openAI-math-dataset / lean /OAI /Analysis /BiholderTransport /Regularity /CompactUniformSecondRemainder.lean
| import Mathlib | |
| import OAI.Analysis.BiholderTransport.LinearAlgebra.Calculus | |
| import OAI.Analysis.BiholderTransport.Coordinates.ChartCostJets | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open Set Filter Manifold Bundle Metric | |
| open scoped Topology ContDiff NNReal | |
| namespace WeakMTWTransport | |
| lemma compact_uniform_second_remainder {Q E : Type*} | |
| [NormedAddCommGroup Q] [NormedSpace ℝ Q] | |
| [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| {P : Set Q} (hP : IsCompact P) {F : Q → E → ℝ} | |
| (hF : ∀ q∈P, ContDiffAt ℝ ∞ (Function.uncurry F) (q,0)) : | |
| ∃ B>0, ∃ d>0, ∀ q∈P, ∀ z : E, ‖z‖<d → | |
| |F q z-F q 0-fderiv ℝ (F q) 0 z|≤B*‖z‖^2 := by | |
| let H := fun w : E×Q => fderiv ℝ (fderiv ℝ (F w.2)) w.1 | |
| have hC : ∀ q∈P, ContinuousAt H (0,q) := by | |
| intro q hq | |
| exact (ContDiffAt.partial_snd_fderiv_two (hF q hq)).continuousAt.comp | |
| (x := (0,q)) (f := fun w : E×Q => (w.2,w.1)) | |
| (continuousAt_snd.prodMk continuousAt_fst) | |
| obtain ⟨B,hB,HB⟩ := compact_eventually_fiberwise_bound hP | |
| (f := fun _ : Q => ()) continuous_const (g := fun _ : E => ()) | |
| (continuousAt_const (x := (0:E))) (h := H) (fun q hq _ => hC q hq) | |
| have HS : ∀ᶠ z : E in 𝓝 0, ∀ q∈P, ()=() → ContDiffAt ℝ 2 (F q) z := by | |
| apply compact_eventually_fiberwise hP (f := fun _ : Q => ()) continuous_const | |
| (g := fun _ : E => ()) (continuousAt_const (x := (0:E))) | |
| intro q hq _ | |
| have He := ((hF q hq).of_le | |
| (m := 2) (ENat.natCast_le_of_coe_top_le_withTop le_rfl 2)).eventually (by simp) | |
| have hc : ContinuousAt (fun w : E×Q => (w.2,w.1)) (0,q) := | |
| continuousAt_snd.prodMk continuousAt_fst | |
| filter_upwards [hc.tendsto.eventually He] with w hw | |
| exact hw.comp w.1 (contDiffAt_const.prodMk contDiffAt_id) | |
| obtain ⟨d,hd,hnear⟩ := Metric.eventually_nhds_iff.mp (HB.and HS) | |
| refine ⟨B,hB,d,hd,?_⟩ | |
| intro q hq z hz | |
| have Hrem := first_order_remainder_le_of_hessian_bound (convex_ball (0:E) d) hB.le | |
| (fun w hw => (hnear hw).2 q hq rfl) (fun w hw => (hnear hw).1 q hq rfl) | |
| (show (0:E)∈ball 0 d by simpa only [mem_ball,dist_self] using hd) | |
| (show z∈ball 0 d by simpa only [mem_ball,dist_zero_right] using hz) | |
| simpa only [sub_zero] using Hrem | |
| end WeakMTWTransport | |
| end | |
| end | |
| end | |
| end OAI | |