openAI-math-dataset / lean /OAI /Analysis /BiholderTransport /CostGeometry /LocalBranchJetBound.lean
| import Mathlib | |
| import OAI.Analysis.BiholderTransport.Contact.SmoothSupportSemibound | |
| import OAI.Analysis.BiholderTransport.Coordinates.BranchChartBaseline | |
| import OAI.Analysis.BiholderTransport.Coordinates.BranchChartJets | |
| import OAI.Analysis.BiholderTransport.Coordinates.LocalOuterJetBound | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open Set Filter Manifold Bundle | |
| open scoped Topology ContDiff BoundedContinuousFunction | |
| namespace WeakMTWTransport | |
| section LocalBranchJetBound | |
| 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] | |
| [MeasurableSpace M] [BorelSpace M] | |
| lemma WeakMTW.exists_local_branch_jet_bound (hmtw : WeakMTW (n := n) (M := M)) | |
| {lam cap : ℝ} (hlam : 0<lam) (hcap : 0≤cap) {a:M} | |
| {r : TangentSpace 𝓘(ℝ,Model n) a} (hr : r∈minimizingVectors a) | |
| {G : M×M → ℝ} | |
| (hG : ContMDiffAt (𝓘(ℝ,Model n).prod 𝓘(ℝ,Model n)) 𝓘(ℝ,ℝ) ∞ G | |
| (a,riemannianExp a r)) | |
| (hagree : ∀ᶠ z in 𝓝 (⟨a,r⟩ : TangentBundle 𝓘(ℝ,Model n) M), | |
| z.2∈injectivityDomain z.1 → | |
| (fun q:M×M => cost q.1 q.2) =ᶠ[𝓝 (z.1,riemannianExp z.1 z.2)] G) | |
| {l:ℝ} (hl : 1<l) : | |
| let c := fun y:M => fun w:Model n => G ((extChartAt 𝓘(ℝ,Model n) a).symm w,y) | |
| ∃ U : Set ((Model n×M)×ℝ), | |
| U∈𝓝 ((extChartAt 𝓘(ℝ,Model n) a a,riemannianExp a r),l) ∧ ∃ C≥0, | |
| ∀ q∈U,∀ (x0:M) (uv : (M →ᵇ ℝ)×(M →ᵇ ℝ)), | |
| uv∈densityDualClass (metricVolume n) lam cap x0 → | |
| ∀ φ:ℝ → ℝ,StrictMono φ → | |
| ContDiffAt ℝ 2 φ (uv.1 ((extChartAt 𝓘(ℝ,Model n) a).symm q.1.1)) → | |
| HasDerivAt φ q.2 (uv.1 ((extChartAt 𝓘(ℝ,Model n) a).symm q.1.1)) → | |
| iteratedDeriv 2 φ (uv.1 ((extChartAt 𝓘(ℝ,Model n) a).symm q.1.1))≤0 → | |
| ∀ H:Model n →L[ℝ] Model n →L[ℝ] ℝ,(∀ d,0≤H d d) → | |
| HasLowerSecondTaylor (fun h => | |
| φ (uv.1 ((extChartAt 𝓘(ℝ,Model n) a).symm (q.1.1+h)))+c q.1.2 (q.1.1+h)) 0 H → | |
| ∀ d,H d d≤C*‖d‖^2 := by | |
| dsimp only | |
| let z := extChartAt 𝓘(ℝ,Model n) a a | |
| let b := riemannianExp a r | |
| let q0 : (Model n×M)×ℝ := ((z,b),l) | |
| let c := fun y:M => fun w:Model n => G ((extChartAt 𝓘(ℝ,Model n) a).symm w,y) | |
| obtain ⟨hc,L,hcD,hpreg,m,hm,Hm⟩ := exists_branch_chart_baseline hr hG hagree hl | |
| have hlp : 0<l := zero_lt_one.trans hl | |
| let δ := m/(4*(l+1)) | |
| have hδ : 0<δ := div_pos hm (by positivity) | |
| obtain ⟨N,hN,C,hC,HC⟩ := hmtw.exists_local_outer_chart_jet_bound hlam hcap a L hpreg hδ | |
| (show 0<l+1 by positivity) | |
| have ha : z∈(extChartAt 𝓘(ℝ,Model n) a).target := mem_extChartAt_target a | |
| have hGa : ContMDiffAt (𝓘(ℝ,Model n).prod 𝓘(ℝ,Model n)) 𝓘(ℝ,ℝ) ∞ G | |
| ((extChartAt 𝓘(ℝ,Model n) a).symm z,b) := by | |
| simpa only [z,b,(extChartAt 𝓘(ℝ,Model n) a).left_inv (mem_extChartAt_source a)] using hG | |
| obtain ⟨hD,hH,hcnear⟩ := branch_chart_jet_continuous ha hGa | |
| let P := fun q:(Model n×M)×ℝ => q.2⁻¹ • (-fderiv ℝ (c q.1.2) q.1.1) | |
| have hP0 : P q0=L := by | |
| dsimp only [P,q0,c,z,b] | |
| rw [hcD] | |
| ext d | |
| simp only [smul_apply,neg_apply,smul_eq_mul,neg_mul,neg_neg] | |
| field_simp [hlp.ne'] | |
| have hP : ContinuousAt P q0 := | |
| (continuousAt_snd.inv₀ hlp.ne').smul ((hD.comp (x := q0) (f := fun q : (Model n×M)×ℝ => q.1) continuousAt_fst).neg) | |
| let J := fun q:(Model n×M)×ℝ => (q.1.1,P q) | |
| have hJ : ContinuousAt J q0 := (continuousAt_fst.comp continuousAt_fst).prodMk hP | |
| have hJ0 : J q0=(z,L) := by simp only [J,q0,hP0] | |
| let Y := fun q:(Model n×M)×ℝ => coordinateBackward a (-1,q.1.1,P q) | |
| have hY : ContinuousAt Y q0 := | |
| ((coordinateBackward_contMDiffAt (q := (-1,z,P q0)) ha).continuousAt).comp (x := q0) | |
| (f := fun q : (Model n×M)×ℝ => (-1,q.1.1,P q)) (continuousAt_const.prodMk hJ) | |
| have hY0 : Y q0=coordinateBackward a (-1,z,L) := by simp only [Y,q0,hP0] | |
| have hcost : ContMDiffAt (𝓘(ℝ,Model n).prod 𝓘(ℝ,Model n)) 𝓘(ℝ,ℝ) ∞ | |
| (fun q:M×M => cost q.1 q.2) ((extChartAt 𝓘(ℝ,Model n) a).symm z,Y q0) := by | |
| rw [hY0,coordinateBackward_eq_exp_chartGradient ha] | |
| exact cost_contMDiffAt_of_injectivityDomain | |
| (⟨(extChartAt 𝓘(ℝ,Model n) a).symm z,chartGradientVector a z L⟩ : TangentBundle 𝓘(ℝ,Model n) M) hpreg | |
| have htrue := (chartCost_jet_continuous ha hcost).1.comp (x := q0) | |
| (f := fun q : (Model n×M)×ℝ => (q.1.1,Y q)) | |
| ((continuousAt_fst.comp continuousAt_fst).prodMk hY) | |
| let W := fun q:(Model n×M)×ℝ => | |
| q.2 • fderiv ℝ (fderiv ℝ (chartCost a (Y q))) q.1.1-fderiv ℝ (fderiv ℝ (c q.1.2)) q.1.1 | |
| let : NormedAddCommGroup (Model n →L[ℝ] ℝ) := ContinuousLinearMap.toNormedAddCommGroup | |
| let : NormedSpace ℝ (Model n →L[ℝ] ℝ) := ContinuousLinearMap.toNormedSpace | |
| let : NormedAddCommGroup (Model n →L[ℝ] Model n →L[ℝ] ℝ) := | |
| ContinuousLinearMap.toNormedAddCommGroup | |
| have hW : ContinuousAt W q0 := (continuousAt_snd.smul htrue).sub (hH.comp (x := q0) (f := fun q : (Model n×M)×ℝ => q.1) continuousAt_fst) | |
| have hW0 (d:Model n) : m*‖d‖^2≤W q0 d d := by | |
| dsimp only [W,q0] | |
| rw [hY0] | |
| exact Hm d | |
| let U := {q:(Model n×M)×ℝ | J q∈N ∧ 0<q.2 ∧ q.2≤l+1 ∧ | |
| ContDiffAt ℝ 2 (c q.1.2) q.1.1 ∧ ‖W q-W q0‖< m/2} | |
| have hU : U∈𝓝 q0 := by | |
| have hn := hJ.preimage_mem_nhds (by rwa [hJ0]) | |
| have hnW : ∀ᶠ q in 𝓝 q0,‖W q-W q0‖< m/2 := by | |
| filter_upwards [hW.tendsto.eventually | |
| (Metric.ball_mem_nhds (W q0) (show 0 < m/2 by positivity))] with q hq | |
| change dist (W q) (W q0) < m/2 at hq | |
| rwa [dist_eq_norm (W q) (W q0)] at hq | |
| filter_upwards [hn,continuousAt_snd.preimage_mem_nhds (lt_mem_nhds hlp), | |
| continuousAt_snd.preimage_mem_nhds (gt_mem_nhds (show l<l+1 by linarith)), | |
| continuousAt_fst.tendsto.eventually hcnear,hnW] with q hq hp hu hs hw | |
| exact ⟨hq,hp,hu.le,hs,hw⟩ | |
| refine ⟨U,hU,C,hC,?_⟩ | |
| intro q hq x0 uv huv φ hmono hφ hd hconc H hHjet hjet d | |
| obtain ⟨hqN,hqpos,hqmax,hqc,hqW⟩ := hq | |
| have hbase (e:Model n) : 2*δ*q.2*‖e‖^2≤W q e e := by | |
| have Hb := hW0 e | |
| have Habs := (abs_le.mp (bilinear_diag_sub_bound (W q) (W q0) e)).1 | |
| have Hsmall := mul_le_mul_of_nonneg_right hqW.le (sq_nonneg ‖e‖) | |
| have Hl : 2*δ*q.2≤ m/2 := by | |
| calc | |
| _ ≤ 2*δ*(l+1) := mul_le_mul_of_nonneg_left hqmax (by positivity) | |
| _ = m/2 := by dsimp [δ]; field_simp [show l+1≠0 by positivity]; ring | |
| have Hδ := mul_le_mul_of_nonneg_right Hl (sq_nonneg ‖e‖) | |
| nlinarith only [Hb,Habs,Hsmall,Hδ] | |
| exact HC x0 uv huv q.1.1 (c q.1.2) q.2 hqpos hqmax hqc hqN hbase | |
| φ hmono hφ hd hconc H hHjet hjet d | |
| end LocalBranchJetBound | |
| end WeakMTWTransport | |
| end | |
| end | |
| end | |
| end OAI | |