| import Mathlib |
| import OAI.Analysis.BiholderTransport.Calculus.FullSet |
|
|
| namespace OAI |
|
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| noncomputable section |
|
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| namespace WeakMTWTransport |
|
|
| section |
| open Set Filter |
| open scoped Topology ContDiff |
|
|
| variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [NormedAddCommGroup F] [NormedSpace ℝ F] |
|
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| 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] |
|
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| 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)) |
|
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| 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 |
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| end |
|
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| end OAI |
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|
