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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 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 | import Mathlib
import OAI.Analysis.BiholderTransport.Calculus.SecondTaylorComposition
namespace OAI
noncomputable section
open Set Filter
open scoped Topology ContDiff
namespace WeakMTWTransport
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]
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
end
end OAI
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