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 can
Taylor-expand to order two the function `f` on the segment `[x + h v, x + h (v + w)]`, giving a
bilinear estimate for `f (x + hv + hw) - f (x + hv)` in terms of `f' w` and of `f'' ⬝ w`, up to
`o(h^2)`.

This is a technical statement used to show that the second derivative is symmetric. -/
theorem Convex.taylor_approx_two_segment {v w : E} (hv : x + v ∈ interior s)
    (hw : x + v + w ∈ interior s) :
    (fun h : ℝ => f (x + h • v + h • w)
        - f (x + h • v) - h • f' x w - h ^ 2 • f'' v w - (h ^ 2 / 2) • f'' w w) =o[𝓝[>] 0]
      fun h => h ^ 2 := by
  -- it suffices to check that the expression is bounded by `ε * ((‖v‖ + ‖w‖) * ‖w‖) * h^2` for
  -- small enough `h`, for any positive `ε`.
  refine' IsLittleO.trans_isBigO
    (isLittleO_iff.2 fun ε εpos => _) (isBigO_const_mul_self ((‖v‖ + ‖w‖) * ‖w‖) _ _)
  -- consider a ball of radius `δ` around `x` in which the Taylor approximation for `f''` is
  -- good up to `δ`.
  rw [HasFDerivWithinAt, hasFDerivAtFilter_iff_isLittleO, isLittleO_iff] at hx
  rcases Metric.mem_nhdsWithin_iff.1 (hx εpos) with ⟨δ, δpos, sδ⟩
  have E1 : ∀ᶠ h in 𝓝[>] (0 : ℝ), h * (‖v‖ + ‖w‖) < δ := by
    have : Filter.Tendsto (fun h => h * (‖v‖ + ‖w‖)) (𝓝[>] (0 : ℝ)) (𝓝 (0 * (‖v‖ + ‖w‖))) :=
      (continuous_id.mul continuous_const).continuousWithinAt
    apply (tendsto_order.1 this).2 δ
    simpa only [zero_mul] using δpos
  have E2 : ∀ᶠ h in 𝓝[>] (0 : ℝ), (h : ℝ) < 1 :=
    mem_nhdsWithin_Ioi_iff_exists_Ioo_subset.2
      ⟨(1 : ℝ), by simp only [mem_Ioi, zero_lt_one], fun x hx => hx.2⟩
  filter_upwards [E1, E2, self_mem_nhdsWithin] with h hδ h_lt_1 hpos
  -- we consider `h` small enough that all points under consideration belong to this ball,
  -- and also with `0 < h < 1`.
  replace hpos : 0 < h := hpos
  have xt_mem : ∀ t ∈ Icc (0 : ℝ) 1, x + h • v + (t * h) • w ∈ interior s := by
    intro t ht
    have : x + h • v ∈ interior s := s_conv.add_smul_mem_interior xs hv ⟨hpos, h_lt_1.le⟩
    rw [← 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 : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : E) in 𝓝[interior s] x, ‖f' x_1 - f' x - f'' (x_1 - x)‖ ≤ c * ‖x_1 - x‖
v w : E
hv : x + v ∈ interior s
hw : x + v + w ∈ interior s
ε : ℝ
εpos : 0 < ε
δ : ℝ
δpos : δ > 0
sδ : Metric.ball x δ ∩ interior s ⊆ {x_1 | (fun x_2 => ‖f' x_2 - f' x - f'' (x_2 - x)‖ ≤ ε * ‖x_2 - x‖) x_1}
E1 : ∀ᶠ (h : ℝ) in 𝓝[>] 0, h * (‖v‖ + ‖w‖) < δ
E2 : ∀ᶠ (h : ℝ) in 𝓝[>] 0, h < 1
h : ℝ
hδ : h * (‖v‖ + ‖w‖) < δ
h_lt_1 : h < 1
hpos : 0 < h
t : ℝ
ht : t ∈ Icc 0 1
this : x + h • v ∈ interior s
⊢ x + h • v + t • h • w ∈ interior s

Next tactic:
apply s_conv.interior.add_smul_mem this _ ht
/-- 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 can
Taylor-expand to order two the function `f` on the segment `[x + h v, x + h (v + w)]`, giving a
bilinear estimate for `f (x + hv + hw) - f (x + hv)` in terms of `f' w` and of `f'' ⬝ w`, up to
`o(h^2)`.

This is a technical statement used to show that the second derivative is symmetric. -/
theorem Convex.taylor_approx_two_segment {v w : E} (hv : x + v ∈ interior s)
    (hw : x + v + w ∈ interior s) :
    (fun h : ℝ => f (x + h • v + h • w)
        - f (x + h • v) - h • f' x w - h ^ 2 • f'' v w - (h ^ 2 / 2) • f'' w w) =o[𝓝[>] 0]
      fun h => h ^ 2 := by
  -- it suffices to check that the expression is bounded by `ε * ((‖v‖ + ‖w‖) * ‖w‖) * h^2` for
  -- small enough `h`, for any positive `ε`.
  refine' IsLittleO.trans_isBigO
    (isLittleO_iff.2 fun ε εpos => _) (isBigO_const_mul_self ((‖v‖ + ‖w‖) * ‖w‖) _ _)
  -- consider a ball of radius `δ` around `x` in which the Taylor approximation for `f''` is
  -- good up to `δ`.
  rw [HasFDerivWithinAt, hasFDerivAtFilter_iff_isLittleO, isLittleO_iff] at hx
  rcases Metric.mem_nhdsWithin_iff.1 (hx εpos) with ⟨δ, δpos, sδ⟩
  have E1 : ∀ᶠ h in 𝓝[>] (0 : ℝ), h * (‖v‖ + ‖w‖) < δ := by
    have : Filter.Tendsto (fun h => h * (‖v‖ + ‖w‖)) (𝓝[>] (0 : ℝ)) (𝓝 (0 * (‖v‖ + ‖w‖))) :=
      (continuous_id.mul continuous_const).continuousWithinAt
    apply (tendsto_order.1 this).2 δ
    simpa only [zero_mul] using δpos
  have E2 : ∀ᶠ h in 𝓝[>] (0 : ℝ), (h : ℝ) < 1 :=
    mem_nhdsWithin_Ioi_iff_exists_Ioo_subset.2
      ⟨(1 : ℝ), by simp only [mem_Ioi, zero_lt_one], fun x hx => hx.2⟩
  filter_upwards [E1, E2, self_mem_nhdsWithin] with h hδ h_lt_1 hpos
  -- we consider `h` small enough that all points under consideration belong to this ball,
  -- and also with `0 < h < 1`.
  replace hpos : 0 < h := hpos
  have xt_mem : ∀ t ∈ Icc (0 : ℝ) 1, x + h • v + (t * h) • w ∈ interior s := by
    intro t ht
    have : x + h • v ∈ interior s := s_conv.add_smul_mem_interior xs hv ⟨hpos, h_lt_1.le⟩
    rw [← smul_smul]
    apply s_conv.interior.add_smul_mem this _ ht
    

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 : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : E) in 𝓝[interior s] x, ‖f' x_1 - f' x - f'' (x_1 - x)‖ ≤ c * ‖x_1 - x‖
v w : E
hv : x + v ∈ interior s
hw : x + v + w ∈ interior s
ε : ℝ
εpos : 0 < ε
δ : ℝ
δpos : δ > 0
sδ : Metric.ball x δ ∩ interior s ⊆ {x_1 | (fun x_2 => ‖f' x_2 - f' x - f'' (x_2 - x)‖ ≤ ε * ‖x_2 - x‖) x_1}
E1 : ∀ᶠ (h : ℝ) in 𝓝[>] 0, h * (‖v‖ + ‖w‖) < δ
E2 : ∀ᶠ (h : ℝ) in 𝓝[>] 0, h < 1
h : ℝ
hδ : h * (‖v‖ + ‖w‖) < δ
h_lt_1 : h < 1
hpos : 0 < h
t : ℝ
ht : t ∈ Icc 0 1
this : x + h • v ∈ interior s
⊢ x + h • v + h • w ∈ interior s

Next tactic:
rw [add_assoc] at hw
/-- 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 can
Taylor-expand to order two the function `f` on the segment `[x + h v, x + h (v + w)]`, giving a
bilinear estimate for `f (x + hv + hw) - f (x + hv)` in terms of `f' w` and of `f'' ⬝ w`, up to
`o(h^2)`.

This is a technical statement used to show that the second derivative is symmetric. -/
theorem Convex.taylor_approx_two_segment {v w : E} (hv : x + v ∈ interior s)
    (hw : x + v + w ∈ interior s) :
    (fun h : ℝ => f (x + h • v + h • w)
        - f (x + h • v) - h • f' x w - h ^ 2 • f'' v w - (h ^ 2 / 2) • f'' w w) =o[𝓝[>] 0]
      fun h => h ^ 2 := by
  -- it suffices to check that the expression is bounded by `ε * ((‖v‖ + ‖w‖) * ‖w‖) * h^2` for
  -- small enough `h`, for any positive `ε`.
  refine' IsLittleO.trans_isBigO
    (isLittleO_iff.2 fun ε εpos => _) (isBigO_const_mul_self ((‖v‖ + ‖w‖) * ‖w‖) _ _)
  -- consider a ball of radius `δ` around `x` in which the Taylor approximation for `f''` is
  -- good up to `δ`.
  rw [HasFDerivWithinAt, hasFDerivAtFilter_iff_isLittleO, isLittleO_iff] at hx
  rcases Metric.mem_nhdsWithin_iff.1 (hx εpos) with ⟨δ, δpos, sδ⟩
  have E1 : ∀ᶠ h in 𝓝[>] (0 : ℝ), h * (‖v‖ + ‖w‖) < δ := by
    have : Filter.Tendsto (fun h => h * (‖v‖ + ‖w‖)) (𝓝[>] (0 : ℝ)) (𝓝 (0 * (‖v‖ + ‖w‖))) :=
      (continuous_id.mul continuous_const).continuousWithinAt
    apply (tendsto_order.1 this).2 δ
    simpa only [zero_mul] using δpos
  have E2 : ∀ᶠ h in 𝓝[>] (0 : ℝ), (h : ℝ) < 1 :=
    mem_nhdsWithin_Ioi_iff_exists_Ioo_subset.2
      ⟨(1 : ℝ), by simp only [mem_Ioi, zero_lt_one], fun x hx => hx.2⟩
  filter_upwards [E1, E2, self_mem_nhdsWithin] with h hδ h_lt_1 hpos
  -- we consider `h` small enough that all points under consideration belong to this ball,
  -- and also with `0 < h < 1`.
  replace hpos : 0 < h := hpos
  have xt_mem : ∀ t ∈ Icc (0 : ℝ) 1, x + h • v + (t * h) • w ∈ interior s := by
    intro t ht
    have : x + h • v ∈ interior s := s_conv.add_smul_mem_interior xs hv ⟨hpos, h_lt_1.le⟩
    rw [← smul_smul]
    apply s_conv.interior.add_smul_mem this _ ht
    rw [add_assoc] at hw
    

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 : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : E) in 𝓝[interior s] x, ‖f' x_1 - f' x - f'' (x_1 - x)‖ ≤ c * ‖x_1 - x‖
v w : E
hv : x + v ∈ interior s
hw : x + (v + w) ∈ interior s
ε : ℝ
εpos : 0 < ε
δ : ℝ
δpos : δ > 0
sδ : Metric.ball x δ ∩ interior s ⊆ {x_1 | (fun x_2 => ‖f' x_2 - f' x - f'' (x_2 - x)‖ ≤ ε * ‖x_2 - x‖) x_1}
E1 : ∀ᶠ (h : ℝ) in 𝓝[>] 0, h * (‖v‖ + ‖w‖) < δ
E2 : ∀ᶠ (h : ℝ) in 𝓝[>] 0, h < 1
h : ℝ
hδ : h * (‖v‖ + ‖w‖) < δ
h_lt_1 : h < 1
hpos : 0 < h
t : ℝ
ht : t ∈ Icc 0 1
this : x + h • v ∈ interior s
⊢ x + h • v + h • w ∈ interior s

Next tactic:
rw [add_assoc, ← smul_add]
/-- 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 can
Taylor-expand to order two the function `f` on the segment `[x + h v, x + h (v + w)]`, giving a
bilinear estimate for `f (x + hv + hw) - f (x + hv)` in terms of `f' w` and of `f'' ⬝ w`, up to
`o(h^2)`.

This is a technical statement used to show that the second derivative is symmetric. -/
theorem Convex.taylor_approx_two_segment {v w : E} (hv : x + v ∈ interior s)
    (hw : x + v + w ∈ interior s) :
    (fun h : ℝ => f (x + h • v + h • w)
        - f (x + h • v) - h • f' x w - h ^ 2 • f'' v w - (h ^ 2 / 2) • f'' w w) =o[𝓝[>] 0]
      fun h => h ^ 2 := by
  -- it suffices to check that the expression is bounded by `ε * ((‖v‖ + ‖w‖) * ‖w‖) * h^2` for
  -- small enough `h`, for any positive `ε`.
  refine' IsLittleO.trans_isBigO
    (isLittleO_iff.2 fun ε εpos => _) (isBigO_const_mul_self ((‖v‖ + ‖w‖) * ‖w‖) _ _)
  -- consider a ball of radius `δ` around `x` in which the Taylor approximation for `f''` is
  -- good up to `δ`.
  rw [HasFDerivWithinAt, hasFDerivAtFilter_iff_isLittleO, isLittleO_iff] at hx
  rcases Metric.mem_nhdsWithin_iff.1 (hx εpos) with ⟨δ, δpos, sδ⟩
  have E1 : ∀ᶠ h in 𝓝[>] (0 : ℝ), h * (‖v‖ + ‖w‖) < δ := by
    have : Filter.Tendsto (fun h => h * (‖v‖ + ‖w‖)) (𝓝[>] (0 : ℝ)) (𝓝 (0 * (‖v‖ + ‖w‖))) :=
      (continuous_id.mul continuous_const).continuousWithinAt
    apply (tendsto_order.1 this).2 δ
    simpa only [zero_mul] using δpos
  have E2 : ∀ᶠ h in 𝓝[>] (0 : ℝ), (h : ℝ) < 1 :=
    mem_nhdsWithin_Ioi_iff_exists_Ioo_subset.2
      ⟨(1 : ℝ), by simp only [mem_Ioi, zero_lt_one], fun x hx => hx.2⟩
  filter_upwards [E1, E2, self_mem_nhdsWithin] with h hδ h_lt_1 hpos
  -- we consider `h` small enough that all points under consideration belong to this ball,
  -- and also with `0 < h < 1`.
  replace hpos : 0 < h := hpos
  have xt_mem : ∀ t ∈ Icc (0 : ℝ) 1, x + h • v + (t * h) • w ∈ interior s := by
    intro t ht
    have : x + h • v ∈ interior s := s_conv.add_smul_mem_interior xs hv ⟨hpos, h_lt_1.le⟩
    rw [← smul_smul]
    apply s_conv.interior.add_smul_mem this _ ht
    rw [add_assoc] at hw
    rw [add_assoc, ← smul_add]
    

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 : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : E) in 𝓝[interior s] x, ‖f' x_1 - f' x - f'' (x_1 - x)‖ ≤ c * ‖x_1 - x‖
v w : E
hv : x + v ∈ interior s
hw : x + (v + w) ∈ interior s
ε : ℝ
εpos : 0 < ε
δ : ℝ
δpos : δ > 0
sδ : Metric.ball x δ ∩ interior s ⊆ {x_1 | (fun x_2 => ‖f' x_2 - f' x - f'' (x_2 - x)‖ ≤ ε * ‖x_2 - x‖) x_1}
E1 : ∀ᶠ (h : ℝ) in 𝓝[>] 0, h * (‖v‖ + ‖w‖) < δ
E2 : ∀ᶠ (h : ℝ) in 𝓝[>] 0, h < 1
h : ℝ
hδ : h * (‖v‖ + ‖w‖) < δ
h_lt_1 : h < 1
hpos : 0 < h
t : ℝ
ht : t ∈ Icc 0 1
this : x + h • v ∈ interior s
⊢ x + h • (v + w) ∈ interior s

Next tactic:
exact s_conv.add_smul_mem_interior xs hw ⟨hpos, h_lt_1.le⟩
/-- One can get `f'' v w` as the limit of `h ^ (-2)` times the alternate sum of the values of `f`
along the vertices of a quadrilateral with sides `h v` and `h w` based at `x`.
In a setting where `f` is not guaranteed to be continuous at `f`, we can still
get this if we use a quadrilateral based at `h v + h w`. -/
theorem Convex.isLittleO_alternate_sum_square {v w : E} (h4v : x + (4 : ℝ) • v ∈ interior s)
    (h4w : x + (4 : ℝ) • w ∈ interior s) :
    (fun h : ℝ => f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w))
        - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) - h ^ 2 • f'' v w) =o[𝓝[>] 0]
      fun h => h ^ 2 := 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
v w : E
h4v : x + 4 • v ∈ interior s
h4w : x + 4 • w ∈ interior s
⊢ (fun h =>
      f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -
        h ^ 2 • (f'' v) w) =o[𝓝[>] 0]
    fun h => h ^ 2

Next tactic:
have A : (1 : ℝ) / 2 ∈ Ioc (0 : ℝ) 1 := ⟨by norm_num, by norm_num⟩
/-- One can get `f'' v w` as the limit of `h ^ (-2)` times the alternate sum of the values of `f`
along the vertices of a quadrilateral with sides `h v` and `h w` based at `x`.
In a setting where `f` is not guaranteed to be continuous at `f`, we can still
get this if we use a quadrilateral based at `h v + h w`. -/
theorem Convex.isLittleO_alternate_sum_square {v w : E} (h4v : x + (4 : ℝ) • v ∈ interior s)
    (h4w : x + (4 : ℝ) • w ∈ interior s) :
    (fun h : ℝ => f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w))
        - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) - h ^ 2 • f'' v w) =o[𝓝[>] 0]
      fun h => h ^ 2 := by
  have A : (1 : ℝ) / 2 ∈ Ioc (0 : ℝ) 1 := ⟨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
v w : E
h4v : x + 4 • v ∈ interior s
h4w : x + 4 • w ∈ interior s
⊢ 0 < 1 / 2

Next tactic:
norm_num
/-- One can get `f'' v w` as the limit of `h ^ (-2)` times the alternate sum of the values of `f`
along the vertices of a quadrilateral with sides `h v` and `h w` based at `x`.
In a setting where `f` is not guaranteed to be continuous at `f`, we can still
get this if we use a quadrilateral based at `h v + h w`. -/
theorem Convex.isLittleO_alternate_sum_square {v w : E} (h4v : x + (4 : ℝ) • v ∈ interior s)
    (h4w : x + (4 : ℝ) • w ∈ interior s) :
    (fun h : ℝ => f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w))
        - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) - h ^ 2 • f'' v w) =o[𝓝[>] 0]
      fun h => h ^ 2 := by
  have A : (1 : ℝ) / 2 ∈ Ioc (0 : ℝ) 1 := ⟨by norm_num, 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
v w : E
h4v : x + 4 • v ∈ interior s
h4w : x + 4 • w ∈ interior s
⊢ 1 / 2 ≤ 1

Next tactic:
norm_num
/-- One can get `f'' v w` as the limit of `h ^ (-2)` times the alternate sum of the values of `f`
along the vertices of a quadrilateral with sides `h v` and `h w` based at `x`.
In a setting where `f` is not guaranteed to be continuous at `f`, we can still
get this if we use a quadrilateral based at `h v + h w`. -/
theorem Convex.isLittleO_alternate_sum_square {v w : E} (h4v : x + (4 : ℝ) • v ∈ interior s)
    (h4w : x + (4 : ℝ) • w ∈ interior s) :
    (fun h : ℝ => f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w))
        - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) - h ^ 2 • f'' v w) =o[𝓝[>] 0]
      fun h => h ^ 2 := by
  have A : (1 : ℝ) / 2 ∈ Ioc (0 : ℝ) 1 := ⟨by norm_num, by norm_num⟩
  

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 : 1 / 2 ∈ Ioc 0 1
⊢ (fun h =>
      f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -
        h ^ 2 • (f'' v) w) =o[𝓝[>] 0]
    fun h => h ^ 2

Next tactic:
have B : (1 : ℝ) / 2 ∈ Icc (0 : ℝ) 1 := ⟨by norm_num, by norm_num⟩
/-- One can get `f'' v w` as the limit of `h ^ (-2)` times the alternate sum of the values of `f`
along the vertices of a quadrilateral with sides `h v` and `h w` based at `x`.
In a setting where `f` is not guaranteed to be continuous at `f`, we can still
get this if we use a quadrilateral based at `h v + h w`. -/
theorem Convex.isLittleO_alternate_sum_square {v w : E} (h4v : x + (4 : ℝ) • v ∈ interior s)
    (h4w : x + (4 : ℝ) • w ∈ interior s) :
    (fun h : ℝ => f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w))
        - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) - h ^ 2 • f'' v w) =o[𝓝[>] 0]
      fun h => h ^ 2 := by
  have A : (1 : ℝ) / 2 ∈ Ioc (0 : ℝ) 1 := ⟨by norm_num, by norm_num⟩
  have B : (1 : ℝ) / 2 ∈ Icc (0 : ℝ) 1 := ⟨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
v w : E
h4v : x + 4 • v ∈ interior s
h4w : x + 4 • w ∈ interior s
A : 1 / 2 ∈ Ioc 0 1
⊢ 0 ≤ 1 / 2

Next tactic:
norm_num
/-- One can get `f'' v w` as the limit of `h ^ (-2)` times the alternate sum of the values of `f`
along the vertices of a quadrilateral with sides `h v` and `h w` based at `x`.
In a setting where `f` is not guaranteed to be continuous at `f`, we can still
get this if we use a quadrilateral based at `h v + h w`. -/
theorem Convex.isLittleO_alternate_sum_square {v w : E} (h4v : x + (4 : ℝ) • v ∈ interior s)
    (h4w : x + (4 : ℝ) • w ∈ interior s) :
    (fun h : ℝ => f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w))
        - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) - h ^ 2 • f'' v w) =o[𝓝[>] 0]
      fun h => h ^ 2 := by
  have A : (1 : ℝ) / 2 ∈ Ioc (0 : ℝ) 1 := ⟨by norm_num, by norm_num⟩
  have B : (1 : ℝ) / 2 ∈ Icc (0 : ℝ) 1 := ⟨by norm_num, 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
v w : E
h4v : x + 4 • v ∈ interior s
h4w : x + 4 • w ∈ interior s
A : 1 / 2 ∈ Ioc 0 1
⊢ 1 / 2 ≤ 1

Next tactic:
norm_num
/-- One can get `f'' v w` as the limit of `h ^ (-2)` times the alternate sum of the values of `f`
along the vertices of a quadrilateral with sides `h v` and `h w` based at `x`.
In a setting where `f` is not guaranteed to be continuous at `f`, we can still
get this if we use a quadrilateral based at `h v + h w`. -/
theorem Convex.isLittleO_alternate_sum_square {v w : E} (h4v : x + (4 : ℝ) • v ∈ interior s)
    (h4w : x + (4 : ℝ) • w ∈ interior s) :
    (fun h : ℝ => f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w))
        - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) - h ^ 2 • f'' v w) =o[𝓝[>] 0]
      fun h => h ^ 2 := by
  have A : (1 : ℝ) / 2 ∈ Ioc (0 : ℝ) 1 := ⟨by norm_num, by norm_num⟩
  have B : (1 : ℝ) / 2 ∈ Icc (0 : ℝ) 1 := ⟨by norm_num, by norm_num⟩
  

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 : 1 / 2 ∈ Ioc 0 1
B : 1 / 2 ∈ Icc 0 1
⊢ (fun h =>
      f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -
        h ^ 2 • (f'' v) w) =o[𝓝[>] 0]
    fun h => h ^ 2

Next tactic:
have C : ∀ w : E, (2 : ℝ) • w = 2 • w := fun w => by simp only [two_smul]
/-- One can get `f'' v w` as the limit of `h ^ (-2)` times the alternate sum of the values of `f`
along the vertices of a quadrilateral with sides `h v` and `h w` based at `x`.
In a setting where `f` is not guaranteed to be continuous at `f`, we can still
get this if we use a quadrilateral based at `h v + h w`. -/
theorem Convex.isLittleO_alternate_sum_square {v w : E} (h4v : x + (4 : ℝ) • v ∈ interior s)
    (h4w : x + (4 : ℝ) • w ∈ interior s) :
    (fun h : ℝ => f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w))
        - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) - h ^ 2 • f'' v w) =o[𝓝[>] 0]
      fun h => h ^ 2 := by
  have A : (1 : ℝ) / 2 ∈ Ioc (0 : ℝ) 1 := ⟨by norm_num, by norm_num⟩
  have B : (1 : ℝ) / 2 ∈ Icc (0 : ℝ) 1 := ⟨by norm_num, by norm_num⟩
  have C : ∀ w : E, (2 : ℝ) • w = 2 • w := fun w => 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
v w✝ : E
h4v : x + 4 • v ∈ interior s
h4w : x + 4 • w✝ ∈ interior s
A : 1 / 2 ∈ Ioc 0 1
B : 1 / 2 ∈ Icc 0 1
w : E
⊢ 2 • w = 2 • w

Next tactic:
simp only [two_smul]
/-- One can get `f'' v w` as the limit of `h ^ (-2)` times the alternate sum of the values of `f`
along the vertices of a quadrilateral with sides `h v` and `h w` based at `x`.
In a setting where `f` is not guaranteed to be continuous at `f`, we can still
get this if we use a quadrilateral based at `h v + h w`. -/
theorem Convex.isLittleO_alternate_sum_square {v w : E} (h4v : x + (4 : ℝ) • v ∈ interior s)
    (h4w : x + (4 : ℝ) • w ∈ interior s) :
    (fun h : ℝ => f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w))
        - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) - h ^ 2 • f'' v w) =o[𝓝[>] 0]
      fun h => h ^ 2 := by
  have A : (1 : ℝ) / 2 ∈ Ioc (0 : ℝ) 1 := ⟨by norm_num, by norm_num⟩
  have B : (1 : ℝ) / 2 ∈ Icc (0 : ℝ) 1 := ⟨by norm_num, by norm_num⟩
  have C : ∀ w : E, (2 : ℝ) • w = 2 • w := fun w => by simp only [two_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 : 1 / 2 ∈ Ioc 0 1
B : 1 / 2 ∈ Icc 0 1
C : ∀ (w : E), 2 • w = 2 • w
⊢ (fun h =>
      f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -
        h ^ 2 • (f'' v) w) =o[𝓝[>] 0]
    fun h => h ^ 2

Next tactic:
have h2v2w : x + (2 : ℝ) • v + (2 : ℝ) • w ∈ interior s := by
    convert s_conv.interior.add_smul_sub_mem h4v h4w B using 1
    simp only [smul_sub, smul_smul, one_div, add_sub_add_left_eq_sub, mul_add, add_smul]
    norm_num
    simp only [show (4 : ℝ) = (2 : ℝ) + (2 : ℝ) by norm_num, _root_.add_smul]
    abel