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110f4f3 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 | import Mathlib
import OAI.Analysis.BiholderTransport.LinearAlgebra.ChartFirstMatrix
import OAI.Analysis.BiholderTransport.Regularity.ActualOutward
namespace OAI
section
noncomputable section
open Set Filter Manifold Bundle
open scoped Topology ContDiff
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]
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) // 0 < J.w₀ i}
∀ 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) // 0 < J.w₀ i}
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
end FirstOutward
end WeakMTWTransport
end
end
end OAI
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