l3lab/ntp-mathlib huggingface dataset:
/-- If a function is differentiable on the right of a point `a : ℝ`, continuous at `a`, and
its derivative also converges at `a`, then `f` is differentiable on the right at `a`. -/
theorem has_deriv_at_interval_left_endpoint_of_tendsto_deriv {s : Set ℝ} {e : E} {a : ℝ} {f : ℝ → E}
    (f_diff : DifferentiableOn ℝ f s) (f_lim : ContinuousWithinAt f s a) (hs : s ∈ 𝓝[>] a)
    (f_lim' : Tendsto (fun x => deriv f x) (𝓝[>] a) (𝓝 e)) : HasDerivWithinAt f e (Ici a) a := by
  /- This is a specialization of `has_fderiv_at_boundary_of_tendsto_fderiv`. To be in the setting of
    this theorem, we need to work on an open interval with closure contained in `s ∪ {a}`, that we
    call `t = (a, b)`. Then, we check all the assumptions of this theorem and we apply it. -/
  obtain ⟨b, ab : a < b, sab : Ioc a b ⊆ s⟩ := mem_nhdsWithin_Ioi_iff_exists_Ioc_subset.1 hs
  let t := Ioo a b
  have ts : t ⊆ s := Subset.trans Ioo_subset_Ioc_self sab
  have t_diff : DifferentiableOn ℝ f t := f_diff.mono ts
  have t_conv : Convex ℝ t := convex_Ioo a b
  have t_open : IsOpen t := isOpen_Ioo
  have t_closure : closure t = Icc a b := closure_Ioo ab.ne
  have t_cont : ∀ y ∈ closure t, ContinuousWithinAt f t y := by
    rw [t_closure]
    

Proof state:
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set ℝ
e : E
a : ℝ
f : ℝ → E
f_diff : DifferentiableOn ℝ f s
f_lim : ContinuousWithinAt f s a
hs : s ∈ 𝓝[>] a
f_lim' : Tendsto (fun x => deriv f x) (𝓝[>] a) (𝓝 e)
b : ℝ
ab : a < b
sab : Ioc a b ⊆ s
t : Set ℝ := Ioo a b
ts : t ⊆ s
t_diff : DifferentiableOn ℝ f t
t_conv : Convex ℝ t
t_open : IsOpen t
t_closure : closure t = Icc a b
⊢ ∀ y ∈ Icc a b, ContinuousWithinAt f t y

Next tactic:
intro y hy
/-- If a function is differentiable on the right of a point `a : ℝ`, continuous at `a`, and
its derivative also converges at `a`, then `f` is differentiable on the right at `a`. -/
theorem has_deriv_at_interval_left_endpoint_of_tendsto_deriv {s : Set ℝ} {e : E} {a : ℝ} {f : ℝ → E}
    (f_diff : DifferentiableOn ℝ f s) (f_lim : ContinuousWithinAt f s a) (hs : s ∈ 𝓝[>] a)
    (f_lim' : Tendsto (fun x => deriv f x) (𝓝[>] a) (𝓝 e)) : HasDerivWithinAt f e (Ici a) a := by
  /- This is a specialization of `has_fderiv_at_boundary_of_tendsto_fderiv`. To be in the setting of
    this theorem, we need to work on an open interval with closure contained in `s ∪ {a}`, that we
    call `t = (a, b)`. Then, we check all the assumptions of this theorem and we apply it. -/
  obtain ⟨b, ab : a < b, sab : Ioc a b ⊆ s⟩ := mem_nhdsWithin_Ioi_iff_exists_Ioc_subset.1 hs
  let t := Ioo a b
  have ts : t ⊆ s := Subset.trans Ioo_subset_Ioc_self sab
  have t_diff : DifferentiableOn ℝ f t := f_diff.mono ts
  have t_conv : Convex ℝ t := convex_Ioo a b
  have t_open : IsOpen t := isOpen_Ioo
  have t_closure : closure t = Icc a b := closure_Ioo ab.ne
  have t_cont : ∀ y ∈ closure t, ContinuousWithinAt f t y := by
    rw [t_closure]
    intro y hy
    

Proof state:
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set ℝ
e : E
a : ℝ
f : ℝ → E
f_diff : DifferentiableOn ℝ f s
f_lim : ContinuousWithinAt f s a
hs : s ∈ 𝓝[>] a
f_lim' : Tendsto (fun x => deriv f x) (𝓝[>] a) (𝓝 e)
b : ℝ
ab : a < b
sab : Ioc a b ⊆ s
t : Set ℝ := Ioo a b
ts : t ⊆ s
t_diff : DifferentiableOn ℝ f t
t_conv : Convex ℝ t
t_open : IsOpen t
t_closure : closure t = Icc a b
y : ℝ
hy : y ∈ Icc a b
⊢ ContinuousWithinAt f t y

Next tactic:
by_cases h : y = a
/-- If a function is differentiable on the right of a point `a : ℝ`, continuous at `a`, and
its derivative also converges at `a`, then `f` is differentiable on the right at `a`. -/
theorem has_deriv_at_interval_left_endpoint_of_tendsto_deriv {s : Set ℝ} {e : E} {a : ℝ} {f : ℝ → E}
    (f_diff : DifferentiableOn ℝ f s) (f_lim : ContinuousWithinAt f s a) (hs : s ∈ 𝓝[>] a)
    (f_lim' : Tendsto (fun x => deriv f x) (𝓝[>] a) (𝓝 e)) : HasDerivWithinAt f e (Ici a) a := by
  /- This is a specialization of `has_fderiv_at_boundary_of_tendsto_fderiv`. To be in the setting of
    this theorem, we need to work on an open interval with closure contained in `s ∪ {a}`, that we
    call `t = (a, b)`. Then, we check all the assumptions of this theorem and we apply it. -/
  obtain ⟨b, ab : a < b, sab : Ioc a b ⊆ s⟩ := mem_nhdsWithin_Ioi_iff_exists_Ioc_subset.1 hs
  let t := Ioo a b
  have ts : t ⊆ s := Subset.trans Ioo_subset_Ioc_self sab
  have t_diff : DifferentiableOn ℝ f t := f_diff.mono ts
  have t_conv : Convex ℝ t := convex_Ioo a b
  have t_open : IsOpen t := isOpen_Ioo
  have t_closure : closure t = Icc a b := closure_Ioo ab.ne
  have t_cont : ∀ y ∈ closure t, ContinuousWithinAt f t y := by
    rw [t_closure]
    intro y hy
    by_cases h : y = a
    · 

Proof state:
case pos
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set ℝ
e : E
a : ℝ
f : ℝ → E
f_diff : DifferentiableOn ℝ f s
f_lim : ContinuousWithinAt f s a
hs : s ∈ 𝓝[>] a
f_lim' : Tendsto (fun x => deriv f x) (𝓝[>] a) (𝓝 e)
b : ℝ
ab : a < b
sab : Ioc a b ⊆ s
t : Set ℝ := Ioo a b
ts : t ⊆ s
t_diff : DifferentiableOn ℝ f t
t_conv : Convex ℝ t
t_open : IsOpen t
t_closure : closure t = Icc a b
y : ℝ
hy : y ∈ Icc a b
h : y = a
⊢ ContinuousWithinAt f t y

Next tactic:
rw [h]
/-- If a function is differentiable on the right of a point `a : ℝ`, continuous at `a`, and
its derivative also converges at `a`, then `f` is differentiable on the right at `a`. -/
theorem has_deriv_at_interval_left_endpoint_of_tendsto_deriv {s : Set ℝ} {e : E} {a : ℝ} {f : ℝ → E}
    (f_diff : DifferentiableOn ℝ f s) (f_lim : ContinuousWithinAt f s a) (hs : s ∈ 𝓝[>] a)
    (f_lim' : Tendsto (fun x => deriv f x) (𝓝[>] a) (𝓝 e)) : HasDerivWithinAt f e (Ici a) a := by
  /- This is a specialization of `has_fderiv_at_boundary_of_tendsto_fderiv`. To be in the setting of
    this theorem, we need to work on an open interval with closure contained in `s ∪ {a}`, that we
    call `t = (a, b)`. Then, we check all the assumptions of this theorem and we apply it. -/
  obtain ⟨b, ab : a < b, sab : Ioc a b ⊆ s⟩ := mem_nhdsWithin_Ioi_iff_exists_Ioc_subset.1 hs
  let t := Ioo a b
  have ts : t ⊆ s := Subset.trans Ioo_subset_Ioc_self sab
  have t_diff : DifferentiableOn ℝ f t := f_diff.mono ts
  have t_conv : Convex ℝ t := convex_Ioo a b
  have t_open : IsOpen t := isOpen_Ioo
  have t_closure : closure t = Icc a b := closure_Ioo ab.ne
  have t_cont : ∀ y ∈ closure t, ContinuousWithinAt f t y := by
    rw [t_closure]
    intro y hy
    by_cases h : y = a
    · rw [h]; 

Proof state:
case pos
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set ℝ
e : E
a : ℝ
f : ℝ → E
f_diff : DifferentiableOn ℝ f s
f_lim : ContinuousWithinAt f s a
hs : s ∈ 𝓝[>] a
f_lim' : Tendsto (fun x => deriv f x) (𝓝[>] a) (𝓝 e)
b : ℝ
ab : a < b
sab : Ioc a b ⊆ s
t : Set ℝ := Ioo a b
ts : t ⊆ s
t_diff : DifferentiableOn ℝ f t
t_conv : Convex ℝ t
t_open : IsOpen t
t_closure : closure t = Icc a b
y : ℝ
hy : y ∈ Icc a b
h : y = a
⊢ ContinuousWithinAt f t a

Next tactic:
exact f_lim.mono ts
/-- If a function is differentiable on the right of a point `a : ℝ`, continuous at `a`, and
its derivative also converges at `a`, then `f` is differentiable on the right at `a`. -/
theorem has_deriv_at_interval_left_endpoint_of_tendsto_deriv {s : Set ℝ} {e : E} {a : ℝ} {f : ℝ → E}
    (f_diff : DifferentiableOn ℝ f s) (f_lim : ContinuousWithinAt f s a) (hs : s ∈ 𝓝[>] a)
    (f_lim' : Tendsto (fun x => deriv f x) (𝓝[>] a) (𝓝 e)) : HasDerivWithinAt f e (Ici a) a := by
  /- This is a specialization of `has_fderiv_at_boundary_of_tendsto_fderiv`. To be in the setting of
    this theorem, we need to work on an open interval with closure contained in `s ∪ {a}`, that we
    call `t = (a, b)`. Then, we check all the assumptions of this theorem and we apply it. -/
  obtain ⟨b, ab : a < b, sab : Ioc a b ⊆ s⟩ := mem_nhdsWithin_Ioi_iff_exists_Ioc_subset.1 hs
  let t := Ioo a b
  have ts : t ⊆ s := Subset.trans Ioo_subset_Ioc_self sab
  have t_diff : DifferentiableOn ℝ f t := f_diff.mono ts
  have t_conv : Convex ℝ t := convex_Ioo a b
  have t_open : IsOpen t := isOpen_Ioo
  have t_closure : closure t = Icc a b := closure_Ioo ab.ne
  have t_cont : ∀ y ∈ closure t, ContinuousWithinAt f t y := by
    rw [t_closure]
    intro y hy
    by_cases h : y = a
    · rw [h]; exact f_lim.mono ts
    · 

Proof state:
case neg
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set ℝ
e : E
a : ℝ
f : ℝ → E
f_diff : DifferentiableOn ℝ f s
f_lim : ContinuousWithinAt f s a
hs : s ∈ 𝓝[>] a
f_lim' : Tendsto (fun x => deriv f x) (𝓝[>] a) (𝓝 e)
b : ℝ
ab : a < b
sab : Ioc a b ⊆ s
t : Set ℝ := Ioo a b
ts : t ⊆ s
t_diff : DifferentiableOn ℝ f t
t_conv : Convex ℝ t
t_open : IsOpen t
t_closure : closure t = Icc a b
y : ℝ
hy : y ∈ Icc a b
h : ¬y = a
⊢ ContinuousWithinAt f t y

Next tactic:
have : y ∈ s := sab ⟨lt_of_le_of_ne hy.1 (Ne.symm h), hy.2⟩
/-- If a function is differentiable on the right of a point `a : ℝ`, continuous at `a`, and
its derivative also converges at `a`, then `f` is differentiable on the right at `a`. -/
theorem has_deriv_at_interval_left_endpoint_of_tendsto_deriv {s : Set ℝ} {e : E} {a : ℝ} {f : ℝ → E}
    (f_diff : DifferentiableOn ℝ f s) (f_lim : ContinuousWithinAt f s a) (hs : s ∈ 𝓝[>] a)
    (f_lim' : Tendsto (fun x => deriv f x) (𝓝[>] a) (𝓝 e)) : HasDerivWithinAt f e (Ici a) a := by
  /- This is a specialization of `has_fderiv_at_boundary_of_tendsto_fderiv`. To be in the setting of
    this theorem, we need to work on an open interval with closure contained in `s ∪ {a}`, that we
    call `t = (a, b)`. Then, we check all the assumptions of this theorem and we apply it. -/
  obtain ⟨b, ab : a < b, sab : Ioc a b ⊆ s⟩ := mem_nhdsWithin_Ioi_iff_exists_Ioc_subset.1 hs
  let t := Ioo a b
  have ts : t ⊆ s := Subset.trans Ioo_subset_Ioc_self sab
  have t_diff : DifferentiableOn ℝ f t := f_diff.mono ts
  have t_conv : Convex ℝ t := convex_Ioo a b
  have t_open : IsOpen t := isOpen_Ioo
  have t_closure : closure t = Icc a b := closure_Ioo ab.ne
  have t_cont : ∀ y ∈ closure t, ContinuousWithinAt f t y := by
    rw [t_closure]
    intro y hy
    by_cases h : y = a
    · rw [h]; exact f_lim.mono ts
    · have : y ∈ s := sab ⟨lt_of_le_of_ne hy.1 (Ne.symm h), hy.2⟩
      

Proof state:
case neg
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set ℝ
e : E
a : ℝ
f : ℝ → E
f_diff : DifferentiableOn ℝ f s
f_lim : ContinuousWithinAt f s a
hs : s ∈ 𝓝[>] a
f_lim' : Tendsto (fun x => deriv f x) (𝓝[>] a) (𝓝 e)
b : ℝ
ab : a < b
sab : Ioc a b ⊆ s
t : Set ℝ := Ioo a b
ts : t ⊆ s
t_diff : DifferentiableOn ℝ f t
t_conv : Convex ℝ t
t_open : IsOpen t
t_closure : closure t = Icc a b
y : ℝ
hy : y ∈ Icc a b
h : ¬y = a
this : y ∈ s
⊢ ContinuousWithinAt f t y

Next tactic:
exact (f_diff.continuousOn y this).mono ts
/-- If a function is differentiable on the right of a point `a : ℝ`, continuous at `a`, and
its derivative also converges at `a`, then `f` is differentiable on the right at `a`. -/
theorem has_deriv_at_interval_left_endpoint_of_tendsto_deriv {s : Set ℝ} {e : E} {a : ℝ} {f : ℝ → E}
    (f_diff : DifferentiableOn ℝ f s) (f_lim : ContinuousWithinAt f s a) (hs : s ∈ 𝓝[>] a)
    (f_lim' : Tendsto (fun x => deriv f x) (𝓝[>] a) (𝓝 e)) : HasDerivWithinAt f e (Ici a) a := by
  /- This is a specialization of `has_fderiv_at_boundary_of_tendsto_fderiv`. To be in the setting of
    this theorem, we need to work on an open interval with closure contained in `s ∪ {a}`, that we
    call `t = (a, b)`. Then, we check all the assumptions of this theorem and we apply it. -/
  obtain ⟨b, ab : a < b, sab : Ioc a b ⊆ s⟩ := mem_nhdsWithin_Ioi_iff_exists_Ioc_subset.1 hs
  let t := Ioo a b
  have ts : t ⊆ s := Subset.trans Ioo_subset_Ioc_self sab
  have t_diff : DifferentiableOn ℝ f t := f_diff.mono ts
  have t_conv : Convex ℝ t := convex_Ioo a b
  have t_open : IsOpen t := isOpen_Ioo
  have t_closure : closure t = Icc a b := closure_Ioo ab.ne
  have t_cont : ∀ y ∈ closure t, ContinuousWithinAt f t y := by
    rw [t_closure]
    intro y hy
    by_cases h : y = a
    · rw [h]; exact f_lim.mono ts
    · have : y ∈ s := sab ⟨lt_of_le_of_ne hy.1 (Ne.symm h), hy.2⟩
      exact (f_diff.continuousOn y this).mono ts
  

Proof state:
case intro.intro
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set ℝ
e : E
a : ℝ
f : ℝ → E
f_diff : DifferentiableOn ℝ f s
f_lim : ContinuousWithinAt f s a
hs : s ∈ 𝓝[>] a
f_lim' : Tendsto (fun x => deriv f x) (𝓝[>] a) (𝓝 e)
b : ℝ
ab : a < b
sab : Ioc a b ⊆ s
t : Set ℝ := Ioo a b
ts : t ⊆ s
t_diff : DifferentiableOn ℝ f t
t_conv : Convex ℝ t
t_open : IsOpen t
t_closure : closure t = Icc a b
t_cont : ∀ y ∈ closure t, ContinuousWithinAt f t y
⊢ HasDerivWithinAt f e (Ici a) a

Next tactic:
have t_diff' : Tendsto (fun x => fderiv ℝ f x) (𝓝[t] a) (𝓝 (smulRight 1 e)) := by
    simp only [deriv_fderiv.symm]
    exact Tendsto.comp
      (isBoundedBilinearMap_smulRight : IsBoundedBilinearMap ℝ _).continuous_right.continuousAt
      (tendsto_nhdsWithin_mono_left Ioo_subset_Ioi_self f_lim')
/-- If a function is differentiable on the right of a point `a : ℝ`, continuous at `a`, and
its derivative also converges at `a`, then `f` is differentiable on the right at `a`. -/
theorem has_deriv_at_interval_left_endpoint_of_tendsto_deriv {s : Set ℝ} {e : E} {a : ℝ} {f : ℝ → E}
    (f_diff : DifferentiableOn ℝ f s) (f_lim : ContinuousWithinAt f s a) (hs : s ∈ 𝓝[>] a)
    (f_lim' : Tendsto (fun x => deriv f x) (𝓝[>] a) (𝓝 e)) : HasDerivWithinAt f e (Ici a) a := by
  /- This is a specialization of `has_fderiv_at_boundary_of_tendsto_fderiv`. To be in the setting of
    this theorem, we need to work on an open interval with closure contained in `s ∪ {a}`, that we
    call `t = (a, b)`. Then, we check all the assumptions of this theorem and we apply it. -/
  obtain ⟨b, ab : a < b, sab : Ioc a b ⊆ s⟩ := mem_nhdsWithin_Ioi_iff_exists_Ioc_subset.1 hs
  let t := Ioo a b
  have ts : t ⊆ s := Subset.trans Ioo_subset_Ioc_self sab
  have t_diff : DifferentiableOn ℝ f t := f_diff.mono ts
  have t_conv : Convex ℝ t := convex_Ioo a b
  have t_open : IsOpen t := isOpen_Ioo
  have t_closure : closure t = Icc a b := closure_Ioo ab.ne
  have t_cont : ∀ y ∈ closure t, ContinuousWithinAt f t y := by
    rw [t_closure]
    intro y hy
    by_cases h : y = a
    · rw [h]; exact f_lim.mono ts
    · have : y ∈ s := sab ⟨lt_of_le_of_ne hy.1 (Ne.symm h), hy.2⟩
      exact (f_diff.continuousOn y this).mono ts
  have t_diff' : Tendsto (fun x => fderiv ℝ f x) (𝓝[t] a) (𝓝 (smulRight 1 e)) := by
    

Proof state:
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set ℝ
e : E
a : ℝ
f : ℝ → E
f_diff : DifferentiableOn ℝ f s
f_lim : ContinuousWithinAt f s a
hs : s ∈ 𝓝[>] a
f_lim' : Tendsto (fun x => deriv f x) (𝓝[>] a) (𝓝 e)
b : ℝ
ab : a < b
sab : Ioc a b ⊆ s
t : Set ℝ := Ioo a b
ts : t ⊆ s
t_diff : DifferentiableOn ℝ f t
t_conv : Convex ℝ t
t_open : IsOpen t
t_closure : closure t = Icc a b
t_cont : ∀ y ∈ closure t, ContinuousWithinAt f t y
⊢ Tendsto (fun x => fderiv ℝ f x) (𝓝[t] a) (𝓝 (smulRight 1 e))

Next tactic:
simp only [deriv_fderiv.symm]
/-- If a function is differentiable on the right of a point `a : ℝ`, continuous at `a`, and
its derivative also converges at `a`, then `f` is differentiable on the right at `a`. -/
theorem has_deriv_at_interval_left_endpoint_of_tendsto_deriv {s : Set ℝ} {e : E} {a : ℝ} {f : ℝ → E}
    (f_diff : DifferentiableOn ℝ f s) (f_lim : ContinuousWithinAt f s a) (hs : s ∈ 𝓝[>] a)
    (f_lim' : Tendsto (fun x => deriv f x) (𝓝[>] a) (𝓝 e)) : HasDerivWithinAt f e (Ici a) a := by
  /- This is a specialization of `has_fderiv_at_boundary_of_tendsto_fderiv`. To be in the setting of
    this theorem, we need to work on an open interval with closure contained in `s ∪ {a}`, that we
    call `t = (a, b)`. Then, we check all the assumptions of this theorem and we apply it. -/
  obtain ⟨b, ab : a < b, sab : Ioc a b ⊆ s⟩ := mem_nhdsWithin_Ioi_iff_exists_Ioc_subset.1 hs
  let t := Ioo a b
  have ts : t ⊆ s := Subset.trans Ioo_subset_Ioc_self sab
  have t_diff : DifferentiableOn ℝ f t := f_diff.mono ts
  have t_conv : Convex ℝ t := convex_Ioo a b
  have t_open : IsOpen t := isOpen_Ioo
  have t_closure : closure t = Icc a b := closure_Ioo ab.ne
  have t_cont : ∀ y ∈ closure t, ContinuousWithinAt f t y := by
    rw [t_closure]
    intro y hy
    by_cases h : y = a
    · rw [h]; exact f_lim.mono ts
    · have : y ∈ s := sab ⟨lt_of_le_of_ne hy.1 (Ne.symm h), hy.2⟩
      exact (f_diff.continuousOn y this).mono ts
  have t_diff' : Tendsto (fun x => fderiv ℝ f x) (𝓝[t] a) (𝓝 (smulRight 1 e)) := by
    simp only [deriv_fderiv.symm]
    

Proof state:
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set ℝ
e : E
a : ℝ
f : ℝ → E
f_diff : DifferentiableOn ℝ f s
f_lim : ContinuousWithinAt f s a
hs : s ∈ 𝓝[>] a
f_lim' : Tendsto (fun x => deriv f x) (𝓝[>] a) (𝓝 e)
b : ℝ
ab : a < b
sab : Ioc a b ⊆ s
t : Set ℝ := Ioo a b
ts : t ⊆ s
t_diff : DifferentiableOn ℝ f t
t_conv : Convex ℝ t
t_open : IsOpen t
t_closure : closure t = Icc a b
t_cont : ∀ y ∈ closure t, ContinuousWithinAt f t y
⊢ Tendsto (fun x => smulRight 1 (deriv f x)) (𝓝[Ioo a b] a) (𝓝 (smulRight 1 e))

Next tactic:
exact Tendsto.comp
      (isBoundedBilinearMap_smulRight : IsBoundedBilinearMap ℝ _).continuous_right.continuousAt
      (tendsto_nhdsWithin_mono_left Ioo_subset_Ioi_self f_lim')
/-- If a function is differentiable on the right of a point `a : ℝ`, continuous at `a`, and
its derivative also converges at `a`, then `f` is differentiable on the right at `a`. -/
theorem has_deriv_at_interval_left_endpoint_of_tendsto_deriv {s : Set ℝ} {e : E} {a : ℝ} {f : ℝ → E}
    (f_diff : DifferentiableOn ℝ f s) (f_lim : ContinuousWithinAt f s a) (hs : s ∈ 𝓝[>] a)
    (f_lim' : Tendsto (fun x => deriv f x) (𝓝[>] a) (𝓝 e)) : HasDerivWithinAt f e (Ici a) a := by
  /- This is a specialization of `has_fderiv_at_boundary_of_tendsto_fderiv`. To be in the setting of
    this theorem, we need to work on an open interval with closure contained in `s ∪ {a}`, that we
    call `t = (a, b)`. Then, we check all the assumptions of this theorem and we apply it. -/
  obtain ⟨b, ab : a < b, sab : Ioc a b ⊆ s⟩ := mem_nhdsWithin_Ioi_iff_exists_Ioc_subset.1 hs
  let t := Ioo a b
  have ts : t ⊆ s := Subset.trans Ioo_subset_Ioc_self sab
  have t_diff : DifferentiableOn ℝ f t := f_diff.mono ts
  have t_conv : Convex ℝ t := convex_Ioo a b
  have t_open : IsOpen t := isOpen_Ioo
  have t_closure : closure t = Icc a b := closure_Ioo ab.ne
  have t_cont : ∀ y ∈ closure t, ContinuousWithinAt f t y := by
    rw [t_closure]
    intro y hy
    by_cases h : y = a
    · rw [h]; exact f_lim.mono ts
    · have : y ∈ s := sab ⟨lt_of_le_of_ne hy.1 (Ne.symm h), hy.2⟩
      exact (f_diff.continuousOn y this).mono ts
  have t_diff' : Tendsto (fun x => fderiv ℝ f x) (𝓝[t] a) (𝓝 (smulRight 1 e)) := by
    simp only [deriv_fderiv.symm]
    exact Tendsto.comp
      (isBoundedBilinearMap_smulRight : IsBoundedBilinearMap ℝ _).continuous_right.continuousAt
      (tendsto_nhdsWithin_mono_left Ioo_subset_Ioi_self f_lim')
  -- now we can apply `has_fderiv_at_boundary_of_differentiable`
  

Proof state:
case intro.intro
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set ℝ
e : E
a : ℝ
f : ℝ → E
f_diff : DifferentiableOn ℝ f s
f_lim : ContinuousWithinAt f s a
hs : s ∈ 𝓝[>] a
f_lim' : Tendsto (fun x => deriv f x) (𝓝[>] a) (𝓝 e)
b : ℝ
ab : a < b
sab : Ioc a b ⊆ s
t : Set ℝ := Ioo a b
ts : t ⊆ s
t_diff : DifferentiableOn ℝ f t
t_conv : Convex ℝ t
t_open : IsOpen t
t_closure : closure t = Icc a b
t_cont : ∀ y ∈ closure t, ContinuousWithinAt f t y
t_diff' : Tendsto (fun x => fderiv ℝ f x) (𝓝[t] a) (𝓝 (smulRight 1 e))
⊢ HasDerivWithinAt f e (Ici a) a

Next tactic:
have : HasDerivWithinAt f e (Icc a b) a := by
    rw [hasDerivWithinAt_iff_hasFDerivWithinAt, ← t_closure]
    exact has_fderiv_at_boundary_of_tendsto_fderiv t_diff t_conv t_open t_cont t_diff'
/-- If a function is differentiable on the right of a point `a : ℝ`, continuous at `a`, and
its derivative also converges at `a`, then `f` is differentiable on the right at `a`. -/
theorem has_deriv_at_interval_left_endpoint_of_tendsto_deriv {s : Set ℝ} {e : E} {a : ℝ} {f : ℝ → E}
    (f_diff : DifferentiableOn ℝ f s) (f_lim : ContinuousWithinAt f s a) (hs : s ∈ 𝓝[>] a)
    (f_lim' : Tendsto (fun x => deriv f x) (𝓝[>] a) (𝓝 e)) : HasDerivWithinAt f e (Ici a) a := by
  /- This is a specialization of `has_fderiv_at_boundary_of_tendsto_fderiv`. To be in the setting of
    this theorem, we need to work on an open interval with closure contained in `s ∪ {a}`, that we
    call `t = (a, b)`. Then, we check all the assumptions of this theorem and we apply it. -/
  obtain ⟨b, ab : a < b, sab : Ioc a b ⊆ s⟩ := mem_nhdsWithin_Ioi_iff_exists_Ioc_subset.1 hs
  let t := Ioo a b
  have ts : t ⊆ s := Subset.trans Ioo_subset_Ioc_self sab
  have t_diff : DifferentiableOn ℝ f t := f_diff.mono ts
  have t_conv : Convex ℝ t := convex_Ioo a b
  have t_open : IsOpen t := isOpen_Ioo
  have t_closure : closure t = Icc a b := closure_Ioo ab.ne
  have t_cont : ∀ y ∈ closure t, ContinuousWithinAt f t y := by
    rw [t_closure]
    intro y hy
    by_cases h : y = a
    · rw [h]; exact f_lim.mono ts
    · have : y ∈ s := sab ⟨lt_of_le_of_ne hy.1 (Ne.symm h), hy.2⟩
      exact (f_diff.continuousOn y this).mono ts
  have t_diff' : Tendsto (fun x => fderiv ℝ f x) (𝓝[t] a) (𝓝 (smulRight 1 e)) := by
    simp only [deriv_fderiv.symm]
    exact Tendsto.comp
      (isBoundedBilinearMap_smulRight : IsBoundedBilinearMap ℝ _).continuous_right.continuousAt
      (tendsto_nhdsWithin_mono_left Ioo_subset_Ioi_self f_lim')
  -- now we can apply `has_fderiv_at_boundary_of_differentiable`
  have : HasDerivWithinAt f e (Icc a b) a := by
    

Proof state:
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set ℝ
e : E
a : ℝ
f : ℝ → E
f_diff : DifferentiableOn ℝ f s
f_lim : ContinuousWithinAt f s a
hs : s ∈ 𝓝[>] a
f_lim' : Tendsto (fun x => deriv f x) (𝓝[>] a) (𝓝 e)
b : ℝ
ab : a < b
sab : Ioc a b ⊆ s
t : Set ℝ := Ioo a b
ts : t ⊆ s
t_diff : DifferentiableOn ℝ f t
t_conv : Convex ℝ t
t_open : IsOpen t
t_closure : closure t = Icc a b
t_cont : ∀ y ∈ closure t, ContinuousWithinAt f t y
t_diff' : Tendsto (fun x => fderiv ℝ f x) (𝓝[t] a) (𝓝 (smulRight 1 e))
⊢ HasDerivWithinAt f e (Icc a b) a

Next tactic:
rw [hasDerivWithinAt_iff_hasFDerivWithinAt, ← t_closure]