import Mathlib import OAI.Geometry.BallPacking.Degree.HomotopyFlux namespace OAI noncomputable section namespace HigherDimensionalBallPacking.Rigidity.Degree section open scoped ContDiff Topology Manifold open Set Function Filter MeasureTheory variable {d k : ℕ} lemma supported_indicator_smooth {Z : Type*} [NormedAddCommGroup Z] [NormedSpace ℝ Z] {S K : Set (CoordinateSpace d)} (hS : IsOpen S) (hK : IsClosed K) (hKS : K⊆S) {f : CoordinateSpace d → Z} (hf : ContDiffOn ℝ ∞ f S) (hz : ∀ x∈S,x∉K →f x=0) : ContDiff ℝ ∞ (S.indicator f) := by classical apply contDiff_iff_contDiffAt.mpr intro x by_cases hx : x∈S · apply (hf.contDiffAt (hS.mem_nhds hx)).congr_of_eventuallyEq filter_upwards [hS.mem_nhds hx] with y hy exact indicator_of_mem hy _ · apply (contDiffAt_const (c := (0:Z))).congr_of_eventuallyEq filter_upwards [hK.isOpen_compl.mem_nhds (fun h => hx (hKS h))] with y hy by_cases hys : y∈S · rw [indicator_of_mem hys,hz y hys hy] · exact indicator_of_notMem hys _ lemma extDeriv_supported_indicator {S K : Set (CoordinateSpace d)} (hS : IsOpen S) (hK : IsClosed K) (hKS : K⊆S) {w : CoordinateSpace d → CoordinateSpace d [⋀^Fin k]→L[ℝ] ℝ} (hz : ∀ x∈S,x∉K →w x=0) : extDeriv (S.indicator w)=S.indicator (extDeriv w) := by classical funext x by_cases hx : x∈S · have he : S.indicator w =ᶠ[𝓝 x] w := by filter_upwards [hS.mem_nhds hx] with y hy exact indicator_of_mem hy _ rw [he.extDeriv_eq,indicator_of_mem hx] · have he : S.indicator w =ᶠ[𝓝 x] fun _ => 0 := by filter_upwards [hK.isOpen_compl.mem_nhds (fun h => hx (hKS h))] with y hy by_cases hys : y∈S · rw [indicator_of_mem hys,hz y hys hy] · exact indicator_of_notMem hys _ rw [he.extDeriv_eq,indicator_of_notMem hx] ext v rw [extDeriv_apply (differentiableAt_const _)] simp variable {M I : Type*} [TopologicalSpace M] [T2Space M] {a : SignedSmoothAtlas M I d} (α : AtlasForm a k) omit [T2Space M] in lemma AtlasForm.supporting_chart_compact [T2Space M] {K : Set M} (hK : IsCompact K) (i : I) (hi : K⊆(a.chart i).source) : IsCompact (a.chart i '' K) := hK.image_of_continuousOn ((a.chart i).continuousOn.mono hi) omit [T2Space M] in lemma AtlasForm.supporting_chart_zero [T2Space M] {K : Set M} (hα : α.SupportedIn K) (i : I) {q : CoordinateSpace d} (hq : q∈(a.chart i).target) (hqn : q∉a.chart i '' K) : α.form i q=0 := hα i q hq (fun hk => hqn ⟨_,hk,(a.chart i).right_inv hq⟩) def AtlasForm.chartExtension (i : I) : CoordinateSpace d → CoordinateSpace d [⋀^Fin k]→L[ℝ] ℝ := (a.chart i).target.indicator (α.form i) lemma AtlasForm.chartExtension_zero {K : Set M} (hα : α.SupportedIn K) (i : I) {q : CoordinateSpace d} (hqn : q∉a.chart i '' K) : α.chartExtension i q=0 := by classical by_cases hq : q∈(a.chart i).target · rw [chartExtension,indicator_of_mem hq] exact α.supporting_chart_zero hα i hq hqn · exact indicator_of_notMem hq _ lemma AtlasForm.chartExtension_smooth {K : Set M} (hK : IsCompact K) (hα : α.SupportedIn K) (i : I) (hi : K⊆(a.chart i).source) : ContDiff ℝ ∞ (α.chartExtension i) := by apply supported_indicator_smooth (a.chart i).open_target (AtlasForm.supporting_chart_compact (a := a) hK i hi).isClosed · rintro _ ⟨x,hx,rfl⟩ exact (a.chart i).map_source (hi hx) · exact α.smooth i · exact fun q hq hn => α.supporting_chart_zero hα i hq hn lemma AtlasForm.chartExtension_support {K : Set M} (hK : IsCompact K) (hα : α.SupportedIn K) (i : I) (hi : K⊆(a.chart i).source) : HasCompactSupport (α.chartExtension i) := HasCompactSupport.intro (AtlasForm.supporting_chart_compact (a := a) hK i hi) (fun _ h => α.chartExtension_zero hα i h) lemma AtlasForm.exterior_chartExtension {K : Set M} (hK : IsCompact K) (hα : α.SupportedIn K) (i : I) (hi : K⊆(a.chart i).source) : α.exterior.chartExtension i=extDeriv (α.chartExtension i) := by symm apply extDeriv_supported_indicator (a.chart i).open_target (AtlasForm.supporting_chart_compact (a := a) hK i hi).isClosed · rintro _ ⟨x,hx,rfl⟩ exact (a.chart i).map_source (hi hx) · exact fun q hq hn => α.supporting_chart_zero hα i hq hn def AtlasForm.chartMass {a : SignedSmoothAtlas M I d} (α : AtlasForm a d) (i : I) : ℝ := ∫ q,a.sign i*α.chartExtension i q coordinateVector lemma AtlasForm.chart_coefficient_integrable {a : SignedSmoothAtlas M I d} (α : AtlasForm a d) {K : Set M} (hK : IsCompact K) (hα : α.SupportedIn K) (i : I) (hi : K⊆(a.chart i).source) : Integrable (fun q => a.sign i*α.chartExtension i q coordinateVector) := by have hc : Continuous (fun q => a.sign i*α.chartExtension i q coordinateVector) := continuous_const.mul ((ContinuousAlternatingMap.apply ℝ _ ℝ coordinateVector).continuous.comp (α.chartExtension_smooth hK hα i hi).continuous) apply hc.integrable_of_hasCompactSupport apply HasCompactSupport.intro (AtlasForm.supporting_chart_compact (a := a) hK i hi) intro q hq rw [α.chartExtension_zero hα i hq] exact mul_zero _ lemma AtlasForm.chartMass_exterior_zero {a : SignedSmoothAtlas M I (k+1)} (α : AtlasForm a k) {K : Set M} (hK : IsCompact K) (hα : α.SupportedIn K) (i : I) (hi : K⊆(a.chart i).source) : α.exterior.chartMass i=0 := by unfold chartMass rw [α.exterior_chartExtension hK hα i hi,integral_const_mul, compact_euclidean_stokes ((α.chartExtension_smooth hK hα i hi).of_le (by exact WithTop.coe_le_coe.mpr le_top)) (α.chartExtension_support hK hα i hi),mul_zero] end section open scoped ContDiff Topology Manifold open Set Function Filter MeasureTheory variable {M I : Type*} [TopologicalSpace M] [T2Space M] {d k : ℕ} {a : SignedSmoothAtlas M I d} lemma AtlasForm.chartMass_eq_compact_integral (α : AtlasForm a d) {K : Set M} (hK : IsCompact K) (hα : α.SupportedIn K) (i : I) (hi : K⊆(a.chart i).source) : α.chartMass i=∫q in a.chart i '' K,a.sign i*α.form i q coordinateVector := by unfold chartMass rw [←setIntegral_eq_integral_of_forall_compl_eq_zero (s := a.chart i '' K)] · apply setIntegral_congr_fun · exact (hK.image_of_continuousOn ((a.chart i).continuousOn.mono hi)).measurableSet · intro q hq obtain ⟨x,hx,rfl⟩ := hq dsimp only rw [chartExtension,indicator_of_mem ((a.chart i).map_source (hi hx))] · intro q hq rw [α.chartExtension_zero hα i hq] exact mul_zero _ lemma AtlasForm.chartMass_change (α : AtlasForm a d) {K : Set M} (hK : IsCompact K) (hα : α.SupportedIn K) (i j : I) (hi : K⊆(a.chart i).source) (hj : K⊆(a.chart j).source) : α.chartMass i=α.chartMass j := by rw [α.chartMass_eq_compact_integral hK hα i hi,α.chartMass_eq_compact_integral hK hα j hj] apply topForm_chartMass_change (a.chart j) (a.chart i) hK hi hj (a.transition_smooth i j) · rintro q ⟨x,hx,rfl⟩ apply α.compatible i j exact ⟨(a.chart i).map_source (hi hx),by change (a.chart i).symm ((a.chart i) x)∈(a.chart j).source rw [(a.chart i).left_inv (hi hx)]; exact hj hx⟩ · rintro q ⟨x,hx,rfl⟩ apply a.orientation i j exact ⟨(a.chart i).map_source (hi hx),by change (a.chart i).symm ((a.chart i) x)∈(a.chart j).source rw [(a.chart i).left_inv (hi hx)]; exact hj hx⟩ omit [T2Space M] in lemma AtlasForm.chartMass_sum [T2Space M] {J : Type*} [Fintype J] (Ω : AtlasForm a d) (β : J → AtlasForm a d) (i : I) (he : ∀ q∈(a.chart i).target,Ω.form i q=∑j,(β j).form i q) (hi : ∀ j,Integrable (fun q => a.sign i*(β j).chartExtension i q coordinateVector)) : Ω.chartMass i=∑j,(β j).chartMass i := by classical change (∫q,a.sign i*Ω.chartExtension i q coordinateVector)=∑j,∫q,a.sign i*(β j).chartExtension i q coordinateVector rw [←integral_finsetSum _ (fun j _ => hi j)] apply integral_congr_ae apply Filter.Eventually.of_forall intro q by_cases hq : q∈(a.chart i).target · simp only [AtlasForm.chartExtension,indicator_of_mem hq,he q hq] rw [ContinuousAlternatingMap.sum_apply,Finset.mul_sum] · simp only [AtlasForm.chartExtension,indicator_of_notMem hq] change a.sign i*0=∑j,a.sign i*0 simp variable [Fintype I] structure AtlasPartition (a : SignedSmoothAtlas M I d) (K : Set M) where weight : I → M → ℝ smooth : ∀ i j,ContDiffOn ℝ ∞ (weight i ∘ (a.chart j).symm) (a.chart j).target sum_one : ∀ x∈K,∑i,weight i x=1 subordinate : ∀ i,tsupport (weight i)⊆(a.chart i).source namespace AtlasPartition variable {K : Set M} (p : AtlasPartition a K) def mass (Ω : AtlasForm a d) : ℝ := ∑i,(Ω.smul (p.weight i) (p.smooth i)).chartMass i omit [T2Space M] in lemma smul_sum [T2Space M] (Ω : AtlasForm a k) (hΩ : Ω.SupportedIn K) (i : I) {q : CoordinateSpace d} (hq : q∈(a.chart i).target) : ∑j,(Ω.smul (p.weight j) (p.smooth j)).form i q=Ω.form i q := by change (∑j,p.weight j ((a.chart i).symm q) • Ω.form i q)=Ω.form i q rw [←Finset.sum_smul] by_cases hx : (a.chart i).symm q∈K · rw [p.sum_one _ hx,one_smul] · rw [hΩ i q hq hx,smul_zero] lemma mass_eq_chartMass {L : Set M} (hL : IsCompact L) (hLK : L⊆K) (Ω : AtlasForm a d) (hΩ : Ω.SupportedIn L) (j : I) (hj : L⊆(a.chart j).source) : p.mass Ω=Ω.chartMass j := by classical let D : I → AtlasForm a d := fun i => Ω.smul (p.weight i) (p.smooth i) let C : I → Set M := fun i => L∩tsupport (p.weight i) have hC (i : I) : IsCompact (C i) := hL.inter_right (isClosed_tsupport _) have hD (i : I) : (D i).SupportedIn (C i) := Ω.smul_supported _ _ hΩ have hci (i : I) : C i⊆(a.chart i).source := fun _ h => p.subordinate i h.2 have hcj (i : I) : C i⊆(a.chart j).source := fun _ h => hj h.1 have hm (i : I) : (D i).chartMass i=(D i).chartMass j := (D i).chartMass_change (hC i) (hD i) i j (hci i) (hcj i) change (∑i,(D i).chartMass i)=_ simp_rw [hm] symm apply Ω.chartMass_sum D j · intro q hq have hΩK : Ω.SupportedIn K := fun i q hq hn => hΩ i q hq (fun hx => hn (hLK hx)) exact (p.smul_sum Ω hΩK j hq).symm · exact fun i => (D i).chart_coefficient_integrable (hC i) (hD i) j (hcj i) end AtlasPartition end section open scoped ContDiff Topology Manifold open Set Function Filter MeasureTheory variable {M I : Type*} [TopologicalSpace M] [T2Space M] [Fintype I] {d k : ℕ} {a : SignedSmoothAtlas M I d} {K : Set M} lemma AtlasPartition.mass_sum (p : AtlasPartition a K) (hK : IsCompact K) {J : Type*} [Fintype J] (Ω : AtlasForm a d) (β : J → AtlasForm a d) (hβ : ∀ j,(β j).SupportedIn K) (he : ∀ i q,q∈(a.chart i).target → Ω.form i q=∑j,(β j).form i q) : p.mass Ω=∑j,p.mass (β j) := by classical unfold mass rw [Finset.sum_comm] apply Finset.sum_congr rfl intro i hi apply AtlasForm.chartMass_sum _ (fun j => (β j).smul (p.weight i) (p.smooth i)) i · intro q hq change p.weight i ((a.chart i).symm q) • Ω.form i q= ∑j,p.weight i ((a.chart i).symm q) • (β j).form i q rw [he i q hq,Finset.smul_sum] · intro j exact ((β j).smul (p.weight i) (p.smooth i)).chart_coefficient_integrable (hK.inter_right (isClosed_tsupport _)) ((β j).smul_supported _ _ (hβ j)) i (fun _ h => p.subordinate i h.2) lemma extDeriv_fintype_sum {J : Type*} [Fintype J] (β : J → CoordinateSpace d → CoordinateSpace d [⋀^Fin k]→L[ℝ] ℝ) (q : CoordinateSpace d) (hβ : ∀ j,DifferentiableAt ℝ (β j) q) : extDeriv (fun y => ∑j,β j y) q=∑j,extDeriv (β j) q := by classical simp only [extDeriv,fderiv_fun_sum (fun j _ => hβ j)] exact map_sum (ContinuousAlternatingMap.alternatizeUncurryFinCLM ℝ _ ℝ) _ _ theorem AtlasPartition.compact_stokes {a : SignedSmoothAtlas M I (k+1)} (p : AtlasPartition a K) (hK : IsCompact K) (α : AtlasForm a k) (hα : α.SupportedIn K) : p.mass α.exterior=0 := by classical let β : I → AtlasForm a k := fun i => α.smul (p.weight i) (p.smooth i) let L : I → Set M := fun i => K∩tsupport (p.weight i) have hL (i : I) : IsCompact (L i) := hK.inter_right (isClosed_tsupport _) have hβ (i : I) : (β i).SupportedIn (L i) := α.smul_supported _ _ hα have hLi (i : I) : L i⊆(a.chart i).source := fun _ h => p.subordinate i h.2 have hdβ (i : I) : (β i).exterior.SupportedIn K := by have hs := (β i).exterior_supported (hL i).isClosed (hβ i) exact fun j q hq hn => hs j q hq (fun h => hn h.1) have hsum (i : I) (q : CoordinateSpace (k+1)) (hq : q∈(a.chart i).target) : α.exterior.form i q=∑j,(β j).exterior.form i q := by have he : α.form i =ᶠ[𝓝 q] fun y => ∑j,(β j).form i y := by filter_upwards [(a.chart i).open_target.mem_nhds hq] with y hy exact (p.smul_sum α hα i hy).symm change extDeriv (α.form i) q=∑j,extDeriv ((β j).form i) q rw [he.extDeriv_eq] apply extDeriv_fintype_sum intro j exact (((β j).smooth i).contDiffAt ((a.chart i).open_target.mem_nhds hq)).differentiableAt (by simp) rw [p.mass_sum hK α.exterior (fun j => (β j).exterior) hdβ hsum] apply Finset.sum_eq_zero intro i hi rw [p.mass_eq_chartMass (hL i) inter_subset_left (β i).exterior ((β i).exterior_supported (hL i).isClosed (hβ i)) i (hLi i)] exact (β i).chartMass_exterior_zero (hL i) (hβ i) i (hLi i) end section open scoped ContDiff Topology open Set Function Filter variable {A B Z : Type*} [NormedAddCommGroup A] [NormedSpace ℝ A] [NormedAddCommGroup B] [NormedSpace ℝ B] [NormedAddCommGroup Z] [NormedSpace ℝ Z] {k : ℕ} def alternationCLM : (ContinuousMultilinearMap ℝ (fun _ : Fin k => A) ℝ) →L[ℝ] (A [⋀^Fin k]→L[ℝ] ℝ) where toFun := ContinuousMultilinearMap.alternatization map_add' := map_add _ map_smul' r f := by ext v simp only [RingHom.id_apply,ContinuousMultilinearMap.alternatization_apply_apply, smul_apply,ContinuousAlternatingMap.smul_apply] rw [Finset.smul_sum] apply Finset.sum_congr rfl intro σ hσ exact smul_comm _ _ _ cont := by rw [ContinuousAlternatingMap.isEmbedding_toContinuousMultilinearMap.continuous_iff] change Continuous (fun f : ContinuousMultilinearMap ℝ (fun _ : Fin k => A) ℝ => ∑σ : Equiv.Perm (Fin k),Equiv.Perm.sign σ • f.domDomCongr σ) apply continuous_finsetSum intro σ hσ exact (ContinuousMultilinearMap.domDomCongrₗᵢ ℝ A ℝ σ).continuous.const_smul _ lemma alternationCLM_alternating (w : A [⋀^Fin k]→L[ℝ] ℝ) : alternationCLM w.toContinuousMultilinearMap=(k.factorial:ℝ) • w := by apply ContinuousAlternatingMap.toAlternatingMap_injective change (ContinuousMultilinearMap.alternatization w.toContinuousMultilinearMap).toAlternatingMap=_ rw [ContinuousMultilinearMap.alternatization_apply_toAlternatingMap] change MultilinearMap.alternatization w.toAlternatingMap.toMultilinearMap= (k.factorial:ℝ) • w.toAlternatingMap simpa only [Fintype.card_fin,Nat.cast_smul_eq_nsmul] using w.toAlternatingMap.coe_alternatization lemma contDiffOn_alternating_of_multilinear {f : Z → A [⋀^Fin k]→L[ℝ] ℝ} {S : Set Z} (hf : ContDiffOn ℝ ∞ (fun z => (f z).toContinuousMultilinearMap) S) : ContDiffOn ℝ ∞ f S := by have h := (alternationCLM (A := A) (k := k)).contDiff.comp_contDiffOn hf have hn : (k.factorial:ℝ)≠0 := Nat.cast_ne_zero.mpr (Nat.factorial_ne_zero _) convert h.const_smul (k.factorial:ℝ)⁻¹ using 1 funext z simp only [Function.comp_apply] rw [alternationCLM_alternating,smul_smul,inv_mul_cancel₀ hn,one_smul] lemma contDiffOn_alternating_comp {f : Z → B [⋀^Fin k]→L[ℝ] ℝ} {g : Z → A →L[ℝ] B} {S : Set Z} (hf : ContDiffOn ℝ ∞ f S) (hg : ContDiffOn ℝ ∞ g S) : ContDiffOn ℝ ∞ (fun z => (f z).compContinuousLinearMap (g z)) S := by apply contDiffOn_alternating_of_multilinear have hp : ContDiffOn ℝ ∞ (fun z => ((fun _ : Fin k => g z),(f z).toContinuousMultilinearMap)) S := (contDiffOn_pi.mpr (fun _ => hg)).prodMk ((ContinuousAlternatingMap.toContinuousMultilinearMapCLM ℝ : (B [⋀^Fin k]→L[ℝ] ℝ) →L[ℝ] ContinuousMultilinearMap ℝ (fun _ : Fin k => B) ℝ).contDiff.comp_contDiffOn hf) have ha : ContDiff ℝ ∞ (fun p : (Fin k → A →L[ℝ] B) × ContinuousMultilinearMap ℝ (fun _ : Fin k => B) ℝ => p.2.compContinuousLinearMap p.1) := (show AnalyticOnNhd ℝ _ univ from ContinuousMultilinearMap.analyticOnNhd_uncurry_compContinuousLinearMap).contDiff exact ha.comp_contDiffOn hp end section open scoped ContDiff Topology Manifold open Set Function Filter variable {M I P : Type*} [TopologicalSpace M] [NormedAddCommGroup P] [NormedSpace ℝ P] {d k : ℕ} {a : SignedSmoothAtlas M I d} structure AtlasSmoothMap (a : SignedSmoothAtlas M I d) (P : Type*) [NormedAddCommGroup P] [NormedSpace ℝ P] where map : M → P smooth : ∀ i,ContDiffOn ℝ ∞ (map ∘ (a.chart i).symm) (a.chart i).target namespace AtlasSmoothMap variable (f : AtlasSmoothMap a P) lemma derivative_transition (i j : I) {q : CoordinateSpace d} (hq : q∈((a.chart i).symm ≫ₕ a.chart j).source) : fderiv ℝ (f.map ∘ (a.chart i).symm) q= (fderiv ℝ (f.map ∘ (a.chart j).symm) (a.chart j ((a.chart i).symm q))).comp (fderiv ℝ (a.chart j ∘ (a.chart i).symm) q) := by let g := a.chart j ∘ (a.chart i).symm have he : (f.map ∘ (a.chart i).symm) =ᶠ[𝓝 q] ((f.map ∘ (a.chart j).symm) ∘ g) := by filter_upwards [((a.chart i).symm ≫ₕ a.chart j).open_source.mem_nhds hq] with r hr simp only [Function.comp_apply,g] exact congrArg f.map ((a.chart j).left_inv hr.2).symm rw [he.fderiv_eq] apply fderiv_comp · exact ((f.smooth j).contDiffAt ((a.chart j).open_target.mem_nhds ((a.chart j).map_source hq.2))).differentiableAt (by simp) · exact ((a.transition_smooth i j).contDiffAt (((a.chart i).symm ≫ₕ a.chart j).open_source.mem_nhds hq)).differentiableAt (by simp) def pullback (w : P → P [⋀^Fin k]→L[ℝ] ℝ) (hw : ContDiff ℝ ∞ w) : AtlasForm a k where form i q := (w (f.map ((a.chart i).symm q))).compContinuousLinearMap (fderiv ℝ (f.map ∘ (a.chart i).symm) q) smooth i := contDiffOn_alternating_comp (hw.comp_contDiffOn (f.smooth i)) ((contDiffOn_infty_iff_fderiv_of_isOpen (a.chart i).open_target).mp (f.smooth i)).2 compatible i j q hq := by change (w (f.map ((a.chart i).symm q))).compContinuousLinearMap _= ((w (f.map ((a.chart j).symm (a.chart j ((a.chart i).symm q))))).compContinuousLinearMap _).compContinuousLinearMap _ have he := (a.chart j).left_inv hq.2 simp only [OpenPartialHomeomorph.symm_symm] at he rw [he,f.derivative_transition i j hq] rfl lemma pullback_exterior (w : P → P [⋀^Fin k]→L[ℝ] ℝ) (hw : ContDiff ℝ ∞ w) (i : I) {q : CoordinateSpace d} (hq : q∈(a.chart i).target) : (f.pullback w hw).exterior.form i q= (extDeriv w (f.map ((a.chart i).symm q))).compContinuousLinearMap (fderiv ℝ (f.map ∘ (a.chart i).symm) q) := by exact extDeriv_pullback (hw.differentiable (by simp) _) ((f.smooth i).contDiffAt ((a.chart i).open_target.mem_nhds hq)) (by rw [minSmoothness_of_isRCLikeNormedField] exact WithTop.coe_le_coe.mpr le_top) lemma pullback_supported (w : P → P [⋀^Fin k]→L[ℝ] ℝ) (hw : ContDiff ℝ ∞ w) {K : Set M} (hK : ∀ x : M,x∉K →w (f.map x)=0) : (f.pullback w hw).SupportedIn K := by intro i q hq hn change (w (f.map ((a.chart i).symm q))).compContinuousLinearMap _=0 rw [hK _ hn] ext v rfl end AtlasSmoothMap end open scoped ContDiff Topology open Set Filter MeasureTheory variable {m : ℕ} def spatialBox (R : ℝ) : Set (CoordinateSpace m) := Icc (fun _ => -R) (fun _ => R) def lowerCorner (R : ℝ) : CoordinateSpace (m+1) := Fin.cons 0 (fun _ => -R) def upperCorner (R : ℝ) : CoordinateSpace (m+1) := Fin.cons 1 (fun _ => R) def exitFaces (R : ℝ) : Set (CoordinateSpace (m+1)) := Icc (lowerCorner R) (upperCorner R) ∩ ({v | v 0=1} ∪ ⋃ i : Fin m,{v | v i.succ = -R} ∪ {v | v i.succ=R}) lemma corners_le {R : ℝ} (hR : 0≤R) : lowerCorner (m := m) R≤upperCorner R := by intro i refine Fin.cases ?_ (fun j => ?_) i · norm_num [lowerCorner,upperCorner] · simpa [lowerCorner,upperCorner] using neg_le_self hR lemma exitFaces_compact (R : ℝ) : IsCompact (exitFaces (m := m) R) := by apply isCompact_Icc.inter_right apply IsClosed.union · exact isClosed_eq (continuous_apply _) continuous_const · exact isClosed_iUnion_of_finite (fun i : Fin m => (isClosed_eq (continuous_apply _) continuous_const).union (isClosed_eq (continuous_apply _) continuous_const)) lemma tail_mem_spatialBox {R : ℝ} {v : CoordinateSpace (m+1)} (hv : v∈Icc (lowerCorner R) (upperCorner R)) : Fin.tail v∈spatialBox R := by exact ⟨fun i => hv.1 i.succ,fun i => hv.2 i.succ⟩ lemma time_mem_interval {R : ℝ} {v : CoordinateSpace (m+1)} (hv : v∈Icc (lowerCorner R) (upperCorner R)) : v 0∈Icc (0:ℝ) 1 := ⟨hv.1 0,hv.2 0⟩ lemma timeFace_mem_box {R : ℝ} {t : ℝ} {x : CoordinateSpace m} (ht : t∈Icc (0:ℝ) 1) (hx : x∈spatialBox R) : timeFace t x∈Icc (lowerCorner R) (upperCorner R) := by constructor · intro i exact Fin.cases ht.1 hx.1 i · intro i exact Fin.cases ht.2 hx.2 i lemma initialFace_box (R : ℝ) : Icc (lowerCorner (m := m) R ∘ (0:Fin (m+1)).succAbove) (upperCorner R ∘ (0:Fin (m+1)).succAbove)=spatialBox R := by simp only [Fin.succAbove_zero,lowerCorner,upperCorner,Function.comp_def,Fin.cons_succ,spatialBox] lemma insertNth_mem_box {a b : CoordinateSpace (m+1)} (_hab : a≤b) (i : Fin (m+1)) {c : ℝ} (hc : c∈Icc (a i) (b i)) {y : CoordinateSpace m} (hy : y∈Icc (a ∘ i.succAbove) (b ∘ i.succAbove)) : i.insertNth c y∈Icc a b := by constructor · change ∀ j,a j ≤ i.insertNth (α := fun _ => ℝ) c y j rw [Fin.forall_iff_succAbove i] exact ⟨by simpa using hc.1,fun j => by simpa using hy.1 j⟩ · change ∀ j,i.insertNth (α := fun _ => ℝ) c y j ≤ b j rw [Fin.forall_iff_succAbove i] exact ⟨by simpa using hc.2,fun j => by simpa using hy.2 j⟩ lemma sideFace_mem_exit (R : ℝ) (hR : 0≤R) (i : Fin (m+1)) (hi : i≠0) {y : CoordinateSpace m} (hy : y∈Icc (lowerCorner R ∘ i.succAbove) (upperCorner R ∘ i.succAbove)) : i.insertNth (upperCorner R i) y∈exitFaces R ∧ i.insertNth (lowerCorner R i) y∈exitFaces R := by have hab := corners_le (m := m) hR have hi' : ∃ j : Fin m,i=j.succ := by rcases Fin.eq_zero_or_eq_succ i with h|⟨j,h⟩ · exact (hi h).elim · exact ⟨j,h⟩ obtain ⟨j,rfl⟩ := hi' constructor · refine ⟨insertNth_mem_box hab j.succ ⟨hab j.succ,le_rfl⟩ hy,Or.inr ?_⟩ exact mem_iUnion.mpr ⟨j,Or.inr (by simp [upperCorner])⟩ · refine ⟨insertNth_mem_box hab j.succ ⟨le_rfl,hab j.succ⟩ hy,Or.inr ?_⟩ exact mem_iUnion.mpr ⟨j,Or.inl (by simp [lowerCorner])⟩ lemma norm_ball_subset_spatialBox {R r : ℝ} (hr : r≤R) : Metric.ball (0:CoordinateSpace m) r⊆spatialBox R := by intro x hx have hnorm : ‖x‖ (abs_le.mp (hb i)).1,fun i => (abs_le.mp (hb i)).2⟩ lemma separating_bump {K : Set (CoordinateSpace m)} (hK : IsCompact K) (h0 : (0:CoordinateSpace m)∉K) {R : ℝ} (hR : 0