| import Mathlib |
| import OAI.Analysis.BiholderTransport.Geodesics.MinimizingCoordinateLimit |
| import OAI.Analysis.BiholderTransport.Regularity.ParametricActive |
| |
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff |
|
|
| namespace WeakMTWTransport |
| section ConfigurationCompact |
| 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 active_configuration_subseq {ι : Type*} [Fintype ι] [Nonempty ι] |
| [TopologicalSpace ι] [DiscreteTopology ι] |
| {a:M} {v:M → ℝ} (hv:Continuous v) {α D:ℝ} {B:ℝ → ℝ} (hB:Continuous B) |
| {β:ℕ → ℝ} {β₀:ℝ} (hβ:Tendsto β atTop (𝓝 β₀)) |
| {bj p:ℕ → Model n} {p₀:Model n} |
| (hb:Tendsto bj atTop (𝓝 (extChartAt 𝓘(ℝ,Model n) a a))) |
| (hp:Tendsto p atTop (𝓝 p₀)) |
| (hbt:∀k,bj k∈(extChartAt 𝓘(ℝ,Model n) a).target) |
| {pj:ℕ → ι → Model n} {w:ℕ → ι → ℝ} (i:ℕ → ι) |
| (ha:∀k j,chartFiberInverse a (bj k) (pj k j)∈ |
| activeLogs (modifiedDatum v α D (β k) B) ((extChartAt 𝓘(ℝ,Model n) a).symm (bj k))) |
| (hw:∀k j,0 ≤ w k j) (hs:∀k,∑j,w k j=1) |
| (hbar:∀k,∑j,w k j • pj k j=p k) : |
| ∃σ:ℕ → ℕ,StrictMono σ ∧ ∃r:ι → Model n,∃m:ι → ℝ,∃i₀:ι, |
| Tendsto (pj ∘ σ) atTop (𝓝 r) ∧ Tendsto (w ∘ σ) atTop (𝓝 m) ∧ |
| (∀ᶠ k in atTop,i (σ k)=i₀) ∧ |
| (∀j,(show TangentSpace 𝓘(ℝ,Model n) a from r j)∈activeLogs (modifiedDatum v α D β₀ B) a) ∧ |
| (∀j,0 ≤ m j) ∧ ∑j,m j=1 ∧ ∑j,m j • r j=p₀ := by |
| let χ:=extChartAt 𝓘(ℝ,Model n) a |
| have hat:χ a∈χ.target := mem_extChartAt_target a |
| have hy:Tendsto (fun k=>χ.symm (bj k)) atTop (𝓝 a) := by |
| have H:=(continuousAt_extChartAt_symm'' hat).tendsto.comp hb |
| rwa [χ.left_inv (mem_extChartAt_source a)] at H |
| obtain ⟨K,hK,hKs,N,hN⟩:=compact_in_chart_tail (n := n) (a := a) hy (mem_extChartAt_source a) |
| let shift:=fun k:ℕ=>k+N |
| have hshift:Tendsto shift atTop atTop := tendsto_add_atTop_nat N |
| let C:=minimizingChartParameters (n := n) a K |
| have hC:IsCompact C := isCompact_minimizingChartParameters hK hKs |
| let S:= {q:EnvelopeData (Model n) ι | (∀j,(q.1.1,q.1.2 j)∈C) ∧ |
| (∀j,0 ≤ q.2.1 j) ∧ ∑j,q.2.1 j=1 ∧ q.2.2∈Icc 0 0} |
| have hc:IsCompact (S×ˢ(univ:Set ι)) := (compact_common_base_configurations hC 0).prod isCompact_univ |
| let f:ℕ → EnvelopeData (Model n) ι × ι := fun k=>(((bj (shift k),pj (shift k)),(w (shift k),0)),i (shift k)) |
| have hf:∀k,f k∈S×ˢ(univ:Set ι) := by |
| intro k |
| refine ⟨⟨?_,hw _,hs _,le_rfl,le_rfl⟩,mem_univ _⟩ |
| intro j |
| exact inverse_mem_minimizingChartParameters (hbt _) (hN k) (ha _ j).1 |
| obtain ⟨q,hq,τ,hτ,hlim⟩:=hc.isSeqCompact hf |
| let σ:=shift ∘ τ |
| have hσ:StrictMono σ := fun k l h=>Nat.add_lt_add_right (hτ h) N |
| have hr:Tendsto (pj ∘ σ) atTop (𝓝 q.1.1.2) := hlim.fst_nhds.fst_nhds.snd_nhds |
| have hm:Tendsto (w ∘ σ) atTop (𝓝 q.1.2.1) := hlim.fst_nhds.snd_nhds.fst_nhds |
| have hi:∀ᶠ k in atTop,i (σ k)=q.2 := by |
| exact hlim.snd_nhds.eventually ((isOpen_discrete {q.2}).mem_nhds rfl) |
| refine ⟨σ,hσ,q.1.1.2,q.1.2.1,q.2,hr,hm,hi,?_,hq.1.2.1,hq.1.2.2.1,?_⟩ |
| · intro j |
| apply modified_active_coordinates_limit hv hB (hβ.comp hσ.tendsto_atTop) |
| (hb.comp hσ.tendsto_atTop) ((tendsto_pi_nhds.mp hr) j) |
| (Eventually.of_forall (fun k=>hbt (σ k))) |
| (Eventually.of_forall (fun k=>ha (σ k) j)) |
| · have H:Tendsto (fun k=>∑j,w (σ k) j • pj (σ k) j) atTop |
| (𝓝 (∑j,q.1.2.1 j • q.1.1.2 j)) := by |
| apply tendsto_finsetSum |
| intro j _ |
| exact ((tendsto_pi_nhds.mp hm) j).smul ((tendsto_pi_nhds.mp hr) j) |
| simp only [hbar] at H |
| exact tendsto_nhds_unique H (hp.comp hσ.tendsto_atTop) |
| end ConfigurationCompact |
| end WeakMTWTransport |
| |
| end |
| end |
| |
| end OAI |
| |
