| import OAI.Geometry.IsometricImmersion.Flows.FlowJointSmooth |
| import Mathlib.Analysis.Calculus.Deriv.MeanValue |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter MeasureTheory Function |
| open scoped ContDiff Topology Interval |
|
|
| namespace SmoothLocal.Flow |
| open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted |
|
|
| theorem exists_cap_vertical_derivative_bound {q : Coord → ℝ} {U : Set Coord} |
| (hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) (hSU : modelSquare ⊆ U) : |
| ∃ M : ℝ, 0 ≤ M ∧ ∀ p ∈ modelSquare, |coordPartial 1 q p| ≤ M := by |
| obtain ⟨M, hM⟩ := modelSquare_isCompact.exists_bound_of_continuousOn |
| ((partial_contDiffOn hq hU 1).continuousOn.mono hSU) |
| refine ⟨max M 0, le_max_right _ _, ?_⟩ |
| intro p hp |
| have hh : |coordPartial 1 q p| ≤ M := by simpa only [Real.norm_eq_abs] using hM p hp |
| exact hh.trans (le_max_left _ _) |
|
|
| variable {q : Coord → ℝ} {U : Set Coord} {Y : ℝ → ℝ → ℝ} |
|
|
| theorem flow_initial_integral_bound |
| (hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) (hSU : modelSquare ⊆ U) |
| (hY : ContinuousOn (uncurry Y) (Icc (-2 : ℝ) 2 ×ˢ Icc (-2 : ℝ) 2)) |
| (hrange : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2, |
| Y s t ∈ Icc (-3 : ℝ) 3) |
| {M : ℝ} (hM : 0 ≤ M) (hMq : ∀ p ∈ modelSquare, |coordPartial 1 q p| ≤ M) |
| {s t : ℝ} (hs : s ∈ Icc (-2 : ℝ) 2) (ht : t ∈ Icc (-2 : ℝ) 2) : |
| IntervalIntegrable (fun r => coordPartial 1 q (coordinatePoint r (Y s r))) volume 0 t ∧ |
| |∫ r in 0..t, coordPartial 1 q (coordinatePoint r (Y s r))| ≤ 2 * M := by |
| have hmem (u : ℝ) (hu : u ∈ Icc (-2 : ℝ) 2) : |
| coordinatePoint u (Y s u) ∈ modelSquare := |
| coordinatePoint_mem_modelSquare ⟨by linarith [hu.1], by linarith [hu.2]⟩ (hrange s hs u hu) |
| have hy : ContinuousOn (Y s) (Icc (-2 : ℝ) 2) := |
| hY.comp (f := fun u : ℝ => (s, u)) (by fun_prop) (fun u hu => ⟨hs, hu⟩) |
| have hmap : ContinuousOn (fun u => coordinatePoint u (Y s u)) (Icc (-2 : ℝ) 2) := |
| (continuousOn_id.smul continuousOn_const).add (hy.smul continuousOn_const) |
| have hc : ContinuousOn (fun u => coordPartial 1 q (coordinatePoint u (Y s u))) |
| (Icc (-2 : ℝ) 2) := |
| (partial_contDiffOn hq hU 1).continuousOn.comp hmap (fun u hu => hSU (hmem u hu)) |
| have hseg : uIcc (0 : ℝ) t ⊆ Icc (-2 : ℝ) 2 := |
| uIcc_subset_Icc ⟨by norm_num, by norm_num⟩ ht |
| refine ⟨(hc.mono hseg).intervalIntegrable, ?_⟩ |
| have hh := intervalIntegral.norm_integral_le_of_norm_le_const (a := 0) (b := t) (C := M) |
| (f := fun u => coordPartial 1 q (coordinatePoint u (Y s u))) |
| (fun u hu => by |
| rw [Real.norm_eq_abs] |
| exact hMq _ (hmem u (hseg (uIoc_subset_uIcc hu)))) |
| have ht2 : |t| ≤ 2 := abs_le.mpr ht |
| calc |
| _ ≤ M * |t| := by simpa only [Real.norm_eq_abs, sub_zero] using hh |
| _ ≤ M * 2 := mul_le_mul_of_nonneg_left ht2 hM |
| _ = 2 * M := mul_comm _ _ |
|
|
| theorem cap_flow_initial_deriv_bounds |
| (hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) (hSU : modelSquare ⊆ U) |
| (hY : ContinuousOn (uncurry Y) (Icc (-2 : ℝ) 2 ×ˢ Icc (-2 : ℝ) 2)) |
| (hrange : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2, |
| Y s t ∈ Icc (-3 : ℝ) 3) |
| (hstart : ∀ s ∈ Icc (-2 : ℝ) 2, Y s 0 = 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) |
| {M : ℝ} (hM : 0 ≤ M) (hMq : ∀ p ∈ modelSquare, |coordPartial 1 q p| ≤ M) |
| {s t : ℝ} (hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Ioo (-2 : ℝ) 2) : |
| Real.exp (-2 * M) ≤ deriv (fun u => Y u t) s ∧ |
| deriv (fun u => Y u t) s ≤ Real.exp (2 * M) := by |
| have hI := (flow_initial_integral_bound hq hU hSU hY hrange hM hMq |
| ⟨hs.1.le, hs.2.le⟩ ⟨ht.1.le, ht.2.le⟩).2 |
| rw [(cap_flow_hasDerivAt_initial hq hU hSU hY hrange hstart hode hs ht).deriv] |
| constructor |
| · apply Real.exp_monotone |
| linarith [(abs_le.mp hI).2] |
| · apply Real.exp_monotone |
| linarith [(abs_le.mp hI).1] |
|
|
| def triangularFlow (Y : ℝ → ℝ → ℝ) (p : ℝ × ℝ) : ℝ × ℝ := (p.1, Y p.2 p.1) |
|
|
| theorem cap_flow_strictMonoOn_initial |
| (hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) (hSU : modelSquare ⊆ U) |
| (hY : ContinuousOn (uncurry Y) (Icc (-2 : ℝ) 2 ×ˢ Icc (-2 : ℝ) 2)) |
| (hrange : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2, |
| Y s t ∈ Icc (-3 : ℝ) 3) |
| (hstart : ∀ s ∈ Icc (-2 : ℝ) 2, Y s 0 = 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) |
| {t : ℝ} (ht : t ∈ Ioo (-2 : ℝ) 2) : |
| StrictMonoOn (fun s => Y s t) (Ioo (-2 : ℝ) 2) := by |
| have hc : ContinuousOn (fun s => Y s t) (Ioo (-2 : ℝ) 2) := |
| HasDerivAt.continuousOn |
| (fun s hs => cap_flow_hasDerivAt_initial hq hU hSU hY hrange hstart hode hs ht) |
| apply strictMonoOn_of_deriv_pos (convex_Ioo (-2 : ℝ) 2) hc |
| intro s hs |
| apply cap_flow_initial_deriv_pos hq hU hSU hY hrange hstart hode _ ht |
| simpa only [interior_Ioo] using hs |
|
|
| theorem triangularFlow_injOn |
| (hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) (hSU : modelSquare ⊆ U) |
| (hY : ContinuousOn (uncurry Y) (Icc (-2 : ℝ) 2 ×ˢ Icc (-2 : ℝ) 2)) |
| (hrange : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2, |
| Y s t ∈ Icc (-3 : ℝ) 3) |
| (hstart : ∀ s ∈ Icc (-2 : ℝ) 2, Y s 0 = 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) : |
| InjOn (triangularFlow Y) (pairRectangle 2 (-2) 2) := by |
| intro p hp z hz he |
| have ht : p.1 = z.1 := by |
| simpa only [triangularFlow] using congrArg (fun v : ℝ × ℝ => v.1) he |
| have hy : Y p.2 p.1 = Y z.2 z.1 := by |
| simpa only [triangularFlow] using congrArg (fun v : ℝ × ℝ => v.2) he |
| rw [ht] at hy |
| exact Prod.ext ht |
| ((cap_flow_strictMonoOn_initial hq hU hSU hY hrange hstart hode hz.1).injOn hp.2 hz.2 hy) |
|
|
| end SmoothLocal.Flow |
|
|
| end |
|
|
| end OAI |
|
|
