l3lab/ntp-mathlib huggingface dataset:
/-- Assume that `f` is differentiable inside a convex set `s`, and that its derivative `f'` is
differentiable at a point `x`. Then, given two vectors `v` and `w` pointing inside `s`, one
has `f'' v w = f'' w v`. Superseded by `Convex.second_derivative_within_at_symmetric`, which
removes the assumption that `v` and `w` point inside `s`. -/
theorem Convex.second_derivative_within_at_symmetric_of_mem_interior {v w : E}
    (h4v : x + (4 : ℝ) • v ∈ interior s) (h4w : x + (4 : ℝ) • w ∈ interior s) :
    f'' w v = f'' v w := by
  have A : (fun h : ℝ => h ^ 2 • (f'' w v - f'' v w)) =o[𝓝[>] 0] fun h => h ^ 2 := by
    convert (s_conv.isLittleO_alternate_sum_square hf xs hx h4v h4w).sub
      (s_conv.isLittleO_alternate_sum_square hf xs hx h4w h4v) using 1
    ext h
    simp only [add_comm, smul_add, smul_sub]
    abel
  have B : (fun _ : ℝ => f'' w v - f'' v w) =o[𝓝[>] 0] fun _ => (1 : ℝ) := by
    have : (fun h : ℝ => 1 / h ^ 2) =O[𝓝[>] 0] fun h => 1 / h ^ 2 := isBigO_refl _ _
    have C := this.smul_isLittleO A
    apply C.congr' _ _
    · filter_upwards [self_mem_nhdsWithin]
      intro h (hpos : 0 < h)
      rw [← one_smul ℝ (f'' w v - f'' v w), smul_smul, smul_smul]
      

Proof state:
case h
E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set E
s_conv : Convex ℝ s
f : E → F
f' : E → E →L[ℝ] F
f'' : E →L[ℝ] E →L[ℝ] F
hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x
x : E
xs : x ∈ s
hx : HasFDerivWithinAt f' f'' (interior s) x
v w : E
h4v : x + 4 • v ∈ interior s
h4w : x + 4 • w ∈ interior s
A : (fun h => h ^ 2 • ((f'' w) v - (f'' v) w)) =o[𝓝[>] 0] fun h => h ^ 2
this : (fun h => 1 / h ^ 2) =O[𝓝[>] 0] fun h => 1 / h ^ 2
C : (fun x => (1 / x ^ 2) • x ^ 2 • ((f'' w) v - (f'' v) w)) =o[𝓝[>] 0] fun x => (1 / x ^ 2) • x ^ 2
h : ℝ
hpos : 0 < h
⊢ (1 / h ^ 2 * h ^ 2 * 1) • ((f'' w) v - (f'' v) w) = 1 • ((f'' w) v - (f'' v) w)

Next tactic:
congr 1
/-- Assume that `f` is differentiable inside a convex set `s`, and that its derivative `f'` is
differentiable at a point `x`. Then, given two vectors `v` and `w` pointing inside `s`, one
has `f'' v w = f'' w v`. Superseded by `Convex.second_derivative_within_at_symmetric`, which
removes the assumption that `v` and `w` point inside `s`. -/
theorem Convex.second_derivative_within_at_symmetric_of_mem_interior {v w : E}
    (h4v : x + (4 : ℝ) • v ∈ interior s) (h4w : x + (4 : ℝ) • w ∈ interior s) :
    f'' w v = f'' v w := by
  have A : (fun h : ℝ => h ^ 2 • (f'' w v - f'' v w)) =o[𝓝[>] 0] fun h => h ^ 2 := by
    convert (s_conv.isLittleO_alternate_sum_square hf xs hx h4v h4w).sub
      (s_conv.isLittleO_alternate_sum_square hf xs hx h4w h4v) using 1
    ext h
    simp only [add_comm, smul_add, smul_sub]
    abel
  have B : (fun _ : ℝ => f'' w v - f'' v w) =o[𝓝[>] 0] fun _ => (1 : ℝ) := by
    have : (fun h : ℝ => 1 / h ^ 2) =O[𝓝[>] 0] fun h => 1 / h ^ 2 := isBigO_refl _ _
    have C := this.smul_isLittleO A
    apply C.congr' _ _
    · filter_upwards [self_mem_nhdsWithin]
      intro h (hpos : 0 < h)
      rw [← one_smul ℝ (f'' w v - f'' v w), smul_smul, smul_smul]
      congr 1
      

Proof state:
case h.e_a
E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set E
s_conv : Convex ℝ s
f : E → F
f' : E → E →L[ℝ] F
f'' : E →L[ℝ] E →L[ℝ] F
hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x
x : E
xs : x ∈ s
hx : HasFDerivWithinAt f' f'' (interior s) x
v w : E
h4v : x + 4 • v ∈ interior s
h4w : x + 4 • w ∈ interior s
A : (fun h => h ^ 2 • ((f'' w) v - (f'' v) w)) =o[𝓝[>] 0] fun h => h ^ 2
this : (fun h => 1 / h ^ 2) =O[𝓝[>] 0] fun h => 1 / h ^ 2
C : (fun x => (1 / x ^ 2) • x ^ 2 • ((f'' w) v - (f'' v) w)) =o[𝓝[>] 0] fun x => (1 / x ^ 2) • x ^ 2
h : ℝ
hpos : 0 < h
⊢ 1 / h ^ 2 * h ^ 2 * 1 = 1

Next tactic:
field_simp [LT.lt.ne' hpos]
/-- Assume that `f` is differentiable inside a convex set `s`, and that its derivative `f'` is
differentiable at a point `x`. Then, given two vectors `v` and `w` pointing inside `s`, one
has `f'' v w = f'' w v`. Superseded by `Convex.second_derivative_within_at_symmetric`, which
removes the assumption that `v` and `w` point inside `s`. -/
theorem Convex.second_derivative_within_at_symmetric_of_mem_interior {v w : E}
    (h4v : x + (4 : ℝ) • v ∈ interior s) (h4w : x + (4 : ℝ) • w ∈ interior s) :
    f'' w v = f'' v w := by
  have A : (fun h : ℝ => h ^ 2 • (f'' w v - f'' v w)) =o[𝓝[>] 0] fun h => h ^ 2 := by
    convert (s_conv.isLittleO_alternate_sum_square hf xs hx h4v h4w).sub
      (s_conv.isLittleO_alternate_sum_square hf xs hx h4w h4v) using 1
    ext h
    simp only [add_comm, smul_add, smul_sub]
    abel
  have B : (fun _ : ℝ => f'' w v - f'' v w) =o[𝓝[>] 0] fun _ => (1 : ℝ) := by
    have : (fun h : ℝ => 1 / h ^ 2) =O[𝓝[>] 0] fun h => 1 / h ^ 2 := isBigO_refl _ _
    have C := this.smul_isLittleO A
    apply C.congr' _ _
    · filter_upwards [self_mem_nhdsWithin]
      intro h (hpos : 0 < h)
      rw [← one_smul ℝ (f'' w v - f'' v w), smul_smul, smul_smul]
      congr 1
      field_simp [LT.lt.ne' hpos]
    · 

Proof state:
E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set E
s_conv : Convex ℝ s
f : E → F
f' : E → E →L[ℝ] F
f'' : E →L[ℝ] E →L[ℝ] F
hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x
x : E
xs : x ∈ s
hx : HasFDerivWithinAt f' f'' (interior s) x
v w : E
h4v : x + 4 • v ∈ interior s
h4w : x + 4 • w ∈ interior s
A : (fun h => h ^ 2 • ((f'' w) v - (f'' v) w)) =o[𝓝[>] 0] fun h => h ^ 2
this : (fun h => 1 / h ^ 2) =O[𝓝[>] 0] fun h => 1 / h ^ 2
C : (fun x => (1 / x ^ 2) • x ^ 2 • ((f'' w) v - (f'' v) w)) =o[𝓝[>] 0] fun x => (1 / x ^ 2) • x ^ 2
⊢ (fun x => (1 / x ^ 2) • x ^ 2) =ᶠ[𝓝[>] 0] fun x => 1

Next tactic:
filter_upwards [self_mem_nhdsWithin] with h (hpos : 0 < h)
/-- Assume that `f` is differentiable inside a convex set `s`, and that its derivative `f'` is
differentiable at a point `x`. Then, given two vectors `v` and `w` pointing inside `s`, one
has `f'' v w = f'' w v`. Superseded by `Convex.second_derivative_within_at_symmetric`, which
removes the assumption that `v` and `w` point inside `s`. -/
theorem Convex.second_derivative_within_at_symmetric_of_mem_interior {v w : E}
    (h4v : x + (4 : ℝ) • v ∈ interior s) (h4w : x + (4 : ℝ) • w ∈ interior s) :
    f'' w v = f'' v w := by
  have A : (fun h : ℝ => h ^ 2 • (f'' w v - f'' v w)) =o[𝓝[>] 0] fun h => h ^ 2 := by
    convert (s_conv.isLittleO_alternate_sum_square hf xs hx h4v h4w).sub
      (s_conv.isLittleO_alternate_sum_square hf xs hx h4w h4v) using 1
    ext h
    simp only [add_comm, smul_add, smul_sub]
    abel
  have B : (fun _ : ℝ => f'' w v - f'' v w) =o[𝓝[>] 0] fun _ => (1 : ℝ) := by
    have : (fun h : ℝ => 1 / h ^ 2) =O[𝓝[>] 0] fun h => 1 / h ^ 2 := isBigO_refl _ _
    have C := this.smul_isLittleO A
    apply C.congr' _ _
    · filter_upwards [self_mem_nhdsWithin]
      intro h (hpos : 0 < h)
      rw [← one_smul ℝ (f'' w v - f'' v w), smul_smul, smul_smul]
      congr 1
      field_simp [LT.lt.ne' hpos]
    · filter_upwards [self_mem_nhdsWithin] with h (hpos : 0 < h)
      

Proof state:
case h
E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set E
s_conv : Convex ℝ s
f : E → F
f' : E → E →L[ℝ] F
f'' : E →L[ℝ] E →L[ℝ] F
hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x
x : E
xs : x ∈ s
hx : HasFDerivWithinAt f' f'' (interior s) x
v w : E
h4v : x + 4 • v ∈ interior s
h4w : x + 4 • w ∈ interior s
A : (fun h => h ^ 2 • ((f'' w) v - (f'' v) w)) =o[𝓝[>] 0] fun h => h ^ 2
this : (fun h => 1 / h ^ 2) =O[𝓝[>] 0] fun h => 1 / h ^ 2
C : (fun x => (1 / x ^ 2) • x ^ 2 • ((f'' w) v - (f'' v) w)) =o[𝓝[>] 0] fun x => (1 / x ^ 2) • x ^ 2
h : ℝ
hpos : 0 < h
⊢ (1 / h ^ 2) • h ^ 2 = 1

Next tactic:
field_simp [LT.lt.ne' hpos, SMul.smul]
/-- Assume that `f` is differentiable inside a convex set `s`, and that its derivative `f'` is
differentiable at a point `x`. Then, given two vectors `v` and `w` pointing inside `s`, one
has `f'' v w = f'' w v`. Superseded by `Convex.second_derivative_within_at_symmetric`, which
removes the assumption that `v` and `w` point inside `s`. -/
theorem Convex.second_derivative_within_at_symmetric_of_mem_interior {v w : E}
    (h4v : x + (4 : ℝ) • v ∈ interior s) (h4w : x + (4 : ℝ) • w ∈ interior s) :
    f'' w v = f'' v w := by
  have A : (fun h : ℝ => h ^ 2 • (f'' w v - f'' v w)) =o[𝓝[>] 0] fun h => h ^ 2 := by
    convert (s_conv.isLittleO_alternate_sum_square hf xs hx h4v h4w).sub
      (s_conv.isLittleO_alternate_sum_square hf xs hx h4w h4v) using 1
    ext h
    simp only [add_comm, smul_add, smul_sub]
    abel
  have B : (fun _ : ℝ => f'' w v - f'' v w) =o[𝓝[>] 0] fun _ => (1 : ℝ) := by
    have : (fun h : ℝ => 1 / h ^ 2) =O[𝓝[>] 0] fun h => 1 / h ^ 2 := isBigO_refl _ _
    have C := this.smul_isLittleO A
    apply C.congr' _ _
    · filter_upwards [self_mem_nhdsWithin]
      intro h (hpos : 0 < h)
      rw [← one_smul ℝ (f'' w v - f'' v w), smul_smul, smul_smul]
      congr 1
      field_simp [LT.lt.ne' hpos]
    · filter_upwards [self_mem_nhdsWithin] with h (hpos : 0 < h)
      field_simp [LT.lt.ne' hpos, SMul.smul]
  

Proof state:
E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set E
s_conv : Convex ℝ s
f : E → F
f' : E → E →L[ℝ] F
f'' : E →L[ℝ] E →L[ℝ] F
hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x
x : E
xs : x ∈ s
hx : HasFDerivWithinAt f' f'' (interior s) x
v w : E
h4v : x + 4 • v ∈ interior s
h4w : x + 4 • w ∈ interior s
A : (fun h => h ^ 2 • ((f'' w) v - (f'' v) w)) =o[𝓝[>] 0] fun h => h ^ 2
B : (fun x => (f'' w) v - (f'' v) w) =o[𝓝[>] 0] fun x => 1
⊢ (f'' w) v = (f'' v) w

Next tactic:
simpa only [sub_eq_zero] using isLittleO_const_const_iff.1 B
/-- If a function is differentiable inside a convex set with nonempty interior, and has a second
derivative at a point of this convex set, then this second derivative is symmetric. -/
theorem Convex.second_derivative_within_at_symmetric {s : Set E} (s_conv : Convex ℝ s)
    (hne : (interior s).Nonempty) {f : E → F} {f' : E → E →L[ℝ] F} {f'' : E →L[ℝ] E →L[ℝ] F}
    (hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x) {x : E} (xs : x ∈ s)
    (hx : HasFDerivWithinAt f' f'' (interior s) x) (v w : E) : f'' v w = f'' w v := by
  /- we work around a point `x + 4 z` in the interior of `s`. For any vector `m`,
    then `x + 4 (z + t m)` also belongs to the interior of `s` for small enough `t`. This means that
    we will be able to apply `second_derivative_within_at_symmetric_of_mem_interior` to show
    that `f''` is symmetric, after cancelling all the contributions due to `z`. -/
  

Proof state:
E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s✝ : Set E
s_conv✝ : Convex ℝ s✝
f✝ : E → F
f'✝ : E → E →L[ℝ] F
f''✝ : E →L[ℝ] E →L[ℝ] F
hf✝ : ∀ x ∈ interior s✝, HasFDerivAt f✝ (f'✝ x) x
x✝ : E
xs✝ : x✝ ∈ s✝
hx✝ : HasFDerivWithinAt f'✝ f''✝ (interior s✝) x✝
s : Set E
s_conv : Convex ℝ s
hne : Set.Nonempty (interior s)
f : E → F
f' : E → E →L[ℝ] F
f'' : E →L[ℝ] E →L[ℝ] F
hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x
x : E
xs : x ∈ s
hx : HasFDerivWithinAt f' f'' (interior s) x
v w : E
⊢ (f'' v) w = (f'' w) v

Next tactic:
rcases hne with ⟨y, hy⟩
/-- If a function is differentiable inside a convex set with nonempty interior, and has a second
derivative at a point of this convex set, then this second derivative is symmetric. -/
theorem Convex.second_derivative_within_at_symmetric {s : Set E} (s_conv : Convex ℝ s)
    (hne : (interior s).Nonempty) {f : E → F} {f' : E → E →L[ℝ] F} {f'' : E →L[ℝ] E →L[ℝ] F}
    (hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x) {x : E} (xs : x ∈ s)
    (hx : HasFDerivWithinAt f' f'' (interior s) x) (v w : E) : f'' v w = f'' w v := by
  /- we work around a point `x + 4 z` in the interior of `s`. For any vector `m`,
    then `x + 4 (z + t m)` also belongs to the interior of `s` for small enough `t`. This means that
    we will be able to apply `second_derivative_within_at_symmetric_of_mem_interior` to show
    that `f''` is symmetric, after cancelling all the contributions due to `z`. -/
  rcases hne with ⟨y, hy⟩
  

Proof state:
case intro
E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s✝ : Set E
s_conv✝ : Convex ℝ s✝
f✝ : E → F
f'✝ : E → E →L[ℝ] F
f''✝ : E →L[ℝ] E →L[ℝ] F
hf✝ : ∀ x ∈ interior s✝, HasFDerivAt f✝ (f'✝ x) x
x✝ : E
xs✝ : x✝ ∈ s✝
hx✝ : HasFDerivWithinAt f'✝ f''✝ (interior s✝) x✝
s : Set E
s_conv : Convex ℝ s
f : E → F
f' : E → E →L[ℝ] F
f'' : E →L[ℝ] E →L[ℝ] F
hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x
x : E
xs : x ∈ s
hx : HasFDerivWithinAt f' f'' (interior s) x
v w y : E
hy : y ∈ interior s
⊢ (f'' v) w = (f'' w) v

Next tactic:
obtain ⟨z, hz⟩ : ∃ z, z = ((1 : ℝ) / 4) • (y - x) := ⟨((1 : ℝ) / 4) • (y - x), rfl⟩
/-- If a function is differentiable inside a convex set with nonempty interior, and has a second
derivative at a point of this convex set, then this second derivative is symmetric. -/
theorem Convex.second_derivative_within_at_symmetric {s : Set E} (s_conv : Convex ℝ s)
    (hne : (interior s).Nonempty) {f : E → F} {f' : E → E →L[ℝ] F} {f'' : E →L[ℝ] E →L[ℝ] F}
    (hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x) {x : E} (xs : x ∈ s)
    (hx : HasFDerivWithinAt f' f'' (interior s) x) (v w : E) : f'' v w = f'' w v := by
  /- we work around a point `x + 4 z` in the interior of `s`. For any vector `m`,
    then `x + 4 (z + t m)` also belongs to the interior of `s` for small enough `t`. This means that
    we will be able to apply `second_derivative_within_at_symmetric_of_mem_interior` to show
    that `f''` is symmetric, after cancelling all the contributions due to `z`. -/
  rcases hne with ⟨y, hy⟩
  obtain ⟨z, hz⟩ : ∃ z, z = ((1 : ℝ) / 4) • (y - x) := ⟨((1 : ℝ) / 4) • (y - x), rfl⟩
  

Proof state:
case intro.intro
E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s✝ : Set E
s_conv✝ : Convex ℝ s✝
f✝ : E → F
f'✝ : E → E →L[ℝ] F
f''✝ : E →L[ℝ] E →L[ℝ] F
hf✝ : ∀ x ∈ interior s✝, HasFDerivAt f✝ (f'✝ x) x
x✝ : E
xs✝ : x✝ ∈ s✝
hx✝ : HasFDerivWithinAt f'✝ f''✝ (interior s✝) x✝
s : Set E
s_conv : Convex ℝ s
f : E → F
f' : E → E →L[ℝ] F
f'' : E →L[ℝ] E →L[ℝ] F
hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x
x : E
xs : x ∈ s
hx : HasFDerivWithinAt f' f'' (interior s) x
v w y : E
hy : y ∈ interior s
z : E
hz : z = (1 / 4) • (y - x)
⊢ (f'' v) w = (f'' w) v

Next tactic:
have A : ∀ m : E, Filter.Tendsto (fun t : ℝ => x + (4 : ℝ) • (z + t • m)) (𝓝 0) (𝓝 y) := by
    intro m
    have : x + (4 : ℝ) • (z + (0 : ℝ) • m) = y := by simp [hz]
    rw [← this]
    refine' tendsto_const_nhds.add <| tendsto_const_nhds.smul <| tendsto_const_nhds.add _
    exact continuousAt_id.smul continuousAt_const
/-- If a function is differentiable inside a convex set with nonempty interior, and has a second
derivative at a point of this convex set, then this second derivative is symmetric. -/
theorem Convex.second_derivative_within_at_symmetric {s : Set E} (s_conv : Convex ℝ s)
    (hne : (interior s).Nonempty) {f : E → F} {f' : E → E →L[ℝ] F} {f'' : E →L[ℝ] E →L[ℝ] F}
    (hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x) {x : E} (xs : x ∈ s)
    (hx : HasFDerivWithinAt f' f'' (interior s) x) (v w : E) : f'' v w = f'' w v := by
  /- we work around a point `x + 4 z` in the interior of `s`. For any vector `m`,
    then `x + 4 (z + t m)` also belongs to the interior of `s` for small enough `t`. This means that
    we will be able to apply `second_derivative_within_at_symmetric_of_mem_interior` to show
    that `f''` is symmetric, after cancelling all the contributions due to `z`. -/
  rcases hne with ⟨y, hy⟩
  obtain ⟨z, hz⟩ : ∃ z, z = ((1 : ℝ) / 4) • (y - x) := ⟨((1 : ℝ) / 4) • (y - x), rfl⟩
  have A : ∀ m : E, Filter.Tendsto (fun t : ℝ => x + (4 : ℝ) • (z + t • m)) (𝓝 0) (𝓝 y) := by
    

Proof state:
E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s✝ : Set E
s_conv✝ : Convex ℝ s✝
f✝ : E → F
f'✝ : E → E →L[ℝ] F
f''✝ : E →L[ℝ] E →L[ℝ] F
hf✝ : ∀ x ∈ interior s✝, HasFDerivAt f✝ (f'✝ x) x
x✝ : E
xs✝ : x✝ ∈ s✝
hx✝ : HasFDerivWithinAt f'✝ f''✝ (interior s✝) x✝
s : Set E
s_conv : Convex ℝ s
f : E → F
f' : E → E →L[ℝ] F
f'' : E →L[ℝ] E →L[ℝ] F
hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x
x : E
xs : x ∈ s
hx : HasFDerivWithinAt f' f'' (interior s) x
v w y : E
hy : y ∈ interior s
z : E
hz : z = (1 / 4) • (y - x)
⊢ ∀ (m : E), Filter.Tendsto (fun t => x + 4 • (z + t • m)) (𝓝 0) (𝓝 y)

Next tactic:
intro m
/-- If a function is differentiable inside a convex set with nonempty interior, and has a second
derivative at a point of this convex set, then this second derivative is symmetric. -/
theorem Convex.second_derivative_within_at_symmetric {s : Set E} (s_conv : Convex ℝ s)
    (hne : (interior s).Nonempty) {f : E → F} {f' : E → E →L[ℝ] F} {f'' : E →L[ℝ] E →L[ℝ] F}
    (hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x) {x : E} (xs : x ∈ s)
    (hx : HasFDerivWithinAt f' f'' (interior s) x) (v w : E) : f'' v w = f'' w v := by
  /- we work around a point `x + 4 z` in the interior of `s`. For any vector `m`,
    then `x + 4 (z + t m)` also belongs to the interior of `s` for small enough `t`. This means that
    we will be able to apply `second_derivative_within_at_symmetric_of_mem_interior` to show
    that `f''` is symmetric, after cancelling all the contributions due to `z`. -/
  rcases hne with ⟨y, hy⟩
  obtain ⟨z, hz⟩ : ∃ z, z = ((1 : ℝ) / 4) • (y - x) := ⟨((1 : ℝ) / 4) • (y - x), rfl⟩
  have A : ∀ m : E, Filter.Tendsto (fun t : ℝ => x + (4 : ℝ) • (z + t • m)) (𝓝 0) (𝓝 y) := by
    intro m
    

Proof state:
E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s✝ : Set E
s_conv✝ : Convex ℝ s✝
f✝ : E → F
f'✝ : E → E →L[ℝ] F
f''✝ : E →L[ℝ] E →L[ℝ] F
hf✝ : ∀ x ∈ interior s✝, HasFDerivAt f✝ (f'✝ x) x
x✝ : E
xs✝ : x✝ ∈ s✝
hx✝ : HasFDerivWithinAt f'✝ f''✝ (interior s✝) x✝
s : Set E
s_conv : Convex ℝ s
f : E → F
f' : E → E →L[ℝ] F
f'' : E →L[ℝ] E →L[ℝ] F
hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x
x : E
xs : x ∈ s
hx : HasFDerivWithinAt f' f'' (interior s) x
v w y : E
hy : y ∈ interior s
z : E
hz : z = (1 / 4) • (y - x)
m : E
⊢ Filter.Tendsto (fun t => x + 4 • (z + t • m)) (𝓝 0) (𝓝 y)

Next tactic:
have : x + (4 : ℝ) • (z + (0 : ℝ) • m) = y := by simp [hz]
/-- If a function is differentiable inside a convex set with nonempty interior, and has a second
derivative at a point of this convex set, then this second derivative is symmetric. -/
theorem Convex.second_derivative_within_at_symmetric {s : Set E} (s_conv : Convex ℝ s)
    (hne : (interior s).Nonempty) {f : E → F} {f' : E → E →L[ℝ] F} {f'' : E →L[ℝ] E →L[ℝ] F}
    (hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x) {x : E} (xs : x ∈ s)
    (hx : HasFDerivWithinAt f' f'' (interior s) x) (v w : E) : f'' v w = f'' w v := by
  /- we work around a point `x + 4 z` in the interior of `s`. For any vector `m`,
    then `x + 4 (z + t m)` also belongs to the interior of `s` for small enough `t`. This means that
    we will be able to apply `second_derivative_within_at_symmetric_of_mem_interior` to show
    that `f''` is symmetric, after cancelling all the contributions due to `z`. -/
  rcases hne with ⟨y, hy⟩
  obtain ⟨z, hz⟩ : ∃ z, z = ((1 : ℝ) / 4) • (y - x) := ⟨((1 : ℝ) / 4) • (y - x), rfl⟩
  have A : ∀ m : E, Filter.Tendsto (fun t : ℝ => x + (4 : ℝ) • (z + t • m)) (𝓝 0) (𝓝 y) := by
    intro m
    have : x + (4 : ℝ) • (z + (0 : ℝ) • m) = y := by 

Proof state:
E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s✝ : Set E
s_conv✝ : Convex ℝ s✝
f✝ : E → F
f'✝ : E → E →L[ℝ] F
f''✝ : E →L[ℝ] E →L[ℝ] F
hf✝ : ∀ x ∈ interior s✝, HasFDerivAt f✝ (f'✝ x) x
x✝ : E
xs✝ : x✝ ∈ s✝
hx✝ : HasFDerivWithinAt f'✝ f''✝ (interior s✝) x✝
s : Set E
s_conv : Convex ℝ s
f : E → F
f' : E → E →L[ℝ] F
f'' : E →L[ℝ] E →L[ℝ] F
hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x
x : E
xs : x ∈ s
hx : HasFDerivWithinAt f' f'' (interior s) x
v w y : E
hy : y ∈ interior s
z : E
hz : z = (1 / 4) • (y - x)
m : E
⊢ x + 4 • (z + 0 • m) = y

Next tactic:
simp [hz]
/-- If a function is differentiable inside a convex set with nonempty interior, and has a second
derivative at a point of this convex set, then this second derivative is symmetric. -/
theorem Convex.second_derivative_within_at_symmetric {s : Set E} (s_conv : Convex ℝ s)
    (hne : (interior s).Nonempty) {f : E → F} {f' : E → E →L[ℝ] F} {f'' : E →L[ℝ] E →L[ℝ] F}
    (hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x) {x : E} (xs : x ∈ s)
    (hx : HasFDerivWithinAt f' f'' (interior s) x) (v w : E) : f'' v w = f'' w v := by
  /- we work around a point `x + 4 z` in the interior of `s`. For any vector `m`,
    then `x + 4 (z + t m)` also belongs to the interior of `s` for small enough `t`. This means that
    we will be able to apply `second_derivative_within_at_symmetric_of_mem_interior` to show
    that `f''` is symmetric, after cancelling all the contributions due to `z`. -/
  rcases hne with ⟨y, hy⟩
  obtain ⟨z, hz⟩ : ∃ z, z = ((1 : ℝ) / 4) • (y - x) := ⟨((1 : ℝ) / 4) • (y - x), rfl⟩
  have A : ∀ m : E, Filter.Tendsto (fun t : ℝ => x + (4 : ℝ) • (z + t • m)) (𝓝 0) (𝓝 y) := by
    intro m
    have : x + (4 : ℝ) • (z + (0 : ℝ) • m) = y := by simp [hz]
    

Proof state:
E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s✝ : Set E
s_conv✝ : Convex ℝ s✝
f✝ : E → F
f'✝ : E → E →L[ℝ] F
f''✝ : E →L[ℝ] E →L[ℝ] F
hf✝ : ∀ x ∈ interior s✝, HasFDerivAt f✝ (f'✝ x) x
x✝ : E
xs✝ : x✝ ∈ s✝
hx✝ : HasFDerivWithinAt f'✝ f''✝ (interior s✝) x✝
s : Set E
s_conv : Convex ℝ s
f : E → F
f' : E → E →L[ℝ] F
f'' : E →L[ℝ] E →L[ℝ] F
hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x
x : E
xs : x ∈ s
hx : HasFDerivWithinAt f' f'' (interior s) x
v w y : E
hy : y ∈ interior s
z : E
hz : z = (1 / 4) • (y - x)
m : E
this : x + 4 • (z + 0 • m) = y
⊢ Filter.Tendsto (fun t => x + 4 • (z + t • m)) (𝓝 0) (𝓝 y)

Next tactic:
rw [← this]
/-- If a function is differentiable inside a convex set with nonempty interior, and has a second
derivative at a point of this convex set, then this second derivative is symmetric. -/
theorem Convex.second_derivative_within_at_symmetric {s : Set E} (s_conv : Convex ℝ s)
    (hne : (interior s).Nonempty) {f : E → F} {f' : E → E →L[ℝ] F} {f'' : E →L[ℝ] E →L[ℝ] F}
    (hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x) {x : E} (xs : x ∈ s)
    (hx : HasFDerivWithinAt f' f'' (interior s) x) (v w : E) : f'' v w = f'' w v := by
  /- we work around a point `x + 4 z` in the interior of `s`. For any vector `m`,
    then `x + 4 (z + t m)` also belongs to the interior of `s` for small enough `t`. This means that
    we will be able to apply `second_derivative_within_at_symmetric_of_mem_interior` to show
    that `f''` is symmetric, after cancelling all the contributions due to `z`. -/
  rcases hne with ⟨y, hy⟩
  obtain ⟨z, hz⟩ : ∃ z, z = ((1 : ℝ) / 4) • (y - x) := ⟨((1 : ℝ) / 4) • (y - x), rfl⟩
  have A : ∀ m : E, Filter.Tendsto (fun t : ℝ => x + (4 : ℝ) • (z + t • m)) (𝓝 0) (𝓝 y) := by
    intro m
    have : x + (4 : ℝ) • (z + (0 : ℝ) • m) = y := by simp [hz]
    rw [← this]
    

Proof state:
E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s✝ : Set E
s_conv✝ : Convex ℝ s✝
f✝ : E → F
f'✝ : E → E →L[ℝ] F
f''✝ : E →L[ℝ] E →L[ℝ] F
hf✝ : ∀ x ∈ interior s✝, HasFDerivAt f✝ (f'✝ x) x
x✝ : E
xs✝ : x✝ ∈ s✝
hx✝ : HasFDerivWithinAt f'✝ f''✝ (interior s✝) x✝
s : Set E
s_conv : Convex ℝ s
f : E → F
f' : E → E →L[ℝ] F
f'' : E →L[ℝ] E →L[ℝ] F
hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x
x : E
xs : x ∈ s
hx : HasFDerivWithinAt f' f'' (interior s) x
v w y : E
hy : y ∈ interior s
z : E
hz : z = (1 / 4) • (y - x)
m : E
this : x + 4 • (z + 0 • m) = y
⊢ Filter.Tendsto (fun t => x + 4 • (z + t • m)) (𝓝 0) (𝓝 (x + 4 • (z + 0 • m)))

Next tactic:
refine' tendsto_const_nhds.add <| tendsto_const_nhds.smul <| tendsto_const_nhds.add _