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