l3lab/ntp-mathlib huggingface dataset:
/-- L'Hôpital's rule for approaching +∞, `HasDerivAt` version -/
theorem lhopital_zero_atTop (hff' : ∀ᶠ x in atTop, HasDerivAt f (f' x) x)
    (hgg' : ∀ᶠ x in atTop, HasDerivAt g (g' x) x) (hg' : ∀ᶠ x in atTop, g' x ≠ 0)
    (hftop : Tendsto f atTop (𝓝 0)) (hgtop : Tendsto g atTop (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) atTop l) : Tendsto (fun x => f x / g x) atTop l := by
  rw [eventually_iff_exists_mem] at *
  rcases hff' with ⟨s₁, hs₁, hff'⟩
  rcases hgg' with ⟨s₂, hs₂, hgg'⟩
  rcases hg' with ⟨s₃, hs₃, hg'⟩
  let s := s₁ ∩ s₂ ∩ s₃
  have hs : s ∈ atTop := inter_mem (inter_mem hs₁ hs₂) hs₃
  rw [mem_atTop_sets] at hs
  rcases hs with ⟨l, hl⟩
  have hl' : Ioi l ⊆ s := fun x hx => hl x (le_of_lt hx)
  refine' lhopital_zero_atTop_on_Ioi _ _ (fun x hx => hg' x <| (hl' hx).2) hftop hgtop hdiv <;>
    

Proof state:
case intro.intro.intro.intro.intro.intro.intro.refine'_2
a b : ℝ
hab : a < b
l✝ : Filter ℝ
f f' g g' : ℝ → ℝ
hftop : Tendsto f atTop (𝓝 0)
hgtop : Tendsto g atTop (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) atTop l✝
s₁ : Set ℝ
hs₁ : s₁ ∈ atTop
hff' : ∀ y ∈ s₁, HasDerivAt f (f' y) y
s₂ : Set ℝ
hs₂ : s₂ ∈ atTop
hgg' : ∀ y ∈ s₂, HasDerivAt g (g' y) y
s₃ : Set ℝ
hs₃ : s₃ ∈ atTop
hg' : ∀ y ∈ s₃, g' y ≠ 0
s : Set ℝ := s₁ ∩ s₂ ∩ s₃
l : ℝ
hl : ∀ b ≥ l, b ∈ s
hl' : Ioi l ⊆ s
⊢ ∀ x ∈ Ioi l, HasDerivAt (fun x => g x) (g' x) x

Next tactic:
intro x hx
/-- L'Hôpital's rule for approaching +∞, `HasDerivAt` version -/
theorem lhopital_zero_atTop (hff' : ∀ᶠ x in atTop, HasDerivAt f (f' x) x)
    (hgg' : ∀ᶠ x in atTop, HasDerivAt g (g' x) x) (hg' : ∀ᶠ x in atTop, g' x ≠ 0)
    (hftop : Tendsto f atTop (𝓝 0)) (hgtop : Tendsto g atTop (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) atTop l) : Tendsto (fun x => f x / g x) atTop l := by
  rw [eventually_iff_exists_mem] at *
  rcases hff' with ⟨s₁, hs₁, hff'⟩
  rcases hgg' with ⟨s₂, hs₂, hgg'⟩
  rcases hg' with ⟨s₃, hs₃, hg'⟩
  let s := s₁ ∩ s₂ ∩ s₃
  have hs : s ∈ atTop := inter_mem (inter_mem hs₁ hs₂) hs₃
  rw [mem_atTop_sets] at hs
  rcases hs with ⟨l, hl⟩
  have hl' : Ioi l ⊆ s := fun x hx => hl x (le_of_lt hx)
  refine' lhopital_zero_atTop_on_Ioi _ _ (fun x hx => hg' x <| (hl' hx).2) hftop hgtop hdiv <;>
    intro x hx <;> 

Proof state:
case intro.intro.intro.intro.intro.intro.intro.refine'_1
a b : ℝ
hab : a < b
l✝ : Filter ℝ
f f' g g' : ℝ → ℝ
hftop : Tendsto f atTop (𝓝 0)
hgtop : Tendsto g atTop (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) atTop l✝
s₁ : Set ℝ
hs₁ : s₁ ∈ atTop
hff' : ∀ y ∈ s₁, HasDerivAt f (f' y) y
s₂ : Set ℝ
hs₂ : s₂ ∈ atTop
hgg' : ∀ y ∈ s₂, HasDerivAt g (g' y) y
s₃ : Set ℝ
hs₃ : s₃ ∈ atTop
hg' : ∀ y ∈ s₃, g' y ≠ 0
s : Set ℝ := s₁ ∩ s₂ ∩ s₃
l : ℝ
hl : ∀ b ≥ l, b ∈ s
hl' : Ioi l ⊆ s
x : ℝ
hx : x ∈ Ioi l
⊢ HasDerivAt (fun x => f x) (f' x) x

Next tactic:
apply_assumption
/-- L'Hôpital's rule for approaching +∞, `HasDerivAt` version -/
theorem lhopital_zero_atTop (hff' : ∀ᶠ x in atTop, HasDerivAt f (f' x) x)
    (hgg' : ∀ᶠ x in atTop, HasDerivAt g (g' x) x) (hg' : ∀ᶠ x in atTop, g' x ≠ 0)
    (hftop : Tendsto f atTop (𝓝 0)) (hgtop : Tendsto g atTop (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) atTop l) : Tendsto (fun x => f x / g x) atTop l := by
  rw [eventually_iff_exists_mem] at *
  rcases hff' with ⟨s₁, hs₁, hff'⟩
  rcases hgg' with ⟨s₂, hs₂, hgg'⟩
  rcases hg' with ⟨s₃, hs₃, hg'⟩
  let s := s₁ ∩ s₂ ∩ s₃
  have hs : s ∈ atTop := inter_mem (inter_mem hs₁ hs₂) hs₃
  rw [mem_atTop_sets] at hs
  rcases hs with ⟨l, hl⟩
  have hl' : Ioi l ⊆ s := fun x hx => hl x (le_of_lt hx)
  refine' lhopital_zero_atTop_on_Ioi _ _ (fun x hx => hg' x <| (hl' hx).2) hftop hgtop hdiv <;>
    intro x hx <;> 

Proof state:
case intro.intro.intro.intro.intro.intro.intro.refine'_2
a b : ℝ
hab : a < b
l✝ : Filter ℝ
f f' g g' : ℝ → ℝ
hftop : Tendsto f atTop (𝓝 0)
hgtop : Tendsto g atTop (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) atTop l✝
s₁ : Set ℝ
hs₁ : s₁ ∈ atTop
hff' : ∀ y ∈ s₁, HasDerivAt f (f' y) y
s₂ : Set ℝ
hs₂ : s₂ ∈ atTop
hgg' : ∀ y ∈ s₂, HasDerivAt g (g' y) y
s₃ : Set ℝ
hs₃ : s₃ ∈ atTop
hg' : ∀ y ∈ s₃, g' y ≠ 0
s : Set ℝ := s₁ ∩ s₂ ∩ s₃
l : ℝ
hl : ∀ b ≥ l, b ∈ s
hl' : Ioi l ⊆ s
x : ℝ
hx : x ∈ Ioi l
⊢ HasDerivAt (fun x => g x) (g' x) x

Next tactic:
apply_assumption
/-- L'Hôpital's rule for approaching +∞, `HasDerivAt` version -/
theorem lhopital_zero_atTop (hff' : ∀ᶠ x in atTop, HasDerivAt f (f' x) x)
    (hgg' : ∀ᶠ x in atTop, HasDerivAt g (g' x) x) (hg' : ∀ᶠ x in atTop, g' x ≠ 0)
    (hftop : Tendsto f atTop (𝓝 0)) (hgtop : Tendsto g atTop (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) atTop l) : Tendsto (fun x => f x / g x) atTop l := by
  rw [eventually_iff_exists_mem] at *
  rcases hff' with ⟨s₁, hs₁, hff'⟩
  rcases hgg' with ⟨s₂, hs₂, hgg'⟩
  rcases hg' with ⟨s₃, hs₃, hg'⟩
  let s := s₁ ∩ s₂ ∩ s₃
  have hs : s ∈ atTop := inter_mem (inter_mem hs₁ hs₂) hs₃
  rw [mem_atTop_sets] at hs
  rcases hs with ⟨l, hl⟩
  have hl' : Ioi l ⊆ s := fun x hx => hl x (le_of_lt hx)
  refine' lhopital_zero_atTop_on_Ioi _ _ (fun x hx => hg' x <| (hl' hx).2) hftop hgtop hdiv <;>
    intro x hx <;> apply_assumption <;> 

Proof state:
case intro.intro.intro.intro.intro.intro.intro.refine'_1.a
a b : ℝ
hab : a < b
l✝ : Filter ℝ
f f' g g' : ℝ → ℝ
hftop : Tendsto f atTop (𝓝 0)
hgtop : Tendsto g atTop (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) atTop l✝
s₁ : Set ℝ
hs₁ : s₁ ∈ atTop
hff' : ∀ y ∈ s₁, HasDerivAt f (f' y) y
s₂ : Set ℝ
hs₂ : s₂ ∈ atTop
hgg' : ∀ y ∈ s₂, HasDerivAt g (g' y) y
s₃ : Set ℝ
hs₃ : s₃ ∈ atTop
hg' : ∀ y ∈ s₃, g' y ≠ 0
s : Set ℝ := s₁ ∩ s₂ ∩ s₃
l : ℝ
hl : ∀ b ≥ l, b ∈ s
hl' : Ioi l ⊆ s
x : ℝ
hx : x ∈ Ioi l
⊢ x ∈ s₁

Next tactic:
first | exact (hl' hx).1.1| exact (hl' hx).1.2
/-- L'Hôpital's rule for approaching +∞, `HasDerivAt` version -/
theorem lhopital_zero_atTop (hff' : ∀ᶠ x in atTop, HasDerivAt f (f' x) x)
    (hgg' : ∀ᶠ x in atTop, HasDerivAt g (g' x) x) (hg' : ∀ᶠ x in atTop, g' x ≠ 0)
    (hftop : Tendsto f atTop (𝓝 0)) (hgtop : Tendsto g atTop (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) atTop l) : Tendsto (fun x => f x / g x) atTop l := by
  rw [eventually_iff_exists_mem] at *
  rcases hff' with ⟨s₁, hs₁, hff'⟩
  rcases hgg' with ⟨s₂, hs₂, hgg'⟩
  rcases hg' with ⟨s₃, hs₃, hg'⟩
  let s := s₁ ∩ s₂ ∩ s₃
  have hs : s ∈ atTop := inter_mem (inter_mem hs₁ hs₂) hs₃
  rw [mem_atTop_sets] at hs
  rcases hs with ⟨l, hl⟩
  have hl' : Ioi l ⊆ s := fun x hx => hl x (le_of_lt hx)
  refine' lhopital_zero_atTop_on_Ioi _ _ (fun x hx => hg' x <| (hl' hx).2) hftop hgtop hdiv <;>
    intro x hx <;> apply_assumption <;> first | 

Proof state:
case intro.intro.intro.intro.intro.intro.intro.refine'_1.a
a b : ℝ
hab : a < b
l✝ : Filter ℝ
f f' g g' : ℝ → ℝ
hftop : Tendsto f atTop (𝓝 0)
hgtop : Tendsto g atTop (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) atTop l✝
s₁ : Set ℝ
hs₁ : s₁ ∈ atTop
hff' : ∀ y ∈ s₁, HasDerivAt f (f' y) y
s₂ : Set ℝ
hs₂ : s₂ ∈ atTop
hgg' : ∀ y ∈ s₂, HasDerivAt g (g' y) y
s₃ : Set ℝ
hs₃ : s₃ ∈ atTop
hg' : ∀ y ∈ s₃, g' y ≠ 0
s : Set ℝ := s₁ ∩ s₂ ∩ s₃
l : ℝ
hl : ∀ b ≥ l, b ∈ s
hl' : Ioi l ⊆ s
x : ℝ
hx : x ∈ Ioi l
⊢ x ∈ s₁

Next tactic:
exact (hl' hx).1.1
/-- L'Hôpital's rule for approaching +∞, `HasDerivAt` version -/
theorem lhopital_zero_atTop (hff' : ∀ᶠ x in atTop, HasDerivAt f (f' x) x)
    (hgg' : ∀ᶠ x in atTop, HasDerivAt g (g' x) x) (hg' : ∀ᶠ x in atTop, g' x ≠ 0)
    (hftop : Tendsto f atTop (𝓝 0)) (hgtop : Tendsto g atTop (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) atTop l) : Tendsto (fun x => f x / g x) atTop l := by
  rw [eventually_iff_exists_mem] at *
  rcases hff' with ⟨s₁, hs₁, hff'⟩
  rcases hgg' with ⟨s₂, hs₂, hgg'⟩
  rcases hg' with ⟨s₃, hs₃, hg'⟩
  let s := s₁ ∩ s₂ ∩ s₃
  have hs : s ∈ atTop := inter_mem (inter_mem hs₁ hs₂) hs₃
  rw [mem_atTop_sets] at hs
  rcases hs with ⟨l, hl⟩
  have hl' : Ioi l ⊆ s := fun x hx => hl x (le_of_lt hx)
  refine' lhopital_zero_atTop_on_Ioi _ _ (fun x hx => hg' x <| (hl' hx).2) hftop hgtop hdiv <;>
    intro x hx <;> apply_assumption <;> 

Proof state:
case intro.intro.intro.intro.intro.intro.intro.refine'_2.a
a b : ℝ
hab : a < b
l✝ : Filter ℝ
f f' g g' : ℝ → ℝ
hftop : Tendsto f atTop (𝓝 0)
hgtop : Tendsto g atTop (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) atTop l✝
s₁ : Set ℝ
hs₁ : s₁ ∈ atTop
hff' : ∀ y ∈ s₁, HasDerivAt f (f' y) y
s₂ : Set ℝ
hs₂ : s₂ ∈ atTop
hgg' : ∀ y ∈ s₂, HasDerivAt g (g' y) y
s₃ : Set ℝ
hs₃ : s₃ ∈ atTop
hg' : ∀ y ∈ s₃, g' y ≠ 0
s : Set ℝ := s₁ ∩ s₂ ∩ s₃
l : ℝ
hl : ∀ b ≥ l, b ∈ s
hl' : Ioi l ⊆ s
x : ℝ
hx : x ∈ Ioi l
⊢ x ∈ s₂

Next tactic:
first | exact (hl' hx).1.1| exact (hl' hx).1.2
/-- L'Hôpital's rule for approaching +∞, `HasDerivAt` version -/
theorem lhopital_zero_atTop (hff' : ∀ᶠ x in atTop, HasDerivAt f (f' x) x)
    (hgg' : ∀ᶠ x in atTop, HasDerivAt g (g' x) x) (hg' : ∀ᶠ x in atTop, g' x ≠ 0)
    (hftop : Tendsto f atTop (𝓝 0)) (hgtop : Tendsto g atTop (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) atTop l) : Tendsto (fun x => f x / g x) atTop l := by
  rw [eventually_iff_exists_mem] at *
  rcases hff' with ⟨s₁, hs₁, hff'⟩
  rcases hgg' with ⟨s₂, hs₂, hgg'⟩
  rcases hg' with ⟨s₃, hs₃, hg'⟩
  let s := s₁ ∩ s₂ ∩ s₃
  have hs : s ∈ atTop := inter_mem (inter_mem hs₁ hs₂) hs₃
  rw [mem_atTop_sets] at hs
  rcases hs with ⟨l, hl⟩
  have hl' : Ioi l ⊆ s := fun x hx => hl x (le_of_lt hx)
  refine' lhopital_zero_atTop_on_Ioi _ _ (fun x hx => hg' x <| (hl' hx).2) hftop hgtop hdiv <;>
    intro x hx <;> apply_assumption <;> first | 

Proof state:
case intro.intro.intro.intro.intro.intro.intro.refine'_2.a
a b : ℝ
hab : a < b
l✝ : Filter ℝ
f f' g g' : ℝ → ℝ
hftop : Tendsto f atTop (𝓝 0)
hgtop : Tendsto g atTop (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) atTop l✝
s₁ : Set ℝ
hs₁ : s₁ ∈ atTop
hff' : ∀ y ∈ s₁, HasDerivAt f (f' y) y
s₂ : Set ℝ
hs₂ : s₂ ∈ atTop
hgg' : ∀ y ∈ s₂, HasDerivAt g (g' y) y
s₃ : Set ℝ
hs₃ : s₃ ∈ atTop
hg' : ∀ y ∈ s₃, g' y ≠ 0
s : Set ℝ := s₁ ∩ s₂ ∩ s₃
l : ℝ
hl : ∀ b ≥ l, b ∈ s
hl' : Ioi l ⊆ s
x : ℝ
hx : x ∈ Ioi l
⊢ x ∈ s₂

Next tactic:
exact (hl' hx).1.1
/-- L'Hôpital's rule for approaching +∞, `HasDerivAt` version -/
theorem lhopital_zero_atTop (hff' : ∀ᶠ x in atTop, HasDerivAt f (f' x) x)
    (hgg' : ∀ᶠ x in atTop, HasDerivAt g (g' x) x) (hg' : ∀ᶠ x in atTop, g' x ≠ 0)
    (hftop : Tendsto f atTop (𝓝 0)) (hgtop : Tendsto g atTop (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) atTop l) : Tendsto (fun x => f x / g x) atTop l := by
  rw [eventually_iff_exists_mem] at *
  rcases hff' with ⟨s₁, hs₁, hff'⟩
  rcases hgg' with ⟨s₂, hs₂, hgg'⟩
  rcases hg' with ⟨s₃, hs₃, hg'⟩
  let s := s₁ ∩ s₂ ∩ s₃
  have hs : s ∈ atTop := inter_mem (inter_mem hs₁ hs₂) hs₃
  rw [mem_atTop_sets] at hs
  rcases hs with ⟨l, hl⟩
  have hl' : Ioi l ⊆ s := fun x hx => hl x (le_of_lt hx)
  refine' lhopital_zero_atTop_on_Ioi _ _ (fun x hx => hg' x <| (hl' hx).2) hftop hgtop hdiv <;>
    intro x hx <;> apply_assumption <;> first | exact (hl' hx).1.1| 

Proof state:
case intro.intro.intro.intro.intro.intro.intro.refine'_2.a
a b : ℝ
hab : a < b
l✝ : Filter ℝ
f f' g g' : ℝ → ℝ
hftop : Tendsto f atTop (𝓝 0)
hgtop : Tendsto g atTop (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) atTop l✝
s₁ : Set ℝ
hs₁ : s₁ ∈ atTop
hff' : ∀ y ∈ s₁, HasDerivAt f (f' y) y
s₂ : Set ℝ
hs₂ : s₂ ∈ atTop
hgg' : ∀ y ∈ s₂, HasDerivAt g (g' y) y
s₃ : Set ℝ
hs₃ : s₃ ∈ atTop
hg' : ∀ y ∈ s₃, g' y ≠ 0
s : Set ℝ := s₁ ∩ s₂ ∩ s₃
l : ℝ
hl : ∀ b ≥ l, b ∈ s
hl' : Ioi l ⊆ s
x : ℝ
hx : x ∈ Ioi l
⊢ x ∈ s₂

Next tactic:
exact (hl' hx).1.2
/-- L'Hôpital's rule for approaching -∞, `HasDerivAt` version -/
theorem lhopital_zero_atBot (hff' : ∀ᶠ x in atBot, HasDerivAt f (f' x) x)
    (hgg' : ∀ᶠ x in atBot, HasDerivAt g (g' x) x) (hg' : ∀ᶠ x in atBot, g' x ≠ 0)
    (hfbot : Tendsto f atBot (𝓝 0)) (hgbot : Tendsto g atBot (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) atBot l) : Tendsto (fun x => f x / g x) atBot l := by
  

Proof state:
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∀ᶠ (x : ℝ) in atBot, HasDerivAt f (f' x) x
hgg' : ∀ᶠ (x : ℝ) in atBot, HasDerivAt g (g' x) x
hg' : ∀ᶠ (x : ℝ) in atBot, g' x ≠ 0
hfbot : Tendsto f atBot (𝓝 0)
hgbot : Tendsto g atBot (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) atBot l
⊢ Tendsto (fun x => f x / g x) atBot l

Next tactic:
rw [eventually_iff_exists_mem] at *
/-- L'Hôpital's rule for approaching -∞, `HasDerivAt` version -/
theorem lhopital_zero_atBot (hff' : ∀ᶠ x in atBot, HasDerivAt f (f' x) x)
    (hgg' : ∀ᶠ x in atBot, HasDerivAt g (g' x) x) (hg' : ∀ᶠ x in atBot, g' x ≠ 0)
    (hfbot : Tendsto f atBot (𝓝 0)) (hgbot : Tendsto g atBot (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) atBot l) : Tendsto (fun x => f x / g x) atBot l := by
  rw [eventually_iff_exists_mem] at *
  

Proof state:
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∃ v ∈ atBot, ∀ y ∈ v, HasDerivAt f (f' y) y
hgg' : ∃ v ∈ atBot, ∀ y ∈ v, HasDerivAt g (g' y) y
hg' : ∃ v ∈ atBot, ∀ y ∈ v, g' y ≠ 0
hfbot : Tendsto f atBot (𝓝 0)
hgbot : Tendsto g atBot (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) atBot l
⊢ Tendsto (fun x => f x / g x) atBot l

Next tactic:
rcases hff' with ⟨s₁, hs₁, hff'⟩
/-- L'Hôpital's rule for approaching -∞, `HasDerivAt` version -/
theorem lhopital_zero_atBot (hff' : ∀ᶠ x in atBot, HasDerivAt f (f' x) x)
    (hgg' : ∀ᶠ x in atBot, HasDerivAt g (g' x) x) (hg' : ∀ᶠ x in atBot, g' x ≠ 0)
    (hfbot : Tendsto f atBot (𝓝 0)) (hgbot : Tendsto g atBot (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) atBot l) : Tendsto (fun x => f x / g x) atBot l := by
  rw [eventually_iff_exists_mem] at *
  rcases hff' with ⟨s₁, hs₁, hff'⟩
  

Proof state:
case intro.intro
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hgg' : ∃ v ∈ atBot, ∀ y ∈ v, HasDerivAt g (g' y) y
hg' : ∃ v ∈ atBot, ∀ y ∈ v, g' y ≠ 0
hfbot : Tendsto f atBot (𝓝 0)
hgbot : Tendsto g atBot (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) atBot l
s₁ : Set ℝ
hs₁ : s₁ ∈ atBot
hff' : ∀ y ∈ s₁, HasDerivAt f (f' y) y
⊢ Tendsto (fun x => f x / g x) atBot l

Next tactic:
rcases hgg' with ⟨s₂, hs₂, hgg'⟩
/-- L'Hôpital's rule for approaching -∞, `HasDerivAt` version -/
theorem lhopital_zero_atBot (hff' : ∀ᶠ x in atBot, HasDerivAt f (f' x) x)
    (hgg' : ∀ᶠ x in atBot, HasDerivAt g (g' x) x) (hg' : ∀ᶠ x in atBot, g' x ≠ 0)
    (hfbot : Tendsto f atBot (𝓝 0)) (hgbot : Tendsto g atBot (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) atBot l) : Tendsto (fun x => f x / g x) atBot l := by
  rw [eventually_iff_exists_mem] at *
  rcases hff' with ⟨s₁, hs₁, hff'⟩
  rcases hgg' with ⟨s₂, hs₂, hgg'⟩
  

Proof state:
case intro.intro.intro.intro
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hg' : ∃ v ∈ atBot, ∀ y ∈ v, g' y ≠ 0
hfbot : Tendsto f atBot (𝓝 0)
hgbot : Tendsto g atBot (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) atBot l
s₁ : Set ℝ
hs₁ : s₁ ∈ atBot
hff' : ∀ y ∈ s₁, HasDerivAt f (f' y) y
s₂ : Set ℝ
hs₂ : s₂ ∈ atBot
hgg' : ∀ y ∈ s₂, HasDerivAt g (g' y) y
⊢ Tendsto (fun x => f x / g x) atBot l

Next tactic:
rcases hg' with ⟨s₃, hs₃, hg'⟩
/-- L'Hôpital's rule for approaching -∞, `HasDerivAt` version -/
theorem lhopital_zero_atBot (hff' : ∀ᶠ x in atBot, HasDerivAt f (f' x) x)
    (hgg' : ∀ᶠ x in atBot, HasDerivAt g (g' x) x) (hg' : ∀ᶠ x in atBot, g' x ≠ 0)
    (hfbot : Tendsto f atBot (𝓝 0)) (hgbot : Tendsto g atBot (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) atBot l) : Tendsto (fun x => f x / g x) atBot l := by
  rw [eventually_iff_exists_mem] at *
  rcases hff' with ⟨s₁, hs₁, hff'⟩
  rcases hgg' with ⟨s₂, hs₂, hgg'⟩
  rcases hg' with ⟨s₃, hs₃, hg'⟩
  

Proof state:
case intro.intro.intro.intro.intro.intro
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hfbot : Tendsto f atBot (𝓝 0)
hgbot : Tendsto g atBot (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) atBot l
s₁ : Set ℝ
hs₁ : s₁ ∈ atBot
hff' : ∀ y ∈ s₁, HasDerivAt f (f' y) y
s₂ : Set ℝ
hs₂ : s₂ ∈ atBot
hgg' : ∀ y ∈ s₂, HasDerivAt g (g' y) y
s₃ : Set ℝ
hs₃ : s₃ ∈ atBot
hg' : ∀ y ∈ s₃, g' y ≠ 0
⊢ Tendsto (fun x => f x / g x) atBot l

Next tactic:
let s := s₁ ∩ s₂ ∩ s₃
/-- L'Hôpital's rule for approaching -∞, `HasDerivAt` version -/
theorem lhopital_zero_atBot (hff' : ∀ᶠ x in atBot, HasDerivAt f (f' x) x)
    (hgg' : ∀ᶠ x in atBot, HasDerivAt g (g' x) x) (hg' : ∀ᶠ x in atBot, g' x ≠ 0)
    (hfbot : Tendsto f atBot (𝓝 0)) (hgbot : Tendsto g atBot (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) atBot l) : Tendsto (fun x => f x / g x) atBot l := by
  rw [eventually_iff_exists_mem] at *
  rcases hff' with ⟨s₁, hs₁, hff'⟩
  rcases hgg' with ⟨s₂, hs₂, hgg'⟩
  rcases hg' with ⟨s₃, hs₃, hg'⟩
  let s := s₁ ∩ s₂ ∩ s₃
  

Proof state:
case intro.intro.intro.intro.intro.intro
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hfbot : Tendsto f atBot (𝓝 0)
hgbot : Tendsto g atBot (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) atBot l
s₁ : Set ℝ
hs₁ : s₁ ∈ atBot
hff' : ∀ y ∈ s₁, HasDerivAt f (f' y) y
s₂ : Set ℝ
hs₂ : s₂ ∈ atBot
hgg' : ∀ y ∈ s₂, HasDerivAt g (g' y) y
s₃ : Set ℝ
hs₃ : s₃ ∈ atBot
hg' : ∀ y ∈ s₃, g' y ≠ 0
s : Set ℝ := s₁ ∩ s₂ ∩ s₃
⊢ Tendsto (fun x => f x / g x) atBot l

Next tactic:
have hs : s ∈ atBot := inter_mem (inter_mem hs₁ hs₂) hs₃
/-- L'Hôpital's rule for approaching -∞, `HasDerivAt` version -/
theorem lhopital_zero_atBot (hff' : ∀ᶠ x in atBot, HasDerivAt f (f' x) x)
    (hgg' : ∀ᶠ x in atBot, HasDerivAt g (g' x) x) (hg' : ∀ᶠ x in atBot, g' x ≠ 0)
    (hfbot : Tendsto f atBot (𝓝 0)) (hgbot : Tendsto g atBot (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) atBot l) : Tendsto (fun x => f x / g x) atBot l := by
  rw [eventually_iff_exists_mem] at *
  rcases hff' with ⟨s₁, hs₁, hff'⟩
  rcases hgg' with ⟨s₂, hs₂, hgg'⟩
  rcases hg' with ⟨s₃, hs₃, hg'⟩
  let s := s₁ ∩ s₂ ∩ s₃
  have hs : s ∈ atBot := inter_mem (inter_mem hs₁ hs₂) hs₃
  

Proof state:
case intro.intro.intro.intro.intro.intro
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hfbot : Tendsto f atBot (𝓝 0)
hgbot : Tendsto g atBot (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) atBot l
s₁ : Set ℝ
hs₁ : s₁ ∈ atBot
hff' : ∀ y ∈ s₁, HasDerivAt f (f' y) y
s₂ : Set ℝ
hs₂ : s₂ ∈ atBot
hgg' : ∀ y ∈ s₂, HasDerivAt g (g' y) y
s₃ : Set ℝ
hs₃ : s₃ ∈ atBot
hg' : ∀ y ∈ s₃, g' y ≠ 0
s : Set ℝ := s₁ ∩ s₂ ∩ s₃
hs : s ∈ atBot
⊢ Tendsto (fun x => f x / g x) atBot l

Next tactic:
rw [mem_atBot_sets] at hs
/-- L'Hôpital's rule for approaching -∞, `HasDerivAt` version -/
theorem lhopital_zero_atBot (hff' : ∀ᶠ x in atBot, HasDerivAt f (f' x) x)
    (hgg' : ∀ᶠ x in atBot, HasDerivAt g (g' x) x) (hg' : ∀ᶠ x in atBot, g' x ≠ 0)
    (hfbot : Tendsto f atBot (𝓝 0)) (hgbot : Tendsto g atBot (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) atBot l) : Tendsto (fun x => f x / g x) atBot l := by
  rw [eventually_iff_exists_mem] at *
  rcases hff' with ⟨s₁, hs₁, hff'⟩
  rcases hgg' with ⟨s₂, hs₂, hgg'⟩
  rcases hg' with ⟨s₃, hs₃, hg'⟩
  let s := s₁ ∩ s₂ ∩ s₃
  have hs : s ∈ atBot := inter_mem (inter_mem hs₁ hs₂) hs₃
  rw [mem_atBot_sets] at hs
  

Proof state:
case intro.intro.intro.intro.intro.intro
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hfbot : Tendsto f atBot (𝓝 0)
hgbot : Tendsto g atBot (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) atBot l
s₁ : Set ℝ
hs₁ : s₁ ∈ atBot
hff' : ∀ y ∈ s₁, HasDerivAt f (f' y) y
s₂ : Set ℝ
hs₂ : s₂ ∈ atBot
hgg' : ∀ y ∈ s₂, HasDerivAt g (g' y) y
s₃ : Set ℝ
hs₃ : s₃ ∈ atBot
hg' : ∀ y ∈ s₃, g' y ≠ 0
s : Set ℝ := s₁ ∩ s₂ ∩ s₃
hs : ∃ a, ∀ b ≤ a, b ∈ s
⊢ Tendsto (fun x => f x / g x) atBot l

Next tactic:
rcases hs with ⟨l, hl⟩
/-- L'Hôpital's rule for approaching -∞, `HasDerivAt` version -/
theorem lhopital_zero_atBot (hff' : ∀ᶠ x in atBot, HasDerivAt f (f' x) x)
    (hgg' : ∀ᶠ x in atBot, HasDerivAt g (g' x) x) (hg' : ∀ᶠ x in atBot, g' x ≠ 0)
    (hfbot : Tendsto f atBot (𝓝 0)) (hgbot : Tendsto g atBot (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) atBot l) : Tendsto (fun x => f x / g x) atBot l := by
  rw [eventually_iff_exists_mem] at *
  rcases hff' with ⟨s₁, hs₁, hff'⟩
  rcases hgg' with ⟨s₂, hs₂, hgg'⟩
  rcases hg' with ⟨s₃, hs₃, hg'⟩
  let s := s₁ ∩ s₂ ∩ s₃
  have hs : s ∈ atBot := inter_mem (inter_mem hs₁ hs₂) hs₃
  rw [mem_atBot_sets] at hs
  rcases hs with ⟨l, hl⟩
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro
a b : ℝ
hab : a < b
l✝ : Filter ℝ
f f' g g' : ℝ → ℝ
hfbot : Tendsto f atBot (𝓝 0)
hgbot : Tendsto g atBot (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) atBot l✝
s₁ : Set ℝ
hs₁ : s₁ ∈ atBot
hff' : ∀ y ∈ s₁, HasDerivAt f (f' y) y
s₂ : Set ℝ
hs₂ : s₂ ∈ atBot
hgg' : ∀ y ∈ s₂, HasDerivAt g (g' y) y
s₃ : Set ℝ
hs₃ : s₃ ∈ atBot
hg' : ∀ y ∈ s₃, g' y ≠ 0
s : Set ℝ := s₁ ∩ s₂ ∩ s₃
l : ℝ
hl : ∀ b ≤ l, b ∈ s
⊢ Tendsto (fun x => f x / g x) atBot l✝

Next tactic:
have hl' : Iio l ⊆ s := fun x hx => hl x (le_of_lt hx)
/-- L'Hôpital's rule for approaching -∞, `HasDerivAt` version -/
theorem lhopital_zero_atBot (hff' : ∀ᶠ x in atBot, HasDerivAt f (f' x) x)
    (hgg' : ∀ᶠ x in atBot, HasDerivAt g (g' x) x) (hg' : ∀ᶠ x in atBot, g' x ≠ 0)
    (hfbot : Tendsto f atBot (𝓝 0)) (hgbot : Tendsto g atBot (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) atBot l) : Tendsto (fun x => f x / g x) atBot l := by
  rw [eventually_iff_exists_mem] at *
  rcases hff' with ⟨s₁, hs₁, hff'⟩
  rcases hgg' with ⟨s₂, hs₂, hgg'⟩
  rcases hg' with ⟨s₃, hs₃, hg'⟩
  let s := s₁ ∩ s₂ ∩ s₃
  have hs : s ∈ atBot := inter_mem (inter_mem hs₁ hs₂) hs₃
  rw [mem_atBot_sets] at hs
  rcases hs with ⟨l, hl⟩
  have hl' : Iio l ⊆ s := fun x hx => hl x (le_of_lt hx)
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro
a b : ℝ
hab : a < b
l✝ : Filter ℝ
f f' g g' : ℝ → ℝ
hfbot : Tendsto f atBot (𝓝 0)
hgbot : Tendsto g atBot (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) atBot l✝
s₁ : Set ℝ
hs₁ : s₁ ∈ atBot
hff' : ∀ y ∈ s₁, HasDerivAt f (f' y) y
s₂ : Set ℝ
hs₂ : s₂ ∈ atBot
hgg' : ∀ y ∈ s₂, HasDerivAt g (g' y) y
s₃ : Set ℝ
hs₃ : s₃ ∈ atBot
hg' : ∀ y ∈ s₃, g' y ≠ 0
s : Set ℝ := s₁ ∩ s₂ ∩ s₃
l : ℝ
hl : ∀ b ≤ l, b ∈ s
hl' : Iio l ⊆ s
⊢ Tendsto (fun x => f x / g x) atBot l✝

Next tactic:
refine' lhopital_zero_atBot_on_Iio _ _ (fun x hx => hg' x <| (hl' hx).2) hfbot hgbot hdiv
/-- L'Hôpital's rule for approaching -∞, `HasDerivAt` version -/
theorem lhopital_zero_atBot (hff' : ∀ᶠ x in atBot, HasDerivAt f (f' x) x)
    (hgg' : ∀ᶠ x in atBot, HasDerivAt g (g' x) x) (hg' : ∀ᶠ x in atBot, g' x ≠ 0)
    (hfbot : Tendsto f atBot (𝓝 0)) (hgbot : Tendsto g atBot (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) atBot l) : Tendsto (fun x => f x / g x) atBot l := by
  rw [eventually_iff_exists_mem] at *
  rcases hff' with ⟨s₁, hs₁, hff'⟩
  rcases hgg' with ⟨s₂, hs₂, hgg'⟩
  rcases hg' with ⟨s₃, hs₃, hg'⟩
  let s := s₁ ∩ s₂ ∩ s₃
  have hs : s ∈ atBot := inter_mem (inter_mem hs₁ hs₂) hs₃
  rw [mem_atBot_sets] at hs
  rcases hs with ⟨l, hl⟩
  have hl' : Iio l ⊆ s := fun x hx => hl x (le_of_lt hx)
  refine' lhopital_zero_atBot_on_Iio _ _ (fun x hx => hg' x <| (hl' hx).2) hfbot hgbot hdiv <;>
    

Proof state:
case intro.intro.intro.intro.intro.intro.intro.refine'_1
a b : ℝ
hab : a < b
l✝ : Filter ℝ
f f' g g' : ℝ → ℝ
hfbot : Tendsto f atBot (𝓝 0)
hgbot : Tendsto g atBot (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) atBot l✝
s₁ : Set ℝ
hs₁ : s₁ ∈ atBot
hff' : ∀ y ∈ s₁, HasDerivAt f (f' y) y
s₂ : Set ℝ
hs₂ : s₂ ∈ atBot
hgg' : ∀ y ∈ s₂, HasDerivAt g (g' y) y
s₃ : Set ℝ
hs₃ : s₃ ∈ atBot
hg' : ∀ y ∈ s₃, g' y ≠ 0
s : Set ℝ := s₁ ∩ s₂ ∩ s₃
l : ℝ
hl : ∀ b ≤ l, b ∈ s
hl' : Iio l ⊆ s
⊢ ∀ x ∈ Iio l, HasDerivAt (fun x => f x) (f' x) x

Next tactic:
intro x hx