| import OAI.Geometry.IsometricImmersion.Flows.FlowInverse | |
| import Mathlib.Topology.IsLocalHomeomorph | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter MeasureTheory Function | |
| open scoped ContDiff Topology Interval | |
| namespace SmoothLocal.Flow | |
| open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted | |
| variable {q : Coord → ℝ} {U : Set Coord} {Y : ℝ → ℝ → ℝ} | |
| theorem exists_global_triangularFlow_chart | |
| (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) : | |
| ∃ e : OpenPartialHomeomorph (ℝ × ℝ) (ℝ × ℝ), | |
| e.source = pairRectangle 2 (-2) 2 ∧ | |
| e.target = triangularFlow Y '' pairRectangle 2 (-2) 2 ∧ | |
| (e : (ℝ × ℝ) → (ℝ × ℝ)) = triangularFlow Y ∧ | |
| ContDiffOn ℝ ∞ e e.source ∧ ContDiffOn ℝ ∞ e.symm e.target := by | |
| let D := pairRectangle 2 (-2) 2 | |
| have hD : IsOpen D := pairRectangle_isOpen 2 (-2) 2 | |
| have hTF : ContDiffOn ℝ ∞ (triangularFlow Y) D := | |
| contDiffOn_fst.prodMk (cap_flow_joint_contDiffOn hq hU hSU hY hrange hstart hode) | |
| have hinj : InjOn (triangularFlow Y) D := triangularFlow_injOn hq hU hSU hY hrange hstart hode | |
| have hlocal : IsLocalHomeomorphOn (triangularFlow Y) D := by | |
| intro x hx | |
| obtain ⟨e, he, hsub, heq, hf, hi⟩ := | |
| exists_triangularFlow_smooth_chart hq hU hSU hY hrange hstart hode hx | |
| exact ⟨e, he, heq.symm⟩ | |
| have hopen : IsOpenMap (D.domRestrict (triangularFlow Y)) := by | |
| apply IsOpenMap.of_nhds_le | |
| intro x | |
| have he : (𝓝 x).map (D.domRestrict (triangularFlow Y)) = | |
| 𝓝 (triangularFlow Y x) := by | |
| change (𝓝 x).map ((triangularFlow Y) ∘ (Subtype.val : D → ℝ × ℝ)) = _ | |
| rw [← Filter.map_map, hD.isOpenEmbedding_subtypeVal.map_nhds_eq, | |
| hlocal.map_nhds_eq x.2] | |
| exact he.ge | |
| let pe := hinj.toPartialEquiv (triangularFlow Y) D | |
| let e := OpenPartialHomeomorph.ofContinuousOpenRestrict pe hTF.continuousOn hopen hD | |
| have heq : (e : (ℝ × ℝ) → (ℝ × ℝ)) = triangularFlow Y := rfl | |
| have hsource : e.source = D := rfl | |
| refine ⟨e, hsource, rfl, heq, ?_, ?_⟩ | |
| · rw [heq, hsource] | |
| exact hTF | |
| · intro y hy | |
| have hx : e.symm y ∈ D := by | |
| rw [← hsource] | |
| exact e.map_target hy | |
| obtain ⟨A, hA⟩ := triangularFlow_hasFDerivAt_equiv hq hU hSU hY hrange hstart hode hx | |
| have hAe : HasFDerivAt e (A : (ℝ × ℝ) →L[ℝ] (ℝ × ℝ)) (e.symm y) := by | |
| rw [heq] | |
| exact hA | |
| have hfe : ContDiffAt ℝ ∞ e (e.symm y) := by | |
| rw [heq] | |
| exact hTF.contDiffAt (hD.mem_nhds hx) | |
| exact (e.contDiffAt_symm hy hAe hfe).contDiffWithinAt | |
| def flowCap (Y : ℝ → ℝ → ℝ) (b : ℝ) : Set (ℝ × ℝ) := | |
| triangularFlow Y '' pairRectangle 2 (-2) b | |
| theorem flowCap_mono (Y : ℝ → ℝ → ℝ) {b1 b2 : ℝ} (hb : b1 ≤ b2) : | |
| flowCap Y b1 ⊆ flowCap Y b2 := by | |
| apply image_mono | |
| intro p hp | |
| exact ⟨hp.1, hp.2.1, hp.2.2.trans_le hb⟩ | |
| theorem flowCap_isOpen_of_chart {e : OpenPartialHomeomorph (ℝ × ℝ) (ℝ × ℝ)} | |
| (hsource : e.source = pairRectangle 2 (-2) 2) | |
| (heq : (e : (ℝ × ℝ) → (ℝ × ℝ)) = triangularFlow Y) | |
| {b : ℝ} (hb : b ≤ 2) : IsOpen (flowCap Y b) := by | |
| let C := pairRectangle 2 (-2) b | |
| have hC : IsOpen C := pairRectangle_isOpen 2 (-2) b | |
| have hsub : C ⊆ e.source := by | |
| rw [hsource] | |
| intro p hp | |
| exact ⟨hp.1, hp.2.1, hp.2.2.trans_le hb⟩ | |
| let eb := e.restrOpen C hC | |
| have hes : eb.source = C := by | |
| change e.source ∩ C = C | |
| exact inter_eq_right.mpr hsub | |
| have het : eb.target = flowCap Y b := by | |
| rw [← eb.image_source_eq_target, hes] | |
| change e '' C = triangularFlow Y '' C | |
| rw [heq] | |
| rw [← het] | |
| exact eb.open_target | |
| end SmoothLocal.Flow | |
| end | |
| end OAI | |