| import Mathlib | |
| import OAI.Analysis.BiholderTransport.Convexity.ActualCenterSemibound | |
| import OAI.Analysis.BiholderTransport.Regularity.CenterSequenceCompact | |
| import OAI.Analysis.BiholderTransport.Regularity.CenterSequenceRegular | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set Filter Manifold Bundle | |
| open scoped Topology ContDiff NNReal | |
| namespace WeakMTWTransport | |
| section CenterSequenceRegular | |
| 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] | |
| lemma CenterSequenceData.regular_compact {a c:M} {u v:M → ℝ} {Φ:ℝ×ℝ → ℝ} {γ l:ℝ} | |
| (hv:Continuous v) (hΦ:ContDiffAt ℝ ∞ Φ (γ,v c)) | |
| (hder:deriv (fun s=>Φ (γ,s)) (v c)=l) (hl:0 < l) (hl1:l < 1) | |
| {γj lj τj tj:ℕ → ℝ} {xj bj qj:ℕ → Model n} {zj:ℕ → M} | |
| {F:ℕ → Model n → ℝ} | |
| (D:CenterSequenceData a c u v Φ γj lj τj tj xj bj qj zj F) | |
| {q:Model n} | |
| (hq:(show TangentSpace 𝓘(ℝ,Model n) a from q)∈minimizingVectors a) | |
| (he:riemannianExp a q=c) | |
| (hγ:Tendsto γj atTop (𝓝 γ)) (hlm:Tendsto lj atTop (𝓝 l)) | |
| (hτ:Tendsto τj atTop (𝓝 0)) (ht:Tendsto tj atTop (𝓝 1)) | |
| (hx:Tendsto xj atTop (𝓝 (extChartAt 𝓘(ℝ,Model n) c c))) | |
| (hb:Tendsto bj atTop (𝓝 (extChartAt 𝓘(ℝ,Model n) a a))) | |
| (hτpos:∀ᶠ i in atTop,0 < τj i) (L:ℝ≥0) | |
| (hLip:∀ᶠ i in atTop,LipschitzWith L (fun y=>Φ (γj i,v y))) : | |
| (show TangentSpace 𝓘(ℝ,Model n) a from q)∈injectivityDomain a := by | |
| obtain ⟨σ,hσ,r,hr,hrr⟩:=center_support_family_subseq D.row hx | |
| exact (D.comp σ hσ.tendsto_atTop).regular hv hΦ hder hl hl1 hr hq he | |
| (hγ.comp hσ.tendsto_atTop) (hlm.comp hσ.tendsto_atTop) | |
| (hτ.comp hσ.tendsto_atTop) (ht.comp hσ.tendsto_atTop) | |
| (hx.comp hσ.tendsto_atTop) (hb.comp hσ.tendsto_atTop) hrr | |
| (hσ.tendsto_atTop.eventually hτpos) L (hσ.tendsto_atTop.eventually hLip) | |
| end CenterSequenceRegular | |
| end WeakMTWTransport | |
| end | |
| end | |
| end OAI | |