| import Mathlib |
| import OAI.Geometry.BallPacking.Flows.FlowSmoothExtension |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace PackingSufficiencySupport.Hamiltonian |
| open scoped ContDiff Manifold Topology |
| open Set Function Manifold |
| section |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] |
| {M : Type*} [TopologicalSpace M] [T2Space M] [CompactSpace M] |
| [ChartedSpace E M] [IsManifold 𝓘(ℝ,E) ∞ M] |
|
|
| theorem exists_nonautonomous_manifold_flow |
| {V : (p : ℝ × M) → TangentSpace 𝓘(ℝ,E) p.2} |
| (hV : ContMDiff ((𝓘(ℝ,ℝ)).prod 𝓘(ℝ,E)) (𝓘(ℝ,E)).tangent ∞ |
| (fun p => (⟨p.2,V p⟩ : TangentBundle 𝓘(ℝ,E) M))) : |
| ∃ Φ Ψ : ℝ × M → M, |
| ContMDiff ((𝓘(ℝ,ℝ)).prod 𝓘(ℝ,E)) 𝓘(ℝ,E) ∞ Φ ∧ |
| ContMDiff ((𝓘(ℝ,ℝ)).prod 𝓘(ℝ,E)) 𝓘(ℝ,E) ∞ Ψ ∧ |
| (∀ x, Φ (0,x) = x) ∧ |
| (∀ t ∈ Icc (0:ℝ) 1, ∀ x, Ψ (t,Φ (t,x)) = x ∧ Φ (t,Ψ (t,x)) = x) ∧ |
| (∀ x t, t ∈ Ioo (-2:ℝ) 2 → HasMFDerivAt 𝓘(ℝ,ℝ) 𝓘(ℝ,E) |
| (fun s => Φ (s,x)) t ((1 : ℝ →L[ℝ] ℝ).smulRight (V (t,Φ (t,x))))) ∧ |
| (∀ x, (∀ s, V (s,x) = 0) → ∀ t, Φ (t,x) = x) := by |
| obtain ⟨χ,G,hχ,hχ1,hGs,hG0,hGa,hGd,hclock⟩ := exists_compact_time_lift hV |
| let Φ : ℝ × M → M := fun p => (G (p.1,(0,p.2))).2 |
| let Ψ : ℝ × M → M := fun p => (G (-p.1,p)).2 |
| have hΦs : ContMDiff ((𝓘(ℝ,ℝ)).prod 𝓘(ℝ,E)) 𝓘(ℝ,E) ∞ Φ := |
| contMDiff_snd.comp (hGs.comp (contMDiff_fst.prodMk (contMDiff_const.prodMk contMDiff_snd))) |
| have hneg : ContMDiff ((𝓘(ℝ,ℝ)).prod 𝓘(ℝ,E)) 𝓘(ℝ,ℝ) ∞ (fun p : ℝ × M => -p.1) := |
| contDiff_neg.contMDiff.comp contMDiff_fst |
| have hΨs : ContMDiff ((𝓘(ℝ,ℝ)).prod 𝓘(ℝ,E)) 𝓘(ℝ,E) ∞ Ψ := |
| contMDiff_snd.comp (hGs.comp (hneg.prodMk contMDiff_id)) |
| have hΦclock (t : ℝ) (ht : t ∈ Ioo (-2:ℝ) 2) (x : M) : G (t,(0,x)) = (t,Φ (t,x)) := by |
| apply Prod.ext |
| · simpa only [zero_add] using hclock 0 (by norm_num) x t ht |
| · rfl |
| refine ⟨Φ,Ψ,hΦs,hΨs,?_,?_,?_,?_⟩ |
| · intro x |
| exact congrArg Prod.snd (hG0 (0,x)) |
| · intro t ht x |
| have ht' : t ∈ Ioo (-2:ℝ) 2 := by constructor <;> linarith [ht.1,ht.2] |
| have hmt : -t ∈ Ioo (-2:ℝ) 2 := by constructor <;> linarith [ht.1,ht.2] |
| constructor |
| · change (G (-t,(t,Φ (t,x)))).2 = x |
| rw [←hΦclock t ht' x,←hGa,neg_add_cancel,hG0] |
| · have hΨclock : G (-t,(t,x)) = (0,Ψ (t,x)) := by |
| apply Prod.ext |
| · simpa only [add_neg_cancel] using hclock t ht x (-t) hmt |
| · rfl |
| change (G (t,(0,Ψ (t,x)))).2 = x |
| rw [←hΨclock,←hGa,add_neg_cancel,hG0] |
| · intro x t ht |
| have hd := timeLiftField_spatial_derivative (hGd (0,x) t) |
| have hh : χ t = 1 := hχ1 t ⟨by linarith [ht.1],by linarith [ht.2]⟩ |
| rw [hΦclock t ht x] at hd |
| simpa only [hh,one_smul] using hd |
| · intro x hx t |
| exact timeLift_spatial_stationary hχ hV hx (hGd (0,x)) (hG0 (0,x)) t |
|
|
| def nonautonomousSlice (Φ Ψ : ℝ × M → M) |
| (hΦ : ContMDiff ((𝓘(ℝ,ℝ)).prod 𝓘(ℝ,E)) 𝓘(ℝ,E) ∞ Φ) |
| (hΨ : ContMDiff ((𝓘(ℝ,ℝ)).prod 𝓘(ℝ,E)) 𝓘(ℝ,E) ∞ Ψ) |
| (t : ℝ) (hinv : ∀ x, Ψ (t,Φ (t,x)) = x ∧ Φ (t,Ψ (t,x)) = x) : |
| M ≃ₘ⟮𝓘(ℝ,E),𝓘(ℝ,E)⟯ M where |
| toFun x := Φ (t,x) |
| invFun x := Ψ (t,x) |
| left_inv x := (hinv x).1 |
| right_inv x := (hinv x).2 |
| contMDiff_toFun := hΦ.comp (contMDiff_const.prodMk contMDiff_id) |
| contMDiff_invFun := hΨ.comp (contMDiff_const.prodMk contMDiff_id) |
|
|
| end |
|
|
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] |
| {M : Type*} [TopologicalSpace M] [T2Space M] |
| [ChartedSpace E M] [IsManifold 𝓘(ℝ,E) ∞ M] |
|
|
| theorem exists_spatially_compact_nonautonomous_flow |
| {V : (p : ℝ × M) → TangentSpace 𝓘(ℝ,E) p.2} |
| (hV : ContMDiff ((𝓘(ℝ,ℝ)).prod 𝓘(ℝ,E)) (𝓘(ℝ,E)).tangent ∞ |
| (fun p => (⟨p.2,V p⟩ : TangentBundle 𝓘(ℝ,E) M))) |
| {K : Set M} (hK : IsCompact K) (hVK : ∀ s x, x ∉ K → V (s,x)=0) : |
| ∃ Φ Ψ : ℝ × M → M, |
| ContMDiff ((𝓘(ℝ,ℝ)).prod 𝓘(ℝ,E)) 𝓘(ℝ,E) ∞ Φ ∧ |
| ContMDiff ((𝓘(ℝ,ℝ)).prod 𝓘(ℝ,E)) 𝓘(ℝ,E) ∞ Ψ ∧ |
| (∀ x, Φ (0,x) = x) ∧ |
| (∀ t ∈ Icc (0:ℝ) 1, ∀ x, Ψ (t,Φ (t,x)) = x ∧ Φ (t,Ψ (t,x)) = x) ∧ |
| (∀ x t, t ∈ Ioo (-2:ℝ) 2 → HasMFDerivAt 𝓘(ℝ,ℝ) 𝓘(ℝ,E) |
| (fun s => Φ (s,x)) t ((1 : ℝ →L[ℝ] ℝ).smulRight (V (t,Φ (t,x))))) ∧ |
| (∀ x, (∀ s, V (s,x) = 0) → ∀ t, Φ (t,x) = x) := by |
| obtain ⟨χ,G,hχ,hχ1,hGs,hG0,hGa,hGd,hclock⟩ := exists_spatially_compact_time_lift hV hK hVK |
| let Φ : ℝ × M → M := fun p => (G (p.1,(0,p.2))).2 |
| let Ψ : ℝ × M → M := fun p => (G (-p.1,p)).2 |
| have hΦs : ContMDiff ((𝓘(ℝ,ℝ)).prod 𝓘(ℝ,E)) 𝓘(ℝ,E) ∞ Φ := |
| contMDiff_snd.comp (hGs.comp (contMDiff_fst.prodMk (contMDiff_const.prodMk contMDiff_snd))) |
| have hneg : ContMDiff ((𝓘(ℝ,ℝ)).prod 𝓘(ℝ,E)) 𝓘(ℝ,ℝ) ∞ (fun p : ℝ × M => -p.1) := |
| contDiff_neg.contMDiff.comp contMDiff_fst |
| have hΨs : ContMDiff ((𝓘(ℝ,ℝ)).prod 𝓘(ℝ,E)) 𝓘(ℝ,E) ∞ Ψ := |
| contMDiff_snd.comp (hGs.comp (hneg.prodMk contMDiff_id)) |
| have hΦclock (t : ℝ) (ht : t ∈ Ioo (-2:ℝ) 2) (x : M) : G (t,(0,x)) = (t,Φ (t,x)) := by |
| apply Prod.ext |
| · simpa only [zero_add] using hclock 0 (by norm_num) x t ht |
| · rfl |
| refine ⟨Φ,Ψ,hΦs,hΨs,?_,?_,?_,?_⟩ |
| · intro x |
| exact congrArg Prod.snd (hG0 (0,x)) |
| · intro t ht x |
| have ht' : t ∈ Ioo (-2:ℝ) 2 := by constructor <;> linarith [ht.1,ht.2] |
| have hmt : -t ∈ Ioo (-2:ℝ) 2 := by constructor <;> linarith [ht.1,ht.2] |
| constructor |
| · change (G (-t,(t,Φ (t,x)))).2 = x |
| rw [←hΦclock t ht' x,←hGa,neg_add_cancel,hG0] |
| · have hΨclock : G (-t,(t,x)) = (0,Ψ (t,x)) := by |
| apply Prod.ext |
| · simpa only [add_neg_cancel] using hclock t ht x (-t) hmt |
| · rfl |
| change (G (t,(0,Ψ (t,x)))).2 = x |
| rw [←hΨclock,←hGa,add_neg_cancel,hG0] |
| · intro x t ht |
| have hd := timeLiftField_spatial_derivative (hGd (0,x) t) |
| have hh : χ t = 1 := hχ1 t ⟨by linarith [ht.1],by linarith [ht.2]⟩ |
| rw [hΦclock t ht x] at hd |
| simpa only [hh,one_smul] using hd |
| · intro x hx t |
| exact timeLift_spatial_stationary hχ hV hx (hGd (0,x)) (hG0 (0,x)) t |
|
|
| end PackingSufficiencySupport.Hamiltonian |
|
|
| namespace PackingSufficiencySupport |
| open Set |
|
|
| theorem constant_above_on_convex {s : Set ℝ} (hs : Convex ℝ s) |
| {f : ℝ → ℝ} (hf : ContinuousOn f s) {a t₀ : ℝ} (ht₀ : t₀ ∈ s) |
| (hzero : ∀ t ∈ s, a < f t → HasDerivWithinAt f 0 s t) |
| (h₀ : a < f t₀) : ∀ t ∈ s, f t = f t₀ := by |
| let : PreconnectedSpace s := isPreconnected_iff_preconnectedSpace.mp hs.isPreconnected |
| let T : Set s := {t | f t = f t₀} |
| have hfc : Continuous (fun t : s => f t) := hf.domRestrict |
| have hc : IsClosed T := isClosed_eq hfc continuous_const |
| have ho : IsOpen T := by |
| apply isOpen_iff_mem_nhds.mpr |
| intro t ht |
| have hft : a < f t := by rw [ht]; exact h₀ |
| obtain ⟨r,hr,hball⟩ := Metric.mem_nhds_iff.mp |
| ((isOpen_lt continuous_const hfc).mem_nhds hft) |
| apply Filter.mem_of_superset (Metric.ball_mem_nhds t hr) |
| intro u hu |
| let U : Set ℝ := s ∩ Metric.ball (t : ℝ) r |
| have hUc : Convex ℝ U := hs.inter (convex_ball (t:ℝ) r) |
| have hUd (v : ℝ) (hv : v ∈ U) : HasFDerivWithinAt f (0 : ℝ →L[ℝ] ℝ) U v := by |
| have habove : a < f v := hball (show (⟨v,hv.1⟩ : s) ∈ Metric.ball t r from hv.2) |
| simpa using (hzero v hv.1 habove).hasFDerivWithinAt.mono inter_subset_left |
| have htU : (t : ℝ) ∈ U := ⟨t.property,Metric.mem_ball_self hr⟩ |
| have huU : (u : ℝ) ∈ U := ⟨u.property,hu⟩ |
| have hb := hUc.norm_image_sub_le_of_norm_hasFDerivWithin_le hUd |
| (fun v hv => by simp : ∀ v ∈ U, ‖(0 : ℝ →L[ℝ] ℝ)‖ ≤ (0:ℝ)) htU huU |
| have he : f u = f t := sub_eq_zero.mp (norm_le_zero_iff.mp (by simpa using hb)) |
| exact he.trans ht |
| have he : T = univ := (show IsClopen T from ⟨hc,ho⟩).eq_univ ⟨⟨t₀,ht₀⟩,rfl⟩ |
| intro t ht |
| exact (show (⟨t,ht⟩ : s) ∈ T from he.symm ▸ mem_univ _) |
|
|
| theorem sublevel_iff_of_zero_in_collar {s : Set ℝ} (hs : Convex ℝ s) |
| {f : ℝ → ℝ} (hf : ContinuousOn f s) {a c t u : ℝ} (hac : a < c) |
| (ht : t ∈ s) (hu : u ∈ s) |
| (hzero : ∀ r ∈ s, a < f r → HasDerivWithinAt f 0 s r) : |
| f t ≤ c ↔ f u ≤ c := by |
| have hdir (t u : ℝ) (ht : t ∈ s) (hu : u ∈ s) (h : f t ≤ c) : f u ≤ c := by |
| by_contra hn |
| have hc : c < f u := lt_of_not_ge hn |
| have he := constant_above_on_convex hs hf hu hzero (hac.trans hc) t ht |
| linarith |
| exact ⟨hdir t u ht hu,hdir u t hu ht⟩ |
|
|
| end PackingSufficiencySupport |
|
|
| namespace PackingSufficiencySupport.Hamiltonian |
| open scoped ContDiff Manifold Topology |
| open Set Function Manifold |
| section |
|
|
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| {M : Type*} [TopologicalSpace M] [ChartedSpace E M] [IsManifold 𝓘(ℝ,E) ∞ M] |
|
|
| omit [IsManifold 𝓘(ℝ,E) ∞ M] in |
|
|
| theorem manifold_flow_sublevels_iff |
| {Φ : ℝ × M → M} {V : (p : ℝ × M) → TangentSpace 𝓘(ℝ,E) p.2} |
| (hΦ : ContMDiff ((𝓘(ℝ,ℝ)).prod 𝓘(ℝ,E)) 𝓘(ℝ,E) ∞ Φ) |
| (hΦ0 : ∀ x, Φ (0,x)=x) |
| (hODE : ∀ x t, t ∈ Icc (0:ℝ) 1 → HasMFDerivAt 𝓘(ℝ,ℝ) 𝓘(ℝ,E) |
| (fun s => Φ (s,x)) t ((1 : ℝ →L[ℝ] ℝ).smulRight (V (t,Φ (t,x))))) |
| {ι : Type*} (μ : ι → M → ℝ) |
| (hμ : ∀ i, ContMDiff 𝓘(ℝ,E) 𝓘(ℝ,ℝ) ∞ (μ i)) |
| (a c : ι → ℝ) (hac : ∀ i, a i<c i) |
| (htan : ∀ i t x, a i<μ i x → mfderiv 𝓘(ℝ,E) 𝓘(ℝ,ℝ) (μ i) x (V (t,x))=0) |
| (t : ℝ) (ht : t ∈ Icc (0:ℝ) 1) (x : M) : |
| (∀ i, μ i (Φ (t,x))≤c i) ↔ (∀ i, μ i x≤c i) := by |
| have he (i : ι) : μ i (Φ (t,x))≤c i ↔ μ i x≤c i := by |
| have hz (s : ℝ) (hs : s ∈ Icc (0:ℝ) 1) (ha : a i<μ i (Φ (s,x))) : |
| HasDerivWithinAt (fun r => μ i (Φ (r,x))) 0 (Icc (0:ℝ) 1) s := by |
| have hd := ((hμ i).mdifferentiable (by simp) (Φ (s,x))).hasMFDerivAt.comp s |
| (hODE x s hs) |
| have heq : (mfderiv 𝓘(ℝ,E) 𝓘(ℝ,ℝ) (μ i) (Φ (s,x))).comp |
| ((1 : ℝ →L[ℝ] ℝ).smulRight (V (s,Φ (s,x))))=(0 : ℝ →L[ℝ] ℝ) := by |
| ext |
| simp only [ContinuousLinearMap.comp_apply,ContinuousLinearMap.smulRight_apply, |
| map_smul,htan i s (Φ (s,x)) ha,smul_zero] |
| rfl |
| have hd' : HasMFDerivAt 𝓘(ℝ,ℝ) 𝓘(ℝ,ℝ) |
| (fun r => μ i (Φ (r,x))) s 0 := hd.congr_mfderiv heq |
| exact (hasDerivAt_iff_hasFDerivAt.mpr |
| (by simpa using! hasMFDerivAt_iff_hasFDerivAt.mp hd')).hasDerivWithinAt |
| have hc : ContinuousOn (fun r => μ i (Φ (r,x))) (Icc (0:ℝ) 1) := |
| ((hμ i).continuous.comp (hΦ.continuous.comp |
| (continuous_id.prodMk continuous_const))).continuousOn |
| simpa only [hΦ0] using sublevel_iff_of_zero_in_collar (convex_Icc (0:ℝ) 1) |
| hc (hac i) ht (show (0:ℝ) ∈ Icc (0:ℝ) 1 by norm_num) hz |
| exact forall_congr' he |
| |
| omit [IsManifold 𝓘(ℝ,E) ∞ M] in |
| |
| theorem manifold_flow_image_sublevels |
| {Φ Ψ : ℝ × M → M} |
| (hΦ : ContMDiff ((𝓘(ℝ,ℝ)).prod 𝓘(ℝ,E)) 𝓘(ℝ,E) ∞ Φ) |
| (hΨ : ContMDiff ((𝓘(ℝ,ℝ)).prod 𝓘(ℝ,E)) 𝓘(ℝ,E) ∞ Ψ) |
| (t : ℝ) (hinv : ∀ x, Ψ (t,Φ (t,x))=x ∧ Φ (t,Ψ (t,x))=x) |
| {D : Set M} (hD : ∀ x, Φ (t,x) ∈ D ↔ x ∈ D) : |
| (fun x => Φ (t,x)) '' D=D ∧ |
| (fun x => Φ (t,x)) '' interior D=interior D := by |
| let f := nonautonomousSlice Φ Ψ hΦ hΨ t hinv |
| have he : (fun x => Φ (t,x)) '' D=D := by |
| ext y |
| constructor |
| · rintro ⟨x,hx,rfl⟩; exact (hD x).mpr hx |
| · intro hy |
| exact ⟨Ψ (t,y),(hD _).mp (by simpa only [(hinv y).2] using hy),(hinv y).2⟩ |
| refine ⟨he,?_⟩ |
| change f.toHomeomorph '' interior D=interior D |
| rw [f.toHomeomorph.image_interior] |
| exact congrArg interior he |
| |
| end |
| section |
| |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] |
| {M : Type*} [TopologicalSpace M] [T2Space M] [NormalSpace M] [SigmaCompactSpace M] |
| [ChartedSpace E M] [IsManifold 𝓘(ℝ,E) ∞ M] |
| |
| theorem manifold_relative_sublevel_moser |
| {Ω : ℝ → ManifoldTwoForm E M} {α : ℝ → ManifoldOneForm E M} |
| (hΩ : ∀ c, ContDiffOn ℝ ∞ (fun q : ℝ × E => chartTwoForm (Ω q.1) c q.2) |
| (univ ×ˢ (extChartAt 𝓘(ℝ,E) c).target)) |
| (hα : ∀ c, ContDiffOn ℝ ∞ (fun q : ℝ × E => chartOneForm (α q.1) c q.2) |
| (univ ×ˢ (extChartAt 𝓘(ℝ,E) c).target)) |
| {ι : Type*} (μ : ι → M → ℝ) |
| (hμ : ∀ i, ContMDiff 𝓘(ℝ,E) 𝓘(ℝ,ℝ) ∞ (μ i)) |
| (a c : ι → ℝ) (hac : ∀ i, a i<c i) |
| (hD : IsCompact {x | ∀ i, μ i x≤c i}) |
| {U : Set (ℝ × M)} (hU : IsOpen U) |
| (hDU : Icc (0:ℝ) 1 ×ˢ {x | ∀ i, μ i x≤c i} ⊆ U) |
| (hinv : ∀ p ∈ U, (Ω p.1 p.2).IsInvertible) |
| (hskew : ∀ p ∈ U, ∀ v w, Ω p.1 p.2 v w = -Ω p.1 p.2 w v) |
| (hclosed : ∀ t b y, y ∈ (extChartAt 𝓘(ℝ,E) b).target → |
| (t,(extChartAt 𝓘(ℝ,E) b).symm y) ∈ U → ∀ u v w, |
| fderiv ℝ (fun q : ℝ × E => chartTwoForm (Ω q.1) b q.2) (t,y) (0,u) v w - |
| fderiv ℝ (fun q : ℝ × E => chartTwoForm (Ω q.1) b q.2) (t,y) (0,v) u w + |
| fderiv ℝ (fun q : ℝ × E => chartTwoForm (Ω q.1) b q.2) (t,y) (0,w) u v = 0) |
| (htime : ∀ t b y, y ∈ (extChartAt 𝓘(ℝ,E) b).target → |
| (t,(extChartAt 𝓘(ℝ,E) b).symm y) ∈ U → ∀ v w, |
| fderiv ℝ (fun q : ℝ × E => chartTwoForm (Ω q.1) b q.2) (t,y) (1,0) v w = |
| fderiv ℝ (fun q : ℝ × E => chartOneForm (α q.1) b q.2) (t,y) (0,v) w - |
| fderiv ℝ (fun q : ℝ × E => chartOneForm (α q.1) b q.2) (t,y) (0,w) v) |
| (Z : ι → M → E) |
| (hface : ∀ i p, p ∈ U → a i<μ i p.2 → |
| α p.1 p.2=0 ∨ |
| (Ω p.1 p.2 (Z i p.2) = -mfderiv 𝓘(ℝ,E) 𝓘(ℝ,ℝ) (μ i) p.2 ∧ |
| α p.1 p.2 (Z i p.2)=0)) : |
| ∃ Φ Ψ : ℝ × M → M, |
| ContMDiff ((𝓘(ℝ,ℝ)).prod 𝓘(ℝ,E)) 𝓘(ℝ,E) ∞ Φ ∧ |
| ContMDiff ((𝓘(ℝ,ℝ)).prod 𝓘(ℝ,E)) 𝓘(ℝ,E) ∞ Ψ ∧ |
| (∀ x, Φ (0,x)=x) ∧ |
| (∀ t ∈ Icc (0:ℝ) 1, ∀ x, Ψ (t,Φ (t,x))=x ∧ Φ (t,Ψ (t,x))=x) ∧ |
| (∃ C : Set M, IsCompact C ∧ ∀ t x, x ∉ C → Φ (t,x)=x) ∧ |
| (∀ x, (∀ t, α t x=0) → ∀ t, Φ (t,x)=x) ∧ |
| (∀ t ∈ Icc (0:ℝ) 1, |
| (fun x => Φ (t,x)) '' {x | ∀ i, μ i x≤c i}={x | ∀ i, μ i x≤c i} ∧ |
| (fun x => Φ (t,x)) '' interior {x | ∀ i, μ i x≤c i}=interior {x | ∀ i, μ i x≤c i}) ∧ |
| (∀ t ∈ Icc (0:ℝ) 1, ∀ x, (∀ i, μ i x≤c i) → ∀ v w, |
| manifoldPullback Φ Ω t x v w=Ω 0 x v w) := by |
| obtain ⟨V,hV,hVe,hVz,⟨C,hC,hVC⟩,hVtan⟩ := exists_joint_localized_manifold_moser_field |
| hΩ hα (isCompact_Icc.prod hD) hU hDU hinv |
| obtain ⟨Φ,Ψ,hΦ,hΨ,hΦ0,hΦinv,hODE,hfix⟩ := |
| exists_spatially_compact_nonautonomous_flow hV hC hVC |
| have htangent (i : ι) (t : ℝ) (x : M) (hx : a i<μ i x) : |
| mfderiv 𝓘(ℝ,E) 𝓘(ℝ,ℝ) (μ i) x (V (t,x))=0 := by |
| apply hVtan |
| intro hp |
| rcases hface i (t,x) hp hx with hz | ⟨hm,hh⟩ |
| · simp only [manifoldMoserField,hz,map_zero,neg_zero] |
| exact (mfderiv 𝓘(ℝ,E) 𝓘(ℝ,ℝ) (μ i) x).map_zero |
| · exact manifoldMoserField_moment_tangent (hinv (t,x) hp) (hskew (t,x) hp) |
| _ (Z i x) hm hh |
| have hiff (t : ℝ) (ht : t ∈ Icc (0:ℝ) 1) (x : M) : |
| (∀ i, μ i (Φ (t,x))≤c i) ↔ (∀ i, μ i x≤c i) := by |
| apply manifold_flow_sublevels_iff hΦ hΦ0 _ μ hμ a c hac htangent t ht x |
| intro y s hs |
| exact hODE y s ⟨by linarith [hs.1],by linarith [hs.2]⟩ |
| refine ⟨Φ,Ψ,hΦ,hΨ,hΦ0,hΦinv,⟨C,hC,?_⟩,?_,?_,?_⟩ |
| · intro t x hx |
| exact hfix x (fun s => hVC s x hx) t |
| · intro x hx t |
| exact hfix x (fun s => hVz (s,x) (hx s)) t |
| · intro t ht |
| exact manifold_flow_image_sublevels hΦ hΨ t (hΦinv t ht) (hiff t ht) |
| · intro t ht x hx v w |
| have hd (s : ℝ) (hs : s ∈ Icc (0:ℝ) 1) : |
| HasDerivAt (fun r => manifoldPullback Φ Ω r x v w) 0 s := by |
| apply local_manifold_form_transport_derivative hΦ hV isOpen_Ioo hODE hΩ |
| (show s ∈ Ioo (-2:ℝ) 2 by constructor <;> linarith [hs.1,hs.2]) x |
| intro b y hy he |
| have hpK : (s,(extChartAt 𝓘(ℝ,E) b).symm y) ∈ |
| Icc (0:ℝ) 1 ×ˢ {x | ∀ i, μ i x≤c i} := |
| ⟨hs,by rw [he]; exact (hiff s hs x).mpr hx⟩ |
| have hpU := hDU hpK |
| have hn : (univ ×ˢ (extChartAt 𝓘(ℝ,E) b).target) ∈ 𝓝 (s,y) := |
| (isOpen_univ.prod (isOpen_extChartAt_target (I := 𝓘(ℝ,E)) b)).mem_nhds ⟨mem_univ _,hy⟩ |
| apply joint_manifold_moser_PDE hV hU hy hpU hinv |
| (hVe.filter_mono (nhds_le_nhdsSet hpK)) ((hΩ b).contDiffAt hn) |
| ((hα b).contDiffAt hn) _ (hclosed s b y hy hpU) (htime s b y hy hpU) |
| intro u z |
| exact hskew _ hpU _ _ |
| have he := (convex_Icc (0:ℝ) 1).norm_image_sub_le_of_norm_hasDerivWithin_le |
| (fun s hs => (hd s hs).hasDerivWithinAt) |
| (fun s hs => (by simp : ‖(0:ℝ)‖ ≤ (0:ℝ))) |
| (show (0:ℝ) ∈ Icc (0:ℝ) 1 by norm_num) ht |
| have hconst : manifoldPullback Φ Ω t x v w=manifoldPullback Φ Ω 0 x v w := by |
| simpa only [zero_mul,norm_le_zero_iff,sub_eq_zero] using he |
| rw [hconst] |
| have hid : (fun y => Φ (0,y))=id := funext hΦ0 |
| have hDifferential : preferredDifferential (E := E) (fun y => Φ (0,y)) x= |
| ContinuousLinearMap.id ℝ E := by |
| change mfderiv 𝓘(ℝ,E) 𝓘(ℝ,E) (fun y => Φ (0,y)) x=_ |
| rw [hid] |
| exact mfderiv_id |
| rw [manifoldPullback,hDifferential] |
| simp only [hΦ0,ContinuousLinearMap.bilinearComp_apply,ContinuousLinearMap.id_apply] |
| |
| theorem manifold_exact_sublevel_moser |
| {Γ : ℝ → ManifoldOneForm E M} (hΓ : SmoothOneFormFamily Γ) |
| {ι : Type*} (μ : ι → M → ℝ) |
| (hμ : ∀ i, ContMDiff 𝓘(ℝ,E) 𝓘(ℝ,ℝ) ∞ (μ i)) |
| (a c : ι → ℝ) (hac : ∀ i, a i<c i) |
| (hD : IsCompact {x | ∀ i, μ i x≤c i}) |
| {U : Set (ℝ × M)} (hU : IsOpen U) |
| (hDU : Icc (0:ℝ) 1 ×ˢ {x | ∀ i, μ i x≤c i} ⊆ U) |
| (hinv : ∀ p ∈ U, (manifoldExteriorOneForm (Γ p.1) p.2).IsInvertible) |
| (Z : ι → M → E) |
| (hface : ∀ i p, p ∈ U → a i<μ i p.2 → |
| (∀ t, Γ t p.2=Γ 0 p.2) ∨ |
| (manifoldExteriorOneForm (Γ p.1) p.2 (Z i p.2) = -mfderiv 𝓘(ℝ,E) 𝓘(ℝ,ℝ) (μ i) p.2 ∧ |
| ∀ t, Γ t p.2 (Z i p.2)=Γ 0 p.2 (Z i p.2))) : |
| ∃ Φ Ψ : ℝ × M → M, |
| ContMDiff ((𝓘(ℝ,ℝ)).prod 𝓘(ℝ,E)) 𝓘(ℝ,E) ∞ Φ ∧ |
| ContMDiff ((𝓘(ℝ,ℝ)).prod 𝓘(ℝ,E)) 𝓘(ℝ,E) ∞ Ψ ∧ |
| (∀ x, Φ (0,x)=x) ∧ |
| (∀ t ∈ Icc (0:ℝ) 1, ∀ x, Ψ (t,Φ (t,x))=x ∧ Φ (t,Ψ (t,x))=x) ∧ |
| (∃ C : Set M, IsCompact C ∧ ∀ t x, x ∉ C → Φ (t,x)=x) ∧ |
| (∀ x, (∀ t, Γ t x=Γ 0 x) → ∀ t, Φ (t,x)=x) ∧ |
| (∀ t ∈ Icc (0:ℝ) 1, |
| (fun x => Φ (t,x)) '' {x | ∀ i, μ i x≤c i}={x | ∀ i, μ i x≤c i} ∧ |
| (fun x => Φ (t,x)) '' interior {x | ∀ i, μ i x≤c i}=interior {x | ∀ i, μ i x≤c i}) ∧ |
| (∀ t ∈ Icc (0:ℝ) 1, ∀ x, (∀ i, μ i x≤c i) → ∀ v w, |
| manifoldPullback Φ (fun s => manifoldExteriorOneForm (Γ s)) t x v w= |
| manifoldExteriorOneForm (Γ 0) x v w) := by |
| have hf : ∀ i p, p ∈ U → a i<μ i p.2 → |
| manifoldTimePrimitive Γ p.1 p.2=0 ∨ |
| (manifoldExteriorOneForm (Γ p.1) p.2 (Z i p.2) = -mfderiv 𝓘(ℝ,E) 𝓘(ℝ,ℝ) (μ i) p.2 ∧ |
| manifoldTimePrimitive Γ p.1 p.2 (Z i p.2)=0) := by |
| intro i p hp ha |
| rcases hface i p hp ha with hz | ⟨hm,hz⟩ |
| · exact Or.inl (manifoldTimePrimitive_stationary hz p.1) |
| · exact Or.inr ⟨hm,manifoldTimePrimitive_eval_constant hΓ hz p.1⟩ |
| have hΩ := manifoldExteriorOneForm_path_smooth hΓ |
| have hα := manifoldTimePrimitive_smooth hΓ |
| have hc := manifoldExteriorOneForm_path_closed hΓ |
| have ht := manifoldExteriorOneForm_path_time hΓ |
| obtain ⟨Φ,Ψ,hΦ,hΨ,h0,hi,hC,hfix,hpres,hpull⟩ := manifold_relative_sublevel_moser |
| (Ω := fun s => manifoldExteriorOneForm (Γ s)) (α := manifoldTimePrimitive Γ) |
| hΩ hα μ hμ a c hac hD hU hDU hinv |
| (fun p _ v w => manifoldExteriorOneForm_skew (Γ p.1) p.2 v w) |
| (fun t b y hy _ u v w => hc t b y hy u v w) |
| (fun t b y hy _ v w => ht t b y hy v w) Z hf |
| refine ⟨Φ,Ψ,hΦ,hΨ,h0,hi,hC,?_,hpres,hpull⟩ |
| intro x hx t |
| exact hfix x (fun s => manifoldTimePrimitive_stationary hx s) t |
| |
| end |
| section |
| |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| {M : Type*} [TopologicalSpace M] [ChartedSpace E M] [IsManifold 𝓘(ℝ,E) ∞ M] |
| |
| def PreservesTwoFormOn (e : M ≃ₘ⟮𝓘(ℝ,E),𝓘(ℝ,E)⟯ M) |
| (Ω₀ Ω₁ : ManifoldTwoForm E M) (D : Set M) : Prop := |
| ∀ x ∈ D, ∀ v w, Ω₁ (e x) |
| (mfderiv 𝓘(ℝ,E) 𝓘(ℝ,E) e x v) (mfderiv 𝓘(ℝ,E) 𝓘(ℝ,E) e x w) = Ω₀ x v w |
| |
| namespace PreservesTwoFormOn |
| variable {e f : M ≃ₘ⟮𝓘(ℝ,E),𝓘(ℝ,E)⟯ M} |
| {Ω₀ Ω₁ Ω₂ : ManifoldTwoForm E M} {D : Set M} |
| |
| omit [IsManifold 𝓘(ℝ,E) ∞ M] in |
| |
| theorem trans (he : PreservesTwoFormOn e Ω₀ Ω₁ D) |
| (hf : PreservesTwoFormOn f Ω₁ Ω₂ D) (hmap : MapsTo e D D) : |
| PreservesTwoFormOn (e.trans f) Ω₀ Ω₂ D := by |
| intro x hx v w |
| have hd := mfderiv_comp x (f.contMDiff.mdifferentiable (by simp) (e x)) |
| (e.contMDiff.mdifferentiable (by simp) x) |
| change Ω₂ (f (e x)) (mfderiv 𝓘(ℝ,E) 𝓘(ℝ,E) (f ∘ e) x v) |
| (mfderiv 𝓘(ℝ,E) 𝓘(ℝ,E) (f ∘ e) x w) = _ |
| rw [hd] |
| exact (hf (e x) (hmap hx) _ _).trans (he x hx v w) |
| |
| omit [IsManifold 𝓘(ℝ,E) ∞ M] in |
| |
| theorem symm (he : PreservesTwoFormOn e Ω₀ Ω₁ D) (hD : e '' D = D) : |
| PreservesTwoFormOn e.symm Ω₁ Ω₀ D := by |
| intro y hy v w |
| have hx : e.symm y ∈ D := by |
| obtain ⟨x,hx,hxy⟩ := hD.symm ▸ hy |
| simpa only [←hxy,e.symm_apply_apply] using hx |
| have hd := mfderiv_comp y (e.contMDiff.mdifferentiable (by simp) (e.symm y)) |
| (e.symm.contMDiff.mdifferentiable (by simp) y) |
| have heq : (e : M → M) ∘ e.symm = id := by funext z; exact e.apply_symm_apply z |
| rw [heq,mfderiv_id] at hd |
| have hv := congrArg (fun L : E →L[ℝ] E => L v) hd.symm |
| have hw := congrArg (fun L : E →L[ℝ] E => L w) hd.symm |
| have h := he (e.symm y) hx (mfderiv 𝓘(ℝ,E) 𝓘(ℝ,E) e.symm y v) |
| (mfderiv 𝓘(ℝ,E) 𝓘(ℝ,E) e.symm y w) |
| change mfderiv 𝓘(ℝ,E) 𝓘(ℝ,E) e (e.symm y) (mfderiv 𝓘(ℝ,E) 𝓘(ℝ,E) e.symm y v)=v at hv |
| change mfderiv 𝓘(ℝ,E) 𝓘(ℝ,E) e (e.symm y) (mfderiv 𝓘(ℝ,E) 𝓘(ℝ,E) e.symm y w)=w at hw |
| rw [hv,hw,e.apply_symm_apply] at h |
| exact h.symm |
| |
| end PreservesTwoFormOn |
| |
| variable [FiniteDimensional ℝ E] [T2Space M] [NormalSpace M] [SigmaCompactSpace M] |
| |
| theorem manifold_exact_moser_endpoint |
| {Γ : ℝ → ManifoldOneForm E M} (hΓ : SmoothOneFormFamily Γ) |
| {ι : Type*} (μ : ι → M → ℝ) |
| (hμ : ∀ i, ContMDiff 𝓘(ℝ,E) 𝓘(ℝ,ℝ) ∞ (μ i)) |
| (a c : ι → ℝ) (hac : ∀ i, a i<c i) |
| (hD : IsCompact {x | ∀ i, μ i x≤c i}) |
| {U : Set (ℝ × M)} (hU : IsOpen U) |
| (hDU : Icc (0:ℝ) 1 ×ˢ {x | ∀ i, μ i x≤c i} ⊆ U) |
| (hinv : ∀ p ∈ U, (manifoldExteriorOneForm (Γ p.1) p.2).IsInvertible) |
| (Z : ι → M → E) |
| (hface : ∀ i p, p ∈ U → a i<μ i p.2 → |
| (∀ t, Γ t p.2=Γ 0 p.2) ∨ |
| (manifoldExteriorOneForm (Γ p.1) p.2 (Z i p.2) = -mfderiv 𝓘(ℝ,E) 𝓘(ℝ,ℝ) (μ i) p.2 ∧ |
| ∀ t, Γ t p.2 (Z i p.2)=Γ 0 p.2 (Z i p.2))) : |
| ∃ e : M ≃ₘ⟮𝓘(ℝ,E),𝓘(ℝ,E)⟯ M, |
| e '' {x | ∀ i, μ i x≤c i}={x | ∀ i, μ i x≤c i} ∧ |
| e '' interior {x | ∀ i, μ i x≤c i}=interior {x | ∀ i, μ i x≤c i} ∧ |
| PreservesTwoFormOn e (manifoldExteriorOneForm (Γ 0)) (manifoldExteriorOneForm (Γ 1)) |
| {x | ∀ i, μ i x≤c i} ∧ |
| (∃ C : Set M, IsCompact C ∧ ∀ x, x ∉ C → e x=x) ∧ |
| (∀ x, (∀ t, Γ t x=Γ 0 x) → e x=x) := by |
| obtain ⟨Φ,Ψ,hΦ,hΨ,_,hi,hC,hfix,hpres,hpull⟩ := |
| manifold_exact_sublevel_moser hΓ μ hμ a c hac hD hU hDU hinv Z hface |
| have h1 : (1:ℝ) ∈ Icc (0:ℝ) 1 := ⟨zero_le_one,le_rfl⟩ |
| let e : M ≃ₘ⟮𝓘(ℝ,E),𝓘(ℝ,E)⟯ M := |
| { toFun := fun x => Φ (1,x) |
| invFun := fun x => Ψ (1,x) |
| left_inv := fun x => (hi 1 h1 x).1 |
| right_inv := fun x => (hi 1 h1 x).2 |
| contMDiff_toFun := hΦ.comp (contMDiff_const.prodMk contMDiff_id) |
| contMDiff_invFun := hΨ.comp (contMDiff_const.prodMk contMDiff_id) } |
| refine ⟨e,(hpres 1 h1).1,(hpres 1 h1).2,?_,?_,?_⟩ |
| · intro x hx v w |
| exact hpull 1 h1 x hx v w |
| · obtain ⟨C,hc,h⟩ := hC |
| exact ⟨C,hc,fun x hx => h 1 x hx⟩ |
| · intro x hx |
| exact hfix x hx 1 |
| |
| end |
| section |
| |
| variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [NormedAddCommGroup F] [NormedSpace ℝ F] |
| {M N : Type*} [TopologicalSpace M] [ChartedSpace E M] |
| [TopologicalSpace N] [ChartedSpace F N] |
| |
| def FormNeighborhoodEmbedding (K : Set N) (σ : ManifoldTwoForm F N) |
| (Ω : ManifoldTwoForm E M) (f : N → M) : Prop := |
| ∃ U : Set N, IsOpen U ∧ K ⊆ U ∧ ContMDiffOn 𝓘(ℝ,F) 𝓘(ℝ,E) ∞ f U ∧ |
| Topology.IsEmbedding (fun x : U => f x) ∧ |
| ∀ x ∈ U, ∀ v w, Ω (f x) |
| (mfderiv 𝓘(ℝ,F) 𝓘(ℝ,E) f x v) (mfderiv 𝓘(ℝ,F) 𝓘(ℝ,E) f x w) = σ x v w |
| |
| theorem FormNeighborhoodEmbedding.transport |
| {K : Set N} {σ : ManifoldTwoForm F N} {Ω₀ Ω₁ : ManifoldTwoForm E M} |
| {f : N → M} (hf : FormNeighborhoodEmbedding K σ Ω₁ f) |
| {D : Set M} (he : M ≃ₘ⟮𝓘(ℝ,E),𝓘(ℝ,E)⟯ M) |
| (hform : PreservesTwoFormOn he Ω₁ Ω₀ D) (himage : MapsTo f K (interior D)) : |
| FormNeighborhoodEmbedding K σ Ω₀ (he ∘ f) := by |
| obtain ⟨U,hU,hKU,hs,hfemb,hσ⟩ := hf |
| let W := U ∩ f ⁻¹' interior D |
| have hW : IsOpen W := hs.continuousOn.isOpen_inter_preimage hU isOpen_interior |
| have hWU : W ⊆ U := inter_subset_left |
| refine ⟨W,hW,fun x hx => ⟨hKU hx,himage hx⟩, |
| he.contMDiff.comp_contMDiffOn (hs.mono hWU),?_,?_⟩ |
| · exact he.toHomeomorph.isEmbedding.comp (hfemb.comp (Topology.IsEmbedding.inclusion hWU)) |
| · intro x hx v w |
| have hd := mfderiv_comp x (he.contMDiff.mdifferentiable (by simp) (f x)) |
| ((hs.contMDiffAt (hU.mem_nhds hx.1)).mdifferentiableAt (by simp)) |
| rw [hd] |
| exact (hform (f x) (interior_subset hx.2) _ _).trans (hσ x hx.1 v w) |
|
|
| theorem transport_form_packing {ι : Type*} (K : ι → Set N) |
| (σ : ManifoldTwoForm F N) (Ω₀ Ω₁ : ManifoldTwoForm E M) {D : Set M} |
| (e : M ≃ₘ⟮𝓘(ℝ,E),𝓘(ℝ,E)⟯ M) |
| (hform : PreservesTwoFormOn e Ω₀ Ω₁ D) |
| (hD : e '' D=D) (hInt : e '' interior D=interior D) |
| (f : ι → N → M) (hf : ∀ i, FormNeighborhoodEmbedding (K i) σ Ω₁ (f i)) |
| (himage : ∀ i, MapsTo (f i) (K i) (interior D)) |
| (hdis : Pairwise (fun i j => Disjoint (f i '' K i) (f j '' K j))) : |
| (∀ i, FormNeighborhoodEmbedding (K i) σ Ω₀ (e.symm ∘ f i)) ∧ |
| (∀ i, MapsTo (e.symm ∘ f i) (K i) (interior D)) ∧ |
| Pairwise (fun i j => Disjoint ((e.symm ∘ f i) '' K i) ((e.symm ∘ f j) '' K j)) := by |
| refine ⟨fun i => (hf i).transport e.symm (hform.symm hD) (himage i),?_,?_⟩ |
| · intro i x hx |
| obtain ⟨y,hy,hyx⟩ := hInt.symm ▸ himage i hx |
| simpa only [Function.comp_apply,←hyx,e.symm_apply_apply] using hy |
| · intro i j hij |
| rw [Set.image_comp,Set.image_comp] |
| exact (hdis hij).image (e.symm.injective.injOn (s := Set.univ)) (subset_univ _) (subset_univ _) |
|
|
| end |
| section |
|
|
| variable {E F G : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup G] [NormedSpace ℝ G] |
| {M N P : Type*} [TopologicalSpace M] [ChartedSpace E M] |
| [TopologicalSpace N] [ChartedSpace F N] [TopologicalSpace P] [ChartedSpace G P] |
|
|
| theorem FormNeighborhoodEmbedding.postcompose {K : Set N} |
| {σ : ManifoldTwoForm F N} {Ω₀ : ManifoldTwoForm E M} {Ω₁ : ManifoldTwoForm G P} |
| {f : N → M} (hf : FormNeighborhoodEmbedding K σ Ω₀ f) |
| {T : Set M} (hT : IsOpen T) {g : M → P} |
| (hg : ContMDiffOn 𝓘(ℝ,E) 𝓘(ℝ,G) ∞ g T) |
| (hemb : Topology.IsEmbedding (fun x : T => g x.val)) |
| (hform : ∀ x ∈ T, ∀ v w, Ω₁ (g x) (mfderiv 𝓘(ℝ,E) 𝓘(ℝ,G) g x v) |
| (mfderiv 𝓘(ℝ,E) 𝓘(ℝ,G) g x w) = Ω₀ x v w) |
| (him : MapsTo f K T) : FormNeighborhoodEmbedding K σ Ω₁ (g ∘ f) := by |
| obtain ⟨U,hU,hKU,hs,he,hσ⟩ := hf |
| let W := U ∩ f ⁻¹' T |
| have hW : IsOpen W := hs.continuousOn.isOpen_inter_preimage hU hT |
| have hWU : W ⊆ U := inter_subset_left |
| have hWT : MapsTo f W T := inter_subset_right |
| refine ⟨W,hW,fun x hx => ⟨hKU hx,him hx⟩,hg.comp (hs.mono hWU) hWT,?_,?_⟩ |
| · have he' := (he.comp (Topology.IsEmbedding.inclusion hWU)).codRestrict T |
| (fun x : W => show f x.val ∈ T from x.property.2) |
| exact hemb.comp he' |
| · intro x hx v w |
| have hd := mfderiv_comp x ((hg.contMDiffAt (hT.mem_nhds hx.2)).mdifferentiableAt (by simp)) |
| ((hs.contMDiffAt (hU.mem_nhds hx.1)).mdifferentiableAt (by simp)) |
| rw [hd] |
| exact (hform (f x) hx.2 _ _).trans (hσ x hx.1 v w) |
| |
| theorem postcompose_form_packing {ι : Type*} (K : ι → Set N) |
| (σ : ManifoldTwoForm F N) (Ω₀ : ManifoldTwoForm E M) (Ω₁ : ManifoldTwoForm G P) |
| {T : Set M} (hT : IsOpen T) (g : M → P) |
| (hg : ContMDiffOn 𝓘(ℝ,E) 𝓘(ℝ,G) ∞ g T) |
| (hemb : Topology.IsEmbedding (fun x : T => g x.val)) |
| (hform : ∀ x ∈ T, ∀ v w, Ω₁ (g x) (mfderiv 𝓘(ℝ,E) 𝓘(ℝ,G) g x v) |
| (mfderiv 𝓘(ℝ,E) 𝓘(ℝ,G) g x w) = Ω₀ x v w) |
| (f : ι → N → M) (hf : ∀ i, FormNeighborhoodEmbedding (K i) σ Ω₀ (f i)) |
| (him : ∀ i, MapsTo (f i) (K i) T) |
| (hd : Pairwise (fun i j => Disjoint (f i '' K i) (f j '' K j))) : |
| (∀ i, FormNeighborhoodEmbedding (K i) σ Ω₁ (g ∘ f i)) ∧ |
| Pairwise (fun i j => Disjoint ((g ∘ f i) '' K i) ((g ∘ f j) '' K j)) := by |
| refine ⟨fun i => (hf i).postcompose hT hg hemb hform (him i),?_⟩ |
| have hinj : InjOn g T := by |
| intro x hx y hy heq |
| exact congrArg Subtype.val (hemb.injective (a₁ := ⟨x,hx⟩) (a₂ := ⟨y,hy⟩) heq) |
| intro i j hij |
| rw [Set.image_comp,Set.image_comp] |
| exact (hd hij).image hinj (image_subset_iff.mpr (him i)) (image_subset_iff.mpr (him j)) |
| |
| end |
| |
| def successorStandardForm (m : ℕ) : Ambient (m+1) →L[ℝ] Ambient (m+1) →L[ℝ] ℝ := |
| (productForm (phaseArea (ι := Fin m))).bilinearComp (complexSplitPhase m).toContinuousLinearMap (complexSplitPhase m).toContinuousLinearMap |
| |
| @[simp] theorem successorStandardForm_apply (m : ℕ) (v w : Ambient (m+1)) : |
| successorStandardForm m v w = standardForm v w := complexSplitPhase_form m v w |
| |
| theorem isBallEmbedding_iff_formNeighborhoodEmbedding {m : ℕ} {r : ℝ} |
| {f : Ambient (m+1) → Ambient (m+1)} : |
| IsBallEmbedding r f ↔ FormNeighborhoodEmbedding (closedBall (m+1) r) |
| (fun _ => successorStandardForm m) (fun _ => successorStandardForm m) f := by |
| constructor |
| · rintro ⟨U,hU,hBU,hs,he,hform⟩ |
| refine ⟨U,hU,hBU,hs.contMDiffOn,he,?_⟩ |
| intro x hx v w |
| rw [mfderiv_eq_fderiv] |
| change successorStandardForm m (fderiv ℝ f x v) (fderiv ℝ f x w) = successorStandardForm m v w |
| erw [successorStandardForm_apply, successorStandardForm_apply] |
| exact hform x hx v w |
| · rintro ⟨U,hU,hBU,hs,he,hform⟩ |
| refine ⟨U,hU,hBU,hs.contDiffOn,he,?_⟩ |
| intro x hx v w |
| have hh := hform x hx v w |
| rw [mfderiv_eq_fderiv] at hh |
| change successorStandardForm m (fderiv ℝ f x v) (fderiv ℝ f x w) = successorStandardForm m v w at hh |
| erw [successorStandardForm_apply, successorStandardForm_apply] at hh |
| exact hh |
| |
| end PackingSufficiencySupport.Hamiltonian |
| end |
| |
| end OAI |
| |
