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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 | import Mathlib
import OAI.Analysis.BiholderTransport.Contact.ProximalSelectorDerivative
import OAI.Analysis.BiholderTransport.Calculus.SecondTaylorComposition
import OAI.Analysis.BiholderTransport.Convexity.ProximalProducer
namespace OAI
noncomputable section
open Set Filter
open scoped Topology ContDiff
namespace WeakMTWTransport
variable {E:Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]
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
end
end OAI
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