| import Mathlib |
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| namespace OAI |
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| noncomputable section |
| open Set Filter Function |
| open scoped Topology ContDiff InnerProductSpace |
| namespace YauCounterexamples |
| variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [NormedAddCommGroup F] [NormedSpace ℝ F] |
| lemma directional_linear_comp (L : E →L[ℝ] F) {f : F → ℂ} |
| (hf : Differentiable ℝ f) (x v : E) : |
| fderiv ℝ (f ∘ L) x v = fderiv ℝ f (L x) (L v) := by |
| rw [fderiv_comp x (hf _) L.differentiableAt,L.fderiv] |
| rfl |
| lemma second_linear_comp (L : E →L[ℝ] F) {f : F → ℂ} |
| (hf : ContDiff ℝ ∞ f) (x v w : E) : |
| fderiv ℝ (fun y => fderiv ℝ (f ∘ L) y w) x v = |
| fderiv ℝ (fun y => fderiv ℝ f y (L w)) (L x) (L v) := by |
| simp_rw [directional_linear_comp L (hf.differentiable (by simp))] |
| exact directional_linear_comp L |
| (((hf.fderiv_right (m:=∞) (by simp)).clm_apply contDiff_const).differentiable (by simp)) x v |
| lemma directional_mul_complex {f g : E → ℂ} (hf : Differentiable ℝ f) |
| (hg : Differentiable ℝ g) (x v : E) : |
| fderiv ℝ (fun y => f y*g y) x v = fderiv ℝ f x v*g x+f x*fderiv ℝ g x v := by |
| have hh := ((hf x).hasFDerivAt.mul (hg x).hasFDerivAt).fderiv |
| change fderiv ℝ (f*g) x v = _ |
| rw [hh] |
| simp [smul_eq_mul,mul_comm,add_comm] |
| lemma second_mul_complex {f g : E → ℂ} (hf : ContDiff ℝ ∞ f) |
| (hg : ContDiff ℝ ∞ g) (x v w : E) : |
| fderiv ℝ (fun y => fderiv ℝ (fun z => f z*g z) y w) x v = |
| fderiv ℝ (fun y => fderiv ℝ f y w) x v*g x + |
| fderiv ℝ f x w*fderiv ℝ g x v + |
| fderiv ℝ f x v*fderiv ℝ g x w + |
| f x*fderiv ℝ (fun y => fderiv ℝ g y w) x v := by |
| have hd := hf.differentiable (by simp) |
| have he := hg.differentiable (by simp) |
| have hd' := ((hf.fderiv_right (m:=∞) (by simp)).clm_apply |
| (contDiff_const : ContDiff ℝ ∞ (fun (_:E) => w))).differentiable (by simp) |
| have he' := ((hg.fderiv_right (m:=∞) (by simp)).clm_apply |
| (contDiff_const : ContDiff ℝ ∞ (fun (_:E) => w))).differentiable (by simp) |
| simp_rw [directional_mul_complex hd he] |
| change fderiv ℝ ((fun y => fderiv ℝ f y w)*g+f*(fun y => fderiv ℝ g y w)) x v = _ |
| rw [fderiv_add ((hd' x).mul (he x)) ((hd x).mul (he' x))] |
| simp only [add_apply] |
| change fderiv ℝ (fun y => fderiv ℝ f y w*g y) x v + fderiv ℝ (fun y => f y*fderiv ℝ g y w) x v = _ |
| rw [directional_mul_complex hd' he,directional_mul_complex hd he'] |
| ring |
| end YauCounterexamples |
| end |
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| end OAI |
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