| import Mathlib |
| import OAI.Analysis.BiholderTransport.Contact.ProximalSelectorDerivative |
| import OAI.Analysis.BiholderTransport.Calculus.SecondTaylorComposition |
| import OAI.Analysis.BiholderTransport.Convexity.ProximalProducer |
|
|
| namespace OAI |
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|
| noncomputable section |
| open Set Filter |
| open scoped Topology ContDiff |
|
|
| namespace WeakMTWTransport |
| variable {E:Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] |
|
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| lemma semiconvex_phase_graph_derivative |
| {f:E → ℝ} {Y:E → E} {P:E → E →L[ℝ] ℝ} |
| {Φ:(E×(E →L[ℝ] ℝ)) → E×(E →L[ℝ] ℝ)} |
| {x p:E} {A R:E →L[ℝ] E} {H:E →L[ℝ] E →L[ℝ] ℝ} |
| {D:(E×(E →L[ℝ] ℝ)) →L[ℝ] E×(E →L[ℝ] ℝ)} |
| (hA:∀v w,inner ℝ (A v) w=inner ℝ v (A w)) |
| (hexp:HasQuadraticExpansion (fun h=>f (Y x+h)) p A) |
| {r:ℝ} (hr:0<r) (K:ℝ) |
| (hY:HasFDerivAt Y R x) (hP:HasFDerivAt P H x) |
| (hΦ:HasFDerivAt Φ D (x,P x)) |
| (hbase:∀ᶠ z in 𝓝 x,(Φ (z,P z)).1=Y z) |
| (hsub:∀ᶠ z in 𝓝 x, |
| IsSemiconvexSubgradientOn f (Metric.ball (Y x) r) K (Y z) |
| ((InnerProductSpace.toDual ℝ E).symm (Φ (z,P z)).2)) : |
| D.comp ((ContinuousLinearMap.id ℝ E).prod H)= |
| R.prod (((innerSL ℝ).comp A).comp R) := by |
| let ξ:=fun z=>(InnerProductSpace.toDual ℝ E).symm (Φ (z,P z)).2 |
| have hξ:ξ x=p:=semiconvex_subgradient_eq_at_expansion hA hexp hr K hsub.self_of_nhds |
| have hd:=semiconvex_subgradient_along_selector hA hexp hr K hY hξ hsub |
| have hs:HasFDerivAt (fun z=>(Φ (z,P z)).2) (((innerSL ℝ).comp A).comp R) x := by |
| have hh:HasFDerivAt (fun z=>(innerSL ℝ) (ξ z)) ((innerSL ℝ).comp (A.comp R)) x := |
| (innerSL ℝ).hasFDerivAt.comp x hd |
| have he:(fun z=>(Φ (z,P z)).2)=(fun z=>(innerSL ℝ) (ξ z)) := by |
| funext z |
| exact ((InnerProductSpace.toDual ℝ E).apply_symm_apply (Φ (z,P z)).2).symm |
| rw [he] |
| simpa only [ContinuousLinearMap.comp_assoc] using hh |
| have hb:HasFDerivAt (fun z=>(Φ (z,P z)).1) R x := hY.congr_of_eventuallyEq hbase |
| have hboth:=hb.prodMk hs |
| have hchain:=hΦ.comp x ((hasFDerivAt_id x).prodMk hP) |
| exact hchain.unique hboth |
|
|
| end WeakMTWTransport |
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| end |
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| end OAI |
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