| import OAI.Geometry.IsometricImmersion.Flows.FlowTransport |
| import OAI.Geometry.IsometricImmersion.Metrics.MetricLocality |
| import Mathlib.Analysis.ODE.ExistUnique |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter Function Metric |
| open scoped ContDiff Topology |
|
|
| namespace SmoothLocal.Flow |
| open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted |
|
|
| theorem cap_trajectory_mem_ball_of_displacement {Y : ℝ → ℝ → ℝ} |
| (hdisp : ∀ p ∈ capChartDomain, |capFlowHeight Y p-p 1| ≤ (1:ℝ)/50) |
| {s t : ℝ} (hs : s ∈ Ioo (-2:ℝ) 2) (ht : t ∈ Ioo (-2:ℝ) 2) : |
| Y s t ∈ closedBall (0:ℝ) 3 := by |
| have hp : coordinatePoint t s ∈ capChartDomain := by |
| change coordinatePoint t s 0 ∈ Ioo (-2:ℝ) 2 ∧ coordinatePoint t s 1 ∈ Ioo (-2:ℝ) 2 |
| simpa [coordinatePoint] using (show t ∈ Ioo (-2:ℝ) 2 ∧ s ∈ Ioo (-2:ℝ) 2 from ⟨ht,hs⟩) |
| have hd : |Y s t-s| ≤ (1:ℝ)/50 := by |
| simpa [capFlowHeight,coordinatePoint] using hdisp (coordinatePoint t s) hp |
| have hy : |Y s t| ≤ 3 := by |
| have hh : |Y s t| ≤ |Y s t-s|+|s| := by |
| simpa only [sub_add_cancel] using abs_add_le (Y s t-s) s |
| have hs' : |s| < 2 := abs_lt.mpr hs |
| linarith |
| simpa only [mem_closedBall,Real.dist_eq,sub_zero] using hy |
|
|
| theorem capChart_eqOn_of_same_ode |
| {q : Coord → ℝ} {U : Set Coord} {Y Z : ℝ → ℝ → ℝ} |
| (hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) (hSU : modelSquare ⊆ U) |
| (hYstart : ∀ s ∈ Icc (-2:ℝ) 2, Y s 0 = s) |
| (hYode : ∀ s ∈ Icc (-2:ℝ) 2, ∀ t ∈ Icc (-2:ℝ) 2, |
| HasDerivWithinAt (Y s) (-q (coordinatePoint t (Y s t))) (Icc (-2:ℝ) 2) t) |
| (hYdisp : ∀ p ∈ capChartDomain, |capFlowHeight Y p-p 1| ≤ (1:ℝ)/50) |
| (hZstart : ∀ s ∈ Icc (-2:ℝ) 2, Z s 0 = s) |
| (hZode : ∀ s ∈ Icc (-2:ℝ) 2, ∀ t ∈ Icc (-2:ℝ) 2, |
| HasDerivWithinAt (Z s) (-q (coordinatePoint t (Z s t))) (Icc (-2:ℝ) 2) t) |
| (hZdisp : ∀ p ∈ capChartDomain, |capFlowHeight Z p-p 1| ≤ (1:ℝ)/50) : |
| EqOn (capChart Y) (capChart Z) capChartDomain := by |
| obtain ⟨K,hK⟩ := exists_flowField_lipschitz hq hU hSU |
| have hsame (s : ℝ) (hs : s ∈ Ioo (-2:ℝ) 2) : EqOn (Y s) (Z s) (Ioo (-2:ℝ) 2) := by |
| have hs' : s ∈ Icc (-2:ℝ) 2 := ⟨hs.1.le,hs.2.le⟩ |
| apply ODE_solution_unique_of_mem_Ioo |
| (v := flowField q) (s := fun _ => closedBall (0:ℝ) 3) |
| (fun t ht => hK t ⟨ht.1.le,ht.2.le⟩) (t₀ := 0) (by norm_num) |
| · intro t ht |
| exact ⟨(hYode s hs' t ⟨ht.1.le,ht.2.le⟩).hasDerivAt (Icc_mem_nhds ht.1 ht.2), |
| cap_trajectory_mem_ball_of_displacement hYdisp hs ht⟩ |
| · intro t ht |
| exact ⟨(hZode s hs' t ⟨ht.1.le,ht.2.le⟩).hasDerivAt (Icc_mem_nhds ht.1 ht.2), |
| cap_trajectory_mem_ball_of_displacement hZdisp hs ht⟩ |
| · rw [hYstart s hs',hZstart s hs'] |
| intro p hp |
| unfold capChart capFlowHeight |
| rw [hsame (p 1) hp.2 hp.1] |
|
|
| theorem capPullback_eventuallyEq_of_chart_eqOn {Y Z : ℝ → ℝ → ℝ} |
| (heq : EqOn (capChart Y) (capChart Z) capChartDomain) |
| (f : Coord → ℝ) {p : Coord} (hp : p ∈ capChartDomain) : |
| capPullback Y f =ᶠ[𝓝 p] capPullback Z f := by |
| filter_upwards [capChartDomain_isOpen.mem_nhds hp] with r hr |
| exact congrArg f (heq hr) |
|
|
| theorem capPullback_all_jets_eq_of_chart_eqOn {Y Z : ℝ → ℝ → ℝ} |
| (heq : EqOn (capChart Y) (capChart Z) capChartDomain) |
| (f : Coord → ℝ) (dirs : List (Fin 2)) {p : Coord} (hp : p ∈ capChartDomain) : |
| iteratedCoordPartial dirs (capPullback Y f) p = iteratedCoordPartial dirs (capPullback Z f) p := |
| (iteratedCoordPartial_eventuallyEq (capPullback_eventuallyEq_of_chart_eqOn heq f hp) dirs).self_of_nhds |
|
|
| theorem capChart_images_eq_of_eqOn {Y Z : ℝ → ℝ → ℝ} |
| (heq : EqOn (capChart Y) (capChart Z) capChartDomain) |
| {S : Set Coord} (hS : S ⊆ capChartDomain) : capChart Y '' S = capChart Z '' S := |
| Set.image_congr (fun _ hp => heq (hS hp)) |
|
|
| theorem capPullback_square_rectangle_eq_of_chart_eqOn {Y Z : ℝ → ℝ → ℝ} |
| (heq : EqOn (capChart Y) (capChart Z) capChartDomain) |
| (f : Coord → ℝ) {tl tr sb st : ℝ} (ht : tl ≤ tr) (hs : sb ≤ st) |
| (hbox : closedRectangle tl tr sb st ⊆ capChartDomain) : |
| rectangleIntegral tl tr sb st (fun p => (capPullback Y f p)^2) = |
| rectangleIntegral tl tr sb st (fun p => (capPullback Z f p)^2) := by |
| apply rectangleIntegral_congr ht hs |
| intro p hp |
| exact congrArg (fun x : Coord => (f x)^2) (heq (hbox hp)) |
|
|
| theorem capPullback_gradient_rectangle_eq_of_chart_eqOn {Y Z : ℝ → ℝ → ℝ} |
| (heq : EqOn (capChart Y) (capChart Z) capChartDomain) |
| (f : Coord → ℝ) {tl tr sb st : ℝ} (ht : tl ≤ tr) (hs : sb ≤ st) |
| (hbox : closedRectangle tl tr sb st ⊆ capChartDomain) : |
| rectangleIntegral tl tr sb st (fun p => (coordPartial 0 (capPullback Y f) p)^2+ |
| (coordPartial 1 (capPullback Y f) p)^2) = |
| rectangleIntegral tl tr sb st (fun p => (coordPartial 0 (capPullback Z f) p)^2+ |
| (coordPartial 1 (capPullback Z f) p)^2) := by |
| apply rectangleIntegral_congr ht hs |
| intro p hp |
| change (coordPartial 0 (capPullback Y f) p)^2 + (coordPartial 1 (capPullback Y f) p)^2 = |
| (coordPartial 0 (capPullback Z f) p)^2 + (coordPartial 1 (capPullback Z f) p)^2 |
| rw [coordPartial_eq_of_eventuallyEq (capPullback_eventuallyEq_of_chart_eqOn heq f (hbox hp)) 0, |
| coordPartial_eq_of_eventuallyEq (capPullback_eventuallyEq_of_chart_eqOn heq f (hbox hp)) 1] |
|
|
| end SmoothLocal.Flow |
|
|
| end |
|
|
| end OAI |
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|
