| import Mathlib |
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| namespace OAI |
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| section |
| section |
| noncomputable section |
| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
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| namespace WeakMTWTransport |
| section CriticalChain |
| variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [NormedAddCommGroup F] [NormedSpace ℝ F] |
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| lemma second_fderiv_comp_at_critical {f : F → ℝ} {g : E → F} {p : E} |
| (hf : ContDiffAt ℝ 2 f (g p)) (hg : ContDiffAt ℝ 2 g p) |
| (hzero : fderiv ℝ f (g p)=0) (v w : E) : |
| fderiv ℝ (fderiv ℝ (f ∘ g)) p v w= |
| fderiv ℝ (fderiv ℝ f) (g p) (fderiv ℝ g p v) (fderiv ℝ g p w) := by |
| have hdf := (hf.fderiv_right (m := 1) (by norm_num)).differentiableAt (by norm_num) |
| have hdg := (hg.fderiv_right (m := 1) (by norm_num)).differentiableAt (by norm_num) |
| have hn : ∀ᶠ a in 𝓝 p, DifferentiableAt ℝ g a ∧ DifferentiableAt ℝ f (g a) := by |
| have hnG := (hg.of_le (m := 1) (by norm_num)).eventually (by norm_num) |
| have hnF := hg.continuousAt.eventually ((hf.of_le (m := 1) (by norm_num)).eventually (by norm_num)) |
| filter_upwards [hnG,hnF] with a ha hb |
| exact ⟨ha.differentiableAt (by norm_num),hb.differentiableAt (by norm_num)⟩ |
| have heq : fderiv ℝ (f ∘ g) =ᶠ[𝓝 p] |
| (fun a => (fderiv ℝ f (g a)).comp (fderiv ℝ g a)) := |
| hn.mono (fun a ha => fderiv_comp a ha.2 ha.1) |
| have hD := (hdf.hasFDerivAt.comp p (hg.differentiableAt (by norm_num)).hasFDerivAt).clm_comp hdg.hasFDerivAt |
| have H := congrArg (fun A : E →L[ℝ] E →L[ℝ] ℝ => A v w) (heq.fderiv_eq.trans hD.fderiv) |
| simpa only [_root_.add_apply,ContinuousLinearMap.comp_apply,ContinuousLinearMap.flip_apply, ContinuousLinearMap.compL_apply, Function.comp_apply, |
| hzero, ContinuousLinearMap.zero_comp,_root_.zero_apply,zero_add] using H |
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| end CriticalChain |
| end WeakMTWTransport |
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| end |
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| end |
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| end |
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| end OAI |
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