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695bd70 | 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 | import Mathlib
namespace OAI
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
end OAI
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