l3lab/ntp-mathlib huggingface dataset:
theorem mem_A_of_differentiable {ε : ℝ} (hε : 0 < ε) {x : ℝ}
    (hx : DifferentiableWithinAt ℝ f (Ici x) x) :
    ∃ R > 0, ∀ r ∈ Ioo (0 : ℝ) R, x ∈ A f (derivWithin f (Ici x) x) r ε := by
  have := hx.hasDerivWithinAt
  simp_rw [hasDerivWithinAt_iff_isLittleO, isLittleO_iff] at this
  rcases mem_nhdsWithin_Ici_iff_exists_Ico_subset.1 (this (half_pos hε)) with ⟨m, xm, hm⟩
  

Proof state:
case intro.intro
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
ε : ℝ
hε : 0 < ε
x : ℝ
hx : DifferentiableWithinAt ℝ f (Ici x) x
this : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : ℝ) in 𝓝[≥] x, ‖f x_1 - f x - (x_1 - x) • derivWithin f (Ici x) x‖ ≤ c * ‖x_1 - x‖
m : ℝ
xm : m ∈ Ioi x
hm : Ico x m ⊆ {x_1 | (fun x_2 => ‖f x_2 - f x - (x_2 - x) • derivWithin f (Ici x) x‖ ≤ ε / 2 * ‖x_2 - x‖) x_1}
⊢ ∃ R > 0, ∀ r ∈ Ioo 0 R, x ∈ A f (derivWithin f (Ici x) x) r ε

Next tactic:
refine' ⟨m - x, by linarith [show x < m from xm], fun r hr => _⟩
theorem mem_A_of_differentiable {ε : ℝ} (hε : 0 < ε) {x : ℝ}
    (hx : DifferentiableWithinAt ℝ f (Ici x) x) :
    ∃ R > 0, ∀ r ∈ Ioo (0 : ℝ) R, x ∈ A f (derivWithin f (Ici x) x) r ε := by
  have := hx.hasDerivWithinAt
  simp_rw [hasDerivWithinAt_iff_isLittleO, isLittleO_iff] at this
  rcases mem_nhdsWithin_Ici_iff_exists_Ico_subset.1 (this (half_pos hε)) with ⟨m, xm, hm⟩
  refine' ⟨m - x, by 

Proof state:
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
ε : ℝ
hε : 0 < ε
x : ℝ
hx : DifferentiableWithinAt ℝ f (Ici x) x
this : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : ℝ) in 𝓝[≥] x, ‖f x_1 - f x - (x_1 - x) • derivWithin f (Ici x) x‖ ≤ c * ‖x_1 - x‖
m : ℝ
xm : m ∈ Ioi x
hm : Ico x m ⊆ {x_1 | (fun x_2 => ‖f x_2 - f x - (x_2 - x) • derivWithin f (Ici x) x‖ ≤ ε / 2 * ‖x_2 - x‖) x_1}
⊢ m - x > 0

Next tactic:
linarith [show x < m from xm]
theorem mem_A_of_differentiable {ε : ℝ} (hε : 0 < ε) {x : ℝ}
    (hx : DifferentiableWithinAt ℝ f (Ici x) x) :
    ∃ R > 0, ∀ r ∈ Ioo (0 : ℝ) R, x ∈ A f (derivWithin f (Ici x) x) r ε := by
  have := hx.hasDerivWithinAt
  simp_rw [hasDerivWithinAt_iff_isLittleO, isLittleO_iff] at this
  rcases mem_nhdsWithin_Ici_iff_exists_Ico_subset.1 (this (half_pos hε)) with ⟨m, xm, hm⟩
  refine' ⟨m - x, by linarith [show x < m from xm], fun r hr => _⟩
  

Proof state:
case intro.intro
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
ε : ℝ
hε : 0 < ε
x : ℝ
hx : DifferentiableWithinAt ℝ f (Ici x) x
this : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : ℝ) in 𝓝[≥] x, ‖f x_1 - f x - (x_1 - x) • derivWithin f (Ici x) x‖ ≤ c * ‖x_1 - x‖
m : ℝ
xm : m ∈ Ioi x
hm : Ico x m ⊆ {x_1 | (fun x_2 => ‖f x_2 - f x - (x_2 - x) • derivWithin f (Ici x) x‖ ≤ ε / 2 * ‖x_2 - x‖) x_1}
r : ℝ
hr : r ∈ Ioo 0 (m - x)
⊢ x ∈ A f (derivWithin f (Ici x) x) r ε

Next tactic:
have : r ∈ Ioc (r / 2) r := ⟨half_lt_self hr.1, le_rfl⟩
theorem mem_A_of_differentiable {ε : ℝ} (hε : 0 < ε) {x : ℝ}
    (hx : DifferentiableWithinAt ℝ f (Ici x) x) :
    ∃ R > 0, ∀ r ∈ Ioo (0 : ℝ) R, x ∈ A f (derivWithin f (Ici x) x) r ε := by
  have := hx.hasDerivWithinAt
  simp_rw [hasDerivWithinAt_iff_isLittleO, isLittleO_iff] at this
  rcases mem_nhdsWithin_Ici_iff_exists_Ico_subset.1 (this (half_pos hε)) with ⟨m, xm, hm⟩
  refine' ⟨m - x, by linarith [show x < m from xm], fun r hr => _⟩
  have : r ∈ Ioc (r / 2) r := ⟨half_lt_self hr.1, le_rfl⟩
  

Proof state:
case intro.intro
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
ε : ℝ
hε : 0 < ε
x : ℝ
hx : DifferentiableWithinAt ℝ f (Ici x) x
this✝ : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : ℝ) in 𝓝[≥] x, ‖f x_1 - f x - (x_1 - x) • derivWithin f (Ici x) x‖ ≤ c * ‖x_1 - x‖
m : ℝ
xm : m ∈ Ioi x
hm : Ico x m ⊆ {x_1 | (fun x_2 => ‖f x_2 - f x - (x_2 - x) • derivWithin f (Ici x) x‖ ≤ ε / 2 * ‖x_2 - x‖) x_1}
r : ℝ
hr : r ∈ Ioo 0 (m - x)
this : r ∈ Ioc (r / 2) r
⊢ x ∈ A f (derivWithin f (Ici x) x) r ε

Next tactic:
refine' ⟨r, this, fun y hy z hz => _⟩
theorem mem_A_of_differentiable {ε : ℝ} (hε : 0 < ε) {x : ℝ}
    (hx : DifferentiableWithinAt ℝ f (Ici x) x) :
    ∃ R > 0, ∀ r ∈ Ioo (0 : ℝ) R, x ∈ A f (derivWithin f (Ici x) x) r ε := by
  have := hx.hasDerivWithinAt
  simp_rw [hasDerivWithinAt_iff_isLittleO, isLittleO_iff] at this
  rcases mem_nhdsWithin_Ici_iff_exists_Ico_subset.1 (this (half_pos hε)) with ⟨m, xm, hm⟩
  refine' ⟨m - x, by linarith [show x < m from xm], fun r hr => _⟩
  have : r ∈ Ioc (r / 2) r := ⟨half_lt_self hr.1, le_rfl⟩
  refine' ⟨r, this, fun y hy z hz => _⟩
  

Proof state:
case intro.intro
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
ε : ℝ
hε : 0 < ε
x : ℝ
hx : DifferentiableWithinAt ℝ f (Ici x) x
this✝ : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : ℝ) in 𝓝[≥] x, ‖f x_1 - f x - (x_1 - x) • derivWithin f (Ici x) x‖ ≤ c * ‖x_1 - x‖
m : ℝ
xm : m ∈ Ioi x
hm : Ico x m ⊆ {x_1 | (fun x_2 => ‖f x_2 - f x - (x_2 - x) • derivWithin f (Ici x) x‖ ≤ ε / 2 * ‖x_2 - x‖) x_1}
r : ℝ
hr : r ∈ Ioo 0 (m - x)
this : r ∈ Ioc (r / 2) r
y : ℝ
hy : y ∈ Icc x (x + r)
z : ℝ
hz : z ∈ Icc x (x + r)
⊢ ‖f z - f y - (z - y) • derivWithin f (Ici x) x‖ ≤ ε * r

Next tactic:
calc
    ‖f z - f y - (z - y) • derivWithin f (Ici x) x‖ =
        ‖f z - f x - (z - x) • derivWithin f (Ici x) x -
            (f y - f x - (y - x) • derivWithin f (Ici x) x)‖ :=
      by congr 1; simp only [sub_smul]; abel
    _ ≤
        ‖f z - f x - (z - x) • derivWithin f (Ici x) x‖ +
          ‖f y - f x - (y - x) • derivWithin f (Ici x) x‖ :=
      (norm_sub_le _ _)
    _ ≤ ε / 2 * ‖z - x‖ + ε / 2 * ‖y - x‖ :=
      (add_le_add (hm ⟨hz.1, hz.2.trans_lt (by linarith [hr.2])⟩)
        (hm ⟨hy.1, hy.2.trans_lt (by linarith [hr.2])⟩))
    _ ≤ ε / 2 * r + ε / 2 * r := by
      gcongr
      · rw [Real.norm_of_nonneg] <;> linarith [hz.1, hz.2]
      · rw [Real.norm_of_nonneg] <;> linarith [hy.1, hy.2]
    _ = ε * r := by ring
theorem mem_A_of_differentiable {ε : ℝ} (hε : 0 < ε) {x : ℝ}
    (hx : DifferentiableWithinAt ℝ f (Ici x) x) :
    ∃ R > 0, ∀ r ∈ Ioo (0 : ℝ) R, x ∈ A f (derivWithin f (Ici x) x) r ε := by
  have := hx.hasDerivWithinAt
  simp_rw [hasDerivWithinAt_iff_isLittleO, isLittleO_iff] at this
  rcases mem_nhdsWithin_Ici_iff_exists_Ico_subset.1 (this (half_pos hε)) with ⟨m, xm, hm⟩
  refine' ⟨m - x, by linarith [show x < m from xm], fun r hr => _⟩
  have : r ∈ Ioc (r / 2) r := ⟨half_lt_self hr.1, le_rfl⟩
  refine' ⟨r, this, fun y hy z hz => _⟩
  calc
    ‖f z - f y - (z - y) • derivWithin f (Ici x) x‖ =
        ‖f z - f x - (z - x) • derivWithin f (Ici x) x -
            (f y - f x - (y - x) • derivWithin f (Ici x) x)‖ :=
      by 

Proof state:
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
ε : ℝ
hε : 0 < ε
x : ℝ
hx : DifferentiableWithinAt ℝ f (Ici x) x
this✝ : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : ℝ) in 𝓝[≥] x, ‖f x_1 - f x - (x_1 - x) • derivWithin f (Ici x) x‖ ≤ c * ‖x_1 - x‖
m : ℝ
xm : m ∈ Ioi x
hm : Ico x m ⊆ {x_1 | (fun x_2 => ‖f x_2 - f x - (x_2 - x) • derivWithin f (Ici x) x‖ ≤ ε / 2 * ‖x_2 - x‖) x_1}
r : ℝ
hr : r ∈ Ioo 0 (m - x)
this : r ∈ Ioc (r / 2) r
y : ℝ
hy : y ∈ Icc x (x + r)
z : ℝ
hz : z ∈ Icc x (x + r)
⊢ ‖f z - f y - (z - y) • derivWithin f (Ici x) x‖ =
    ‖f z - f x - (z - x) • derivWithin f (Ici x) x - (f y - f x - (y - x) • derivWithin f (Ici x) x)‖

Next tactic:
congr 1
theorem mem_A_of_differentiable {ε : ℝ} (hε : 0 < ε) {x : ℝ}
    (hx : DifferentiableWithinAt ℝ f (Ici x) x) :
    ∃ R > 0, ∀ r ∈ Ioo (0 : ℝ) R, x ∈ A f (derivWithin f (Ici x) x) r ε := by
  have := hx.hasDerivWithinAt
  simp_rw [hasDerivWithinAt_iff_isLittleO, isLittleO_iff] at this
  rcases mem_nhdsWithin_Ici_iff_exists_Ico_subset.1 (this (half_pos hε)) with ⟨m, xm, hm⟩
  refine' ⟨m - x, by linarith [show x < m from xm], fun r hr => _⟩
  have : r ∈ Ioc (r / 2) r := ⟨half_lt_self hr.1, le_rfl⟩
  refine' ⟨r, this, fun y hy z hz => _⟩
  calc
    ‖f z - f y - (z - y) • derivWithin f (Ici x) x‖ =
        ‖f z - f x - (z - x) • derivWithin f (Ici x) x -
            (f y - f x - (y - x) • derivWithin f (Ici x) x)‖ :=
      by congr 1; 

Proof state:
case e_a
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
ε : ℝ
hε : 0 < ε
x : ℝ
hx : DifferentiableWithinAt ℝ f (Ici x) x
this✝ : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : ℝ) in 𝓝[≥] x, ‖f x_1 - f x - (x_1 - x) • derivWithin f (Ici x) x‖ ≤ c * ‖x_1 - x‖
m : ℝ
xm : m ∈ Ioi x
hm : Ico x m ⊆ {x_1 | (fun x_2 => ‖f x_2 - f x - (x_2 - x) • derivWithin f (Ici x) x‖ ≤ ε / 2 * ‖x_2 - x‖) x_1}
r : ℝ
hr : r ∈ Ioo 0 (m - x)
this : r ∈ Ioc (r / 2) r
y : ℝ
hy : y ∈ Icc x (x + r)
z : ℝ
hz : z ∈ Icc x (x + r)
⊢ f z - f y - (z - y) • derivWithin f (Ici x) x =
    f z - f x - (z - x) • derivWithin f (Ici x) x - (f y - f x - (y - x) • derivWithin f (Ici x) x)

Next tactic:
simp only [sub_smul]
theorem mem_A_of_differentiable {ε : ℝ} (hε : 0 < ε) {x : ℝ}
    (hx : DifferentiableWithinAt ℝ f (Ici x) x) :
    ∃ R > 0, ∀ r ∈ Ioo (0 : ℝ) R, x ∈ A f (derivWithin f (Ici x) x) r ε := by
  have := hx.hasDerivWithinAt
  simp_rw [hasDerivWithinAt_iff_isLittleO, isLittleO_iff] at this
  rcases mem_nhdsWithin_Ici_iff_exists_Ico_subset.1 (this (half_pos hε)) with ⟨m, xm, hm⟩
  refine' ⟨m - x, by linarith [show x < m from xm], fun r hr => _⟩
  have : r ∈ Ioc (r / 2) r := ⟨half_lt_self hr.1, le_rfl⟩
  refine' ⟨r, this, fun y hy z hz => _⟩
  calc
    ‖f z - f y - (z - y) • derivWithin f (Ici x) x‖ =
        ‖f z - f x - (z - x) • derivWithin f (Ici x) x -
            (f y - f x - (y - x) • derivWithin f (Ici x) x)‖ :=
      by congr 1; simp only [sub_smul]; 

Proof state:
case e_a
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
ε : ℝ
hε : 0 < ε
x : ℝ
hx : DifferentiableWithinAt ℝ f (Ici x) x
this✝ : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : ℝ) in 𝓝[≥] x, ‖f x_1 - f x - (x_1 - x) • derivWithin f (Ici x) x‖ ≤ c * ‖x_1 - x‖
m : ℝ
xm : m ∈ Ioi x
hm : Ico x m ⊆ {x_1 | (fun x_2 => ‖f x_2 - f x - (x_2 - x) • derivWithin f (Ici x) x‖ ≤ ε / 2 * ‖x_2 - x‖) x_1}
r : ℝ
hr : r ∈ Ioo 0 (m - x)
this : r ∈ Ioc (r / 2) r
y : ℝ
hy : y ∈ Icc x (x + r)
z : ℝ
hz : z ∈ Icc x (x + r)
⊢ f z - f y - (z • derivWithin f (Ici x) x - y • derivWithin f (Ici x) x) =
    f z - f x - (z • derivWithin f (Ici x) x - x • derivWithin f (Ici x) x) -
      (f y - f x - (y • derivWithin f (Ici x) x - x • derivWithin f (Ici x) x))

Next tactic:
abel
theorem mem_A_of_differentiable {ε : ℝ} (hε : 0 < ε) {x : ℝ}
    (hx : DifferentiableWithinAt ℝ f (Ici x) x) :
    ∃ R > 0, ∀ r ∈ Ioo (0 : ℝ) R, x ∈ A f (derivWithin f (Ici x) x) r ε := by
  have := hx.hasDerivWithinAt
  simp_rw [hasDerivWithinAt_iff_isLittleO, isLittleO_iff] at this
  rcases mem_nhdsWithin_Ici_iff_exists_Ico_subset.1 (this (half_pos hε)) with ⟨m, xm, hm⟩
  refine' ⟨m - x, by linarith [show x < m from xm], fun r hr => _⟩
  have : r ∈ Ioc (r / 2) r := ⟨half_lt_self hr.1, le_rfl⟩
  refine' ⟨r, this, fun y hy z hz => _⟩
  calc
    ‖f z - f y - (z - y) • derivWithin f (Ici x) x‖ =
        ‖f z - f x - (z - x) • derivWithin f (Ici x) x -
            (f y - f x - (y - x) • derivWithin f (Ici x) x)‖ :=
      by congr 1; simp only [sub_smul]; abel
    _ ≤
        ‖f z - f x - (z - x) • derivWithin f (Ici x) x‖ +
          ‖f y - f x - (y - x) • derivWithin f (Ici x) x‖ :=
      (norm_sub_le _ _)
    _ ≤ ε / 2 * ‖z - x‖ + ε / 2 * ‖y - x‖ :=
      (add_le_add (hm ⟨hz.1, hz.2.trans_lt (by 

Proof state:
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
ε : ℝ
hε : 0 < ε
x : ℝ
hx : DifferentiableWithinAt ℝ f (Ici x) x
this✝ : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : ℝ) in 𝓝[≥] x, ‖f x_1 - f x - (x_1 - x) • derivWithin f (Ici x) x‖ ≤ c * ‖x_1 - x‖
m : ℝ
xm : m ∈ Ioi x
hm : Ico x m ⊆ {x_1 | (fun x_2 => ‖f x_2 - f x - (x_2 - x) • derivWithin f (Ici x) x‖ ≤ ε / 2 * ‖x_2 - x‖) x_1}
r : ℝ
hr : r ∈ Ioo 0 (m - x)
this : r ∈ Ioc (r / 2) r
y : ℝ
hy : y ∈ Icc x (x + r)
z : ℝ
hz : z ∈ Icc x (x + r)
⊢ x + r < m

Next tactic:
linarith [hr.2]
theorem mem_A_of_differentiable {ε : ℝ} (hε : 0 < ε) {x : ℝ}
    (hx : DifferentiableWithinAt ℝ f (Ici x) x) :
    ∃ R > 0, ∀ r ∈ Ioo (0 : ℝ) R, x ∈ A f (derivWithin f (Ici x) x) r ε := by
  have := hx.hasDerivWithinAt
  simp_rw [hasDerivWithinAt_iff_isLittleO, isLittleO_iff] at this
  rcases mem_nhdsWithin_Ici_iff_exists_Ico_subset.1 (this (half_pos hε)) with ⟨m, xm, hm⟩
  refine' ⟨m - x, by linarith [show x < m from xm], fun r hr => _⟩
  have : r ∈ Ioc (r / 2) r := ⟨half_lt_self hr.1, le_rfl⟩
  refine' ⟨r, this, fun y hy z hz => _⟩
  calc
    ‖f z - f y - (z - y) • derivWithin f (Ici x) x‖ =
        ‖f z - f x - (z - x) • derivWithin f (Ici x) x -
            (f y - f x - (y - x) • derivWithin f (Ici x) x)‖ :=
      by congr 1; simp only [sub_smul]; abel
    _ ≤
        ‖f z - f x - (z - x) • derivWithin f (Ici x) x‖ +
          ‖f y - f x - (y - x) • derivWithin f (Ici x) x‖ :=
      (norm_sub_le _ _)
    _ ≤ ε / 2 * ‖z - x‖ + ε / 2 * ‖y - x‖ :=
      (add_le_add (hm ⟨hz.1, hz.2.trans_lt (by linarith [hr.2])⟩)
        (hm ⟨hy.1, hy.2.trans_lt (by 

Proof state:
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
ε : ℝ
hε : 0 < ε
x : ℝ
hx : DifferentiableWithinAt ℝ f (Ici x) x
this✝ : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : ℝ) in 𝓝[≥] x, ‖f x_1 - f x - (x_1 - x) • derivWithin f (Ici x) x‖ ≤ c * ‖x_1 - x‖
m : ℝ
xm : m ∈ Ioi x
hm : Ico x m ⊆ {x_1 | (fun x_2 => ‖f x_2 - f x - (x_2 - x) • derivWithin f (Ici x) x‖ ≤ ε / 2 * ‖x_2 - x‖) x_1}
r : ℝ
hr : r ∈ Ioo 0 (m - x)
this : r ∈ Ioc (r / 2) r
y : ℝ
hy : y ∈ Icc x (x + r)
z : ℝ
hz : z ∈ Icc x (x + r)
⊢ x + r < m

Next tactic:
linarith [hr.2]
theorem mem_A_of_differentiable {ε : ℝ} (hε : 0 < ε) {x : ℝ}
    (hx : DifferentiableWithinAt ℝ f (Ici x) x) :
    ∃ R > 0, ∀ r ∈ Ioo (0 : ℝ) R, x ∈ A f (derivWithin f (Ici x) x) r ε := by
  have := hx.hasDerivWithinAt
  simp_rw [hasDerivWithinAt_iff_isLittleO, isLittleO_iff] at this
  rcases mem_nhdsWithin_Ici_iff_exists_Ico_subset.1 (this (half_pos hε)) with ⟨m, xm, hm⟩
  refine' ⟨m - x, by linarith [show x < m from xm], fun r hr => _⟩
  have : r ∈ Ioc (r / 2) r := ⟨half_lt_self hr.1, le_rfl⟩
  refine' ⟨r, this, fun y hy z hz => _⟩
  calc
    ‖f z - f y - (z - y) • derivWithin f (Ici x) x‖ =
        ‖f z - f x - (z - x) • derivWithin f (Ici x) x -
            (f y - f x - (y - x) • derivWithin f (Ici x) x)‖ :=
      by congr 1; simp only [sub_smul]; abel
    _ ≤
        ‖f z - f x - (z - x) • derivWithin f (Ici x) x‖ +
          ‖f y - f x - (y - x) • derivWithin f (Ici x) x‖ :=
      (norm_sub_le _ _)
    _ ≤ ε / 2 * ‖z - x‖ + ε / 2 * ‖y - x‖ :=
      (add_le_add (hm ⟨hz.1, hz.2.trans_lt (by linarith [hr.2])⟩)
        (hm ⟨hy.1, hy.2.trans_lt (by linarith [hr.2])⟩))
    _ ≤ ε / 2 * r + ε / 2 * r := by
      

Proof state:
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
ε : ℝ
hε : 0 < ε
x : ℝ
hx : DifferentiableWithinAt ℝ f (Ici x) x
this✝ : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : ℝ) in 𝓝[≥] x, ‖f x_1 - f x - (x_1 - x) • derivWithin f (Ici x) x‖ ≤ c * ‖x_1 - x‖
m : ℝ
xm : m ∈ Ioi x
hm : Ico x m ⊆ {x_1 | (fun x_2 => ‖f x_2 - f x - (x_2 - x) • derivWithin f (Ici x) x‖ ≤ ε / 2 * ‖x_2 - x‖) x_1}
r : ℝ
hr : r ∈ Ioo 0 (m - x)
this : r ∈ Ioc (r / 2) r
y : ℝ
hy : y ∈ Icc x (x + r)
z : ℝ
hz : z ∈ Icc x (x + r)
⊢ ε / 2 * ‖z - x‖ + ε / 2 * ‖y - x‖ ≤ ε / 2 * r + ε / 2 * r

Next tactic:
gcongr
theorem mem_A_of_differentiable {ε : ℝ} (hε : 0 < ε) {x : ℝ}
    (hx : DifferentiableWithinAt ℝ f (Ici x) x) :
    ∃ R > 0, ∀ r ∈ Ioo (0 : ℝ) R, x ∈ A f (derivWithin f (Ici x) x) r ε := by
  have := hx.hasDerivWithinAt
  simp_rw [hasDerivWithinAt_iff_isLittleO, isLittleO_iff] at this
  rcases mem_nhdsWithin_Ici_iff_exists_Ico_subset.1 (this (half_pos hε)) with ⟨m, xm, hm⟩
  refine' ⟨m - x, by linarith [show x < m from xm], fun r hr => _⟩
  have : r ∈ Ioc (r / 2) r := ⟨half_lt_self hr.1, le_rfl⟩
  refine' ⟨r, this, fun y hy z hz => _⟩
  calc
    ‖f z - f y - (z - y) • derivWithin f (Ici x) x‖ =
        ‖f z - f x - (z - x) • derivWithin f (Ici x) x -
            (f y - f x - (y - x) • derivWithin f (Ici x) x)‖ :=
      by congr 1; simp only [sub_smul]; abel
    _ ≤
        ‖f z - f x - (z - x) • derivWithin f (Ici x) x‖ +
          ‖f y - f x - (y - x) • derivWithin f (Ici x) x‖ :=
      (norm_sub_le _ _)
    _ ≤ ε / 2 * ‖z - x‖ + ε / 2 * ‖y - x‖ :=
      (add_le_add (hm ⟨hz.1, hz.2.trans_lt (by linarith [hr.2])⟩)
        (hm ⟨hy.1, hy.2.trans_lt (by linarith [hr.2])⟩))
    _ ≤ ε / 2 * r + ε / 2 * r := by
      gcongr
      · 

Proof state:
case h₁.h
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
ε : ℝ
hε : 0 < ε
x : ℝ
hx : DifferentiableWithinAt ℝ f (Ici x) x
this✝ : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : ℝ) in 𝓝[≥] x, ‖f x_1 - f x - (x_1 - x) • derivWithin f (Ici x) x‖ ≤ c * ‖x_1 - x‖
m : ℝ
xm : m ∈ Ioi x
hm : Ico x m ⊆ {x_1 | (fun x_2 => ‖f x_2 - f x - (x_2 - x) • derivWithin f (Ici x) x‖ ≤ ε / 2 * ‖x_2 - x‖) x_1}
r : ℝ
hr : r ∈ Ioo 0 (m - x)
this : r ∈ Ioc (r / 2) r
y : ℝ
hy : y ∈ Icc x (x + r)
z : ℝ
hz : z ∈ Icc x (x + r)
⊢ ‖z - x‖ ≤ r

Next tactic:
rw [Real.norm_of_nonneg]
theorem mem_A_of_differentiable {ε : ℝ} (hε : 0 < ε) {x : ℝ}
    (hx : DifferentiableWithinAt ℝ f (Ici x) x) :
    ∃ R > 0, ∀ r ∈ Ioo (0 : ℝ) R, x ∈ A f (derivWithin f (Ici x) x) r ε := by
  have := hx.hasDerivWithinAt
  simp_rw [hasDerivWithinAt_iff_isLittleO, isLittleO_iff] at this
  rcases mem_nhdsWithin_Ici_iff_exists_Ico_subset.1 (this (half_pos hε)) with ⟨m, xm, hm⟩
  refine' ⟨m - x, by linarith [show x < m from xm], fun r hr => _⟩
  have : r ∈ Ioc (r / 2) r := ⟨half_lt_self hr.1, le_rfl⟩
  refine' ⟨r, this, fun y hy z hz => _⟩
  calc
    ‖f z - f y - (z - y) • derivWithin f (Ici x) x‖ =
        ‖f z - f x - (z - x) • derivWithin f (Ici x) x -
            (f y - f x - (y - x) • derivWithin f (Ici x) x)‖ :=
      by congr 1; simp only [sub_smul]; abel
    _ ≤
        ‖f z - f x - (z - x) • derivWithin f (Ici x) x‖ +
          ‖f y - f x - (y - x) • derivWithin f (Ici x) x‖ :=
      (norm_sub_le _ _)
    _ ≤ ε / 2 * ‖z - x‖ + ε / 2 * ‖y - x‖ :=
      (add_le_add (hm ⟨hz.1, hz.2.trans_lt (by linarith [hr.2])⟩)
        (hm ⟨hy.1, hy.2.trans_lt (by linarith [hr.2])⟩))
    _ ≤ ε / 2 * r + ε / 2 * r := by
      gcongr
      · rw [Real.norm_of_nonneg] <;> 

Proof state:
case h₁.h
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
ε : ℝ
hε : 0 < ε
x : ℝ
hx : DifferentiableWithinAt ℝ f (Ici x) x
this✝ : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : ℝ) in 𝓝[≥] x, ‖f x_1 - f x - (x_1 - x) • derivWithin f (Ici x) x‖ ≤ c * ‖x_1 - x‖
m : ℝ
xm : m ∈ Ioi x
hm : Ico x m ⊆ {x_1 | (fun x_2 => ‖f x_2 - f x - (x_2 - x) • derivWithin f (Ici x) x‖ ≤ ε / 2 * ‖x_2 - x‖) x_1}
r : ℝ
hr : r ∈ Ioo 0 (m - x)
this : r ∈ Ioc (r / 2) r
y : ℝ
hy : y ∈ Icc x (x + r)
z : ℝ
hz : z ∈ Icc x (x + r)
⊢ z - x ≤ r

Next tactic:
linarith [hz.1, hz.2]
theorem mem_A_of_differentiable {ε : ℝ} (hε : 0 < ε) {x : ℝ}
    (hx : DifferentiableWithinAt ℝ f (Ici x) x) :
    ∃ R > 0, ∀ r ∈ Ioo (0 : ℝ) R, x ∈ A f (derivWithin f (Ici x) x) r ε := by
  have := hx.hasDerivWithinAt
  simp_rw [hasDerivWithinAt_iff_isLittleO, isLittleO_iff] at this
  rcases mem_nhdsWithin_Ici_iff_exists_Ico_subset.1 (this (half_pos hε)) with ⟨m, xm, hm⟩
  refine' ⟨m - x, by linarith [show x < m from xm], fun r hr => _⟩
  have : r ∈ Ioc (r / 2) r := ⟨half_lt_self hr.1, le_rfl⟩
  refine' ⟨r, this, fun y hy z hz => _⟩
  calc
    ‖f z - f y - (z - y) • derivWithin f (Ici x) x‖ =
        ‖f z - f x - (z - x) • derivWithin f (Ici x) x -
            (f y - f x - (y - x) • derivWithin f (Ici x) x)‖ :=
      by congr 1; simp only [sub_smul]; abel
    _ ≤
        ‖f z - f x - (z - x) • derivWithin f (Ici x) x‖ +
          ‖f y - f x - (y - x) • derivWithin f (Ici x) x‖ :=
      (norm_sub_le _ _)
    _ ≤ ε / 2 * ‖z - x‖ + ε / 2 * ‖y - x‖ :=
      (add_le_add (hm ⟨hz.1, hz.2.trans_lt (by linarith [hr.2])⟩)
        (hm ⟨hy.1, hy.2.trans_lt (by linarith [hr.2])⟩))
    _ ≤ ε / 2 * r + ε / 2 * r := by
      gcongr
      · rw [Real.norm_of_nonneg] <;> 

Proof state:
case h₁.h
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
ε : ℝ
hε : 0 < ε
x : ℝ
hx : DifferentiableWithinAt ℝ f (Ici x) x
this✝ : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : ℝ) in 𝓝[≥] x, ‖f x_1 - f x - (x_1 - x) • derivWithin f (Ici x) x‖ ≤ c * ‖x_1 - x‖
m : ℝ
xm : m ∈ Ioi x
hm : Ico x m ⊆ {x_1 | (fun x_2 => ‖f x_2 - f x - (x_2 - x) • derivWithin f (Ici x) x‖ ≤ ε / 2 * ‖x_2 - x‖) x_1}
r : ℝ
hr : r ∈ Ioo 0 (m - x)
this : r ∈ Ioc (r / 2) r
y : ℝ
hy : y ∈ Icc x (x + r)
z : ℝ
hz : z ∈ Icc x (x + r)
⊢ 0 ≤ z - x

Next tactic:
linarith [hz.1, hz.2]
theorem mem_A_of_differentiable {ε : ℝ} (hε : 0 < ε) {x : ℝ}
    (hx : DifferentiableWithinAt ℝ f (Ici x) x) :
    ∃ R > 0, ∀ r ∈ Ioo (0 : ℝ) R, x ∈ A f (derivWithin f (Ici x) x) r ε := by
  have := hx.hasDerivWithinAt
  simp_rw [hasDerivWithinAt_iff_isLittleO, isLittleO_iff] at this
  rcases mem_nhdsWithin_Ici_iff_exists_Ico_subset.1 (this (half_pos hε)) with ⟨m, xm, hm⟩
  refine' ⟨m - x, by linarith [show x < m from xm], fun r hr => _⟩
  have : r ∈ Ioc (r / 2) r := ⟨half_lt_self hr.1, le_rfl⟩
  refine' ⟨r, this, fun y hy z hz => _⟩
  calc
    ‖f z - f y - (z - y) • derivWithin f (Ici x) x‖ =
        ‖f z - f x - (z - x) • derivWithin f (Ici x) x -
            (f y - f x - (y - x) • derivWithin f (Ici x) x)‖ :=
      by congr 1; simp only [sub_smul]; abel
    _ ≤
        ‖f z - f x - (z - x) • derivWithin f (Ici x) x‖ +
          ‖f y - f x - (y - x) • derivWithin f (Ici x) x‖ :=
      (norm_sub_le _ _)
    _ ≤ ε / 2 * ‖z - x‖ + ε / 2 * ‖y - x‖ :=
      (add_le_add (hm ⟨hz.1, hz.2.trans_lt (by linarith [hr.2])⟩)
        (hm ⟨hy.1, hy.2.trans_lt (by linarith [hr.2])⟩))
    _ ≤ ε / 2 * r + ε / 2 * r := by
      gcongr
      · rw [Real.norm_of_nonneg] <;> linarith [hz.1, hz.2]
      · 

Proof state:
case h₂.h
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
ε : ℝ
hε : 0 < ε
x : ℝ
hx : DifferentiableWithinAt ℝ f (Ici x) x
this✝ : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : ℝ) in 𝓝[≥] x, ‖f x_1 - f x - (x_1 - x) • derivWithin f (Ici x) x‖ ≤ c * ‖x_1 - x‖
m : ℝ
xm : m ∈ Ioi x
hm : Ico x m ⊆ {x_1 | (fun x_2 => ‖f x_2 - f x - (x_2 - x) • derivWithin f (Ici x) x‖ ≤ ε / 2 * ‖x_2 - x‖) x_1}
r : ℝ
hr : r ∈ Ioo 0 (m - x)
this : r ∈ Ioc (r / 2) r
y : ℝ
hy : y ∈ Icc x (x + r)
z : ℝ
hz : z ∈ Icc x (x + r)
⊢ ‖y - x‖ ≤ r

Next tactic:
rw [Real.norm_of_nonneg]
theorem mem_A_of_differentiable {ε : ℝ} (hε : 0 < ε) {x : ℝ}
    (hx : DifferentiableWithinAt ℝ f (Ici x) x) :
    ∃ R > 0, ∀ r ∈ Ioo (0 : ℝ) R, x ∈ A f (derivWithin f (Ici x) x) r ε := by
  have := hx.hasDerivWithinAt
  simp_rw [hasDerivWithinAt_iff_isLittleO, isLittleO_iff] at this
  rcases mem_nhdsWithin_Ici_iff_exists_Ico_subset.1 (this (half_pos hε)) with ⟨m, xm, hm⟩
  refine' ⟨m - x, by linarith [show x < m from xm], fun r hr => _⟩
  have : r ∈ Ioc (r / 2) r := ⟨half_lt_self hr.1, le_rfl⟩
  refine' ⟨r, this, fun y hy z hz => _⟩
  calc
    ‖f z - f y - (z - y) • derivWithin f (Ici x) x‖ =
        ‖f z - f x - (z - x) • derivWithin f (Ici x) x -
            (f y - f x - (y - x) • derivWithin f (Ici x) x)‖ :=
      by congr 1; simp only [sub_smul]; abel
    _ ≤
        ‖f z - f x - (z - x) • derivWithin f (Ici x) x‖ +
          ‖f y - f x - (y - x) • derivWithin f (Ici x) x‖ :=
      (norm_sub_le _ _)
    _ ≤ ε / 2 * ‖z - x‖ + ε / 2 * ‖y - x‖ :=
      (add_le_add (hm ⟨hz.1, hz.2.trans_lt (by linarith [hr.2])⟩)
        (hm ⟨hy.1, hy.2.trans_lt (by linarith [hr.2])⟩))
    _ ≤ ε / 2 * r + ε / 2 * r := by
      gcongr
      · rw [Real.norm_of_nonneg] <;> linarith [hz.1, hz.2]
      · rw [Real.norm_of_nonneg] <;> 

Proof state:
case h₂.h
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
ε : ℝ
hε : 0 < ε
x : ℝ
hx : DifferentiableWithinAt ℝ f (Ici x) x
this✝ : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : ℝ) in 𝓝[≥] x, ‖f x_1 - f x - (x_1 - x) • derivWithin f (Ici x) x‖ ≤ c * ‖x_1 - x‖
m : ℝ
xm : m ∈ Ioi x
hm : Ico x m ⊆ {x_1 | (fun x_2 => ‖f x_2 - f x - (x_2 - x) • derivWithin f (Ici x) x‖ ≤ ε / 2 * ‖x_2 - x‖) x_1}
r : ℝ
hr : r ∈ Ioo 0 (m - x)
this : r ∈ Ioc (r / 2) r
y : ℝ
hy : y ∈ Icc x (x + r)
z : ℝ
hz : z ∈ Icc x (x + r)
⊢ y - x ≤ r

Next tactic:
linarith [hy.1, hy.2]