l3lab/ntp-mathlib huggingface dataset:
theorem finrank_hom_simple_simple_eq_zero_iff (X Y : C) [FiniteDimensional 𝕜 (X ⟶ X)]
    [FiniteDimensional 𝕜 (X ⟶ Y)] [Simple X] [Simple Y] :
    finrank 𝕜 (X ⟶ Y) = 0 ↔ IsEmpty (X ≅ Y) := by
  rw [← not_nonempty_iff, ← not_congr (finrank_hom_simple_simple_eq_one_iff 𝕜 X Y)]
  refine' ⟨fun h => by 

Proof state:
C : Type u_1
inst✝⁹ : Category.{u_3, u_1} C
inst✝⁸ : Preadditive C
𝕜 : Type u_2
inst✝⁷ : Field 𝕜
inst✝⁶ : IsAlgClosed 𝕜
inst✝⁵ : Linear 𝕜 C
inst✝⁴ : HasKernels C
X Y : C
inst✝³ : FiniteDimensional 𝕜 (X ⟶ X)
inst✝² : FiniteDimensional 𝕜 (X ⟶ Y)
inst✝¹ : Simple X
inst✝ : Simple Y
h : finrank 𝕜 (X ⟶ Y) = 0
⊢ ¬finrank 𝕜 (X ⟶ Y) = 1

Next tactic:
rw [h]
theorem finrank_hom_simple_simple_eq_zero_iff (X Y : C) [FiniteDimensional 𝕜 (X ⟶ X)]
    [FiniteDimensional 𝕜 (X ⟶ Y)] [Simple X] [Simple Y] :
    finrank 𝕜 (X ⟶ Y) = 0 ↔ IsEmpty (X ≅ Y) := by
  rw [← not_nonempty_iff, ← not_congr (finrank_hom_simple_simple_eq_one_iff 𝕜 X Y)]
  refine' ⟨fun h => by rw [h]; 

Proof state:
C : Type u_1
inst✝⁹ : Category.{u_3, u_1} C
inst✝⁸ : Preadditive C
𝕜 : Type u_2
inst✝⁷ : Field 𝕜
inst✝⁶ : IsAlgClosed 𝕜
inst✝⁵ : Linear 𝕜 C
inst✝⁴ : HasKernels C
X Y : C
inst✝³ : FiniteDimensional 𝕜 (X ⟶ X)
inst✝² : FiniteDimensional 𝕜 (X ⟶ Y)
inst✝¹ : Simple X
inst✝ : Simple Y
h : finrank 𝕜 (X ⟶ Y) = 0
⊢ ¬0 = 1

Next tactic:
simp
theorem finrank_hom_simple_simple_eq_zero_iff (X Y : C) [FiniteDimensional 𝕜 (X ⟶ X)]
    [FiniteDimensional 𝕜 (X ⟶ Y)] [Simple X] [Simple Y] :
    finrank 𝕜 (X ⟶ Y) = 0 ↔ IsEmpty (X ≅ Y) := by
  rw [← not_nonempty_iff, ← not_congr (finrank_hom_simple_simple_eq_one_iff 𝕜 X Y)]
  refine' ⟨fun h => by rw [h]; simp, fun h => _⟩
  

Proof state:
C : Type u_1
inst✝⁹ : Category.{u_3, u_1} C
inst✝⁸ : Preadditive C
𝕜 : Type u_2
inst✝⁷ : Field 𝕜
inst✝⁶ : IsAlgClosed 𝕜
inst✝⁵ : Linear 𝕜 C
inst✝⁴ : HasKernels C
X Y : C
inst✝³ : FiniteDimensional 𝕜 (X ⟶ X)
inst✝² : FiniteDimensional 𝕜 (X ⟶ Y)
inst✝¹ : Simple X
inst✝ : Simple Y
h : ¬finrank 𝕜 (X ⟶ Y) = 1
⊢ finrank 𝕜 (X ⟶ Y) = 0

Next tactic:
have := finrank_hom_simple_simple_le_one 𝕜 X Y
theorem finrank_hom_simple_simple_eq_zero_iff (X Y : C) [FiniteDimensional 𝕜 (X ⟶ X)]
    [FiniteDimensional 𝕜 (X ⟶ Y)] [Simple X] [Simple Y] :
    finrank 𝕜 (X ⟶ Y) = 0 ↔ IsEmpty (X ≅ Y) := by
  rw [← not_nonempty_iff, ← not_congr (finrank_hom_simple_simple_eq_one_iff 𝕜 X Y)]
  refine' ⟨fun h => by rw [h]; simp, fun h => _⟩
  have := finrank_hom_simple_simple_le_one 𝕜 X Y
  

Proof state:
C : Type u_1
inst✝⁹ : Category.{u_3, u_1} C
inst✝⁸ : Preadditive C
𝕜 : Type u_2
inst✝⁷ : Field 𝕜
inst✝⁶ : IsAlgClosed 𝕜
inst✝⁵ : Linear 𝕜 C
inst✝⁴ : HasKernels C
X Y : C
inst✝³ : FiniteDimensional 𝕜 (X ⟶ X)
inst✝² : FiniteDimensional 𝕜 (X ⟶ Y)
inst✝¹ : Simple X
inst✝ : Simple Y
h : ¬finrank 𝕜 (X ⟶ Y) = 1
this : finrank 𝕜 (X ⟶ Y) ≤ 1
⊢ finrank 𝕜 (X ⟶ Y) = 0

Next tactic:
interval_cases finrank 𝕜 (X ⟶ Y)
theorem finrank_hom_simple_simple_eq_zero_iff (X Y : C) [FiniteDimensional 𝕜 (X ⟶ X)]
    [FiniteDimensional 𝕜 (X ⟶ Y)] [Simple X] [Simple Y] :
    finrank 𝕜 (X ⟶ Y) = 0 ↔ IsEmpty (X ≅ Y) := by
  rw [← not_nonempty_iff, ← not_congr (finrank_hom_simple_simple_eq_one_iff 𝕜 X Y)]
  refine' ⟨fun h => by rw [h]; simp, fun h => _⟩
  have := finrank_hom_simple_simple_le_one 𝕜 X Y
  interval_cases finrank 𝕜 (X ⟶ Y)
  · 

Proof state:
case «0»
C : Type u_1
inst✝⁹ : Category.{u_3, u_1} C
inst✝⁸ : Preadditive C
𝕜 : Type u_2
inst✝⁷ : Field 𝕜
inst✝⁶ : IsAlgClosed 𝕜
inst✝⁵ : Linear 𝕜 C
inst✝⁴ : HasKernels C
X Y : C
inst✝³ : FiniteDimensional 𝕜 (X ⟶ X)
inst✝² : FiniteDimensional 𝕜 (X ⟶ Y)
inst✝¹ : Simple X
inst✝ : Simple Y
h : ¬0 = 1
this : 0 ≤ 1
⊢ 0 = 0

Next tactic:
rfl
theorem finrank_hom_simple_simple_eq_zero_iff (X Y : C) [FiniteDimensional 𝕜 (X ⟶ X)]
    [FiniteDimensional 𝕜 (X ⟶ Y)] [Simple X] [Simple Y] :
    finrank 𝕜 (X ⟶ Y) = 0 ↔ IsEmpty (X ≅ Y) := by
  rw [← not_nonempty_iff, ← not_congr (finrank_hom_simple_simple_eq_one_iff 𝕜 X Y)]
  refine' ⟨fun h => by rw [h]; simp, fun h => _⟩
  have := finrank_hom_simple_simple_le_one 𝕜 X Y
  interval_cases finrank 𝕜 (X ⟶ Y)
  · rfl
  · 

Proof state:
case «1»
C : Type u_1
inst✝⁹ : Category.{u_3, u_1} C
inst✝⁸ : Preadditive C
𝕜 : Type u_2
inst✝⁷ : Field 𝕜
inst✝⁶ : IsAlgClosed 𝕜
inst✝⁵ : Linear 𝕜 C
inst✝⁴ : HasKernels C
X Y : C
inst✝³ : FiniteDimensional 𝕜 (X ⟶ X)
inst✝² : FiniteDimensional 𝕜 (X ⟶ Y)
inst✝¹ : Simple X
inst✝ : Simple Y
h : ¬1 = 1
this : 1 ≤ 1
⊢ 1 = 0

Next tactic:
exact False.elim (h rfl)
theorem finrank_hom_simple_simple (X Y : C) [∀ X Y : C, FiniteDimensional 𝕜 (X ⟶ Y)] [Simple X]
    [Simple Y] : finrank 𝕜 (X ⟶ Y) = if Nonempty (X ≅ Y) then 1 else 0 := by
  

Proof state:
C : Type u_1
inst✝⁸ : Category.{u_3, u_1} C
inst✝⁷ : Preadditive C
𝕜 : Type u_2
inst✝⁶ : Field 𝕜
inst✝⁵ : IsAlgClosed 𝕜
inst✝⁴ : Linear 𝕜 C
inst✝³ : HasKernels C
X Y : C
inst✝² : ∀ (X Y : C), FiniteDimensional 𝕜 (X ⟶ Y)
inst✝¹ : Simple X
inst✝ : Simple Y
⊢ finrank 𝕜 (X ⟶ Y) = if Nonempty (X ≅ Y) then 1 else 0

Next tactic:
split_ifs with h
theorem finrank_hom_simple_simple (X Y : C) [∀ X Y : C, FiniteDimensional 𝕜 (X ⟶ Y)] [Simple X]
    [Simple Y] : finrank 𝕜 (X ⟶ Y) = if Nonempty (X ≅ Y) then 1 else 0 := by
  split_ifs with h
  

Proof state:
case pos
C : Type u_1
inst✝⁸ : Category.{u_3, u_1} C
inst✝⁷ : Preadditive C
𝕜 : Type u_2
inst✝⁶ : Field 𝕜
inst✝⁵ : IsAlgClosed 𝕜
inst✝⁴ : Linear 𝕜 C
inst✝³ : HasKernels C
X Y : C
inst✝² : ∀ (X Y : C), FiniteDimensional 𝕜 (X ⟶ Y)
inst✝¹ : Simple X
inst✝ : Simple Y
h : Nonempty (X ≅ Y)
⊢ finrank 𝕜 (X ⟶ Y) = 1
case neg
C : Type u_1
inst✝⁸ : Category.{u_3, u_1} C
inst✝⁷ : Preadditive C
𝕜 : Type u_2
inst✝⁶ : Field 𝕜
inst✝⁵ : IsAlgClosed 𝕜
inst✝⁴ : Linear 𝕜 C
inst✝³ : HasKernels C
X Y : C
inst✝² : ∀ (X Y : C), FiniteDimensional 𝕜 (X ⟶ Y)
inst✝¹ : Simple X
inst✝ : Simple Y
h : ¬Nonempty (X ≅ Y)
⊢ finrank 𝕜 (X ⟶ Y) = 0

Next tactic:
exact (finrank_hom_simple_simple_eq_one_iff 𝕜 X Y).2 h
theorem finrank_hom_simple_simple (X Y : C) [∀ X Y : C, FiniteDimensional 𝕜 (X ⟶ Y)] [Simple X]
    [Simple Y] : finrank 𝕜 (X ⟶ Y) = if Nonempty (X ≅ Y) then 1 else 0 := by
  split_ifs with h
  exact (finrank_hom_simple_simple_eq_one_iff 𝕜 X Y).2 h
  

Proof state:
case neg
C : Type u_1
inst✝⁸ : Category.{u_3, u_1} C
inst✝⁷ : Preadditive C
𝕜 : Type u_2
inst✝⁶ : Field 𝕜
inst✝⁵ : IsAlgClosed 𝕜
inst✝⁴ : Linear 𝕜 C
inst✝³ : HasKernels C
X Y : C
inst✝² : ∀ (X Y : C), FiniteDimensional 𝕜 (X ⟶ Y)
inst✝¹ : Simple X
inst✝ : Simple Y
h : ¬Nonempty (X ≅ Y)
⊢ finrank 𝕜 (X ⟶ Y) = 0

Next tactic:
exact (finrank_hom_simple_simple_eq_zero_iff 𝕜 X Y).2 (not_nonempty_iff.mp h)
theorem hasStrictDerivAt_log_of_pos (hx : 0 < x) : HasStrictDerivAt log x⁻¹ x := by
  

Proof state:
x : ℝ
hx : 0 < x
⊢ HasStrictDerivAt log x⁻¹ x

Next tactic:
have : HasStrictDerivAt log (exp <| log x)⁻¹ x :=
    (hasStrictDerivAt_exp <| log x).of_local_left_inverse (continuousAt_log hx.ne')
        (ne_of_gt <| exp_pos _) <|
      Eventually.mono (lt_mem_nhds hx) @exp_log
theorem hasStrictDerivAt_log_of_pos (hx : 0 < x) : HasStrictDerivAt log x⁻¹ x := by
  have : HasStrictDerivAt log (exp <| log x)⁻¹ x :=
    (hasStrictDerivAt_exp <| log x).of_local_left_inverse (continuousAt_log hx.ne')
        (ne_of_gt <| exp_pos _) <|
      Eventually.mono (lt_mem_nhds hx) @exp_log
  

Proof state:
x : ℝ
hx : 0 < x
this : HasStrictDerivAt log (rexp (log x))⁻¹ x
⊢ HasStrictDerivAt log x⁻¹ x

Next tactic:
rwa [exp_log hx] at this
theorem hasStrictDerivAt_log (hx : x ≠ 0) : HasStrictDerivAt log x⁻¹ x := by
  

Proof state:
x : ℝ
hx : x ≠ 0
⊢ HasStrictDerivAt log x⁻¹ x

Next tactic:
cases' hx.lt_or_lt with hx hx
theorem hasStrictDerivAt_log (hx : x ≠ 0) : HasStrictDerivAt log x⁻¹ x := by
  cases' hx.lt_or_lt with hx hx
  · 

Proof state:
case inl
x : ℝ
hx✝ : x ≠ 0
hx : x < 0
⊢ HasStrictDerivAt log x⁻¹ x

Next tactic:
convert (hasStrictDerivAt_log_of_pos (neg_pos.mpr hx)).comp x (hasStrictDerivAt_neg x) using 1
theorem hasStrictDerivAt_log (hx : x ≠ 0) : HasStrictDerivAt log x⁻¹ x := by
  cases' hx.lt_or_lt with hx hx
  · convert (hasStrictDerivAt_log_of_pos (neg_pos.mpr hx)).comp x (hasStrictDerivAt_neg x) using 1
    · 

Proof state:
case h.e'_6
x : ℝ
hx✝ : x ≠ 0
hx : x < 0
⊢ log = log ∘ Neg.neg

Next tactic:
ext y
theorem hasStrictDerivAt_log (hx : x ≠ 0) : HasStrictDerivAt log x⁻¹ x := by
  cases' hx.lt_or_lt with hx hx
  · convert (hasStrictDerivAt_log_of_pos (neg_pos.mpr hx)).comp x (hasStrictDerivAt_neg x) using 1
    · ext y; 

Proof state:
case h.e'_6.h
x : ℝ
hx✝ : x ≠ 0
hx : x < 0
y : ℝ
⊢ log y = (log ∘ Neg.neg) y

Next tactic:
exact (log_neg_eq_log y).symm
theorem hasStrictDerivAt_log (hx : x ≠ 0) : HasStrictDerivAt log x⁻¹ x := by
  cases' hx.lt_or_lt with hx hx
  · convert (hasStrictDerivAt_log_of_pos (neg_pos.mpr hx)).comp x (hasStrictDerivAt_neg x) using 1
    · ext y; exact (log_neg_eq_log y).symm
    · 

Proof state:
case h.e'_7
x : ℝ
hx✝ : x ≠ 0
hx : x < 0
⊢ x⁻¹ = (-x)⁻¹ * -1

Next tactic:
field_simp [hx.ne]
theorem hasStrictDerivAt_log (hx : x ≠ 0) : HasStrictDerivAt log x⁻¹ x := by
  cases' hx.lt_or_lt with hx hx
  · convert (hasStrictDerivAt_log_of_pos (neg_pos.mpr hx)).comp x (hasStrictDerivAt_neg x) using 1
    · ext y; exact (log_neg_eq_log y).symm
    · field_simp [hx.ne]
  · 

Proof state:
case inr
x : ℝ
hx✝ : x ≠ 0
hx : 0 < x
⊢ HasStrictDerivAt log x⁻¹ x

Next tactic:
exact hasStrictDerivAt_log_of_pos hx
theorem deriv_log (x : ℝ) : deriv log x = x⁻¹ :=
  if hx : x = 0 then by
    

Proof state:
x✝ x : ℝ
hx : x = 0
⊢ deriv log x = x⁻¹

Next tactic:
rw [deriv_zero_of_not_differentiableAt (differentiableAt_log_iff.not_left.2 hx), hx, inv_zero]
theorem contDiffOn_log {n : ℕ∞} : ContDiffOn ℝ n log {0}ᶜ := by
  

Proof state:
x : ℝ
n : ℕ∞
⊢ ContDiffOn ℝ n log {0}ᶜ

Next tactic:
suffices : ContDiffOn ℝ ⊤ log {0}ᶜ
theorem contDiffOn_log {n : ℕ∞} : ContDiffOn ℝ n log {0}ᶜ := by
  suffices : ContDiffOn ℝ ⊤ log {0}ᶜ; 

Proof state:
x : ℝ
n : ℕ∞
this : ContDiffOn ℝ ⊤ log {0}ᶜ
⊢ ContDiffOn ℝ n log {0}ᶜ
case this x : ℝ n : ℕ∞ ⊢ ContDiffOn ℝ ⊤ log {0}ᶜ

Next tactic:
exact this.of_le le_top
theorem contDiffOn_log {n : ℕ∞} : ContDiffOn ℝ n log {0}ᶜ := by
  suffices : ContDiffOn ℝ ⊤ log {0}ᶜ; exact this.of_le le_top
  

Proof state:
case this
x : ℝ
n : ℕ∞
⊢ ContDiffOn ℝ ⊤ log {0}ᶜ

Next tactic:
refine' (contDiffOn_top_iff_deriv_of_isOpen isOpen_compl_singleton).2 _
theorem contDiffOn_log {n : ℕ∞} : ContDiffOn ℝ n log {0}ᶜ := by
  suffices : ContDiffOn ℝ ⊤ log {0}ᶜ; exact this.of_le le_top
  refine' (contDiffOn_top_iff_deriv_of_isOpen isOpen_compl_singleton).2 _
  

Proof state:
case this
x : ℝ
n : ℕ∞
⊢ DifferentiableOn ℝ log {0}ᶜ ∧ ContDiffOn ℝ ⊤ (deriv log) {0}ᶜ

Next tactic:
simp [differentiableOn_log, contDiffOn_inv]
theorem HasDerivWithinAt.log (hf : HasDerivWithinAt f f' s x) (hx : f x ≠ 0) :
    HasDerivWithinAt (fun y => log (f y)) (f' / f x) s x := by
  

Proof state:
f : ℝ → ℝ
x f' : ℝ
s : Set ℝ
hf : HasDerivWithinAt f f' s x
hx : f x ≠ 0
⊢ HasDerivWithinAt (fun y => Real.log (f y)) (f' / f x) s x

Next tactic:
rw [div_eq_inv_mul]
theorem HasDerivWithinAt.log (hf : HasDerivWithinAt f f' s x) (hx : f x ≠ 0) :
    HasDerivWithinAt (fun y => log (f y)) (f' / f x) s x := by
  rw [div_eq_inv_mul]
  

Proof state:
f : ℝ → ℝ
x f' : ℝ
s : Set ℝ
hf : HasDerivWithinAt f f' s x
hx : f x ≠ 0
⊢ HasDerivWithinAt (fun y => Real.log (f y)) ((f x)⁻¹ * f') s x

Next tactic:
exact (hasDerivAt_log hx).comp_hasDerivWithinAt x hf
theorem HasDerivAt.log (hf : HasDerivAt f f' x) (hx : f x ≠ 0) :
    HasDerivAt (fun y => log (f y)) (f' / f x) x := by
  

Proof state:
f : ℝ → ℝ
x f' : ℝ
s : Set ℝ
hf : HasDerivAt f f' x
hx : f x ≠ 0
⊢ HasDerivAt (fun y => Real.log (f y)) (f' / f x) x

Next tactic:
rw [← hasDerivWithinAt_univ] at *
theorem HasDerivAt.log (hf : HasDerivAt f f' x) (hx : f x ≠ 0) :
    HasDerivAt (fun y => log (f y)) (f' / f x) x := by
  rw [← hasDerivWithinAt_univ] at *
  

Proof state:
f : ℝ → ℝ
x f' : ℝ
s : Set ℝ
hf : HasDerivWithinAt f f' Set.univ x
hx : f x ≠ 0
⊢ HasDerivWithinAt (fun y => Real.log (f y)) (f' / f x) Set.univ x

Next tactic:
exact hf.log hx
theorem HasStrictDerivAt.log (hf : HasStrictDerivAt f f' x) (hx : f x ≠ 0) :
    HasStrictDerivAt (fun y => log (f y)) (f' / f x) x := by
  

Proof state:
f : ℝ → ℝ
x f' : ℝ
s : Set ℝ
hf : HasStrictDerivAt f f' x
hx : f x ≠ 0
⊢ HasStrictDerivAt (fun y => Real.log (f y)) (f' / f x) x

Next tactic:
rw [div_eq_inv_mul]
theorem HasStrictDerivAt.log (hf : HasStrictDerivAt f f' x) (hx : f x ≠ 0) :
    HasStrictDerivAt (fun y => log (f y)) (f' / f x) x := by
  rw [div_eq_inv_mul]
  

Proof state:
f : ℝ → ℝ
x f' : ℝ
s : Set ℝ
hf : HasStrictDerivAt f f' x
hx : f x ≠ 0
⊢ HasStrictDerivAt (fun y => Real.log (f y)) ((f x)⁻¹ * f') x

Next tactic:
exact (hasStrictDerivAt_log hx).comp x hf
/-- The function `x * log (1 + t / x)` tends to `t` at `+∞`. -/
theorem tendsto_mul_log_one_plus_div_atTop (t : ℝ) :
    Tendsto (fun x => x * log (1 + t / x)) atTop (𝓝 t) := by
  

Proof state:
t : ℝ
⊢ Tendsto (fun x => x * log (1 + t / x)) atTop (𝓝 t)

Next tactic:
have h₁ : Tendsto (fun h => h⁻¹ * log (1 + t * h)) (𝓝[≠] 0) (𝓝 t) := by
    simpa [hasDerivAt_iff_tendsto_slope, slope_fun_def] using
      (((hasDerivAt_id (0 : ℝ)).const_mul t).const_add 1).log (by simp)
/-- The function `x * log (1 + t / x)` tends to `t` at `+∞`. -/
theorem tendsto_mul_log_one_plus_div_atTop (t : ℝ) :
    Tendsto (fun x => x * log (1 + t / x)) atTop (𝓝 t) := by
  have h₁ : Tendsto (fun h => h⁻¹ * log (1 + t * h)) (𝓝[≠] 0) (𝓝 t) := by
    

Proof state:
t : ℝ
⊢ Tendsto (fun h => h⁻¹ * log (1 + t * h)) (𝓝[≠] 0) (𝓝 t)

Next tactic:
simpa [hasDerivAt_iff_tendsto_slope, slope_fun_def] using
      (((hasDerivAt_id (0 : ℝ)).const_mul t).const_add 1).log (by simp)
/-- The function `x * log (1 + t / x)` tends to `t` at `+∞`. -/
theorem tendsto_mul_log_one_plus_div_atTop (t : ℝ) :
    Tendsto (fun x => x * log (1 + t / x)) atTop (𝓝 t) := by
  have h₁ : Tendsto (fun h => h⁻¹ * log (1 + t * h)) (𝓝[≠] 0) (𝓝 t) := by
    simpa [hasDerivAt_iff_tendsto_slope, slope_fun_def] using
      (((hasDerivAt_id (0 : ℝ)).const_mul t).const_add 1).log (by 

Proof state:
t : ℝ
⊢ 1 + t * id 0 ≠ 0

Next tactic:
simp
/-- The function `x * log (1 + t / x)` tends to `t` at `+∞`. -/
theorem tendsto_mul_log_one_plus_div_atTop (t : ℝ) :
    Tendsto (fun x => x * log (1 + t / x)) atTop (𝓝 t) := by
  have h₁ : Tendsto (fun h => h⁻¹ * log (1 + t * h)) (𝓝[≠] 0) (𝓝 t) := by
    simpa [hasDerivAt_iff_tendsto_slope, slope_fun_def] using
      (((hasDerivAt_id (0 : ℝ)).const_mul t).const_add 1).log (by simp)
  

Proof state:
t : ℝ
h₁ : Tendsto (fun h => h⁻¹ * log (1 + t * h)) (𝓝[≠] 0) (𝓝 t)
⊢ Tendsto (fun x => x * log (1 + t / x)) atTop (𝓝 t)

Next tactic:
have h₂ : Tendsto (fun x : ℝ => x⁻¹) atTop (𝓝[≠] 0) :=
    tendsto_inv_atTop_zero'.mono_right (nhdsWithin_mono _ fun x hx => (Set.mem_Ioi.mp hx).ne')
/-- The function `x * log (1 + t / x)` tends to `t` at `+∞`. -/
theorem tendsto_mul_log_one_plus_div_atTop (t : ℝ) :
    Tendsto (fun x => x * log (1 + t / x)) atTop (𝓝 t) := by
  have h₁ : Tendsto (fun h => h⁻¹ * log (1 + t * h)) (𝓝[≠] 0) (𝓝 t) := by
    simpa [hasDerivAt_iff_tendsto_slope, slope_fun_def] using
      (((hasDerivAt_id (0 : ℝ)).const_mul t).const_add 1).log (by simp)
  have h₂ : Tendsto (fun x : ℝ => x⁻¹) atTop (𝓝[≠] 0) :=
    tendsto_inv_atTop_zero'.mono_right (nhdsWithin_mono _ fun x hx => (Set.mem_Ioi.mp hx).ne')
  

Proof state:
t : ℝ
h₁ : Tendsto (fun h => h⁻¹ * log (1 + t * h)) (𝓝[≠] 0) (𝓝 t)
h₂ : Tendsto (fun x => x⁻¹) atTop (𝓝[≠] 0)
⊢ Tendsto (fun x => x * log (1 + t / x)) atTop (𝓝 t)

Next tactic:
simpa only [(· ∘ ·), inv_inv] using h₁.comp h₂
/-- A crude lemma estimating the difference between `log (1-x)` and its Taylor series at `0`,
where the main point of the bound is that it tends to `0`. The goal is to deduce the series
expansion of the logarithm, in `hasSum_pow_div_log_of_abs_lt_1`.

Porting note: TODO: use one of generic theorems about Taylor's series to prove this estimate.
-/
theorem abs_log_sub_add_sum_range_le {x : ℝ} (h : |x| < 1) (n : ℕ) :
    |(∑ i in range n, x ^ (i + 1) / (i + 1)) + log (1 - x)| ≤ |x| ^ (n + 1) / (1 - |x|) := by
  /- For the proof, we show that the derivative of the function to be estimated is small,
    and then apply the mean value inequality. -/
  

Proof state:
x : ℝ
h : |x| < 1
n : ℕ
⊢ |∑ i in Finset.range n, x ^ (i + 1) / (↑i + 1) + log (1 - x)| ≤ |x| ^ (n + 1) / (1 - |x|)

Next tactic:
let F : ℝ → ℝ := fun x => (∑ i in range n, x ^ (i + 1) / (i + 1)) + log (1 - x)
/-- A crude lemma estimating the difference between `log (1-x)` and its Taylor series at `0`,
where the main point of the bound is that it tends to `0`. The goal is to deduce the series
expansion of the logarithm, in `hasSum_pow_div_log_of_abs_lt_1`.

Porting note: TODO: use one of generic theorems about Taylor's series to prove this estimate.
-/
theorem abs_log_sub_add_sum_range_le {x : ℝ} (h : |x| < 1) (n : ℕ) :
    |(∑ i in range n, x ^ (i + 1) / (i + 1)) + log (1 - x)| ≤ |x| ^ (n + 1) / (1 - |x|) := by
  /- For the proof, we show that the derivative of the function to be estimated is small,
    and then apply the mean value inequality. -/
  let F : ℝ → ℝ := fun x => (∑ i in range n, x ^ (i + 1) / (i + 1)) + log (1 - x)
  

Proof state:
x : ℝ
h : |x| < 1
n : ℕ
F : ℝ → ℝ := fun x => ∑ i in Finset.range n, x ^ (i + 1) / (↑i + 1) + log (1 - x)
⊢ |∑ i in Finset.range n, x ^ (i + 1) / (↑i + 1) + log (1 - x)| ≤ |x| ^ (n + 1) / (1 - |x|)

Next tactic:
let F' : ℝ → ℝ := fun x ↦ -x ^ n / (1 - x)
/-- A crude lemma estimating the difference between `log (1-x)` and its Taylor series at `0`,
where the main point of the bound is that it tends to `0`. The goal is to deduce the series
expansion of the logarithm, in `hasSum_pow_div_log_of_abs_lt_1`.

Porting note: TODO: use one of generic theorems about Taylor's series to prove this estimate.
-/
theorem abs_log_sub_add_sum_range_le {x : ℝ} (h : |x| < 1) (n : ℕ) :
    |(∑ i in range n, x ^ (i + 1) / (i + 1)) + log (1 - x)| ≤ |x| ^ (n + 1) / (1 - |x|) := by
  /- For the proof, we show that the derivative of the function to be estimated is small,
    and then apply the mean value inequality. -/
  let F : ℝ → ℝ := fun x => (∑ i in range n, x ^ (i + 1) / (i + 1)) + log (1 - x)
  let F' : ℝ → ℝ := fun x ↦ -x ^ n / (1 - x)
  -- Porting note: In `mathlib3`, the proof used `deriv`/`DifferentiableAt`. `simp` failed to
  -- compute `deriv`, so I changed the proof to use `HasDerivAt` instead
  -- First step: compute the derivative of `F`
  

Proof state:
x : ℝ
h : |x| < 1
n : ℕ
F : ℝ → ℝ := fun x => ∑ i in Finset.range n, x ^ (i + 1) / (↑i + 1) + log (1 - x)
F' : ℝ → ℝ := fun x => -x ^ n / (1 - x)
⊢ |∑ i in Finset.range n, x ^ (i + 1) / (↑i + 1) + log (1 - x)| ≤ |x| ^ (n + 1) / (1 - |x|)

Next tactic:
have A : ∀ y ∈ Ioo (-1 : ℝ) 1, HasDerivAt F (F' y) y := fun y hy ↦ by
    have : HasDerivAt F ((∑ i in range n, ↑(i + 1) * y ^ i / (↑i + 1)) + (-1) / (1 - y)) y :=
      .add (.sum fun i _ ↦ (hasDerivAt_pow (i + 1) y).div_const ((i : ℝ) + 1))
        (((hasDerivAt_id y).const_sub _).log <| sub_ne_zero.2 hy.2.ne')
    convert this using 1
    calc
      -y ^ n / (1 - y) = ∑ i in Finset.range n, y ^ i + -1 / (1 - y) := by
        field_simp [geom_sum_eq hy.2.ne, sub_ne_zero.2 hy.2.ne, sub_ne_zero.2 hy.2.ne']
        ring
      _ = ∑ i in Finset.range n, ↑(i + 1) * y ^ i / (↑i + 1) + -1 / (1 - y) := by
        congr with i
        rw [Nat.cast_succ, mul_div_cancel_left _ (Nat.cast_add_one_pos i).ne']
/-- A crude lemma estimating the difference between `log (1-x)` and its Taylor series at `0`,
where the main point of the bound is that it tends to `0`. The goal is to deduce the series
expansion of the logarithm, in `hasSum_pow_div_log_of_abs_lt_1`.

Porting note: TODO: use one of generic theorems about Taylor's series to prove this estimate.
-/
theorem abs_log_sub_add_sum_range_le {x : ℝ} (h : |x| < 1) (n : ℕ) :
    |(∑ i in range n, x ^ (i + 1) / (i + 1)) + log (1 - x)| ≤ |x| ^ (n + 1) / (1 - |x|) := by
  /- For the proof, we show that the derivative of the function to be estimated is small,
    and then apply the mean value inequality. -/
  let F : ℝ → ℝ := fun x => (∑ i in range n, x ^ (i + 1) / (i + 1)) + log (1 - x)
  let F' : ℝ → ℝ := fun x ↦ -x ^ n / (1 - x)
  -- Porting note: In `mathlib3`, the proof used `deriv`/`DifferentiableAt`. `simp` failed to
  -- compute `deriv`, so I changed the proof to use `HasDerivAt` instead
  -- First step: compute the derivative of `F`
  have A : ∀ y ∈ Ioo (-1 : ℝ) 1, HasDerivAt F (F' y) y := fun y hy ↦ by
    

Proof state:
x : ℝ
h : |x| < 1
n : ℕ
F : ℝ → ℝ := fun x => ∑ i in Finset.range n, x ^ (i + 1) / (↑i + 1) + log (1 - x)
F' : ℝ → ℝ := fun x => -x ^ n / (1 - x)
y : ℝ
hy : y ∈ Set.Ioo (-1) 1
⊢ HasDerivAt F (F' y) y

Next tactic:
have : HasDerivAt F ((∑ i in range n, ↑(i + 1) * y ^ i / (↑i + 1)) + (-1) / (1 - y)) y :=
      .add (.sum fun i _ ↦ (hasDerivAt_pow (i + 1) y).div_const ((i : ℝ) + 1))
        (((hasDerivAt_id y).const_sub _).log <| sub_ne_zero.2 hy.2.ne')
/-- A crude lemma estimating the difference between `log (1-x)` and its Taylor series at `0`,
where the main point of the bound is that it tends to `0`. The goal is to deduce the series
expansion of the logarithm, in `hasSum_pow_div_log_of_abs_lt_1`.

Porting note: TODO: use one of generic theorems about Taylor's series to prove this estimate.
-/
theorem abs_log_sub_add_sum_range_le {x : ℝ} (h : |x| < 1) (n : ℕ) :
    |(∑ i in range n, x ^ (i + 1) / (i + 1)) + log (1 - x)| ≤ |x| ^ (n + 1) / (1 - |x|) := by
  /- For the proof, we show that the derivative of the function to be estimated is small,
    and then apply the mean value inequality. -/
  let F : ℝ → ℝ := fun x => (∑ i in range n, x ^ (i + 1) / (i + 1)) + log (1 - x)
  let F' : ℝ → ℝ := fun x ↦ -x ^ n / (1 - x)
  -- Porting note: In `mathlib3`, the proof used `deriv`/`DifferentiableAt`. `simp` failed to
  -- compute `deriv`, so I changed the proof to use `HasDerivAt` instead
  -- First step: compute the derivative of `F`
  have A : ∀ y ∈ Ioo (-1 : ℝ) 1, HasDerivAt F (F' y) y := fun y hy ↦ by
    have : HasDerivAt F ((∑ i in range n, ↑(i + 1) * y ^ i / (↑i + 1)) + (-1) / (1 - y)) y :=
      .add (.sum fun i _ ↦ (hasDerivAt_pow (i + 1) y).div_const ((i : ℝ) + 1))
        (((hasDerivAt_id y).const_sub _).log <| sub_ne_zero.2 hy.2.ne')
    

Proof state:
x : ℝ
h : |x| < 1
n : ℕ
F : ℝ → ℝ := fun x => ∑ i in Finset.range n, x ^ (i + 1) / (↑i + 1) + log (1 - x)
F' : ℝ → ℝ := fun x => -x ^ n / (1 - x)
y : ℝ
hy : y ∈ Set.Ioo (-1) 1
this : HasDerivAt F (∑ i in Finset.range n, ↑(i + 1) * y ^ i / (↑i + 1) + -1 / (1 - y)) y
⊢ HasDerivAt F (F' y) y

Next tactic:
convert this using 1