| import Mathlib |
| import OAI.Analysis.BiholderTransport.Coordinates.UniformShort |
| import OAI.Analysis.BiholderTransport.Geodesics.ShortGradient |
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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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| namespace WeakMTWTransport |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] |
| {M : Type*} [MetricSpace M] [CompactSpace M] [ChartedSpace E M] |
| [IsManifold 𝓘(ℝ,E) ∞ M] |
| [RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,E) x)] |
| [IsContMDiffRiemannianBundle 𝓘(ℝ,E) ∞ E (fun x : M => TangentSpace 𝓘(ℝ,E) x)] |
| [IsRiemannianManifold 𝓘(ℝ,E) M] |
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| lemma exists_two_short_legs (x : M) (p : TangentSpace 𝓘(ℝ,E) x) |
| {t T δ : ℝ} (ht : 0<t) (hT : t<T) (hδ : 0<δ) : |
| ∃ ε : ℝ, 0<ε ∧ ε<t ∧ t+ε<T ∧ |
| (sprayFlow (t+ε) (⟨x,p⟩ : TangentBundle 𝓘(ℝ,E) M)).1 ∈ |
| Metric.ball (sprayFlow t (⟨x,p⟩ : TangentBundle 𝓘(ℝ,E) M)).1 δ ∧ |
| ∀ᶠ v : TangentSpace 𝓘(ℝ,E) x in 𝓝 p, |
| (sprayFlow (t-ε) (⟨x,v⟩ : TangentBundle 𝓘(ℝ,E) M)).1 ∈ |
| Metric.ball (sprayFlow t (⟨x,p⟩ : TangentBundle 𝓘(ℝ,E) M)).1 δ ∧ |
| (sprayFlow t (⟨x,v⟩ : TangentBundle 𝓘(ℝ,E) M)).1 ∈ |
| Metric.ball (sprayFlow t (⟨x,p⟩ : TangentBundle 𝓘(ℝ,E) M)).1 δ ∧ |
| HasMFDerivAt 𝓘(ℝ,E) 𝓘(ℝ,ℝ) |
| (fun b => dist b (sprayFlow t (⟨x,v⟩ : TangentBundle 𝓘(ℝ,E) M)).1^2/2) |
| (sprayFlow (t-ε) (⟨x,v⟩ : TangentBundle 𝓘(ℝ,E) M)).1 |
| (innerSL ℝ ((-ε) • (sprayFlow (t-ε) (⟨x,v⟩ : TangentBundle 𝓘(ℝ,E) M)).2)) ∧ |
| HasMFDerivAt 𝓘(ℝ,E) 𝓘(ℝ,ℝ) |
| (fun q => dist (sprayFlow (t-ε) (⟨x,v⟩ : TangentBundle 𝓘(ℝ,E) M)).1 q^2/2) |
| (sprayFlow t (⟨x,v⟩ : TangentBundle 𝓘(ℝ,E) M)).1 |
| (innerSL ℝ (ε • (sprayFlow t (⟨x,v⟩ : TangentBundle 𝓘(ℝ,E) M)).2)) := by |
| let a := sprayFlow t (⟨x,p⟩ : TangentBundle 𝓘(ℝ,E) M) |
| obtain ⟨δs,hδs,Hs⟩ := exists_local_spray_cost_start_gradient (E := E) a.1 |
| obtain ⟨δe,hδe,He⟩ := exists_local_spray_cost_gradient (E := E) a.1 |
| let r := min δ (min δs δe) |
| have hr : 0<r := lt_min hδ (lt_min hδs hδe) |
| have hι : Continuous (fun u : ℝ × TangentSpace 𝓘(ℝ,E) x => |
| (⟨x,u.2⟩ : TangentBundle 𝓘(ℝ,E) M)) := |
| (contMDiff_fiber_embedding x).continuous.comp continuous_snd |
| let F : ℝ × TangentSpace 𝓘(ℝ,E) x → TangentBundle 𝓘(ℝ,E) M := |
| fun u => sprayFlow (t-u.1) ⟨x,u.2⟩ |
| have hF : Continuous F := contMDiff_sprayFlow.continuous.comp |
| ((continuous_const.sub continuous_fst).prodMk hι) |
| have hF₀ : F (0,p)=a := by simp [F,a] |
| have hpr : Continuous (fun z : TangentBundle 𝓘(ℝ,E) M => z.1) := |
| FiberBundle.continuous_proj E (fun b : M => TangentSpace 𝓘(ℝ,E) b) |
| have hFin := (hpr.comp hF).continuousAt.preimage_mem_nhds |
| (show Metric.ball a.1 r ∈ 𝓝 (F (0,p)).1 by rw [hF₀]; exact Metric.ball_mem_nhds _ hr) |
| have hQ : Continuous (fun u : ℝ × TangentSpace 𝓘(ℝ,E) x => |
| (sprayFlow t (⟨x,u.2⟩ : TangentBundle 𝓘(ℝ,E) M)).1) := |
| hpr.comp ((contMDiff_spray_fiber x t).continuous.comp continuous_snd) |
| have hQin := (hQ.continuousAt (x := (0,p))).preimage_mem_nhds (Metric.ball_mem_nhds a.1 hr) |
| have hY : Continuous (fun u : ℝ × TangentSpace 𝓘(ℝ,E) x => |
| (sprayFlow (t+u.1) (⟨x,p⟩ : TangentBundle 𝓘(ℝ,E) M)).1) := |
| hpr.comp (contMDiff_sprayFlow.continuous.comp |
| ((continuous_const.add continuous_fst).prodMk continuous_const)) |
| have hYin := (hY.continuousAt (x := (0,p))).preimage_mem_nhds |
| (show Metric.ball a.1 r ∈ 𝓝 (sprayFlow (t+(0:ℝ)) (⟨x,p⟩ : TangentBundle 𝓘(ℝ,E) M)).1 by |
| simpa only [add_zero] using Metric.ball_mem_nhds a.1 hr) |
| have htend : Tendsto (fun u : ℝ × TangentSpace 𝓘(ℝ,E) x => (u.1,F u)) (𝓝 (0,p)) (𝓝 (0,a)) := by |
| simpa only [ContinuousAt,hF₀] using (continuous_fst.prodMk hF).continuousAt (x := (0,p)) |
| have hmin := htend.eventually (sprayFlow_uniformly_short_minimizing a) |
| have hAll : ∀ᶠ u : ℝ × TangentSpace 𝓘(ℝ,E) x in 𝓝 (0,p), |
| (F u).1 ∈ Metric.ball a.1 r ∧ |
| (sprayFlow t (⟨x,u.2⟩ : TangentBundle 𝓘(ℝ,E) M)).1 ∈ Metric.ball a.1 r ∧ |
| (sprayFlow (t+u.1) (⟨x,p⟩ : TangentBundle 𝓘(ℝ,E) M)).1 ∈ Metric.ball a.1 r ∧ |
| dist (F u).1 (sprayFlow u.1 (F u)).1 = |u.1| *‖(F u).2‖ := by |
| filter_upwards [hFin,hQin,hYin,hmin] with u h₁ h₂ h₃ h₄ |
| exact ⟨h₁,h₂,h₃,h₄⟩ |
| obtain ⟨A,hA,B,hB,hAB⟩ := mem_nhds_prod_iff.mp hAll |
| obtain ⟨η,hη,hηA⟩ := Metric.mem_nhds_iff.mp hA |
| let ε := min η (min t (T-t))/2 |
| have hε : 0<ε := div_pos (lt_min hη (lt_min ht (sub_pos.mpr hT))) (by norm_num) |
| have hεη : ε<η := lt_of_lt_of_le (half_lt_self (lt_min hη (lt_min ht (sub_pos.mpr hT)))) (min_le_left _ _) |
| have hεt : ε<t := lt_of_lt_of_le (half_lt_self (lt_min hη (lt_min ht (sub_pos.mpr hT)))) |
| ((min_le_right _ _).trans (min_le_left _ _)) |
| have hεT : ε<T-t := lt_of_lt_of_le (half_lt_self (lt_min hη (lt_min ht (sub_pos.mpr hT)))) |
| ((min_le_right _ _).trans (min_le_right _ _)) |
| have hεA : ε∈A := hηA (by simpa only [Metric.mem_ball,dist_zero_right,Real.norm_eq_abs,abs_of_pos hε] using hεη) |
| have hP := hAB (show (ε,p) ∈ A ×ˢ B from ⟨hεA,mem_of_mem_nhds hB⟩) |
| have hsubδ : Metric.ball a.1 r ⊆ Metric.ball a.1 δ := Metric.ball_subset_ball (min_le_left _ _) |
| have hsubs : Metric.ball a.1 r ⊆ Metric.ball a.1 δs := |
| Metric.ball_subset_ball ((min_le_right _ _).trans (min_le_left _ _)) |
| have hsube : Metric.ball a.1 r ⊆ Metric.ball a.1 δe := |
| Metric.ball_subset_ball ((min_le_right _ _).trans (min_le_right _ _)) |
| refine ⟨ε,hε,hεt,by linarith,hsubδ hP.2.2.1,?_⟩ |
| filter_upwards [hB] with v hv |
| have H := hAB (show (ε,v) ∈ A ×ˢ B from ⟨hεA,hv⟩) |
| have hflow : sprayFlow ε (F (ε,v)) = sprayFlow t (⟨x,v⟩ : TangentBundle 𝓘(ℝ,E) M) := by |
| dsimp [F] |
| have htime : ε + (t - ε) = t := by ring |
| rw [← sprayFlow_add, htime] |
| have hm : dist (F (ε,v)).1 (sprayFlow ε (F (ε,v))).1 = ε*‖(F (ε,v)).2‖ := by |
| simpa only [abs_of_pos hε] using H.2.2.2 |
| have hs := Hs (F (ε,v)) ε hε (hsubs H.1) (by rw [hflow]; exact hsubs H.2.1) hm |
| have he := He (F (ε,v)) ε hε (hsube H.1) (by rw [hflow]; exact hsube H.2.1) hm |
| rw [hflow] at hs he |
| exact ⟨hsubδ H.1,hsubδ H.2.1,hs,he⟩ |
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| end WeakMTWTransport |
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| end |
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| end OAI |
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