| import Mathlib |
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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 F:Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [NormedAddCommGroup F] [NormedSpace ℝ F] |
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| lemma fderiv_translate_at_zero {f:E → F} {x:E} (hf:DifferentiableAt ℝ f x) (c:F) : |
| fderiv ℝ (fun h=>f (x+h)-c) 0=fderiv ℝ f x := by |
| rw [fderiv_sub_const] |
| have H:=(show HasFDerivAt f (fderiv ℝ f x) (x+id 0) from by simpa only [id_eq,add_zero] using hf.hasFDerivAt).comp 0 ((hasFDerivAt_id (0:E)).const_add x) |
| simpa only [Function.comp_def,add_zero,ContinuousLinearMap.comp_id] using H.fderiv |
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| lemma second_fderiv_translate_at_zero {f:E → F} {x:E} (hf:ContDiffAt ℝ 2 f x) (c:F) : |
| fderiv ℝ (fderiv ℝ (fun h=>f (x+h)-c)) 0=fderiv ℝ (fderiv ℝ f) x := by |
| have ht:Tendsto (fun h:E=>x+h) (𝓝 0) (𝓝 x):=by |
| have hc : ContinuousAt (fun h:E=>x+h) 0 := continuousAt_const.add continuousAt_id |
| have H:=hc.tendsto |
| rw [add_zero] at H |
| exact H |
| have hd:∀ᶠ y in 𝓝 x,DifferentiableAt ℝ f y:= |
| (hf.eventually (by norm_num)).mono (fun _ h=>h.differentiableAt (by norm_num)) |
| have he:fderiv ℝ (fun h=>f (x+h)-c)=ᶠ[𝓝 (0:E)] (fun h=>fderiv ℝ f (x+h)):=by |
| filter_upwards [ht.eventually hd] with h hh |
| rw [fderiv_sub_const] |
| have H:=hh.hasFDerivAt.comp h ((hasFDerivAt_id h).const_add x) |
| simpa only [Function.comp_def,ContinuousLinearMap.comp_id] using H.fderiv |
| rw [he.fderiv_eq] |
| have hfd:DifferentiableAt ℝ (fderiv ℝ f) x:= |
| (hf.fderiv_right (m:=1) (by norm_num)).differentiableAt (by norm_num) |
| have H:=(show HasFDerivAt (fderiv ℝ f) (fderiv ℝ (fderiv ℝ f) x) (x+id 0) from by simpa only [id_eq,add_zero] using hfd.hasFDerivAt).comp 0 ((hasFDerivAt_id (0:E)).const_add x) |
| simpa only [Function.comp_def,add_zero,ContinuousLinearMap.comp_id] using H.fderiv |
| end WeakMTWTransport |
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| end |
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| end OAI |
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