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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 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 | import Mathlib
import OAI.Analysis.BiholderTransport.LinearAlgebra.MovingHessianLimit
namespace OAI
noncomputable section
open Set Filter
open scoped Topology ContDiff
namespace WeakMTWTransport
variable {Q E F : Type*} [NormedAddCommGroup Q] [NormedSpace ℝ Q]
[NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F]
lemma secondJetPullback_sum_nonpos
{j k : F →L[ℝ] ℝ} {A B : F →L[ℝ] F →L[ℝ] ℝ}
{m : E →L[ℝ] F} {r : E →L[ℝ] E →L[ℝ] F}
(hj : j+k=0) (hAB : ∀ d:F,A d d+B d d≤0) (e:E) :
secondJetPullback j A m r e e+secondJetPullback k B m r e e≤0 := by
have hs := congrArg (fun L:F →L[ℝ] ℝ=>L (r e e)) hj
simp only [add_apply,zero_apply] at hs
simp only [secondJetPullback_apply]
linarith only [hs,hAB (m e)]
lemma first_fixed_prefix_matrix_limit {ι : Type*}
{W : Q → E → ℝ} {g : Q → E → F} {f h : F → ℝ}
{b : ℕ → Q} {p : ℕ → E} {b₀ : Q} {p₀ : E}
{v : ℕ → ι → E} {v₀ : ι → E}
{jf jh : F →L[ℝ] ℝ} {Af Ah : F →L[ℝ] F →L[ℝ] ℝ}
(hW : ContDiffAt ℝ ∞ (Function.uncurry W) (b₀,p₀))
(hg : ContDiffAt ℝ ∞ (Function.uncurry g) (b₀,p₀))
(hb : Tendsto b atTop (𝓝 b₀)) (hp : Tendsto p atTop (𝓝 p₀))
(hv : ∀ i,Tendsto (fun k=>v k i) atTop (𝓝 (v₀ i)))
(hjf : Tendsto (fun k=>fderiv ℝ f (g (b k) (p k))) atTop (𝓝 jf))
(hjh : Tendsto (fun k=>fderiv ℝ h (g (b k) (p k))) atTop (𝓝 jh))
(hAf : Tendsto (fun k=>fderiv ℝ (fderiv ℝ f) (g (b k) (p k))) atTop (𝓝 Af))
(hAh : Tendsto (fun k=>fderiv ℝ (fderiv ℝ h) (g (b k) (p k))) atTop (𝓝 Ah))
(hDf : ∀ᶠ k in atTop,(∀ᶠ y in 𝓝 (g (b k) (p k)),DifferentiableAt ℝ f y) ∧
DifferentiableAt ℝ (fderiv ℝ f) (g (b k) (p k)))
(hDh : ∀ᶠ k in atTop,(∀ᶠ y in 𝓝 (g (b k) (p k)),DifferentiableAt ℝ h y) ∧
DifferentiableAt ℝ (fderiv ℝ h) (g (b k) (p k)))
(hgrad : jf+jh=0) (hquad : ∀ d:F,Af d d+Ah d d≤0)
(hpos : ∀ᶠ k in atTop,∀ d:E,
0≤(fderiv ℝ (fderiv ℝ (W (b k))) (p k)-
fderiv ℝ (fderiv ℝ (fun q=>f (g (b k) q))) (p k)) d d)
(hker : ∀ᶠ k in atTop,∀ i:ι,∀ d:E,
(fderiv ℝ (fderiv ℝ (W (b k))) (p k)-
fderiv ℝ (fderiv ℝ (fun q=>f (g (b k) q))) (p k)) (v k i-p k) d=0) :
∃ H L : E →L[ℝ] E →L[ℝ] ℝ,
Tendsto (fun k=>fderiv ℝ (fderiv ℝ (W (b k))) (p k)+
fderiv ℝ (fderiv ℝ (fun q=>h (g (b k) q))) (p k)) atTop (𝓝 H) ∧
Tendsto (fun k=>fderiv ℝ (fderiv ℝ (W (b k))) (p k)-
fderiv ℝ (fderiv ℝ (fun q=>f (g (b k) q))) (p k)) atTop (𝓝 L) ∧
(∀ d:E,0≤L d d) ∧ (∀ i:ι,∀ d:E,L (v₀ i-p₀) d=0) ∧
(∀ d:E,H d d≤L d d) := by
let T := fderiv ℝ (fderiv ℝ (W b₀)) p₀
let m := fderiv ℝ (g b₀) p₀
let r := fderiv ℝ (fderiv ℝ (g b₀)) p₀
let F₀ := secondJetPullback jf Af m r
let G₀ := secondJetPullback jh Ah m r
have hT : Tendsto (fun k=>fderiv ℝ (fderiv ℝ (W (b k))) (p k)) atTop (𝓝 T) :=
((ContDiffAt.partial_snd_fderiv_two hW).continuousAt.tendsto).comp (hb.prodMk_nhds hp)
have hF := moving_rough_hessian_limit hg hb hp hjf hAf hDf
have hG := moving_rough_hessian_limit hg hb hp hjh hAh hDh
have hH := hT.add hG
have hL := hT.sub hF
refine ⟨T+G₀,T-F₀,hH,hL,fun d=>bilinear_limit_nonneg hL hpos d,?_,?_⟩
· intro i d
exact bilinear_limit_kernel hL ((hv i).sub hp) (hker.mono (fun k hk=>hk i)) d
· intro d
have H := secondJetPullback_sum_nonpos (m := m) (r := r) hgrad hquad d
change T d d+G₀ d d≤T d d-F₀ d d
linarith only [H]
end WeakMTWTransport
end
end OAI
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