openAI-math-dataset / lean /OAI /Analysis /BiholderTransport /CostGeometry /UniformRegularSplit.lean
| import Mathlib | |
| import OAI.Analysis.BiholderTransport.Coordinates.TotalInjectivity | |
| import OAI.Analysis.BiholderTransport.Coordinates.CompactEventuallyFiberwise | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter Manifold Bundle | |
| open scoped Topology ContDiff | |
| namespace WeakMTWTransport | |
| variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace 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 uniformly_shifted_injectivityDomain_near_one (R : ℝ) : | |
| ∀ᶠ h in 𝓝 (1:ℝ), ∀ x : M, ∀ p : TangentSpace 𝓘(ℝ,Model n) x, | |
| ‖p‖ ≤ R → (1-h) • (sprayFlow h (⟨x,p⟩ : TangentBundle 𝓘(ℝ,Model n) M)).2 ∈ | |
| injectivityDomain (sprayFlow h (⟨x,p⟩ : TangentBundle 𝓘(ℝ,Model n) M)).1 := by | |
| have : IsContinuousRiemannianBundle (Model n) | |
| (fun x : M => TangentSpace 𝓘(ℝ,Model n) x) := | |
| continuousRiemannianBundle_of_smooth (IB := 𝓘(ℝ,Model n)) | |
| let Z : ℝ × TangentBundle 𝓘(ℝ,Model n) M → TangentBundle 𝓘(ℝ,Model n) M := | |
| fun q => tangentScale (1-q.1) (sprayFlow q.1 q.2) | |
| have hZ : ContMDiff (𝓘(ℝ,ℝ).prod (𝓘(ℝ,Model n).prod 𝓘(ℝ,Model n))) | |
| (𝓘(ℝ,Model n).prod 𝓘(ℝ,Model n)) ∞ Z := | |
| contMDiff_tangentScale.comp ((contMDiff_const.sub contMDiff_fst).prodMk contMDiff_sprayFlow) | |
| have H := compact_eventually_fiberwise | |
| (isCompact_bundle_disk (B := M) (F := Model n) | |
| (E := fun x => TangentSpace 𝓘(ℝ,Model n) x) R) | |
| (f := fun _ => ()) continuous_const (g := fun _ : ℝ => ()) | |
| (continuousAt_const (x := 1)) (P := fun h p => (Z (h,p)).2∈injectivityDomain (Z (h,p)).1) | |
| (by | |
| intro p _ _ | |
| change ∀ᶠ q in 𝓝 (1,p), Z q ∈ {z : TangentBundle 𝓘(ℝ,Model n) M | z.2∈injectivityDomain z.1} | |
| apply hZ.continuous.continuousAt.eventually | |
| apply isOpen_total_injectivityDomain.mem_nhds | |
| change (1-(1:ℝ)) • (sprayFlow 1 p).2∈injectivityDomain (sprayFlow 1 p).1 | |
| rw [sub_self,zero_smul] | |
| exact zero_mem_injectivityDomain _) | |
| exact H.mono fun _ hshift base velocity hbound => | |
| hshift ⟨base,velocity⟩ hbound rfl | |
| lemma exists_uniform_regular_split (R : ℝ) : | |
| ∃ h : ℝ, 0 < h ∧ h < 1 ∧ ∀ x : M, | |
| ∀ p : TangentSpace 𝓘(ℝ,Model n) x, ‖p‖ ≤ R → p∈minimizingVectors x → | |
| h • p∈injectivityDomain x ∧ | |
| (1-h) • (sprayFlow h (⟨x,p⟩ : TangentBundle 𝓘(ℝ,Model n) M)).2 ∈ | |
| injectivityDomain (sprayFlow h (⟨x,p⟩ : TangentBundle 𝓘(ℝ,Model n) M)).1 := by | |
| have H : ∀ᶠ h in 𝓝[<] (1:ℝ), 0 < h ∧ h < 1 ∧ ∀ x : M, | |
| ∀ p : TangentSpace 𝓘(ℝ,Model n) x, ‖p‖ ≤ R → | |
| (1-h) • (sprayFlow h (⟨x,p⟩ : TangentBundle 𝓘(ℝ,Model n) M)).2 ∈ | |
| injectivityDomain (sprayFlow h (⟨x,p⟩ : TangentBundle 𝓘(ℝ,Model n) M)).1 := by | |
| filter_upwards [(uniformly_shifted_injectivityDomain_near_one (n := n) (M := M) R).filter_mono nhdsWithin_le_nhds, | |
| (eventually_gt_nhds (show (0:ℝ)<1 by norm_num)).filter_mono nhdsWithin_le_nhds, | |
| self_mem_nhdsWithin] with h hh h0 h1 | |
| exact ⟨h0,h1,hh⟩ | |
| obtain ⟨h,h0,h1,hh⟩ := H.exists | |
| exact ⟨h,h0,h1,fun x p hp hm => | |
| ⟨contracted_minimizer_mem_injectivityDomain hm h0 h1,hh x p hp⟩⟩ | |
| end WeakMTWTransport | |
| end | |
| end OAI | |