| import Mathlib |
| import OAI.Analysis.BiholderTransport.Coordinates.BranchInverse |
| import OAI.Analysis.BiholderTransport.Contact.ActiveContinuationLocal |
| import OAI.Analysis.BiholderTransport.Contact.SubgradientUpper |
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| namespace OAI |
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| noncomputable section |
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| open Set MeasureTheory Manifold Bundle |
| open scoped ContDiff Manifold ENNReal NNReal Topology |
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| open Set Filter |
| open scoped Topology NNReal |
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| open Set Filter |
| open scoped Topology |
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| open Set Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set |
| open scoped Topology |
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| open Set Filter Manifold Bundle ContinuousLinearMap |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter ContinuousLinearMap InnerProductSpace |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
| open scoped NNReal |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology |
| open MeasureTheory |
| open scoped ContDiff ENNReal |
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| open Set Filter Manifold Bundle ContinuousLinearMap MeasureTheory |
| open scoped Topology ContDiff Manifold Bundle ENNReal |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology Manifold |
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| open Set Filter Manifold Bundle ContinuousLinearMap |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter Bundle |
| open scoped Topology Bundle |
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| open scoped Topology |
| open Function Manifold Set |
| open Manifold Bundle |
| open scoped Manifold Bundle |
| open Set |
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| open Set Filter |
| open scoped Topology ContDiff |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter |
| open scoped ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter |
| open scoped ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter |
| open scoped ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter |
| open scoped ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter |
| open scoped ContDiff Topology |
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| open Filter Set |
| open scoped Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
|
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
|
|
| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
|
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
|
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
|
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
|
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
|
|
| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
|
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter |
| open scoped Topology |
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| open Set Filter |
| open scoped Topology ContDiff |
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| open Set Filter |
| open scoped Topology ContDiff |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
|
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
|
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
|
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
|
|
| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
|
|
| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
|
|
| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
|
|
| open Set Filter |
| open scoped ContDiff Topology |
|
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| open Set Filter |
| open scoped Topology |
|
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
|
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
|
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| open Set Filter |
| open scoped Topology |
|
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| 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] |
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| lemma WeakMTW.local_projection_surjective (hmtw : WeakMTW (n := n) (M := M)) |
| {v : M → ℝ} (hv : Continuous v) {x : M} {p : TangentSpace 𝓘(ℝ,Model n) x} |
| (hp : p∈normalSubdifferential (n := n) (cTransform v) x) |
| {t : ℝ} (ht : 0<t) (ht1 : t<1) |
| (hID : ∀ q∈normalSubdifferential (n := n) (cTransform v) x, t • q∈injectivityDomain x) : |
| ∀ O∈𝓝 (⟨x,p⟩ : TangentBundle 𝓘(ℝ,Model n) M), |
| ∀ᶠ y in 𝓝 (riemannianExp x (t • p)), |
| ∃ z∈O, z.2∈normalSubdifferential (n := n) (cTransform v) z.1 ∧ |
| riemannianExp z.1 (t • z.2)=y := by |
| intro O hO |
| let y₀ := riemannianExp x (t • p) |
| have htp := hID p hp |
| obtain ⟨P,hP,hP0,_,hPinv⟩ := exists_smooth_exp_branch |
| (⟨x,t • p⟩ : TangentBundle 𝓘(ℝ,Model n) M) (riemannianExp_interior_nonconjugate htp) |
| let Q := fun q : M×M => tangentScale t⁻¹ (P q) |
| have hQ : ContinuousAt Q (x,y₀) := |
| (contMDiff_tangentScale (E := Model n) (M := M)).continuous.continuousAt.comp |
| (continuousAt_const.prodMk hP.continuousAt) |
| have hQ0 : Q (x,y₀)=(⟨x,p⟩ : TangentBundle 𝓘(ℝ,Model n) M) := by |
| dsimp only [Q,y₀] |
| rw [hP0] |
| simp only [tangentScale,smul_smul,inv_mul_cancel₀ ht.ne',one_smul] |
| have hQO : ∀ᶠ q : M×M in 𝓝 (x,y₀), Q q∈O := |
| hQ.preimage_mem_nhds (hQ0.symm ▸ hO) |
| have hPID : ∀ᶠ q : M×M in 𝓝 (x,y₀), (P q).2∈injectivityDomain (P q).1 := by |
| have hmem : {z : TangentBundle 𝓘(ℝ,Model n) M | z.2∈injectivityDomain z.1} ∈ |
| 𝓝 (P (x,y₀)) := by |
| apply (isOpen_total_injectivityDomain (n := n) (M := M)).mem_nhds |
| rw [hP0] |
| exact htp |
| exact hP.continuousAt.preimage_mem_nhds hmem |
| have H := (hQO.and hPID).and hPinv |
| obtain ⟨U,hU,V,hV,hUV⟩ := mem_nhds_prod_iff.mp H |
| have hpH : p∈convexHull ℝ (activeLogs (n := n) v x) := |
| normalSubdifferential_subset_active_hull hv x hp |
| have hIDH : ∀ q∈convexHull ℝ (activeLogs (n := n) v x), t • q∈injectivityDomain x := |
| fun q hq => hID q (active_hull_subset_normalSubdifferential hv x hq) |
| have hmin := hmtw.active_hull_nearby_local_min hv hpH ht ht1 hIDH U hU |
| filter_upwards [hV,hmin] with y hy hh |
| obtain ⟨a,ha,hmin⟩ := hh |
| have hq := hUV (show (a,y)∈U ×ˢ V from ⟨ha,hy⟩) |
| rcases hPa : P (a,y) with ⟨b,r⟩ |
| change (tangentScale t⁻¹ (P (a,y))∈O ∧ |
| (P (a,y)).2∈injectivityDomain (P (a,y)).1) ∧ |
| (P (a,y)).1=a ∧ riemannianExp (P (a,y)).1 (P (a,y)).2=y at hq |
| rw [hPa] at hq |
| have hb : b=a := hq.2.1 |
| subst b |
| have hr : r∈injectivityDomain a := hq.1.2 |
| have hE : riemannianExp a r=y := hq.2.2 |
| have htr : t • (t⁻¹ • r)=r := by rw [smul_smul,mul_inv_cancel₀ ht.ne',one_smul] |
| refine ⟨⟨a,t⁻¹ • r⟩,?_,?_,?_⟩ |
| · exact hq.1.1 |
| · apply local_min_divided_cost_subgradient ht.ne' |
| · simpa only [htr] using hr |
| · simpa only [htr,hE] using hmin |
| · simpa only [htr] using hE |
| |
| end WeakMTWTransport |
| |
| end |
| |
| end OAI |
| |
