import Mathlib import OAI.Analysis.BiholderTransport.Coordinates.BranchInverse import OAI.Analysis.BiholderTransport.Contact.ActiveContinuationLocal import OAI.Analysis.BiholderTransport.Contact.SubgradientUpper namespace OAI noncomputable section open Set MeasureTheory Manifold Bundle open scoped ContDiff Manifold ENNReal NNReal Topology open Set Filter open scoped Topology NNReal open Set Filter open scoped Topology open Set Manifold MeasureTheory Bundle open scoped ENNReal ContDiff Topology open Set open scoped Topology open Set Filter Manifold Bundle ContinuousLinearMap open scoped Topology ContDiff Manifold Bundle open Set Filter ContinuousLinearMap InnerProductSpace open scoped Topology ContDiff open Set Filter ContinuousLinearMap open scoped Topology ContDiff open Set Filter ContinuousLinearMap open scoped Topology ContDiff open Set Filter ContinuousLinearMap open scoped Topology ContDiff open scoped NNReal open Set Filter ContinuousLinearMap open scoped Topology ContDiff open Set Filter ContinuousLinearMap open scoped Topology open MeasureTheory open scoped ContDiff ENNReal open Set Filter Manifold Bundle ContinuousLinearMap MeasureTheory open scoped Topology ContDiff Manifold Bundle ENNReal open Set Filter Manifold MeasureTheory Bundle open scoped ENNReal ContDiff Topology Manifold open Set Filter Manifold Bundle ContinuousLinearMap open scoped Topology ContDiff Manifold Bundle open Set Filter Manifold Bundle open scoped Topology ContDiff Manifold Bundle open Set Filter Manifold Bundle open scoped Topology ContDiff Manifold Bundle open Set Filter Bundle open scoped Topology Bundle open scoped Topology open Function Manifold Set open Manifold Bundle open scoped Manifold Bundle open Set open Set Filter open scoped Topology ContDiff 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 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 open Set Filter Manifold MeasureTheory Bundle open scoped ENNReal ContDiff Topology open Set Filter open scoped 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 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 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 open Filter Set open scoped 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 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 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 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 Topology open Set Filter open scoped Topology ContDiff open Set Filter open scoped Topology ContDiff 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 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 open Set Filter open scoped 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 Topology open Set Filter Manifold MeasureTheory Bundle open scoped ENNReal ContDiff Topology open Set Filter Manifold MeasureTheory Bundle open scoped ENNReal ContDiff Topology 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 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 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