| import Mathlib |
| import OAI.Geometry.BallPacking.Models.DisjointClocks |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace PackingSufficiencySupport.MomentPolytope |
| open scoped ContDiff Topology BigOperators Pointwise |
| open Set Function MeasureTheory |
| open Hamiltonian Comparison |
|
|
| theorem compact_positive_lower {X : Type*} [TopologicalSpace X] |
| {D : Set X} (hc : IsCompact D) (hne : D.Nonempty) {f : X → ℝ} |
| (hf : ContinuousOn f D) (hp : ∀ x ∈ D,0<f x) : |
| ∃ g>0, ∀ x ∈ D,g ≤ f x := by |
| obtain ⟨x,hx,hm⟩ := hc.exists_isMinOn hne hf |
| exact ⟨f x,hp x hx,hm⟩ |
|
|
| theorem inverse_comparison_gap {m N : ℕ} |
| (b : Fin N → Moments m) (c : Fin N → ℝ) (hc : ∀ ν,0 ≤ c ν) |
| (hcomp : IsCompact (region b c)) |
| (K P : Moments m → ℝ) (hK : Continuous K) (hP : ContinuousOn P (region b c)) |
| (Φ : CompactHamiltonianIsotopy (@phaseArea (Fin m) _)) |
| (hmem : ∀ t z, Φ.map t z ∈ planeRegion b c ↔ z ∈ planeRegion b c) |
| (hgap : ∀ z ∈ planeRegion b c, |
| K (planeMoments (Φ.map 1 z))-P (planeMoments (Φ.map 1 z)) < K (planeMoments z)) : |
| ∃ g>0, ∀ v ∈ planeRegion b c, |
| g ≤ K (planeMoments ((Φ.map 1).symm v))-K (planeMoments v)+P (planeMoments v) := by |
| apply compact_positive_lower (planeRegion_isCompact b c hcomp) (planeRegion_nonempty b c hc) |
| · exact ((hK.comp (planeMoments_smooth.continuous.comp (Φ.map 1).symm.continuous)).continuousOn.sub |
| (hK.comp planeMoments_smooth.continuous).continuousOn).add |
| (hP.comp planeMoments_smooth.continuous.continuousOn (fun _ hv => hv)) |
| · intro v hv |
| have hg := hgap ((Φ.map 1).symm v) ((hmem 1 _).1 (by simpa using hv)) |
| simp only [Homeomorph.apply_symm_apply] at hg |
| linarith |
|
|
| theorem finite_surface_comparison {m N : ℕ} {ι : Type*} [Fintype ι] [Nonempty ι] |
| (b : Fin N → Moments m) (c : Fin N → ℝ) |
| (hb : ∀ ν j,0 ≤ b ν j) (hc : ∀ ν,0<c ν) (hcomp : IsCompact (region b c)) |
| (H : Moments m → ℝ) (hH : ContinuousOn H (region b c)) |
| (hHpos : ∀ p ∈ region b c,0<H p) (r : ι → ℝ) |
| (hconc : ConcaveOn ℝ (region b c) (fun p => H p-∑ i,max (r i-∑ j,p j) 0)) |
| (hmean : 0 < ∫ p in region b c,H p-∑ i,max (r i-∑ j,p j) 0) |
| {τ a : ℝ} (hτ : 0<τ) (hτ1 : τ<1) (ha : 0<a) |
| (houter : ∀ p ∈ region b c,p ∉ τ • region b c → a ≤ H p-∑ i,max (r i-∑ j,p j) 0) : |
| ∃ (K : Moments m → ℝ) (Φ : CompactHamiltonianIsotopy (@phaseArea (Fin m) _)) |
| (g ε : ℝ) (h : ι → Moments m → ℝ), |
| ContDiff ℝ ∞ K ∧ 0<g ∧ 0<ε ∧ 8*ε<g ∧ |
| (∀ p ∈ region b c,8*ε<H p) ∧ |
| (∀ i,ContDiff ℝ ∞ (h i)) ∧ |
| (∀ i p,max (r i-∑ j,p j) 0<h i p) ∧ |
| (∀ p,0<(∑ i,h i p)-(∑ i,max (r i-∑ j,p j) 0) ∧ |
| (∑ i,h i p)-(∑ i,max (r i-∑ j,p j) 0)<ε) ∧ |
| (∀ t z,Φ.map t z ∈ planeRegion b c ↔ z ∈ planeRegion b c) ∧ |
| (∀ ν,∃ ρ>0,∀ t z,c ν-weightedMoment (b ν) z<ρ → |
| fderiv ℝ (weightedMoment (b ν)) z (hamiltonianField phaseArea Φ.hamiltonian (t,z))=0) ∧ |
| (∀ Hstar : Moments m → ℝ,(∀ p ∈ region b c,|Hstar p-H p|<2*ε) → |
| (∀ p ∈ region b c,0<Hstar p) ∧ |
| ∀ v ∈ planeRegion b c,g/2 < surfaceRemainder (K ∘ planeMoments) |
| (Hstar ∘ planeMoments) (fun i => h i ∘ planeMoments) (Φ.map 1).symm v) := by |
| have hcap : Continuous (fun p : Moments m => ∑ i,max (r i-∑ j,p j) 0) := by fun_prop |
| obtain ⟨K,Φ,hK,hmem,hgap,hface⟩ := positive_hamiltonian_comparison b c hb hc hcomp |
| (fun p => H p-∑ i,max (r i-∑ j,p j) 0) (hH.sub hcap.continuousOn) |
| hconc hmean hτ hτ1 ha houter |
| obtain ⟨g,hg,hgall⟩ := inverse_comparison_gap b c (fun ν => (hc ν).le) hcomp K _ |
| hK.continuous (hH.sub hcap.continuousOn) Φ hmem hgap |
| have hne : (region b c).Nonempty := by |
| refine ⟨0,?_,?_⟩ |
| · intro j; simp |
| · intro ν; simpa using (hc ν).le |
| obtain ⟨u,hu,huall⟩ := compact_positive_lower hcomp hne hH hHpos |
| let ε := min g u/16 |
| have hε : 0<ε := div_pos (lt_min hg hu) (by norm_num) |
| have he1 : 8*ε<g := by dsimp [ε]; have := min_le_left g u; nlinarith [lt_min hg hu] |
| have he2 : 8*ε<u := by dsimp [ε]; have := min_le_right g u; nlinarith [lt_min hg hu] |
| obtain ⟨h,hhs,hhu,hhe⟩ := exists_smooth_positive_caps (κ := Fin m) r hε |
| refine ⟨K,Φ,g,ε,h,hK,hg,hε,he1,(fun p hp => he2.trans_le (huall p hp)),hhs,hhu,hhe,hmem,hface,?_⟩ |
| intro Hstar hstar |
| refine ⟨?_,?_⟩ |
| · intro p hp |
| have hs := (abs_lt.mp (hstar p hp)).1 |
| have hu' := huall p hp |
| linarith |
| · intro v hv |
| have hg' := hgall v hv |
| simp only [Pi.sub_apply] at hg' |
| have heq : g ≤ K (planeMoments ((Φ.map 1).symm v))-K (planeMoments v)+ |
| H (planeMoments v)-(∑ i,max (r i-∑ j,planeMoments v j) 0) := by linarith |
| have hr := surfaceRemainder_gap (ψ := (Φ.map 1).symm) |
| (K := K ∘ planeMoments) (H := Hstar ∘ planeMoments) |
| (H₀ := H ∘ planeMoments) (C := fun v => ∑ i,max (r i-∑ j,planeMoments v j) 0) |
| (h := fun i => h i ∘ planeMoments) heq (hstar _ hv) (hhe (planeMoments v)).2 |
| exact (by linarith : g/2<g-3*ε).trans hr |
| |
| theorem finite_uniform_area_error {m : ℕ} {D : Set (Moments m)} |
| {A : ℕ → Moments m → ℝ} {H : Moments m → ℝ} |
| (hA : TendstoUniformlyOn A H Filter.atTop D) {ε : ℝ} (hε : 0<ε) : |
| ∃ L : ℕ,∀ ℓ ≥ L,∀ p ∈ D,|A ℓ p-H p|<ε := by |
| obtain ⟨L,hL⟩ := Filter.eventually_atTop.mp ((Metric.tendstoUniformlyOn_iff.mp hA) ε hε) |
| refine ⟨L,fun ℓ hℓ p hp => ?_⟩ |
| simpa only [Real.dist_eq,abs_sub_comm] using hL ℓ hℓ p hp |
| |
| def strictRegion {m N : ℕ} (b : Fin N → Moments m) (c : Fin N → ℝ) : Set (Moments m) := |
| {p | (∀ j,0≤p j) ∧ ∀ ν,(∑ j,b ν j*p j)<c ν} |
| |
| theorem strictRegion_subset {m N : ℕ} (b : Fin N → Moments m) (c : Fin N → ℝ) : |
| strictRegion b c⊆region b c := fun _ hp => ⟨hp.1,fun ν => (hp.2 ν).le⟩ |
| |
| theorem strictRegion_phase_open {m N : ℕ} (b : Fin N → Moments m) (c : Fin N → ℝ) : |
| IsOpen (planeMoments ⁻¹' strictRegion b c) := by |
| have he : planeMoments ⁻¹' strictRegion b c = ⋂ ν,{z | weightedMoment (b ν) z<c ν} := by |
| ext z |
| simp only [mem_preimage,strictRegion,mem_ofPred_eq,mem_iInter] |
| exact ⟨fun h => h.2,fun h => ⟨planeMoments_nonneg z,h⟩⟩ |
| rw [he] |
| exact isOpen_iInter_of_finite (fun ν => isOpen_lt (weightedMoment_smooth (b ν)).continuous continuous_const) |
| |
| theorem planeRegion_outer_constraints {m N : ℕ} (b : Fin N → Moments m) (c : Fin N → ℝ) : |
| planeMoments ⁻¹' region b c = {z | ∀ ν,(fun p => ∑ j,b ν j*p j) (planeMoments z)≤c ν} := by |
| ext z |
| change ((∀ j,0≤planeMoments z j) ∧ _) ↔ _ |
| exact ⟨fun h => h.2,fun h => ⟨planeMoments_nonneg z,h⟩⟩ |
|
|
| theorem momentFace_smooth {m : ℕ} (b : Moments m) : |
| ContDiff ℝ ∞ (fun p : Moments m => ∑ j,b j*p j) := by fun_prop |
|
|
| theorem face_collars_of_comparison {m N : ℕ} (b : Fin N → Moments m) (c : Fin N → ℝ) |
| (Φ : CompactHamiltonianIsotopy (phaseArea (ι := Fin m))) |
| (hface : ∀ ν,∃ ρ>0,∀ t z,c ν-weightedMoment (b ν) z<ρ → |
| fderiv ℝ (weightedMoment (b ν)) z (hamiltonianField phaseArea Φ.hamiltonian (t,z))=0) : |
| ∃ α : Fin N → ℝ,(∀ ν,α ν<c ν) ∧ ∀ ν t z, |
| α ν<(fun p => ∑ j,b ν j*p j) (planeMoments z) → |
| fderiv ℝ ((fun p => ∑ j,b ν j*p j) ∘ planeMoments) z |
| (hamiltonianField phaseArea Φ.hamiltonian (t,z))=0 := by |
| classical |
| choose ρ hρ hzero using hface |
| refine ⟨fun ν => c ν-ρ ν,fun ν => sub_lt_self _ (hρ ν),?_⟩ |
| intro ν t z hz |
| apply hzero ν t z |
| change c ν-(∑ j,b ν j*planeMoments z j)<ρ ν |
| linarith |
|
|
| theorem comparison_preserves_region {m N : ℕ} (b : Fin N → Moments m) (c : Fin N → ℝ) |
| (Φ : CompactHamiltonianIsotopy (phaseArea (ι := Fin m))) |
| (hmem : ∀ t z,Φ.map t z∈planeRegion b c ↔ z∈planeRegion b c) : |
| ∀ t,MapsTo (Φ.map t) (planeMoments ⁻¹' region b c) (planeMoments ⁻¹' region b c) ∧ |
| MapsTo (Φ.map t).symm (planeMoments ⁻¹' region b c) (planeMoments ⁻¹' region b c) := by |
| intro t |
| refine ⟨fun z hz => (hmem t z).2 hz,?_⟩ |
| intro z hz |
| apply (hmem t ((Φ.map t).symm z)).1 |
| change planeMoments ((Φ.map t) ((Φ.map t).symm z))∈region b c |
| rw [Homeomorph.apply_symm_apply] |
| exact hz |
|
|
| end PackingSufficiencySupport.MomentPolytope |
|
|
| namespace PackingSufficiencySupport.MomentPolytope.Annular |
| open scoped ContDiff Manifold Topology BigOperators Pointwise |
| open Set Function Manifold MeasureTheory |
| open Hamiltonian Hamiltonian.AnnularHandleData |
|
|
| variable {M : Type} [TopologicalSpace M] [ChartedSpace Plane M] |
| [IsManifold 𝓘(ℝ,Plane) ∞ M] [T2Space M] [NormalSpace M] [SigmaCompactSpace M] |
| [TopologicalSpace.PseudoMetrizableSpace M] [MeasurableSpace M] [BorelSpace M] |
|
|
| theorem surface_packing_tolerance {m q N : ℕ} [Nonempty (Fin N)] |
| (bF : Fin q → Moments m) (cF : Fin q → ℝ) |
| (hbF : ∀ ν j,0≤bF ν j) (hcF : ∀ ν,0<cF ν) (hcomp : IsCompact (region bF cF)) |
| (A : Moments m → ℝ) (hA : ContinuousOn A (region bF cF)) |
| (hAp : ∀ p∈region bF cF,0<A p) (r r' : Fin N → ℝ) |
| (hr' : ∀ i,0≤r' i) (hrr : ∀ i,r' i<r i) |
| (hsimplex : ∀ i,∀ p : Moments m,(∀ j,0≤p j) → (∑ j,p j)≤r i → p∈region bF cF) |
| (hface : ∀ i,∀ p : Moments m,(∀ j,0≤p j) → (∑ j,p j)≤r i → |
| ∀ ν,(∑ j,bF ν j*p j)≠cF ν) |
| (hconc : ConcaveOn ℝ (region bF cF) (fun p => A p-∑ i,max (r i-∑ j,p j) 0)) |
| (hmean : 0<∫ p in region bF cF,A p-∑ i,max (r i-∑ j,p j) 0) |
| {τ d₀ : ℝ} (hτ : 0<τ) (hτ1 : τ<1) (hd₀ : 0<d₀) |
| (houter : ∀ p∈region bF cF,p∉τ • region bF cF → d₀≤A p-∑ i,max (r i-∑ j,p j) 0) : |
| ∃ ε>0,∀ {I : Type*} [Fintype I],∀ (Kbase : Set M) (_hD : IsCompact Kbase) (_hcD : IsPreconnected (interior Kbase)) |
| (_hne : (interior Kbase).Nonempty) (_hor : PositivePlaneTransitions M) |
| (B : I → SurfaceCoordinateBox M) (π : SmoothPartitionOfUnity I 𝓘(ℝ,Plane) M Kbase), |
| π.IsSubordinate (fun i => (B i).carrier) → |
| (∀ i,volume ((extChartAt 𝓘(ℝ,Plane) (B i).center).target ∩ |
| (extChartAt 𝓘(ℝ,Plane) (B i).center).symm ⁻¹' frontier Kbase)=0) → |
| ∀ (D : AnnularHandleData Plane M) |
| {b δ : ℝ}, 0<b → b<D.width → 0<δ → δ<1/2 → |
| closure D.chart.target⊆interior Kbase → tsupport D.dual⊆interior Kbase → |
| PositivePartialChart D.densityChart → |
| D.clock=positiveCircleClock δ → |
| ∀ {Λ : ManifoldTwoForm Plane M}, SmoothTwoForm Λ → |
| (∀ x u v,Λ x u v= -Λ x v u) → |
| (∀ c y,y∈(extChartAt 𝓘(ℝ,Plane) c).target → 0<chartTwoForm Λ c y (1,0) (0,1)) → |
| ∀ {d : Moments m × M → ℝ}, |
| ContMDiff (𝓘(ℝ,Moments m).prod 𝓘(ℝ,Plane)) 𝓘(ℝ,ℝ) ∞ d → |
| (∀ p∈region bF cF,∀ x∈Kbase,0<d (p,x)) → |
| ∀ {Γ₀ : Moments m → ManifoldOneForm Plane M}, SmoothOneFormFamily Γ₀ → |
| (∀ p x,manifoldExteriorOneForm (Γ₀ p) x=d (p,x) • Λ x) → |
| (∀ p∈region bF cF,|restrictedPartitionFormMass B π Kbase (fun x => d (p,x) • Λ x)-A p|<ε) → |
| ∃ φ : Fin N → Ambient (m+1) → M × PlanePhase (Fin m), |
| (∀ i,FormNeighborhoodEmbedding (closedBall (m+1) (r' i)) (fun _ => successorStandardForm m) |
| (globalHorizontalCoupling phaseArea (Γ₀ ∘ planeMoments)) (φ i)) ∧ |
| (∀ i,MapsTo (φ i) (closedBall (m+1) (r' i)) (interior (Kbase ×ˢ planeRegion bF cF))) ∧ |
| Pairwise (fun i j => Disjoint (φ i '' closedBall (m+1) (r' i)) (φ j '' closedBall (m+1) (r' j))) := by |
| obtain ⟨K,Φ,g,ε,h,hK,hg,hε,_,hAε,hh,hhcap,_,hmem,hfaceΦ,hres⟩ := |
| finite_surface_comparison bF cF hbF hcF hcomp A hA hAp r hconc hmean hτ hτ1 hd₀ houter |
| obtain ⟨α,hαc,hzero⟩ := face_collars_of_comparison bF cF Φ hfaceΦ |
| refine ⟨ε,hε,?_⟩ |
| intro I _ Kbase hD hcD hne hor B π hπ hnull D b δ hb hbw hδ hsmall heD hβD heor hclock |
| Λ hΛ hsΛ hpΛ d hd hdp Γ₀ hΓ₀ hdΓ₀ herr |
| have hSm : ∀ i,∀ p : Moments m,(∀ j,0≤p j) → (∑ j,p j)≤r' i → p∈strictRegion bF cF := by |
| intro i p hp hsum |
| have hrp : (∑ j,p j)≤r i := hsum.trans (hrr i).le |
| exact ⟨hp,fun ν => lt_of_le_of_ne ((hsimplex i p hp hrp).2 ν) (hface i p hp hrp ν)⟩ |
| exact exists_surface_initial_packing hD hcD hne hor B π hπ hnull D hb hbw hδ hsmall heD hβD heor hclock |
| hΛ hsΛ hpΛ hd hcomp hdp hΓ₀ hdΓ₀ hε (fun p hp => by have := hAε p hp; linarith) herr |
| hK Φ (strictRegion_phase_open bF cF) (strictRegion_subset bF cF) |
| (fun t _ => comparison_preserves_region bF cF Φ hmem t) h hh |
| (fun i p _ => (le_max_right _ _).trans (hhcap i p).le) |
| (fun H hH v hv => (by linarith : (0:ℝ)<g/2).le.trans ((hres H hH).2 v hv).le) |
| r r' hr' hrr hSm (fun i p _ _ => (le_max_left _ _).trans (hhcap i p).le) |
| (fun ν p => ∑ j,bF ν j*p j) (fun ν => momentFace_smooth (bF ν)) α cF hαc |
| (planeRegion_outer_constraints bF cF) (planeRegion_isCompact bF cF hcomp) hzero |
|
|
| theorem intrinsic_surface_packing_tolerance {m q N : ℕ} [Nonempty (Fin N)] |
| (bF : Fin q → Moments m) (cF : Fin q → ℝ) |
| (hbF : ∀ ν j,0≤bF ν j) (hcF : ∀ ν,0<cF ν) (hcomp : IsCompact (region bF cF)) |
| (A : Moments m → ℝ) (hA : ContinuousOn A (region bF cF)) |
| (hAp : ∀ p∈region bF cF,0<A p) (r r' : Fin N → ℝ) |
| (hr' : ∀ i,0≤r' i) (hrr : ∀ i,r' i<r i) |
| (hsimplex : ∀ i,∀ p : Moments m,(∀ j,0≤p j) → (∑ j,p j)≤r i → p∈region bF cF) |
| (hface : ∀ i,∀ p : Moments m,(∀ j,0≤p j) → (∑ j,p j)≤r i → |
| ∀ ν,(∑ j,bF ν j*p j)≠cF ν) |
| (hconc : ConcaveOn ℝ (region bF cF) (fun p => A p-∑ i,max (r i-∑ j,p j) 0)) |
| (hmean : 0<∫ p in region bF cF,A p-∑ i,max (r i-∑ j,p j) 0) |
| {τ d₀ : ℝ} (hτ : 0<τ) (hτ1 : τ<1) (hd₀ : 0<d₀) |
| (houter : ∀ p∈region bF cF,p∉τ • region bF cF → d₀≤A p-∑ i,max (r i-∑ j,p j) 0) : |
| ∃ ε>0,∀ (Kbase : Set M) (hD : IsCompact Kbase) (_hcD : IsPreconnected (interior Kbase)) |
| (_hne : (interior Kbase).Nonempty) (_hor : PositivePlaneTransitions M) |
| (_hb : HasSmoothSurfaceBoundary Kbase), |
| ∀ (D : AnnularHandleData Plane M) |
| {b δ : ℝ}, 0<b → b<D.width → 0<δ → δ<1/2 → |
| closure D.chart.target⊆interior Kbase → tsupport D.dual⊆interior Kbase → |
| PositivePartialChart D.densityChart → |
| D.clock=positiveCircleClock δ → |
| ∀ {Λ : ManifoldTwoForm Plane M}, SmoothTwoForm Λ → |
| (∀ x u v,Λ x u v= -Λ x v u) → |
| (∀ c y,y∈(extChartAt 𝓘(ℝ,Plane) c).target → 0<chartTwoForm Λ c y (1,0) (0,1)) → |
| ∀ {d : Moments m × M → ℝ}, |
| ContMDiff (𝓘(ℝ,Moments m).prod 𝓘(ℝ,Plane)) 𝓘(ℝ,ℝ) ∞ d → |
| (∀ p∈region bF cF,∀ x∈Kbase,0<d (p,x)) → |
| ∀ {Γ₀ : Moments m → ManifoldOneForm Plane M}, SmoothOneFormFamily Γ₀ → |
| (∀ p x,manifoldExteriorOneForm (Γ₀ p) x=d (p,x) • Λ x) → |
| (∀ p∈region bF cF,|compactSurfaceFormIntegral hD (fun x => d (p,x) • Λ x)-A p|<ε) → |
| ∃ φ : Fin N → Ambient (m+1) → M × PlanePhase (Fin m), |
| (∀ i,FormNeighborhoodEmbedding (closedBall (m+1) (r' i)) (fun _ => successorStandardForm m) |
| (globalHorizontalCoupling phaseArea (Γ₀ ∘ planeMoments)) (φ i)) ∧ |
| (∀ i,MapsTo (φ i) (closedBall (m+1) (r' i)) (interior (Kbase ×ˢ planeRegion bF cF))) ∧ |
| Pairwise (fun i j => Disjoint (φ i '' closedBall (m+1) (r' i)) (φ j '' closedBall (m+1) (r' j))) := by |
| obtain ⟨ε,hε,hpack⟩ := surface_packing_tolerance (M := M) bF cF hbF hcF hcomp A hA hAp r r' hr' hrr |
| hsimplex hface hconc hmean hτ hτ1 hd₀ houter |
| refine ⟨ε,hε,?_⟩ |
| intro Kbase hD hcD hne hor hb D b δ hbp hbw hδ hsmall heD hβD heor hclock |
| Λ hΛ hsΛ hpΛ d hd hdp Γ₀ hΓ₀ hdΓ₀ herr |
| exact hpack Kbase hD hcD hne hor |
| (fun B : compactSurfaceBoxes hD => B.1) (compactSurfacePartition hD) |
| (compactSurfacePartition_subordinate hD) |
| (fun B => smooth_surface_boundary_chart_null hD hb B.1.center) |
| D hbp hbw hδ hsmall heD hβD heor hclock |
| hΛ hsΛ hpΛ hd hdp hΓ₀ hdΓ₀ herr |
|
|
| theorem exhausted_surface_packing {m q N : ℕ} [Nonempty (Fin N)] |
| (bF : Fin q → Moments m) (cF : Fin q → ℝ) |
| (hbF : ∀ ν j,0≤bF ν j) (hcF : ∀ ν,0<cF ν) (hcomp : IsCompact (region bF cF)) |
| (A : Moments m → ℝ) (hA : ContinuousOn A (region bF cF)) |
| (hAp : ∀ p∈region bF cF,0<A p) (r r' : Fin N → ℝ) |
| (hr' : ∀ i,0≤r' i) (hrr : ∀ i,r' i<r i) |
| (hsimplex : ∀ i,∀ p : Moments m,(∀ j,0≤p j) → (∑ j,p j)≤r i → p∈region bF cF) |
| (hface : ∀ i,∀ p : Moments m,(∀ j,0≤p j) → (∑ j,p j)≤r i → |
| ∀ ν,(∑ j,bF ν j*p j)≠cF ν) |
| (hconc : ConcaveOn ℝ (region bF cF) (fun p => A p-∑ i,max (r i-∑ j,p j) 0)) |
| (hmean : 0<∫ p in region bF cF,A p-∑ i,max (r i-∑ j,p j) 0) |
| {τ d₀ : ℝ} (hτ : 0<τ) (hτ1 : τ<1) (hd₀ : 0<d₀) |
| (houter : ∀ p∈region bF cF,p∉τ • region bF cF → d₀≤A p-∑ i,max (r i-∑ j,p j) 0) |
| (Kbase : ℕ → Set M) (hD : ∀ ℓ,IsCompact (Kbase ℓ)) |
| (hmono : ∀ ℓ,Kbase ℓ⊆interior (Kbase (ℓ+1))) (hcover : (⋃ ℓ,Kbase ℓ)=univ) |
| (hcD : ∀ ℓ,IsPreconnected (interior (Kbase ℓ))) (hne : ∀ ℓ,(interior (Kbase ℓ)).Nonempty) |
| (hb : ∀ ℓ,HasSmoothSurfaceBoundary (Kbase ℓ)) (hor : PositivePlaneTransitions M) |
| (D : AnnularHandleData Plane M) |
| {b δ : ℝ} (hbp : 0<b) (hbw : b<D.width) (hδ : 0<δ) (hsmall : δ<1/2) |
| (heor : PositivePartialChart D.densityChart) |
| (hclock : D.clock=positiveCircleClock δ) |
| {Λ : ManifoldTwoForm Plane M} (hΛ : SmoothTwoForm Λ) |
| (hsΛ : ∀ x u v,Λ x u v= -Λ x v u) |
| (hpΛ : ∀ c y,y∈(extChartAt 𝓘(ℝ,Plane) c).target → 0<chartTwoForm Λ c y (1,0) (0,1)) |
| (d : ℕ → Moments m × M → ℝ) |
| (hd : ∀ ℓ,ContMDiff (𝓘(ℝ,Moments m).prod 𝓘(ℝ,Plane)) 𝓘(ℝ,ℝ) ∞ (d ℓ)) |
| (hdp : ∀ ℓ,∀ p∈region bF cF,∀ x∈Kbase ℓ,0<d ℓ (p,x)) |
| (Γ : ℕ → Moments m → ManifoldOneForm Plane M) (hΓ : ∀ ℓ,SmoothOneFormFamily (Γ ℓ)) |
| (hdΓ : ∀ ℓ p x,manifoldExteriorOneForm (Γ ℓ p) x=d ℓ (p,x) • Λ x) |
| (hlim : TendstoUniformlyOn (fun ℓ p => compactSurfaceFormIntegral (hD ℓ) |
| (fun x => d ℓ (p,x) • Λ x)) A Filter.atTop (region bF cF)) : |
| ∃ ℓ,∃ φ : Fin N → Ambient (m+1) → M × PlanePhase (Fin m), |
| (∀ i,FormNeighborhoodEmbedding (closedBall (m+1) (r' i)) (fun _ => successorStandardForm m) |
| (globalHorizontalCoupling phaseArea (Γ ℓ ∘ planeMoments)) (φ i)) ∧ |
| (∀ i,MapsTo (φ i) (closedBall (m+1) (r' i)) (interior (Kbase ℓ ×ˢ planeRegion bF cF))) ∧ |
| Pairwise (fun i j => Disjoint (φ i '' closedBall (m+1) (r' i)) (φ j '' closedBall (m+1) (r' j))) := by |
| obtain ⟨ε,hε,hpack⟩ := intrinsic_surface_packing_tolerance (M := M) bF cF hbF hcF hcomp |
| A hA hAp r r' hr' hrr hsimplex hface hconc hmean hτ hτ1 hd₀ houter |
| obtain ⟨L,hL⟩ := finite_uniform_area_error hlim hε |
| have hmo : Monotone Kbase := monotone_nat_of_le_succ (fun ℓ => (hmono ℓ).trans interior_subset) |
| have hco : (closure D.chart.target ∪ tsupport D.dual)⊆⋃ ℓ,interior (Kbase ℓ) := by |
| intro x _ |
| obtain ⟨ℓ,hx⟩ := mem_iUnion.mp (show x∈⋃ ℓ,Kbase ℓ by rw [hcover]; trivial) |
| exact mem_iUnion.mpr ⟨ℓ+1,hmono ℓ hx⟩ |
| obtain ⟨J,hJ⟩ := (D.compact_target.union D.dual_compact).elim_directed_cover (fun ℓ => interior (Kbase ℓ)) (fun _ => isOpen_interior) |
| hco (fun i j => ⟨max i j,interior_mono (hmo (le_max_left i j)), |
| interior_mono (hmo (le_max_right i j))⟩) |
| refine ⟨max L J,?_⟩ |
| exact hpack (Kbase (max L J)) (hD _) (hcD _) (hne _) hor (hb _) D hbp hbw hδ hsmall |
| ((subset_union_left).trans (hJ.trans (interior_mono (hmo (le_max_right L J))))) |
| ((subset_union_right).trans (hJ.trans (interior_mono (hmo (le_max_right L J))))) heor hclock |
| hΛ hsΛ hpΛ (hd _) (hdp _) (hΓ _) (hdΓ _) (hL _ (le_max_left L J)) |
|
|
| end PackingSufficiencySupport.MomentPolytope.Annular |
|
|
| namespace PackingSufficiencySupport.Hamiltonian |
| open scoped ContDiff Manifold Topology |
| open Set Function Manifold |
|
|
| variable {M : Type*} [TopologicalSpace M] [ChartedSpace Plane M] |
| [IsManifold 𝓘(ℝ,Plane) ∞ M] |
|
|
| def surfaceFormRatio (Λ Ω : ManifoldTwoForm Plane M) (x : M) : ℝ := |
| chartTwoForm Ω x (extChartAt 𝓘(ℝ,Plane) x x) (1,0) (0,1) / |
| chartTwoForm Λ x (extChartAt 𝓘(ℝ,Plane) x x) (1,0) (0,1) |
|
|
| theorem surface_form_scalar_in_chart {Λ Ω : ManifoldTwoForm Plane M} |
| (hΩ : ∀ x u v,Ω x u v= -Ω x v u) (hΛ : ∀ x u v,Λ x u v= -Λ x v u) |
| {c x : M} (hx : x∈(extChartAt 𝓘(ℝ,Plane) c).source) |
| (hpos : chartTwoForm Λ c (extChartAt 𝓘(ℝ,Plane) c x) (1,0) (0,1)≠0) : |
| Ω x=(chartTwoForm Ω c (extChartAt 𝓘(ℝ,Plane) c x) (1,0) (0,1) / |
| chartTwoForm Λ c (extChartAt 𝓘(ℝ,Plane) c x) (1,0) (0,1)) • Λ x := by |
| apply ContinuousLinearMap.ext |
| intro u |
| apply ContinuousLinearMap.ext |
| intro v |
| change Ω x u v=_*Λ x u v |
| rw [← chartTwoForm_apply_chart hx u v (Ω := Ω),← chartTwoForm_apply_chart hx u v (Ω := Λ)] |
| change chartTwoForm Ω c (extChartAt 𝓘(ℝ,Plane) c x) (chartDifferential c x u) (chartDifferential c x v) = |
| (chartTwoForm Ω c (extChartAt 𝓘(ℝ,Plane) c x) (1,0) (0,1) / |
| chartTwoForm Λ c (extChartAt 𝓘(ℝ,Plane) c x) (1,0) (0,1)) * |
| chartTwoForm Λ c (extChartAt 𝓘(ℝ,Plane) c x) (chartDifferential c x u) (chartDifferential c x v) |
| rw [planar_skew_coefficient _ (chartTwoForm_skew hΩ _ _) (chartDifferential c x u) (chartDifferential c x v), planar_skew_coefficient _ (chartTwoForm_skew hΛ _ _) (chartDifferential c x u) (chartDifferential c x v)] |
| field_simp [hpos] |
|
|
| theorem surface_form_scalar {Λ Ω : ManifoldTwoForm Plane M} |
| (hΩ : ∀ x u v,Ω x u v= -Ω x v u) (hΛ : ∀ x u v,Λ x u v= -Λ x v u) |
| (hpos : ∀ c y,y∈(extChartAt 𝓘(ℝ,Plane) c).target → |
| 0<chartTwoForm Λ c y (1,0) (0,1)) (x : M) : |
| Ω x=surfaceFormRatio Λ Ω x • Λ x := |
| surface_form_scalar_in_chart hΩ hΛ (mem_extChartAt_source x) |
| (ne_of_gt (hpos x _ (mem_extChartAt_target x))) |
|
|
| theorem surfaceFormRatio_eq_chart {Λ Ω : ManifoldTwoForm Plane M} |
| (hΩ : ∀ x u v,Ω x u v= -Ω x v u) (hΛ : ∀ x u v,Λ x u v= -Λ x v u) |
| (hpos : ∀ c y,y∈(extChartAt 𝓘(ℝ,Plane) c).target → |
| 0<chartTwoForm Λ c y (1,0) (0,1)) {c x : M} |
| (hx : x∈(extChartAt 𝓘(ℝ,Plane) c).source) : |
| surfaceFormRatio Λ Ω x=chartTwoForm Ω c (extChartAt 𝓘(ℝ,Plane) c x) (1,0) (0,1) / |
| chartTwoForm Λ c (extChartAt 𝓘(ℝ,Plane) c x) (1,0) (0,1) := by |
| have he : Ω=fun z => surfaceFormRatio Λ Ω z • Λ z := |
| funext (surface_form_scalar hΩ hΛ hpos) |
| conv_rhs => rw [he,chartTwoForm_spatial_smul] |
| simp only [smul_apply,smul_eq_mul,(extChartAt 𝓘(ℝ,Plane) c).left_inv hx] |
| exact (mul_div_cancel_right₀ _ (ne_of_gt (hpos c _ ((extChartAt 𝓘(ℝ,Plane) c).map_source hx)))).symm |
|
|
| variable {P : Type*} [NormedAddCommGroup P] [NormedSpace ℝ P] |
| theorem surfaceFormRatio_smooth {Λ : ManifoldTwoForm Plane M} |
| (hΛ : SmoothTwoForm Λ) (hsΛ : ∀ x u v,Λ x u v= -Λ x v u) |
| (hpΛ : ∀ c y,y∈(extChartAt 𝓘(ℝ,Plane) c).target → 0<chartTwoForm Λ c y (1,0) (0,1)) |
| {Ω : P → ManifoldTwoForm Plane M} (hΩ : SmoothTwoFormFamily Ω) |
| (hsΩ : ∀ p x u v,Ω p x u v= -Ω p x v u) : |
| ContMDiff (𝓘(ℝ,P).prod 𝓘(ℝ,Plane)) 𝓘(ℝ,ℝ) ∞ |
| (fun q : P × M => surfaceFormRatio Λ (Ω q.1) q.2) := by |
| intro q |
| let c := q.2 |
| let y := extChartAt 𝓘(ℝ,Plane) c c |
| have hy : y∈(extChartAt 𝓘(ℝ,Plane) c).target := mem_extChartAt_target c |
| have hn := (isOpen_univ.prod (isOpen_extChartAt_target (I := 𝓘(ℝ,Plane)) c)).mem_nhds |
| (show (q.1,y)∈univ ×ˢ (extChartAt 𝓘(ℝ,Plane) c).target from ⟨mem_univ _,hy⟩) |
| have hnum : ContDiffAt ℝ ∞ (fun z : P × Plane => chartTwoForm (Ω z.1) c z.2 (1,0) (0,1)) (q.1,y) := |
| (((hΩ c).contDiffAt hn).clm_apply contDiffAt_const).clm_apply contDiffAt_const |
| have hden : ContDiffAt ℝ ∞ (fun z : P × Plane => chartTwoForm Λ c z.2 (1,0) (0,1)) (q.1,y) := |
| ((((SmoothTwoFormFamily.const (P := P) hΛ) c).contDiffAt hn).clm_apply |
| contDiffAt_const).clm_apply contDiffAt_const |
| have hrat : ContDiffAt ℝ ∞ (fun z : P × Plane => |
| chartTwoForm (Ω z.1) c z.2 (1,0) (0,1)/chartTwoForm Λ c z.2 (1,0) (0,1)) (q.1,y) := |
| hnum.div hden (ne_of_gt (hpΛ c y hy)) |
| have hc : ContMDiffAt 𝓘(ℝ,Plane) 𝓘(ℝ,Plane) ∞ (extChartAt 𝓘(ℝ,Plane) c) c := |
| contMDiffAt_extChartAt |
| have hin : ContMDiffAt (𝓘(ℝ,P).prod 𝓘(ℝ,Plane)) 𝓘(ℝ,P × Plane) ∞ |
| (fun z : P × M => (z.1,extChartAt 𝓘(ℝ,Plane) c z.2)) q := |
| contMDiffAt_fst.prodMk_space (hc.comp q contMDiffAt_snd) |
| have hlocal := hrat.contMDiffAt.comp q hin |
| apply hlocal.congr_of_eventuallyEq |
| have hnsource : ∀ᶠ z : P × M in 𝓝 q,z.2∈(extChartAt 𝓘(ℝ,Plane) c).source := |
| continuousAt_snd.preimage_mem_nhds (extChartAt_source_mem_nhds c) |
| filter_upwards [hnsource] with z hz |
| exact surfaceFormRatio_eq_chart (hsΩ z.1) hsΛ hpΛ hz |
|
|
| end PackingSufficiencySupport.Hamiltonian |
|
|
| namespace PackingSufficiencySupport.MomentPolytope.Annular |
| open scoped ContDiff Manifold Topology BigOperators Pointwise |
| open Set Function Manifold MeasureTheory |
| open Hamiltonian |
|
|
| variable {M : Type} [TopologicalSpace M] [ChartedSpace Plane M] |
| [IsManifold 𝓘(ℝ,Plane) ∞ M] [T2Space M] [NormalSpace M] [SigmaCompactSpace M] |
| [TopologicalSpace.PseudoMetrizableSpace M] [MeasurableSpace M] [BorelSpace M] |
|
|
| theorem exhausted_curvature_surface_packing {m q N : ℕ} [Nonempty (Fin N)] |
| (bF : Fin q → Moments m) (cF : Fin q → ℝ) |
| (hbF : ∀ ν j,0≤bF ν j) (hcF : ∀ ν,0<cF ν) (hcomp : IsCompact (region bF cF)) |
| (A : Moments m → ℝ) (hA : ContinuousOn A (region bF cF)) |
| (hAp : ∀ p∈region bF cF,0<A p) (r r' : Fin N → ℝ) |
| (hr' : ∀ i,0≤r' i) (hrr : ∀ i,r' i<r i) |
| (hsimplex : ∀ i,∀ p : Moments m,(∀ j,0≤p j) → (∑ j,p j)≤r i → p∈region bF cF) |
| (hface : ∀ i,∀ p : Moments m,(∀ j,0≤p j) → (∑ j,p j)≤r i → |
| ∀ ν,(∑ j,bF ν j*p j)≠cF ν) |
| (hconc : ConcaveOn ℝ (region bF cF) (fun p => A p-∑ i,max (r i-∑ j,p j) 0)) |
| (hmean : 0<∫ p in region bF cF,A p-∑ i,max (r i-∑ j,p j) 0) |
| {τ d₀ : ℝ} (hτ : 0<τ) (hτ1 : τ<1) (hd₀ : 0<d₀) |
| (houter : ∀ p∈region bF cF,p∉τ • region bF cF → d₀≤A p-∑ i,max (r i-∑ j,p j) 0) |
| (D : ℕ → Set M) (hD : ∀ ℓ,IsCompact (D ℓ)) |
| (hmono : ∀ ℓ,D ℓ⊆interior (D (ℓ+1))) (hcover : (⋃ ℓ,D ℓ)=univ) |
| (hcD : ∀ ℓ,IsPreconnected (interior (D ℓ))) (hne : ∀ ℓ,(interior (D ℓ)).Nonempty) |
| (hb : ∀ ℓ,HasSmoothSurfaceBoundary (D ℓ)) (hor : PositivePlaneTransitions M) |
| (H : AnnularHandleData Plane M) |
| {b δ : ℝ} (hbp : 0<b) (hbw : b<H.width) (hδ : 0<δ) (hsmall : δ<1/2) |
| (heor : PositivePartialChart H.densityChart) |
| (hclock : H.clock=positiveCircleClock δ) |
| {Λ : ManifoldTwoForm Plane M} (hΛ : SmoothTwoForm Λ) |
| (hsΛ : ∀ x u v,Λ x u v= -Λ x v u) |
| (hpΛ : ∀ c y,y∈(extChartAt 𝓘(ℝ,Plane) c).target → 0<chartTwoForm Λ c y (1,0) (0,1)) |
| (Γ : ℕ → Moments m → ManifoldOneForm Plane M) (hΓ : ∀ ℓ,SmoothOneFormFamily (Γ ℓ)) |
| (hpos : ∀ ℓ,∀ p∈region bF cF,∀ c y,y∈(extChartAt 𝓘(ℝ,Plane) c).target → |
| (extChartAt 𝓘(ℝ,Plane) c).symm y∈D ℓ → |
| 0<chartTwoForm (manifoldExteriorOneForm (Γ ℓ p)) c y (1,0) (0,1)) |
| (hlim : TendstoUniformlyOn (fun ℓ p => compactSurfaceFormIntegral (hD ℓ) |
| (manifoldExteriorOneForm (Γ ℓ p))) A Filter.atTop (region bF cF)) : |
| ∃ ℓ,∃ φ : Fin N → Ambient (m+1) → M × PlanePhase (Fin m), |
| (∀ i,FormNeighborhoodEmbedding (closedBall (m+1) (r' i)) (fun _ => successorStandardForm m) |
| (globalHorizontalCoupling phaseArea (Γ ℓ ∘ planeMoments)) (φ i)) ∧ |
| (∀ i,MapsTo (φ i) (closedBall (m+1) (r' i)) (interior (D ℓ ×ˢ planeRegion bF cF))) ∧ |
| Pairwise (fun i j => Disjoint (φ i '' closedBall (m+1) (r' i)) (φ j '' closedBall (m+1) (r' j))) := by |
| let κ : ℕ → Moments m → ManifoldTwoForm Plane M := fun ℓ p => manifoldExteriorOneForm (Γ ℓ p) |
| have hκ ℓ : SmoothTwoFormFamily (κ ℓ) := manifoldExteriorOneForm_family_smooth (hΓ ℓ) |
| have hsκ ℓ p x u v : κ ℓ p x u v= -κ ℓ p x v u := manifoldExteriorOneForm_skew _ _ _ _ |
| let d : ℕ → Moments m × M → ℝ := fun ℓ q => surfaceFormRatio Λ (κ ℓ q.1) q.2 |
| have hd ℓ : ContMDiff (𝓘(ℝ,Moments m).prod 𝓘(ℝ,Plane)) 𝓘(ℝ,ℝ) ∞ (d ℓ) := |
| surfaceFormRatio_smooth hΛ hsΛ hpΛ (hκ ℓ) (hsκ ℓ) |
| have heq ℓ p x : κ ℓ p x=d ℓ (p,x) • Λ x := surface_form_scalar (hsκ ℓ p) hsΛ hpΛ x |
| have hdp ℓ p (hp : p∈region bF cF) x (hx : x∈D ℓ) : 0<d ℓ (p,x) := |
| div_pos (hpos ℓ p hp x _ (mem_extChartAt_target x) (by simpa using hx)) |
| (hpΛ x _ (mem_extChartAt_target x)) |
| have hlim' : TendstoUniformlyOn (fun ℓ p => compactSurfaceFormIntegral (hD ℓ) |
| (fun x => d ℓ (p,x) • Λ x)) A Filter.atTop (region bF cF) := by |
| convert hlim using 1 |
| exact funext (fun ℓ => funext (fun p => congrArg (compactSurfaceFormIntegral (hD ℓ)) |
| (funext (fun x => (heq ℓ p x).symm)))) |
| exact exhausted_surface_packing bF cF hbF hcF hcomp A hA hAp r r' hr' hrr hsimplex hface |
| hconc hmean hτ hτ1 hd₀ houter D hD hmono hcover hcD hne hb hor H hbp hbw hδ hsmall |
| heor hclock hΛ hsΛ hpΛ d hd hdp Γ hΓ heq hlim' |
|
|
| end PackingSufficiencySupport.MomentPolytope.Annular |
| end |
|
|
| end OAI |
|
|
