| import Mathlib |
| import OAI.Analysis.BiholderTransport.Calculus.SecondTaylorComposition |
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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 {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] |
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| lemma twist_hasFDerivWithinAt |
| {B : E×E → ℝ} {q g : E → E} {p : E} {S A : E →L[ℝ] E} {s : Set E} |
| (hB : ContDiffAt ℝ 2 B (0,p)) |
| (hxx : ∀ u v : E, fderiv ℝ (fderiv ℝ B) (0,p) (u,0) (v,0)=inner ℝ (S u) v) |
| (hqx : ∀ u v : E, fderiv ℝ (fderiv ℝ B) (0,p) (0,u) (v,0)= -inner ℝ u v) |
| (hq0 : q 0=p) (hqc : ContinuousAt q 0) (hg : HasFDerivWithinAt g A s 0) |
| (hcontact : ∀ᶠ x in 𝓝[s] 0, (fderiv ℝ B (x,q x)).comp |
| (ContinuousLinearMap.inl ℝ E E)= -(innerSL ℝ (g x))) |
| (hcontact0 : (fderiv ℝ B (0,p)).comp |
| (ContinuousLinearMap.inl ℝ E E)= -(innerSL ℝ (g 0))) : |
| HasFDerivWithinAt q (S+A) s 0 := by |
| let J : (E →L[ℝ] ℝ) →L[ℝ] E := |
| (InnerProductSpace.toDual ℝ E).symm.toContinuousLinearEquiv.toContinuousLinearMap |
| let D : E×E → (E →L[ℝ] ℝ) := fun z => |
| (fderiv ℝ B z).comp (ContinuousLinearMap.inl ℝ E E) |
| let H : E×E → E×E := fun z => (z.1,-J (D z)) |
| let L : E×E →L[ℝ] E×E := |
| (ContinuousLinearMap.fst ℝ E E).prod |
| (ContinuousLinearMap.snd ℝ E E-S.comp (ContinuousLinearMap.fst ℝ E E)) |
| let R : E×E →L[ℝ] E×E := |
| (ContinuousLinearMap.fst ℝ E E).prod |
| (ContinuousLinearMap.snd ℝ E E+S.comp (ContinuousLinearMap.fst ℝ E E)) |
| have hRL : Function.LeftInverse R L := by |
| intro z |
| change (z.1,z.2-S z.1+S z.1)=z |
| simp |
| have hLR : Function.RightInverse R L := by |
| intro z |
| change (z.1,z.2+S z.1-S z.1)=z |
| simp |
| let e : (E×E) ≃L[ℝ] (E×E) := ContinuousLinearEquiv.equivOfInverse L R hRL hLR |
| have hDc : ContDiffAt ℝ 1 D (0,p) := |
| (hB.fderiv_right (m := 1) (by norm_num)).clm_comp contDiffAt_const |
| have hHc : ContDiffAt ℝ 1 H (0,p) := contDiffAt_fst.prodMk |
| (J.contDiff.contDiffAt.comp (0,p) hDc).neg |
| have hDd := ((hB.fderiv_right (m := 1) (by norm_num)).differentiableAt |
| (by norm_num)).hasFDerivAt.clm_comp |
| (hasFDerivAt_const (ContinuousLinearMap.inl ℝ E E) (0,p)) |
| have hHd := (hasFDerivAt_fst (p := (0,p))).prodMk |
| (((J.hasFDerivAt).comp (0,p) hDd).neg) |
| have hHD : HasFDerivAt H (e : (E×E) →L[ℝ] (E×E)) (0,p) := by |
| apply hHd.congr_fderiv |
| apply ContinuousLinearMap.ext |
| intro z |
| apply Prod.ext |
| · rfl |
| · apply (InnerProductSpace.toDual ℝ E).injective |
| change (InnerProductSpace.toDual ℝ E) (-J _) = _ |
| rw [map_neg,show ∀ ξ : E →L[ℝ] ℝ, (InnerProductSpace.toDual ℝ E) (J ξ)=ξ from |
| fun ξ => (InnerProductSpace.toDual ℝ E).apply_symm_apply ξ] |
| ext v |
| change -((fderiv ℝ B (0,p)) (0,0)+fderiv ℝ (fderiv ℝ B) (0,p) z (v,0)) = |
| inner ℝ (z.2-S z.1) v |
| rw [show ((0:E),(0:E))=(0:E×E) by rfl,map_zero,zero_add] |
| rw [show z=(z.1,0)+(0,z.2) by ext <;> simp,map_add,add_apply,hxx,hqx] |
| simp only [inner_sub_left,Prod.fst_add,Prod.snd_add,add_zero,zero_add] |
| ring |
| have hstrict := hHc.hasStrictFDerivAt' hHD (by norm_num) |
| have J_inner (a : E) : J (innerSL ℝ a)=a := by |
| apply (InnerProductSpace.toDual ℝ E).injective |
| rw [show (InnerProductSpace.toDual ℝ E) (J (innerSL ℝ a))=innerSL ℝ a from |
| (InnerProductSpace.toDual ℝ E).apply_symm_apply _] |
| rfl |
| have hH0 : H (0,p)=(0,g 0) := by |
| change (0,-J ((fderiv ℝ B (0,p)).comp (ContinuousLinearMap.inl ℝ E E)))=(0,g 0) |
| rw [hcontact0,map_neg,J_inner,neg_neg] |
| let inv := hstrict.localInverse H e (0,p) |
| have hi : HasFDerivAt inv (e.symm : (E×E) →L[ℝ] (E×E)) (0,g 0) := by |
| rw [←hH0] |
| exact hstrict.to_localInverse.hasFDerivAt |
| have hcomp := hi.comp_hasFDerivWithinAt 0 ((hasFDerivAt_id (𝕜 := ℝ) (0:E)).hasFDerivWithinAt.prodMk hg) |
| have hfinal := (hasFDerivAt_snd (p := inv (0,g 0))).comp_hasFDerivWithinAt 0 hcomp |
| have heq : ∀ᶠ x in 𝓝[s] 0, q x=(inv (x,g x)).2 := by |
| have hnear := (continuousAt_id.prodMk hqc).tendsto.eventually |
| (by rw [hq0]; exact hstrict.eventually_left_inverse) |
| filter_upwards [hcontact,mem_nhdsWithin_of_mem_nhds hnear] with x hx hxi |
| have Hx : H (x,q x)=(x,g x) := by |
| change (x,-J ((fderiv ℝ B (x,q x)).comp (ContinuousLinearMap.inl ℝ E E)))=(x,g x) |
| rw [hx,map_neg,J_inner,neg_neg] |
| change inv (H (x,q x))=(x,q x) at hxi |
| rw [Hx] at hxi |
| exact (congrArg Prod.snd hxi).symm |
| have heq0 : q 0=(inv (0,g 0)).2 := by |
| change q 0=(hstrict.localInverse H e (0,p) (0,g 0)).2 |
| rw [←hH0,hstrict.localInverse_apply_image,hq0] |
| apply (hfinal.congr_of_eventuallyEq heq heq0).congr_fderiv |
| ext h |
| change A h+S h=S h+A h |
| exact add_comm _ _ |
|
|
| end WeakMTWTransport |
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| end |
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| end OAI |
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