openAI-math-dataset / lean /OAI /Analysis /BiholderTransport /Regularity /CenterSequenceCompact.lean
| 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 | |