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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 | import Mathlib
import OAI.Analysis.BiholderTransport.Convexity.ActualCenterSemibound
namespace OAI
section
noncomputable section
open Set Filter Manifold Bundle
open scoped Topology ContDiff NNReal
namespace WeakMTWTransport
section CenterSequenceCompact
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]
def CenterSequenceData.comp {a c : M} {u v : M → ℝ} {Φ : ℝ×ℝ → ℝ}
{γj lj τj tj : ℕ → ℝ} {xj bj qj : ℕ → Model n} {zj : ℕ → M}
{F : ℕ → Model n → ℝ}
(D : CenterSequenceData a c u v Φ γj lj τj tj xj bj qj zj F)
(σ : ℕ → ℕ) (hσ : Tendsto σ atTop atTop) :
CenterSequenceData a c u v Φ (γj ∘ σ) (lj ∘ σ) (τj ∘ σ) (tj ∘ σ)
(xj ∘ σ) (bj ∘ σ) (qj ∘ σ) (zj ∘ σ) (F ∘ σ) where
row i := D.row (σ i)
endpoint i := D.endpoint (σ i)
derivative := hσ.eventually D.derivative
gradient := D.gradient.comp hσ
backward := hσ.eventually D.backward
lemma exists_actual_center_gain :
∃ c0>0,∀ᶠ l : ℝ in 𝓝 1, 0<l → l<1 →
∀ (a c : M) (u v : M → ℝ) (Φ : ℝ×ℝ → ℝ) (γ : ℝ),
Continuous v → ContDiffAt ℝ ∞ Φ (γ,v c) →
deriv (fun s=>Φ (γ,s)) (v c)=l →
∀ (γj lj τj tj : ℕ → ℝ) (xj bj qj : ℕ → Model n) (zj : ℕ → M)
(F : ℕ → Model n → ℝ),
CenterSequenceData a c u v Φ γj lj τj tj xj bj qj zj F →
∀ q : Model n,
(show TangentSpace 𝓘(ℝ,Model n) a from q)∈minimizingVectors a →
riemannianExp a q=c →
Tendsto γj atTop (𝓝 γ) → Tendsto lj atTop (𝓝 l) →
Tendsto τj atTop (𝓝 0) → Tendsto tj atTop (𝓝 1) →
Tendsto xj atTop (𝓝 (extChartAt 𝓘(ℝ,Model n) c c)) →
Tendsto bj atTop (𝓝 (extChartAt 𝓘(ℝ,Model n) a a)) →
Tendsto qj atTop (𝓝 q) →
(∀ᶠ i in atTop,0<τj i) → ∀ L : ℝ≥0,
(∀ᶠ i in atTop,LipschitzWith L (fun y=>Φ (γj i,v y))) →
∀ ε : ℝ,0<ε →
∀ᶠ i in atTop,∀ d : Model n,
c0*(1-l)*‖show TangentSpace 𝓘(ℝ,Model n) a from d‖^2+
(iteratedDeriv 2 (fun s=>Φ (γ,s)) (v c)/l^2)*
(inner ℝ (show TangentSpace 𝓘(ℝ,Model n) a from q) d)^2-ε*‖d‖^2 ≤
coordinateCenterMatrix a (tj i) (hopfLax (τj i) (fun y=>Φ (γj i,v y)))
(bj i) (qj i) d d := by
obtain ⟨c0,hc0,H⟩ := exists_actual_center_sequential_gain (n := n) (M := M)
refine ⟨c0,hc0,?_⟩
filter_upwards [H] with l hgain
intro hl hl1 a c u v Φ γ hv hΦ hder γj lj τj tj xj bj qj zj F D q hq he
hγ hlm hτ ht hx hb hqq hτpos L hLip ε hε
by_contra h
have hf := not_eventually.mp h
choose σ hσge hbad using frequently_atTop.mp hf
have hσ : Tendsto σ atTop atTop := tendsto_atTop_mono hσge tendsto_id
let D1 := D.comp σ hσ
obtain ⟨κ,hκ,r,hr,hrr⟩ := center_support_family_subseq D1.row (hx.comp hσ)
let D2 := D1.comp κ hκ.tendsto_atTop
have Hlow := hgain hl hl1 a c u v Φ γ hv hΦ hder
((γj ∘ σ) ∘ κ) ((lj ∘ σ) ∘ κ) ((τj ∘ σ) ∘ κ) ((tj ∘ σ) ∘ κ)
((xj ∘ σ) ∘ κ) ((bj ∘ σ) ∘ κ) ((qj ∘ σ) ∘ κ) ((zj ∘ σ) ∘ κ)
((F ∘ σ) ∘ κ) D2 r q hr hq he
((hγ.comp hσ).comp hκ.tendsto_atTop) ((hlm.comp hσ).comp hκ.tendsto_atTop)
((hτ.comp hσ).comp hκ.tendsto_atTop) ((ht.comp hσ).comp hκ.tendsto_atTop)
((hx.comp hσ).comp hκ.tendsto_atTop) ((hb.comp hσ).comp hκ.tendsto_atTop)
((hqq.comp hσ).comp hκ.tendsto_atTop) hrr
(hκ.tendsto_atTop.eventually (hσ.eventually hτpos)) L
(hκ.tendsto_atTop.eventually (hσ.eventually hLip)) ε hε
obtain ⟨i,hi⟩ := Hlow.exists
exact hbad (κ i) hi
end CenterSequenceCompact
end WeakMTWTransport
end
end
end OAI
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