| import Mathlib | |
| import OAI.Analysis.BiholderTransport.Geodesics.UniformFixedCrossing | |
| import OAI.Analysis.BiholderTransport.Calculus.SamplePoleDerivative | |
| import OAI.Analysis.BiholderTransport.Regularity.NonnegativeTransfer | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter Manifold Bundle | |
| open scoped Topology ContDiff | |
| namespace WeakMTWTransport | |
| variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace M] [Nonempty M] | |
| [ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ,Model n) ∞ M] | |
| [RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] | |
| [IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n) | |
| (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] | |
| [IsRiemannianManifold 𝓘(ℝ,Model n) M] | |
| omit [Nonempty M] in | |
| lemma exists_actual_uniform_regular_split_after {t:ℝ} (ht : 0<t) (ht1 : t<1) : | |
| ∃ s:ℝ,t<s ∧ s<1 ∧ ∀ x:M,∀ p:TangentSpace 𝓘(ℝ,Model n) x, | |
| p∈minimizingVectors x → s • p∈injectivityDomain x ∧ | |
| (1-s) • (sprayFlow s (⟨x,p⟩:TangentBundle 𝓘(ℝ,Model n) M)).2∈ | |
| injectivityDomain (sprayFlow s (⟨x,p⟩:TangentBundle 𝓘(ℝ,Model n) M)).1 := by | |
| have H : ∀ᶠ s in 𝓝[<] (1:ℝ),t<s ∧ s<1 ∧ ∀ x:M, | |
| ∀ p:TangentSpace 𝓘(ℝ,Model n) x,‖p‖≤Metric.diam (univ:Set M) → | |
| (1-s) • (sprayFlow s (⟨x,p⟩:TangentBundle 𝓘(ℝ,Model n) M)).2∈ | |
| injectivityDomain (sprayFlow s (⟨x,p⟩:TangentBundle 𝓘(ℝ,Model n) M)).1 := by | |
| filter_upwards [(uniformly_shifted_injectivityDomain_near_one | |
| (n := n) (M := M) (Metric.diam (univ:Set M))).filter_mono nhdsWithin_le_nhds, | |
| (eventually_gt_nhds ht1).filter_mono nhdsWithin_le_nhds,self_mem_nhdsWithin] | |
| with s hs hst hs1 | |
| exact ⟨hst,hs1,hs⟩ | |
| obtain ⟨s,hst,hs1,hs⟩ := H.exists | |
| exact ⟨s,hst,hs1,fun x p hp=>⟨contracted_minimizer_mem_injectivityDomain hp | |
| (ht.trans hst) hs1,hs x p (minimizing_norm_le_diam hp)⟩⟩ | |
| lemma WeakMTW.actual_sample_pole_matrix (hmtw : WeakMTW (n := n) (M := M)) | |
| {v : M → ℝ} (hv : Continuous v) {t : ℝ} (ht : 0<t) (ht1 : t<1) | |
| {x a c : M} {p : TangentSpace 𝓘(ℝ,Model n) x} | |
| (hp : p∈normalSubdifferential (n := n) (cTransform v) x) | |
| (hz : riemannianExp x (t • p)∈(extChartAt 𝓘(ℝ,Model n) a).source) | |
| (hx : x∈(extChartAt 𝓘(ℝ,Model n) c).source) | |
| (hD : DifferentiableAt ℝ | |
| (fun w : Model n => extChartAt 𝓘(ℝ,Model n) c | |
| (hopfPole (n := n) t (cTransform v) ((extChartAt 𝓘(ℝ,Model n) a).symm w))) | |
| (extChartAt 𝓘(ℝ,Model n) a (riemannianExp x (t • p)))) : | |
| let V := TangentSpace 𝓘(ℝ,Model n) x | |
| let I : V →L[ℝ] V →L[ℝ] ℝ := innerSL ℝ | |
| let L : V →L[ℝ] V →L[ℝ] ℝ := t • I - fderiv ℝ (fderiv ℝ | |
| (fun q:V=>hopfLax t (cTransform v) (riemannianExp x (t • q)))) p | |
| DifferentiableAt ℝ (fderiv ℝ | |
| (fun q:V=>hopfLax t (cTransform v) (riemannianExp x (t • q)))) p ∧ | |
| (∀ d:V,0≤L d d) ∧ (∀ d e:V,L d e=L e d) ∧ | |
| (∀ q∈activeLogs v x,∀ d:V,L (q-p) d=0) ∧ | |
| (∀ (ι:Type) [Fintype ι] (pj:ι → V) (w:ι → ℝ), | |
| (∀ j,0≤w j) → (∑ j,w j=1) → (∑ j,w j • pj j=p) → | |
| (∀ j,pj j∈activeLogs v x) → ∀ i,0<w i → | |
| HasLowerSecondTaylor (fun d:V=>v (riemannianExp x (pj i+d))+‖pj i+d‖^2/2) | |
| 0 (w i • L)) := by | |
| dsimp only | |
| let V := TangentSpace 𝓘(ℝ,Model n) x | |
| let : NormedAddCommGroup (V →L[ℝ] ℝ) := ContinuousLinearMap.toNormedAddCommGroup | |
| let : NormedSpace ℝ (V →L[ℝ] ℝ) := ContinuousLinearMap.toNormedSpace | |
| let : NormedAddCommGroup (V →L[ℝ] V →L[ℝ] ℝ) := ContinuousLinearMap.toNormedAddCommGroup | |
| let q : subgradientGraph (n := n) (cTransform v) := ⟨⟨x,p⟩,hp⟩ | |
| have hpole : hopfPole (n := n) t (cTransform v) (riemannianExp x (t • p))=x := by | |
| exact congrArg Bundle.TotalSpace.proj (hmtw.hopfLax_backward_flow hv ht ht1 q) | |
| obtain ⟨e,R,he,he0,herep⟩ := hmtw.exists_actual_normal_pole_derivative hv ht ht1 hz hx hpole hD | |
| have hpre := hmtw.active_hull_precut hv ht ht1 x p | |
| (normalSubdifferential_subset_active_hull hv x hp) | |
| have hdual : IsCostDualPair (cTransform v) (cTransform (cTransform v)) := | |
| ⟨(cTransform_triple hv).symm,rfl⟩ | |
| have HD' := hmtw.actualPole_second_derivative (continuous_cTransform hv) | |
| (continuous_cTransform (continuous_cTransform hv)) hdual ht ht1 hpre he he0 herep | |
| have HD := HD'.fderiv | |
| rw [HD,sub_sub_cancel] | |
| have H := hmtw.actualPole_taylor (continuous_cTransform hv) | |
| (continuous_cTransform (continuous_cTransform hv)) hdual ht ht1 hpre he he0 herep | |
| obtain ⟨s,hts,hs1,hs⟩ := exists_actual_uniform_regular_split_after (n := n) (M := M) ht ht1 | |
| refine ⟨HD'.differentiableAt,H.2.1,?_,?_,?_⟩ | |
| · intro d f | |
| change inner ℝ (R d) f=inner ℝ (R f) d | |
| exact (H.2.2 d f).trans (real_inner_comm (R f) d) | |
| · intro r hr d | |
| have hk := hmtw.actualPole_kernel (continuous_cTransform hv) | |
| (continuous_cTransform (continuous_cTransform hv)) hdual ht ht1 (ht.trans hts) hs1 | |
| hpre hr.1 (active_contact_biconjugate hv hr.2).2 (hs x r hr.1).1 (hs x r hr.1).2 | |
| he he0 herep | |
| simp only [ContinuousLinearMap.comp_apply,hk,map_zero,zero_apply] | |
| · intro ι _ pj w hw hw1 hbar ha i hi | |
| exact hmtw.modified_active_lower_jet_nonneg hv ht hts hs1 pj w hw hw1 i hi | |
| (fun j=>(ha j).1) (fun j=>(ha j).2) (fun j=>(hs x (pj j) (ha j).1).1) | |
| (fun j=>(hs x (pj j) (ha j).1).2) (hbar ▸ he) (hbar ▸ he0) (hbar ▸ herep) | |
| end WeakMTWTransport | |
| end | |
| end OAI | |