |
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| 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 |
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| end |
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| end OAI |
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