File size: 4,190 Bytes
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 | import Mathlib
import OAI.Analysis.BiholderTransport.Calculus.FullSet
namespace OAI
noncomputable section
namespace WeakMTWTransport
section
open Set Filter
open scoped Topology ContDiff
variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F]
lemma second_fderiv_comp_affine {f : F → ℝ} (L : E →L[ℝ] F) (b : F) (x : E)
(hf : ContDiffAt ℝ 2 f (b+L x)) (v w : E) :
fderiv ℝ (fderiv ℝ (fun h => f (b+L h))) x v w =
fderiv ℝ (fderiv ℝ f) (b+L x) (L v) (L w) := by
have hL : ∀ y:E, HasFDerivAt (fun h : E => b+L h) L y :=
fun y => (L.hasFDerivAt (x := y)).const_add b
have hd := ((hf.fderiv_right (m := 1) (by norm_num)).differentiableAt
(by norm_num)).hasFDerivAt.comp (f := fun h : E => b+L h) x (hL x)
have H := hd.clm_comp (hasFDerivAt_const L x)
have heq : (fun h => fderiv ℝ f (b+L h) |>.comp L) =ᶠ[𝓝 x]
(fderiv ℝ (fun h => f (b+L h))) := by
have hn := (hL x).continuousAt.eventually (hf.eventually (by norm_num))
filter_upwards [hn] with h hh
exact ((hh.differentiableAt (by norm_num)).hasFDerivAt.comp
(f := fun h : E => b+L h) h (hL h)).fderiv.symm
have he := congrArg (fun B : E →L[ℝ] E →L[ℝ] ℝ => B v w)
(H.congr_of_eventuallyEq heq.symm).fderiv
simpa only [ContinuousLinearMap.comp_apply,ContinuousLinearMap.compL_apply,
ContinuousLinearMap.flip_apply,map_zero,ContinuousLinearMap.comp_zero,
zero_add,add_apply,zero_apply] using he
lemma hasSecondTaylor_shift {f : E → ℝ} {x : E} (hf : ContDiffAt ℝ 2 f x) :
HasSecondTaylor (fun h => f (x+h)) (fderiv ℝ f x)
(fderiv ℝ (fderiv ℝ f) x) := by
have hc : ContDiffAt ℝ 2 (fun h => f (x+h)) 0 := by
exact (show ContDiffAt ℝ 2 f (x+(0:E)) by simpa using hf).comp 0
(contDiffAt_const.add contDiffAt_id)
have H := hasSecondTaylor_of_contDiffAt hc
have hD : fderiv ℝ (fun h => f (x+h)) 0=fderiv ℝ f x := by
have hd := (hf.differentiableAt (by norm_num)).hasFDerivAt
have hl := (hasFDerivAt_id (𝕜 := ℝ) (0:E)).const_add x
simpa only [add_zero,ContinuousLinearMap.comp_id,Function.comp_def] using
((show HasFDerivAt f (fderiv ℝ f x) (x+(0:E)) by simpa using hd).comp
(f := fun h : E => x+h) 0 hl).fderiv
have hDD : fderiv ℝ (fderiv ℝ (fun h => f (x+h))) 0=
fderiv ℝ (fderiv ℝ f) x := by
ext v w
simpa only [ContinuousLinearMap.id_apply,add_zero] using
second_fderiv_comp_affine (ContinuousLinearMap.id ℝ E) x 0
(by simpa using hf) v w
rwa [hD,hDD] at H
end
open Set Filter
open scoped Topology ContDiff
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
lemma half_norm_sq_hasFDerivAt (p : E) :
HasFDerivAt (fun v : E => ‖v‖^2/2) (innerSL ℝ p) p := by
have H := ((hasFDerivAt_id (𝕜 := ℝ) p).norm_sq).const_smul (1/2:ℝ)
convert! H using 1
· ext v; simp only [Pi.smul_apply,smul_eq_mul,id_eq]; ring
· ext a
simp only [ContinuousLinearMap.comp_apply,ContinuousLinearMap.id_apply,
smul_apply,smul_eq_mul,innerSL_apply_apply,id_eq]
ring
lemma half_norm_sq_fderiv :
fderiv ℝ (fun v : E => ‖v‖^2/2) = innerSL ℝ := by
funext p
exact (half_norm_sq_hasFDerivAt p).fderiv
lemma half_norm_sq_second_fderiv (p a d : E) :
fderiv ℝ (fderiv ℝ (fun v : E => ‖v‖^2/2)) p a d=inner ℝ a d := by
rw [half_norm_sq_fderiv]
rw [(innerSL ℝ).hasFDerivAt.fderiv]
rfl
lemma half_norm_sq_contDiff : ContDiff ℝ ∞ (fun v : E => ‖v‖^2/2) := by
simpa only [one_div,Pi.smul_apply,smul_eq_mul,div_eq_mul_inv,mul_comm,one_mul] using
((contDiff_norm_sq (𝕜 := ℝ) (E := E) (n := ∞)).const_smul (1/2:ℝ))
lemma norm_square_hasSecondTaylor (p : E) :
HasSecondTaylor (fun h : E => ‖p+h‖^2/2) (innerSL ℝ p) (innerSL ℝ) := by
have hc : ContDiffAt ℝ 2 (fun v : E => ‖v‖^2/2) p :=
((half_norm_sq_contDiff (E := E)).of_le
(m := 2) (ENat.natCast_le_of_coe_top_le_withTop le_rfl 2)).contDiffAt
have H := hasSecondTaylor_shift hc
simpa only [half_norm_sq_fderiv,(innerSL ℝ).hasFDerivAt.fderiv] using H
end WeakMTWTransport
end
end OAI
|