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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 | 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
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