| import Mathlib |
| import OAI.Analysis.BiholderTransport.LinearAlgebra.ChartFirstMatrix |
| import OAI.Analysis.BiholderTransport.Regularity.ActualOutward |
|
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| namespace OAI |
|
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| section |
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| noncomputable section |
| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff |
|
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| namespace WeakMTWTransport |
| section FirstOutward |
| 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] |
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| lemma ChartFixedPrefixLimit.outward_endpoint |
| (hmtw:WeakMTW (n := n) (M := M)) |
| {u v G:M → ℝ} (hu:Continuous u) (hv:Continuous v) (hdual:IsCostDualPair u v) |
| {α D β H:ℝ} {Bc Bo:ℝ → ℝ} (ho:Continuous Bo) (hβ:0 ≤ β) |
| (hob:∀s,0 ≤ Bo s ∧ Bo s ≤ H) (hcb:∀s,-1 ≤ Bc s ∧ Bc s ≤ 1) |
| (hright:∀s,3/4 ≤ s → Bc s ≤ 0) |
| {a:M} {t:ℝ} {q:ℕ → subgradientGraph (n := n) (cTransform (modifiedDatum v α D β Bo))} |
| {q₀:subgradientGraph (n := n) (cTransform (modifiedDatum v α D β Bo))} |
| (J:ChartFixedPrefixLimit a (modifiedDatum v α D β Bo) G t q q₀) : |
| let b:=graphBaseCoordinate a q₀.1 |
| let x:=(extChartAt 𝓘(ℝ,Model n) a).symm b |
| let E:=TangentSpace 𝓘(ℝ,Model n) x |
| let P:E:=chartFiberInverse a b (graphVelocityCoordinate a q₀.1) |
| let pj:=fun i=>chartFiberInverse a b (J.pj₀ i) |
| let κ:Type:={i:Fin (Module.finrank ℝ (Model n)+1) |
| ∀ e:E,‖e‖=1 → 0 < inner ℝ P e → |
| Submodule.span ℝ (range (fun i:κ=>pj i-P))=Submodule.span ℝ {e} → |
| ∀ c eta delta zeta a0:ℝ,0 < D → 0 < c → 0 < a0 → |
| a0*D ≤ (inner ℝ P e)^2 → eta ≤ 1/128 → delta ≤ zeta → delta ≤ eta → zeta ≤ 1/16 → |
| (∀s,s < 1-eta → 1 ≤ Bo s) → |
| 0 < modifiedExcess v α D β Bc Bo (riemannianExp x P) → |
| 2*(H+2)*β ≤ zeta*D → sectionOscillation u v x ((H+2)*β) ≤ (1+delta)*D → |
| c ≤ Bc ((v (riemannianExp x P)-α)/D)-∑ i,J.w₀ i*Bo ((v (riemannianExp x (pj i))-α)/D) → |
| let si:=fun i=>(v (riemannianExp x (pj i))-α)/D |
| let mstar:=c/(12*(Module.finrank ℝ (Model n)+1:ℝ)) |
| let theta0:=min (mstar/2) (a0/3) |
| ∃ i d,0 < J.w₀ i ∧ 0 < d ∧ pj i=P+d • e ∧ mstar ≤ J.w₀ i ∧ |
| c*D/(8*(Module.finrank ℝ (Model n)+1:ℝ)*(inner ℝ P e)) ≤ J.w₀ i*d ∧ |
| d ≤ 3*D/(2*(inner ℝ P e)) ∧ -2*eta ≤ si i ∧ si i < 1-eta ∧ |
| Bo (si i) ≤ 12*(Module.finrank ℝ (Model n)+1:ℝ)/c ∧ 0 < theta0 ∧ |
| ∀ theta∈Icc (-theta0) 1, |
| P+theta • (pj i-P)∈convexHull ℝ (activeLogs (modifiedDatum v α D β Bo) x) ∧ |
| (inner ℝ P e)/2 ≤ inner ℝ (P+theta • (pj i-P)) e ∧ |
| ‖pj i-P‖^2/(inner ℝ (P+theta • (pj i-P)) (pj i-P)) ≤ 3/a0 := by |
| dsimp only |
| intro e he hq hline c eta delta zeta a0 hD hc ha0 hqa heta hdelta hdeltaEta hzeta |
| hleft hexcess herror hosc hparameter |
| classical |
| let b:=graphBaseCoordinate a q₀.1 |
| let x:=(extChartAt 𝓘(ℝ,Model n) a).symm b |
| let P:=chartFiberInverse a b (graphVelocityCoordinate a q₀.1) |
| let pj:=fun i=>chartFiberInverse a b (J.pj₀ i) |
| let κ:Type:={i:Fin (Module.finrank ℝ (Model n)+1) |
| have hr:=positive_barycentric_restriction pj J.w₀ J.nonnegLimit J.totalLimit |
| have hbar:∑ i:κ,J.w₀ i • pj i=P := by |
| rw [hr.2.2.1] |
| change ∑ i,J.w₀ i • chartFiberInverse a b (J.pj₀ i)=P |
| simp only [←map_smul] |
| rw [←map_sum,J.baryLimit] |
| have hN:Fintype.card κ ≤ Module.finrank ℝ (Model n)+1 := by |
| exact (Fintype.card_subtype_le _).trans_eq (Fintype.card_fin _) |
| have hparam:c ≤ Bc ((v (riemannianExp x P)-α)/D)- |
| ∑ i:κ,J.w₀ i*Bo ((v (riemannianExp x (pj i))-α)/D) := by |
| rw [hr.2.2.2 (fun i=>Bo ((v (riemannianExp x (pj i))-α)/D))] |
| exact hparameter |
| obtain ⟨i,d,hd,hpd,hm,hmom,hdmax,hsi,hsi',hcap,htheta,Htheta⟩:= |
| hmtw.actual_outward_endpoint hu hv hdual ho hβ hob hcb hright hN |
| (fun i:κ=>i.2) hr.2.1 hbar (fun i=>J.activeLimit i) |
| he hq hline hD hc ha0 hqa heta hdelta hdeltaEta hzeta hleft hexcess herror hosc hparam |
| refine ⟨i.1,d,i.2,hd,hpd,hm,hmom,hdmax,hsi,hsi',hcap,htheta,?_⟩ |
| intro theta htheta |
| obtain ⟨hconv,hq',hratio⟩:=Htheta theta htheta |
| refine ⟨convexHull_mono ?_ hconv,hq',hratio⟩ |
| rintro _ ⟨j,rfl⟩ |
| exact J.activeLimit j |
|
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| end FirstOutward |
| end WeakMTWTransport |
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| end |
| end |
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| end OAI |
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