| import Mathlib |
| import OAI.Analysis.BiholderTransport.LinearAlgebra.MovingHessianLimit |
|
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| namespace OAI |
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| noncomputable section |
| open Set Filter |
| open scoped Topology ContDiff |
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| namespace WeakMTWTransport |
| variable {Q E F : Type*} [NormedAddCommGroup Q] [NormedSpace ℝ Q] |
| [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [NormedAddCommGroup F] [NormedSpace ℝ F] |
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| 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)] |
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| 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] |
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| end WeakMTWTransport |
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| end |
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| end OAI |
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