| import OAI.Geometry.IsometricImmersion.Flows.FlowLowBounds |
| import Mathlib.Analysis.Calculus.Deriv.MeanValue |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Function Filter |
| open scoped ContDiff Topology |
|
|
| namespace SmoothLocal.Flow |
| open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted |
|
|
| variable {q : Coord → ℝ} {U : Set Coord} {Y : ℝ → ℝ → ℝ} |
| variable (hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) (hSU : modelSquare ⊆ U) |
| variable (hY : ContinuousOn (uncurry Y) (Icc (-2 : ℝ) 2 ×ˢ Icc (-2 : ℝ) 2)) |
| variable (hrange : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2, |
| Y s t ∈ Icc (-3 : ℝ) 3) |
| variable (hstart : ∀ s ∈ Icc (-2 : ℝ) 2, Y s 0 = s) |
| variable (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) |
| include hq hU hSU hY hrange hstart hode |
|
|
| theorem cap_flow_seed_gap {M s0 s1 t : ℝ} (hM : 0 ≤ M) |
| (hq1 : ∀ p ∈ modelSquare, |coordPartial 1 q p| ≤ M) |
| (hs0 : s0 ∈ Ioo (-2 : ℝ) 2) (hs1 : s1 ∈ Ioo (-2 : ℝ) 2) |
| (ht : t ∈ Ioo (-2 : ℝ) 2) (hss : s0 ≤ s1) : |
| Real.exp (-2 * M) * (s1 - s0) ≤ Y s1 t - Y s0 t := by |
| have hd (s : ℝ) (hs : s ∈ Ioo (-2 : ℝ) 2) : |
| DifferentiableAt ℝ (fun r => Y r t) s := |
| (cap_flow_hasDerivAt_initial hq hU hSU hY hrange hstart hode hs ht).differentiableAt |
| have hc : ContinuousOn (fun r => Y r t) (Ioo (-2 : ℝ) 2) := |
| fun s hs => (hd s hs).continuousAt.continuousWithinAt |
| have hdi : DifferentiableOn ℝ (fun r => Y r t) (interior (Ioo (-2 : ℝ) 2)) := by |
| intro s hs |
| exact (hd s (interior_subset hs)).differentiableWithinAt |
| have hbound : ∀ s ∈ interior (Ioo (-2 : ℝ) 2), |
| Real.exp (-2 * M) ≤ deriv (fun r => Y r t) s := by |
| intro s hs |
| exact (cap_flow_initial_deriv_bounds hq hU hSU hY hrange hstart hode hM hq1 |
| (interior_subset hs) ht).1 |
| exact (convex_Ioo (-2 : ℝ) 2).mul_sub_le_image_sub_of_le_deriv hc hdi hbound |
| s0 hs0 s1 hs1 hss |
|
|
| theorem cap_flow_time_lipschitz |
| (hq0 : ∀ p ∈ modelSquare, |q p| ≤ (1 : ℝ) / 100) |
| {s t0 t1 : ℝ} (hs : s ∈ Ioo (-2 : ℝ) 2) |
| (ht0 : t0 ∈ Ioo (-2 : ℝ) 2) (ht1 : t1 ∈ Ioo (-2 : ℝ) 2) : |
| |Y s t1 - Y s t0| ≤ ((1 : ℝ) / 100) * |t1 - t0| := by |
| have hd (t : ℝ) (ht : t ∈ Ioo (-2 : ℝ) 2) : DifferentiableAt ℝ (Y s) t := |
| (cap_flow_time_hasDerivAt hq hU hSU hY hrange hstart hode hs ht).differentiableAt |
| have hb (t : ℝ) (ht : t ∈ Ioo (-2 : ℝ) 2) : ‖deriv (Y s) t‖ ≤ (1 : ℝ) / 100 := by |
| simpa only [Real.norm_eq_abs, timeFlowDerivative] using |
| cap_flow_time_bound hq hU hSU hY hrange hstart hode hq0 hs ht |
| simpa only [Real.norm_eq_abs] using |
| (convex_Ioo (-2 : ℝ) 2).norm_image_sub_le_of_norm_deriv_le hd hb ht0 ht1 |
|
|
| end SmoothLocal.Flow |
|
|
| end |
|
|
| end OAI |
|
|
