| import OAI.Geometry.IsometricImmersion.Caps.CapAreaFormula |
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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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| theorem cap_rectangle_density_integral_bounds |
| {Y : ℝ → ℝ → ℝ} {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) |
| (hsource : e.source = pairRectangle 2 (-2) 2) |
| (heq : (e : (ℝ × ℝ) → (ℝ × ℝ)) = triangularFlow Y) |
| {tl tr sb st M : ℝ} (ht : tl ≤ tr) (hs : sb ≤ st) |
| (hbox : closedRectangle tl tr sb st ⊆ capChartDomain) |
| (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)) |
| {f : Coord → ℝ} (hf : ContinuousOn f (capChart Y '' closedRectangle tl tr sb st)) |
| (hf0 : ∀ p ∈ capChart Y '' closedRectangle tl tr sb st, 0 ≤ f p) : |
| Real.exp (-2*M)*rectangleIntegral tl tr sb st (capPullback Y f) ≤ |
| (∫ p in capChart Y '' closedRectangle tl tr sb st, f p) ∧ |
| (∫ p in capChart Y '' closedRectangle tl tr sb st, f p) ≤ |
| Real.exp (2*M)*rectangleIntegral tl tr sb st (capPullback Y f) := by |
| have hfc : ContinuousOn (capPullback Y f) (closedRectangle tl tr sb st) := |
| hf.comp ((capChart_contDiffOn hYs).continuousOn.mono hbox) (fun p hp => ⟨p,hp,rfl⟩) |
| have hJc : ContinuousOn (coordPartial 1 (capFlowHeight Y)) (closedRectangle tl tr sb st) := |
| (partial_contDiffOn (capFlowHeight_contDiffOn hYs) capChartDomain_isOpen 1).continuousOn.mono hbox |
| rw [cap_image_integral_formula hYs hvar hsource heq ht hs hbox hf] |
| constructor |
| · have hh := rectangleIntegral_mono ht hs (continuousOn_const.mul hfc) (hJc.mul hfc) |
| (fun p hp => mul_le_mul_of_nonneg_right (hJ p hp).1 (hf0 _ ⟨p,hp,rfl⟩)) |
| simpa only [Pi.mul_def, rectangleIntegral_const_mul] using hh |
| · have hh := rectangleIntegral_mono ht hs (hJc.mul hfc) (continuousOn_const.mul hfc) |
| (fun p hp => mul_le_mul_of_nonneg_right (hJ p hp).2 (hf0 _ ⟨p,hp,rfl⟩)) |
| simpa only [Pi.mul_def, rectangleIntegral_const_mul] using hh |
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| theorem cap_pullback_density_integral_le_image |
| {Y : ℝ → ℝ → ℝ} {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) |
| (hsource : e.source = pairRectangle 2 (-2) 2) |
| (heq : (e : (ℝ × ℝ) → (ℝ × ℝ)) = triangularFlow Y) |
| {tl tr sb st M : ℝ} (ht : tl ≤ tr) (hs : sb ≤ st) |
| (hbox : closedRectangle tl tr sb st ⊆ capChartDomain) |
| (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)) |
| {f : Coord → ℝ} (hf : ContinuousOn f (capChart Y '' closedRectangle tl tr sb st)) |
| (hf0 : ∀ p ∈ capChart Y '' closedRectangle tl tr sb st, 0 ≤ f p) : |
| rectangleIntegral tl tr sb st (capPullback Y f) ≤ |
| Real.exp (2*M)*(∫ p in capChart Y '' closedRectangle tl tr sb st, f p) := by |
| have hh := (cap_rectangle_density_integral_bounds hYs hvar hsource heq ht hs hbox hJ hf hf0).1 |
| have hscale : Real.exp (2*M)*Real.exp (-2*M) = 1 := by |
| rw [←Real.exp_add,show 2*M+(-2*M) = 0 by ring,Real.exp_zero] |
| calc |
| _ = Real.exp (2*M)*(Real.exp (-2*M)*rectangleIntegral tl tr sb st (capPullback Y f)) := by |
| rw [←mul_assoc,hscale,one_mul] |
| _ ≤ _ := mul_le_mul_of_nonneg_left hh (Real.exp_pos _).le |
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| end SmoothLocal.Flow |
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| end |
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| end OAI |
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