| import Mathlib | |
| import OAI.Geometry.BallPacking.Necessity.FiniteContinuation | |
| namespace OAI | |
| noncomputable section | |
| open scoped ContDiff Topology | |
| open Set Function Filter | |
| open scoped ContDiff Topology Manifold | |
| open Set Function Filter MeasureTheory | |
| open Set Function MeasureTheory | |
| open Set Function | |
| open SymplecticBallPacking.Hamiltonian (Plane planarCurl) | |
| open SymplecticBallPacking.Hamiltonian (Plane planarCurl angularOneForm radiusSq planarArea planarArea_apply) | |
| open SymplecticBallPacking.Hamiltonian (Plane planarCurl angularOneForm) | |
| open SymplecticBallPacking.Hamiltonian (Plane angularOneForm) | |
| open SymplecticBallPacking.Hamiltonian | |
| open SymplecticBallPacking.Hamiltonian (Plane) | |
| open Set Filter Function | |
| open Set Filter MeasureTheory | |
| open scoped Topology | |
| open Set Filter Finset | |
| open scoped ContDiff Topology Classical | |
| open Set Filter | |
| open scoped BoundedContinuousFunction ContDiff Topology | |
| open Set Function Filter Topology | |
| open scoped NNReal | |
| open scoped ContDiff Topology BoundedContinuousFunction | |
| open Function | |
| open scoped Topology ContDiff | |
| open scoped ContDiff Topology Convolution | |
| open Set Filter Function MeasureTheory ContinuousLinearMap | |
| open Set Filter Function MeasureTheory | |
| open scoped ContDiff Topology | |
| open Set Filter Function | |
| namespace TwoPointContinuation | |
| theorem finite_space_continuation {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| [FiniteDimensional ℝ E] | |
| (F : ℝ × E → E) (hF : Continuous F) (hzero : ∀ x, F (0,x)=x) | |
| (Ω : Set E) (ho : IsOpen Ω) (hb : Bornology.IsBounded Ω) (h0 : 0∈Ω) | |
| (hside : ∀ t∈Icc (0:ℝ) 1, ∀ x∈closure Ω, x∉Ω → F (t,x)≠0) : | |
| ∃ x∈Ω, F (1,x)=0 := by | |
| let n := Module.finrank ℝ E | |
| let e : E ≃L[ℝ] (Fin n → ℝ) := ContinuousLinearEquiv.ofFinrankEq (by simp [n]) | |
| let G : ℝ × (Fin n → ℝ) → (Fin n → ℝ) := fun y => e (F (y.1,e.symm y.2)) | |
| have hG : Continuous G := e.continuous.comp (hF.comp (continuous_fst.prodMk (e.symm.continuous.comp continuous_snd))) | |
| have hG0 (x : Fin n → ℝ) : G (0,x)=x := by simp [G,hzero] | |
| let U := e.symm ⁻¹' Ω | |
| have hU : U=e '' Ω := by | |
| ext x | |
| constructor | |
| · intro hx | |
| exact ⟨e.symm x,hx,e.apply_symm_apply x⟩ | |
| · rintro ⟨x,hx,rfl⟩ | |
| simpa only [U,Set.mem_preimage,ContinuousLinearEquiv.symm_apply_apply] using hx | |
| have hu : IsOpen U := ho.preimage e.symm.continuous | |
| have hbu : Bornology.IsBounded U := by rw [hU]; exact e.lipschitzWith.isBounded_image hb | |
| have h0u : 0∈U := by simpa only [U,Set.mem_preimage,ContinuousLinearEquiv.map_zero] using h0 | |
| have hsu (t : ℝ) (ht : t∈Icc (0:ℝ) 1) (x : Fin n → ℝ) (hx : x∈closure U) (hn : x∉U) : G (t,x)≠0 := by | |
| have hcl : e.symm x∈closure Ω := e.symm.continuous.closure_preimage_subset Ω hx | |
| intro he | |
| have hh : F (t,e.symm x)=0 := e.injective (by simpa only [G,ContinuousLinearEquiv.map_zero] using he) | |
| exact hside t ht (e.symm x) hcl hn hh | |
| obtain ⟨x,hx,hx0⟩ := continuous_localized_continuation n G hG hG0 U hu hbu h0u hsu | |
| refine ⟨e.symm x,hx,e.injective ?_⟩ | |
| simpa only [G,ContinuousLinearEquiv.map_zero] using hx0 | |
| open scoped ContDiff Topology | |
| open Set Filter Function | |
| theorem compact_residual_gap {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| (T : ℝ × E → E) (hT : Continuous T) (D : Set (ℝ × E)) (hD : IsClosed D) | |
| (ht : ∀ y∈D, y.1∈Icc (0:ℝ) 1) | |
| (S : Set E) (hS : IsCompact S) (hTS : ∀ y∈D, T y∈S) | |
| (hno : ∀ y∈D, y.2-T y≠0) : | |
| ∃ δ : ℝ, 0<δ ∧ ∀ y∈D, δ≤‖y.2-T y‖ := by | |
| classical | |
| by_contra hn | |
| push Not at hn | |
| have hh (n : ℕ) : ∃ y∈D, ‖y.2-T y‖<1/((n:ℝ)+1) := hn _ (by positivity) | |
| choose y hy hyr using hh | |
| have hr : Tendsto (fun n => (y n).2-T (y n)) atTop (𝓝 0) := | |
| squeeze_zero_norm (fun n => (hyr n).le) tendsto_one_div_add_atTop_nhds_zero_nat | |
| obtain ⟨a,ha,φ,hφ,hm⟩ := (isCompact_Icc.prod hS).tendsto_subseq | |
| (x := fun n => ((y n).1,T (y n))) (fun n => ⟨ht (y n) (hy n),hTS (y n) (hy n)⟩) | |
| have hx : Tendsto (fun n => (y (φ n)).2) atTop (𝓝 a.2) := by | |
| have hh := hm.snd_nhds.add (hr.comp hφ.tendsto_atTop) | |
| simpa only [comp_apply,add_sub_cancel,add_zero] using hh | |
| have hyconv : Tendsto (y ∘ φ) atTop (𝓝 a) := hm.fst_nhds.prodMk_nhds hx | |
| have had : a∈D := hD.mem_of_tendsto hyconv (Eventually.of_forall (fun n => hy (φ n))) | |
| have he : T a=a.2 := tendsto_nhds_unique (hT.continuousAt.tendsto.comp hyconv) hm.snd_nhds | |
| exact hno a had (sub_eq_zero.mpr he.symm) | |
| open scoped ContDiff Topology | |
| open Set Filter Function | |
| theorem compact_continuation {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| (T : ℝ × E → E) (hT : Continuous T) (hzero : ∀ x, T (0,x)=0) | |
| (Ω : Set E) (ho : IsOpen Ω) (hb : Bornology.IsBounded Ω) (h0 : 0∈Ω) | |
| (S : Set E) (hS : IsCompact S) | |
| (hTS : ∀ t∈Icc (0:ℝ) 1, ∀ x∈closure Ω, T (t,x)∈S) | |
| (hside : ∀ t∈Icc (0:ℝ) 1, ∀ x∈closure Ω, x∉Ω → T (t,x)≠x) : | |
| ∃ x∈Ω, T (1,x)=x := by | |
| classical | |
| by_contra hn | |
| have hfinal (x : E) (hx : x∈closure Ω) : T (1,x)≠x := by | |
| by_cases hm : x∈Ω | |
| · exact fun he => hn ⟨x,hm,he⟩ | |
| · exact hside 1 ⟨zero_le_one,le_rfl⟩ x hx hm | |
| let D := (Icc (0:ℝ) 1 ×ˢ (closure Ω\Ω)) ∪ ({1} ×ˢ closure Ω) | |
| have hD : IsClosed D := (isClosed_Icc.prod (isClosed_closure.inter ho.isClosed_compl)).union | |
| (isClosed_singleton.prod isClosed_closure) | |
| have htd (y : ℝ × E) (hy : y∈D) : y.1∈Icc (0:ℝ) 1 := by | |
| rcases hy with hy|hy | |
| · exact hy.1 | |
| · rw [show y.1=1 from hy.1]; exact ⟨zero_le_one,le_rfl⟩ | |
| have hxd (y : ℝ × E) (hy : y∈D) : y.2∈closure Ω := by | |
| rcases hy with hy|hy | |
| · exact hy.2.1 | |
| · exact hy.2 | |
| have hTSd (y : ℝ × E) (hy : y∈D) : T y∈S := hTS y.1 (htd y hy) y.2 (hxd y hy) | |
| have hno (y : ℝ × E) (hy : y∈D) : y.2-T y≠0 := by | |
| intro he | |
| have he' : T y=y.2 := (sub_eq_zero.mp he).symm | |
| rcases hy with hy|hy | |
| · exact hside y.1 hy.1 y.2 hy.2.1 hy.2.2 he' | |
| · have ht : y.1=1 := hy.1 | |
| exact hfinal y.2 hy.2 (by simpa only [←ht] using he') | |
| obtain ⟨δ,hδ,hgap⟩ := compact_residual_gap T hT D hD htd S hS hTSd hno | |
| obtain ⟨V,hV,P,hP,happ⟩ := finite_rank_approx (S∪{0}) (hS.union isCompact_singleton) | |
| (δ/4) (by positivity) | |
| let : FiniteDimensional ℝ V := hV | |
| let F : ℝ × V → V := fun y => y.2-(P (T (y.1,(y.2:E)))-P 0) | |
| have hF : Continuous F := continuous_snd.sub | |
| ((hP.comp (hT.comp (continuous_fst.prodMk (continuous_subtype_val.comp continuous_snd)))).sub continuous_const) | |
| have hF0 (x : V) : F (0,x)=x := by simp [F,hzero] | |
| let U := ((↑) : V → E) ⁻¹' Ω | |
| have hu : IsOpen U := ho.preimage continuous_subtype_val | |
| have hbu : Bornology.IsBounded U := by | |
| obtain ⟨R,hR,hbR⟩ := hb.exists_pos_norm_le | |
| apply (Metric.isBounded_iff_subset_closedBall (0:V)).mpr | |
| refine ⟨R,fun x hx => ?_⟩ | |
| change dist x 0≤R | |
| rw [dist_zero_right] | |
| change ‖(x:E)‖≤R | |
| exact hbR (x:E) hx | |
| have h0u : (0:V)∈U := h0 | |
| have hclose (x : V) (hx : x∈closure U) : (x:E)∈closure Ω := | |
| continuous_subtype_val.closure_preimage_subset Ω hx | |
| have hnear (t : ℝ) (x : V) (hmem : T (t,(x:E))∈S) (hf : F (t,x)=0) : | |
| ‖(x:E)-T (t,(x:E))‖<δ := by | |
| have he : (x:E)=(P (T (t,(x:E))) : E)-(P 0:E) := | |
| congrArg (fun u : V => (u:E)) (sub_eq_zero.mp hf) | |
| have h1 := happ (T (t,(x:E))) (Or.inl hmem) | |
| have h2 := happ 0 (Or.inr rfl) | |
| simp only [sub_zero] at h2 | |
| have he' : (x:E)-T (t,(x:E))=((P (T (t,(x:E))):E)-T (t,(x:E)))-(P 0:E) := by | |
| calc | |
| (x:E)-T (t,(x:E))=((P (T (t,(x:E))):E)-(P 0:E))-T (t,(x:E)) := | |
| congrArg (fun v : E => v-T (t,(x:E))) he | |
| _ = _ := by abel | |
| rw [he'] | |
| have hh := norm_sub_le ((P (T (t,(x:E))):E)-T (t,(x:E))) (P 0:E) | |
| linarith | |
| have hsideF (t : ℝ) (ht : t∈Icc (0:ℝ) 1) (x : V) (hx : x∈closure U) (hn : x∉U) : F (t,x)≠0 := by | |
| intro hf | |
| exact (not_lt_of_ge (hgap (t,(x:E)) (Or.inl ⟨ht,hclose x hx,hn⟩))) | |
| (hnear t x (hTS t ht (x:E) (hclose x hx)) hf) | |
| obtain ⟨x,hx,hfx⟩ := finite_space_continuation F hF hF0 U hu hbu h0u hsideF | |
| have hxcl : (x:E)∈closure Ω := subset_closure hx | |
| exact (not_lt_of_ge (hgap (1,(x:E)) (Or.inr ⟨rfl,hxcl⟩))) | |
| (hnear 1 x (hTS 1 ⟨zero_le_one,le_rfl⟩ (x:E) hxcl) hfx) | |
| open scoped ContDiff Topology | |
| open Set Filter Function | |
| theorem local_compact_continuation {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| (T : ℝ × E → E) (U : Set E) (hU : IsOpen U) | |
| (hT : ∀ t∈Icc (0:ℝ) 1, ∀ x∈U, ContinuousAt T (t,x)) | |
| (hzero : ∀ x∈U, T (0,x)=0) | |
| (S : Set E) (hS : IsCompact S) (hTS : ∀ t∈Icc (0:ℝ) 1, ∀ x∈U, T (t,x)∈S) | |
| (K : Set E) (hK : IsCompact K) (hKU : K⊆U) (h0 : 0∈K) | |
| (hfix : ∀ t∈Icc (0:ℝ) 1, ∀ x∈U, T (t,x)=x → x∈K) : | |
| ∃ x∈U, T (1,x)=x := by | |
| classical | |
| obtain ⟨χ,hχ0,hχ1,hχb⟩ := exists_continuous_zero_one_of_isClosed | |
| hU.isClosed_compl hK.isClosed (Set.disjoint_left.mpr (fun x hx hk => hx (hKU hk))) | |
| let ρ : E → ℝ := fun x => min 1 (max 0 (4*(χ x-1/4))) | |
| have hρ : Continuous ρ := continuous_const.min (continuous_const.max | |
| (continuous_const.mul (χ.continuous.sub continuous_const))) | |
| have hρb (x : E) : ρ x∈Icc (0:ℝ) 1 := by | |
| dsimp [ρ] | |
| exact ⟨le_min (by norm_num) (le_max_left _ _),min_le_left _ _⟩ | |
| have hρone (x : E) (hx : (1/2:ℝ)≤χ x) : ρ x=1 := by | |
| dsimp [ρ] | |
| rw [min_eq_left] | |
| exact (by linarith : (1:ℝ)≤4*(χ x-1/4)).trans (le_max_right _ _) | |
| have hρzero (x : E) (hx : χ x≤(1/4:ℝ)) : ρ x=0 := by | |
| dsimp [ρ] | |
| rw [max_eq_left (by linarith),min_eq_right (by norm_num)] | |
| let τ : ℝ → ℝ := fun t => min 1 (max 0 t) | |
| have hτ : Continuous τ := continuous_const.min (continuous_const.max continuous_id) | |
| have hτb (t : ℝ) : τ t∈Icc (0:ℝ) 1 := | |
| ⟨le_min (by norm_num) (le_max_left _ _),min_le_left _ _⟩ | |
| have hτe (t : ℝ) (ht : t∈Icc (0:ℝ) 1) : τ t=t := by | |
| dsimp [τ]; rw [max_eq_right ht.1,min_eq_right ht.2] | |
| let G : ℝ × E → E := fun y => ρ y.2 • T (τ y.1,y.2) | |
| have hG : Continuous G := by | |
| rw [continuous_iff_continuousAt] | |
| intro y | |
| by_cases hx : y.2∈U | |
| · exact (hρ.comp continuous_snd).continuousAt.smul | |
| ((hT (τ y.1) (hτb y.1) y.2 hx).comp (f := fun z : ℝ × E => (τ z.1,z.2)) ((hτ.comp continuous_fst).prodMk continuous_snd).continuousAt) | |
| · have hc : χ y.2=0 := hχ0 hx | |
| have hn : ∀ᶠ z : ℝ × E in 𝓝 y, χ z.2<(1/4:ℝ) := | |
| (χ.continuous.comp continuous_snd).continuousAt.eventually (eventually_lt_nhds (by change χ y.2<(1/4:ℝ); rw [hc]; norm_num)) | |
| apply (continuousAt_const (y := (0:E))).congr_of_eventuallyEq | |
| filter_upwards [hn] with z hz | |
| change ρ z.2 • T (τ z.1,z.2)=(0:E) | |
| rw [hρzero z.2 hz.le,zero_smul] | |
| have hG0 (x : E) : G (0,x)=0 := by | |
| by_cases hx : x∈U | |
| · simp only [G,hτe 0 ⟨le_rfl,zero_le_one⟩,hzero x hx,smul_zero] | |
| · simp only [G,hρzero x (by rw [hχ0 hx]; norm_num),zero_smul] | |
| obtain ⟨R,hR,hKR⟩ := hK.isBounded.subset_ball_lt 0 (0:E) | |
| let Ω := {x : E | (3/4:ℝ)<χ x} ∩ Metric.ball 0 R | |
| have ho : IsOpen Ω := (isOpen_lt continuous_const χ.continuous).inter Metric.isOpen_ball | |
| have hb : Bornology.IsBounded Ω := Metric.isBounded_ball.subset inter_subset_right | |
| have hKΩ : K⊆Ω := by | |
| intro x hx | |
| exact ⟨by change (3/4:ℝ)<χ x; rw [hχ1 hx]; norm_num,hKR hx⟩ | |
| have hcl (x : E) (hx : x∈closure Ω) : (3/4:ℝ)≤χ x := by | |
| exact closure_minimal (t := {x : E | (3/4:ℝ)≤χ x}) (by intro x (hx : x∈Ω); change (3/4:ℝ)≤χ x; exact hx.1.le) (isClosed_le continuous_const χ.continuous) hx | |
| have hclU (x : E) (hx : x∈closure Ω) : x∈U := by | |
| by_contra hn | |
| have hh := hcl x hx | |
| rw [hχ0 hn] at hh | |
| norm_num at hh | |
| have hGe (t : ℝ) (ht : t∈Icc (0:ℝ) 1) (x : E) (hx : x∈closure Ω) : G (t,x)=T (t,x) := by | |
| simp only [G,hτe t ht,hρone x (by have hh := hcl x hx; linarith),one_smul] | |
| obtain ⟨x,hx,he⟩ := compact_continuation G hG hG0 Ω ho hb (hKΩ h0) S hS | |
| (fun t ht x hx => by rw [hGe t ht x hx]; exact hTS t ht x (hclU x hx)) | |
| (by | |
| intro t ht x hx hn he | |
| rw [hGe t ht x hx] at he | |
| exact hn (hKΩ (hfix t ht x (hclU x hx) he))) | |
| exact ⟨x,hclU x (subset_closure hx),by rwa [hGe 1 ⟨zero_le_one,le_rfl⟩ x (subset_closure hx)] at he⟩ | |
| open scoped Topology | |
| open Set Filter Function | |
| def LocallyPrecompact {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] | |
| (f : X → Y) (x : X) : Prop := ∃ V∈𝓝 x, ∃ K : Set Y, IsCompact K ∧ f '' V⊆K | |
| theorem locallyPrecompact_finite {X Y : Type*} [TopologicalSpace X] | |
| [MetricSpace Y] [ProperSpace Y] (f : X → Y) (x : X) (hf : ContinuousAt f x) : | |
| LocallyPrecompact f x := by | |
| refine ⟨f ⁻¹' Metric.closedBall (f x) 1,hf.preimage_mem_nhds (Metric.closedBall_mem_nhds _ (by norm_num)), | |
| Metric.closedBall (f x) 1,isCompact_closedBall _ _,image_preimage_subset _ _⟩ | |
| theorem locallyPrecompact_compactLinear {X Y : Type*} | |
| [NormedAddCommGroup X] [NormedSpace ℝ X] [NormedAddCommGroup Y] [NormedSpace ℝ Y] | |
| (L : X →L[ℝ] Y) (hL : IsCompactOperator L) (x : X) : LocallyPrecompact L x := by | |
| obtain ⟨K,hK,hLK⟩ := hL.image_closedBall_subset_compact (‖x‖+1) | |
| refine ⟨Metric.closedBall 0 (‖x‖+1),?_,K,hK,hLK⟩ | |
| apply mem_of_superset (Metric.ball_mem_nhds x (by norm_num : (0:ℝ)<1)) | |
| intro y hy | |
| rw [Metric.mem_closedBall,dist_zero_right] | |
| have hh := norm_le_norm_sub_add y x | |
| have hy' : ‖y-x‖<1 := by simpa only [Metric.mem_ball,dist_eq_norm] using hy | |
| linarith | |
| theorem LocallyPrecompact.prod {X Y Z : Type*} [TopologicalSpace X] [TopologicalSpace Y] | |
| [TopologicalSpace Z] {f : X → Y} {g : X → Z} {x : X} | |
| (hf : LocallyPrecompact f x) (hg : LocallyPrecompact g x) : | |
| LocallyPrecompact (fun x => (f x,g x)) x := by | |
| obtain ⟨V,hV,K,hK,hfK⟩ := hf | |
| obtain ⟨W,hW,L,hL,hgL⟩ := hg | |
| exact ⟨V∩W,inter_mem hV hW,K×ˢL,hK.prod hL,by rintro _ ⟨y,hy,rfl⟩; exact ⟨hfK ⟨y,hy.1,rfl⟩,hgL ⟨y,hy.2,rfl⟩⟩⟩ | |
| theorem LocallyPrecompact.comp {X Y Z : Type*} [TopologicalSpace X] [MetricSpace Y] | |
| [TopologicalSpace Z] {f : X → Y} {x : X} (hf : LocallyPrecompact f x) | |
| (hfc : ContinuousAt f x) (g : Y → Z) (U : Set Y) (ho : IsOpen U) | |
| (hgu : ContinuousOn g U) (hx : f x∈U) : LocallyPrecompact (g∘f) x := by | |
| obtain ⟨V,hV,K,hK,hfK⟩ := hf | |
| obtain ⟨r,hr,hru⟩ := Metric.isOpen_iff.mp ho (f x) hx | |
| let C := K∩Metric.closedBall (f x) (r/2) | |
| have hC : IsCompact C := hK.inter_right Metric.isClosed_closedBall | |
| have hCU : C⊆U := by | |
| intro y hy | |
| apply hru | |
| exact lt_of_le_of_lt hy.2 (half_lt_self hr) | |
| refine ⟨V∩f ⁻¹' Metric.closedBall (f x) (r/2),inter_mem hV | |
| (hfc.preimage_mem_nhds (Metric.closedBall_mem_nhds _ (half_pos hr))),g '' C,hC.image_of_continuousOn (hgu.mono hCU),?_⟩ | |
| rintro _ ⟨y,hy,rfl⟩ | |
| exact ⟨f y,⟨hfK ⟨y,hy.1,rfl⟩,hy.2⟩,rfl⟩ | |
| theorem compact_tube_precompact {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| [TopologicalSpace F] (T : ℝ × E → F) (K : Set E) (hK : IsCompact K) | |
| (hlocal : ∀ y∈Icc (0:ℝ) 1 ×ˢ K, LocallyPrecompact T y) : | |
| ∃ U : Set E, IsOpen U ∧ K⊆U ∧ ∃ S : Set F, IsCompact S ∧ | |
| ∀ t∈Icc (0:ℝ) 1, ∀ x∈U, T (t,x)∈S := by | |
| classical | |
| let Y := Icc (0:ℝ) 1 ×ˢ K | |
| have hY : IsCompact Y := isCompact_Icc.prod hK | |
| have hchoice : ∀ y : Y, ∃ W : Set (ℝ×E), IsOpen W ∧ (y : ℝ×E)∈W ∧ | |
| ∃ C : Set F, IsCompact C ∧ T '' W⊆C := by | |
| intro y | |
| obtain ⟨V,hV,C,hC,hTC⟩ := hlocal y y.2 | |
| obtain ⟨W,hWV,ho,hyW⟩ := mem_nhds_iff.mp hV | |
| exact ⟨W,ho,hyW,C,hC,(image_mono hWV).trans hTC⟩ | |
| choose W hWo hyW C hC hTC using hchoice | |
| obtain ⟨s,hs⟩ := hY.elim_finite_subcover W hWo (by | |
| intro y hy | |
| exact mem_iUnion.mpr ⟨⟨y,hy⟩,hyW ⟨y,hy⟩⟩) | |
| let O := ⋃ y∈s, W y | |
| have ho : IsOpen O := isOpen_biUnion (fun y hy => hWo y) | |
| obtain ⟨δ,hδ,hδO⟩ := hY.exists_thickening_subset_open ho hs | |
| refine ⟨Metric.thickening δ K,Metric.isOpen_thickening,Metric.self_subset_thickening hδ _, | |
| ⋃ y∈s, C y,?_ ,?_⟩ | |
| · exact s.finite_toSet.isCompact_biUnion (fun y hy => hC y) | |
| · intro t ht x hx | |
| obtain ⟨k,hk,hxk⟩ := Metric.mem_thickening_iff.mp hx | |
| have hyO : (t,x)∈O := hδO (Metric.mem_thickening_iff.mpr ⟨(t,k),⟨ht,hk⟩,by | |
| simpa only [Prod.dist_eq,dist_self,max_eq_right (dist_nonneg : (0:ℝ)≤dist x k)] using hxk⟩) | |
| obtain ⟨y,hy,hyW⟩ := mem_iUnion₂.mp hyO | |
| exact mem_iUnion₂.mpr ⟨y,hy,hTC y ⟨(t,x),hyW,rfl⟩⟩ | |
| theorem local_degree_continuation {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| (T : ℝ × E → E) (V : Set E) (hV : IsOpen V) | |
| (hT : ∀ t∈Icc (0:ℝ) 1, ∀ x∈V, ContinuousAt T (t,x)) | |
| (hzero : ∀ x∈V, T (0,x)=0) | |
| (K : Set E) (hK : IsCompact K) (hKV : K⊆V) (h0 : 0∈K) | |
| (hfix : ∀ t∈Icc (0:ℝ) 1, ∀ x∈V, T (t,x)=x → x∈K) | |
| (hloc : ∀ t∈Icc (0:ℝ) 1, ∀ x∈K, LocallyPrecompact T (t,x)) : | |
| ∃ x∈V, T (1,x)=x := by | |
| obtain ⟨U,hU,hKU,S,hS,hTS⟩ := compact_tube_precompact T K hK (fun y hy => hloc y.1 hy.1 y.2 hy.2) | |
| obtain ⟨x,hx,he⟩ := local_compact_continuation T (U∩V) (hU.inter hV) | |
| (fun t ht x hx => hT t ht x hx.2) (fun x hx => hzero x hx.2) S hS | |
| (fun t ht x hx => hTS t ht x hx.1) K hK (fun x hx => ⟨hKU hx,hKV hx⟩) h0 | |
| (fun t ht x hx he => hfix t ht x hx.2 he) | |
| exact ⟨x,hx.2,he⟩ | |
| end TwoPointContinuation | |
| end | |
| end OAI | |