Аналіз I, Розділ 10.5: Правило Лопіталя

Я (прим. перекл. Терренс Тао) намагався зробити переклад якомога точнішим перефразуванням оригінального тексту. Коли є вибір між більш ідіоматичним підходом Lean та більш точним перекладом, я зазвичай обирав останній. Зокрема, будуть місця, де код Lean можна було б "підправити", щоб зробити його більш елегантним та ідіоматичним, але я свідомо уникав цього вибору.

Основні конструкції та результати цього розділу:

Правило Лопіталя.

open Chapter9namespace Chapter10

Твердження 10.5.1 (Правило Лопіталя, I) / Вправа 10.5.1

theorem declaration uses 'sorry'_root_.Filter.Tendsto.of_div {X: Set ℝ} {f g: ℝ → ℝ} {x₀ f'x₀ g'x₀:ℝ}
  (hfx₀: f x₀ = 0) (hgx₀: g x₀ = 0) (hg_non: g'x₀ ≠ 0)
  (hf'x₀: HasDerivWithinAt f f'x₀ X x₀) (hg'x₀: HasDerivWithinAt g g'x₀ X x₀) :
  (∃ δ > 0, ∀ x ∈ X \ {x₀} ∩ .Ioo (x₀ - δ) (x₀ + δ), g x ≠ 0) ∧
  (nhdsWithin x₀ (X \ {x₀})).Tendsto (fun x ↦ f x / g x) (nhds (f'x₀ / g'x₀))
  := byX:Set ℝf:ℝ → ℝg:ℝ → ℝx₀:ℝf'x₀:ℝg'x₀:ℝhfx₀:f x₀ = 0hgx₀:g x₀ = 0hg_non:g'x₀ ≠ 0hf'x₀:HasDerivWithinAt f f'x₀ X x₀hg'x₀:HasDerivWithinAt g g'x₀ X x₀⊢ (∃ δ > 0, ∀ x ∈ X \ {x₀} ∩ Set.Ioo (x₀ - δ) (x₀ + δ), g x ≠ 0) ∧
  Filter.Tendsto (fun x => f x / g x) (nhdsWithin x₀ (X \ {x₀})) (nhds (f'x₀ / g'x₀))
  sorryAll goals completed! 🐙

Твердження 10.5.2 (Правило Лопіталя, II)

theorem _root_.Filter.Tendsto.of_div' {a b L:ℝ} (hab: a < b) {f g f' g': ℝ → ℝ}
  (hf: DifferentiableOn ℝ f (.Icc a b)) (hg: DifferentiableOn ℝ g (.Icc a b))
  (hf': f' = derivWithin f (.Icc a b)) (hg': g' = derivWithin g (.Icc a b))
  (hfa: f a = 0) (hga: g a = 0) (hgnon: ∀ x ∈ Set.Icc a b, g' x ≠ 0)
  (hderiv: (nhdsWithin a (.Icc a b)).Tendsto (fun x ↦ f' x / g' x) (nhds L)) :
  (∀ x ∈ Set.Ioc a b, g x ≠ 0) ∧
  (nhdsWithin a (.Ioc a b)).Tendsto (fun x ↦ f x / g x) (nhds L) := bya:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)⊢ (∀ x ∈ Set.Ioc a b, g x ≠ 0) ∧ Filter.Tendsto (fun x => f x / g x) (nhdsWithin a (Set.Ioc a b)) (nhds L)
  -- Доведення написане так, щоб відповідати структурі оригінального тексту.
  observe hfcon : ContinuousOn f (.Icc a b)a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)⊢ (∀ x ∈ Set.Ioc a b, g x ≠ 0) ∧ Filter.Tendsto (fun x => f x / g x) (nhdsWithin a (Set.Ioc a b)) (nhds L)
  observe hgcon : ContinuousOn g (.Icc a b)a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)⊢ (∀ x ∈ Set.Ioc a b, g x ≠ 0) ∧ Filter.Tendsto (fun x => f x / g x) (nhdsWithin a (Set.Ioc a b)) (nhds L)
  have (x:ℝ) (hx: x ∈ Set.Ioc a b) : g x ≠ 0 := bya:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)⊢ (∀ x ∈ Set.Ioc a b, g x ≠ 0) ∧ Filter.Tendsto (fun x => f x / g x) (nhdsWithin a (Set.Ioc a b)) (nhds L)
    by_contra thisa:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝhx:x ∈ Set.Ioc a bthis:g x = 0⊢ False
    simp at hxa:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis:g x = 0hx:a < x ∧ x ≤ b⊢ False
    have := HasDerivWithinAt.exist_zero hx.1 (hgcon.mono ?_) (hg.mono ?_) ?_refine_4a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis✝:g x = 0hx:a < x ∧ x ≤ bthis:?_mvar.59717 := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)⊢ Falserefine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis:g x = 0hx:a < x ∧ x ≤ b⊢ Set.Icc a x ⊆ Set.Icc a brefine_2a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis:g x = 0hx:a < x ∧ x ≤ b⊢ Set.Ioo a x ⊆ Set.Icc a brefine_3a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis:g x = 0hx:a < x ∧ x ≤ b⊢ g a = g x
    choose y hy hgy using thisrefine_4a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis:g x = 0hx:a < x ∧ x ≤ by:ℝhy:y ∈ Set.Ioo a xhgy:HasDerivWithinAt g 0 (Set.Ioo a x) y⊢ Falserefine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis:g x = 0hx:a < x ∧ x ≤ b⊢ Set.Icc a x ⊆ Set.Icc a brefine_2a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis:g x = 0hx:a < x ∧ x ≤ b⊢ Set.Ioo a x ⊆ Set.Icc a brefine_3a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis:g x = 0hx:a < x ∧ x ≤ b⊢ g a = g x; simp at hyrefine_4a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis:g x = 0hx:a < x ∧ x ≤ by:ℝhgy:HasDerivWithinAt g 0 (Set.Ioo a x) yhy:a < y ∧ y < x⊢ Falserefine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis:g x = 0hx:a < x ∧ x ≤ b⊢ Set.Icc a x ⊆ Set.Icc a brefine_2a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis:g x = 0hx:a < x ∧ x ≤ b⊢ Set.Ioo a x ⊆ Set.Icc a brefine_3a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis:g x = 0hx:a < x ∧ x ≤ b⊢ g a = g x
    have : y ∈ Set.Icc a b := ?_refine_4.refine_2a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis✝:g x = 0hx:a < x ∧ x ≤ by:ℝhgy:HasDerivWithinAt g 0 (Set.Ioo a x) yhy:a < y ∧ y < xthis:_fvar.60008 ∈ Set.Icc _fvar.4271 _fvar.4272 := ?_mvar.61846⊢ Falserefine_4.refine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis:g x = 0hx:a < x ∧ x ≤ by:ℝhgy:HasDerivWithinAt g 0 (Set.Ioo a x) yhy:a < y ∧ y < x⊢ y ∈ Set.Icc a brefine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis:g x = 0hx:a < x ∧ x ≤ b⊢ Set.Icc a x ⊆ Set.Icc a brefine_2a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis:g x = 0hx:a < x ∧ x ≤ b⊢ Set.Ioo a x ⊆ Set.Icc a brefine_3a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis:g x = 0hx:a < x ∧ x ≤ b⊢ g a = g x
    specialize hgnon y thisrefine_4.refine_2a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis✝:g x = 0hx:a < x ∧ x ≤ by:ℝhgy:HasDerivWithinAt g 0 (Set.Ioo a x) yhy:a < y ∧ y < xthis:_fvar.60008 ∈ Set.Icc _fvar.4271 _fvar.4272 := ?_mvar.61846hgnon:g' y ≠ 0⊢ Falserefine_4.refine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis:g x = 0hx:a < x ∧ x ≤ by:ℝhgy:HasDerivWithinAt g 0 (Set.Ioo a x) yhy:a < y ∧ y < x⊢ y ∈ Set.Icc a brefine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis:g x = 0hx:a < x ∧ x ≤ b⊢ Set.Icc a x ⊆ Set.Icc a brefine_2a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis:g x = 0hx:a < x ∧ x ≤ b⊢ Set.Ioo a x ⊆ Set.Icc a brefine_3a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis:g x = 0hx:a < x ∧ x ≤ b⊢ g a = g x
    rw [DifferentiableOn.eq_1] at hf hgrefine_4.refine_2a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ f (Set.Icc a b) xhg:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ g (Set.Icc a b) xhf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis✝:g x = 0hx:a < x ∧ x ≤ by:ℝhgy:HasDerivWithinAt g 0 (Set.Ioo a x) yhy:a < y ∧ y < xthis:y ∈ Set.Icc a bhgnon:g' y ≠ 0⊢ Falserefine_4.refine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis:g x = 0hx:a < x ∧ x ≤ by:ℝhgy:HasDerivWithinAt g 0 (Set.Ioo a x) yhy:a < y ∧ y < x⊢ y ∈ Set.Icc a brefine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis:g x = 0hx:a < x ∧ x ≤ b⊢ Set.Icc a x ⊆ Set.Icc a brefine_2a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis:g x = 0hx:a < x ∧ x ≤ b⊢ Set.Ioo a x ⊆ Set.Icc a brefine_3a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis:g x = 0hx:a < x ∧ x ≤ b⊢ g a = g x; specialize hg y thisrefine_4.refine_2a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ f (Set.Icc a b) xhf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis✝:g x = 0hx:a < x ∧ x ≤ by:ℝhgy:HasDerivWithinAt g 0 (Set.Ioo a x) yhy:a < y ∧ y < xthis:y ∈ Set.Icc a bhgnon:g' y ≠ 0hg:DifferentiableWithinAt ℝ g (Set.Icc a b) y⊢ Falserefine_4.refine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis:g x = 0hx:a < x ∧ x ≤ by:ℝhgy:HasDerivWithinAt g 0 (Set.Ioo a x) yhy:a < y ∧ y < x⊢ y ∈ Set.Icc a brefine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis:g x = 0hx:a < x ∧ x ≤ b⊢ Set.Icc a x ⊆ Set.Icc a brefine_2a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis:g x = 0hx:a < x ∧ x ≤ b⊢ Set.Ioo a x ⊆ Set.Icc a brefine_3a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis:g x = 0hx:a < x ∧ x ≤ b⊢ g a = g x
    replace hg : HasDerivWithinAt g (g' y) (.Ioo a x) y := bya:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)⊢ (∀ x ∈ Set.Ioc a b, g x ≠ 0) ∧ Filter.Tendsto (fun x => f x / g x) (nhdsWithin a (Set.Ioc a b)) (nhds L)
      rw [hg']a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ f (Set.Icc a b) xhf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis✝:g x = 0hx:a < x ∧ x ≤ by:ℝhgy:HasDerivWithinAt g 0 (Set.Ioo a x) yhy:a < y ∧ y < xthis:y ∈ Set.Icc a bhgnon:g' y ≠ 0hg:DifferentiableWithinAt ℝ g (Set.Icc a b) y⊢ HasDerivWithinAt g (derivWithin g (Set.Icc a b) y) (Set.Ioo a x) y; apply hg.hasDerivWithinAt.monoa:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ f (Set.Icc a b) xhf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis✝:g x = 0hx:a < x ∧ x ≤ by:ℝhgy:HasDerivWithinAt g 0 (Set.Ioo a x) yhy:a < y ∧ y < xthis:y ∈ Set.Icc a bhgnon:g' y ≠ 0hg:DifferentiableWithinAt ℝ g (Set.Icc a b) y⊢ Set.Ioo a x ⊆ Set.Icc a b; grindAll goals completed! 🐙
    have hd := derivative_unique ?_ hg hgyrefine_4.refine_2.refine_2a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ f (Set.Icc a b) xhf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis✝:g x = 0hx:a < x ∧ x ≤ by:ℝhgy:HasDerivWithinAt g 0 (Set.Ioo a x) yhy:a < y ∧ y < xthis:y ∈ Set.Icc a bhgnon:g' y ≠ 0hg:HasDerivWithinAt _fvar.4276 (@_fvar.4278 _fvar.62000) (Set.Ioo _fvar.4271 _fvar.61997) _fvar.62000 := ?_mvar.63115hd:?_mvar.66110 := Chapter10.derivative_unique ?_mvar.66113 _fvar.63116 _fvar.62001⊢ Falserefine_4.refine_2.refine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ f (Set.Icc a b) xhf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis✝:g x = 0hx:a < x ∧ x ≤ by:ℝhgy:HasDerivWithinAt g 0 (Set.Ioo a x) yhy:a < y ∧ y < xthis:y ∈ Set.Icc a bhgnon:g' y ≠ 0hg:HasDerivWithinAt _fvar.4276 (@_fvar.4278 _fvar.62000) (Set.Ioo _fvar.4271 _fvar.61997) _fvar.62000 := ?_mvar.63115⊢ ClusterPt y (Filter.principal (Set.Ioo a x \ {y}))refine_4.refine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis:g x = 0hx:a < x ∧ x ≤ by:ℝhgy:HasDerivWithinAt g 0 (Set.Ioo a x) yhy:a < y ∧ y < x⊢ y ∈ Set.Icc a brefine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis:g x = 0hx:a < x ∧ x ≤ b⊢ Set.Icc a x ⊆ Set.Icc a brefine_2a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis:g x = 0hx:a < x ∧ x ≤ b⊢ Set.Ioo a x ⊆ Set.Icc a brefine_3a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis:g x = 0hx:a < x ∧ x ≤ b⊢ g a = g x
    .refine_4.refine_2.refine_2a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ f (Set.Icc a b) xhf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis✝:g x = 0hx:a < x ∧ x ≤ by:ℝhgy:HasDerivWithinAt g 0 (Set.Ioo a x) yhy:a < y ∧ y < xthis:y ∈ Set.Icc a bhgnon:g' y ≠ 0hg:HasDerivWithinAt _fvar.4276 (@_fvar.4278 _fvar.62000) (Set.Ioo _fvar.4271 _fvar.61997) _fvar.62000 := ?_mvar.63115hd:?_mvar.66110 := Chapter10.derivative_unique ?_mvar.66113 _fvar.63116 _fvar.62001⊢ False contradictionAll goals completed! 🐙
    apply ClusterPt.mono _ ((Filter.principal_mono (s := .Ioo a y)).mpr  _)a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ f (Set.Icc a b) xhf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis✝:g x = 0hx:a < x ∧ x ≤ by:ℝhgy:HasDerivWithinAt g 0 (Set.Ioo a x) yhy:a < y ∧ y < xthis:y ∈ Set.Icc a bhgnon:g' y ≠ 0hg:HasDerivWithinAt _fvar.4276 (@_fvar.4278 _fvar.62000) (Set.Ioo _fvar.4271 _fvar.61997) _fvar.62000 := ?_mvar.63115⊢ ClusterPt y (Filter.principal (Set.Ioo a y))a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ f (Set.Icc a b) xhf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis✝:g x = 0hx:a < x ∧ x ≤ by:ℝhgy:HasDerivWithinAt g 0 (Set.Ioo a x) yhy:a < y ∧ y < xthis:y ∈ Set.Icc a bhgnon:g' y ≠ 0hg:HasDerivWithinAt _fvar.4276 (@_fvar.4278 _fvar.62000) (Set.Ioo _fvar.4271 _fvar.61997) _fvar.62000 := ?_mvar.63115⊢ Set.Ioo a y ⊆ Set.Ioo a x \ {y}refine_4.refine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis:g x = 0hx:a < x ∧ x ≤ by:ℝhgy:HasDerivWithinAt g 0 (Set.Ioo a x) yhy:a < y ∧ y < x⊢ y ∈ Set.Icc a brefine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis:g x = 0hx:a < x ∧ x ≤ b⊢ Set.Icc a x ⊆ Set.Icc a brefine_2a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis:g x = 0hx:a < x ∧ x ≤ b⊢ Set.Ioo a x ⊆ Set.Icc a brefine_3a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis:g x = 0hx:a < x ∧ x ≤ b⊢ g a = g x
    .a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ f (Set.Icc a b) xhf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)x:ℝthis✝:g x = 0hx:a < x ∧ x ≤ by:ℝhgy:HasDerivWithinAt g 0 (Set.Ioo a x) yhy:a < y ∧ y < xthis:y ∈ Set.Icc a bhgnon:g' y ≠ 0hg:HasDerivWithinAt _fvar.4276 (@_fvar.4278 _fvar.62000) (Set.Ioo _fvar.4271 _fvar.61997) _fvar.62000 := ?_mvar.63115⊢ ClusterPt y (Filter.principal (Set.Ioo a y)) simp [←mem_closure_iff_clusterPt, closure_Ioo (show a ≠ y by linarith), le_of_lt hy.1]All goals completed! 🐙
    all_goals grindAll goals completed! 🐙
  refine ⟨ this, ?_ ⟩a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hx⊢ Filter.Tendsto (fun x => f x / g x) (nhdsWithin a (Set.Ioc a b)) (nhds L)
  rw [←Convergesto.iff, Convergesto.iff_conv]a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hx⊢ ∀ (a_1 : ℕ → ℝ),
  (∀ (n : ℕ), a_1 n ∈ Set.Ioc a b) →
    Filter.Tendsto a_1 Filter.atTop (nhds a) → Filter.Tendsto (fun n => f (a_1 n) / g (a_1 n)) Filter.atTop (nhds L)ha:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hx⊢ AdherentPt a (Set.Ioc a b)
  .a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hx⊢ ∀ (a_1 : ℕ → ℝ),
  (∀ (n : ℕ), a_1 n ∈ Set.Ioc a b) →
    Filter.Tendsto a_1 Filter.atTop (nhds a) → Filter.Tendsto (fun n => f (a_1 n) / g (a_1 n)) Filter.atTop (nhds L) intro x hx hconva:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhx:∀ (n : ℕ), x n ∈ Set.Ioc a bhconv:Filter.Tendsto x Filter.atTop (nhds a)⊢ Filter.Tendsto (fun n => f (x n) / g (x n)) Filter.atTop (nhds L)
    have hxy (n:ℕ) : ∃ yn ∈ Set.Ioo a (x n), (f (x n))/(g (x n)) = f' yn / (g' yn) := bya:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)⊢ (∀ x ∈ Set.Ioc a b, g x ≠ 0) ∧ Filter.Tendsto (fun x => f x / g x) (nhdsWithin a (Set.Ioc a b)) (nhds L)
      set h : ℝ → ℝ := fun x' ↦ (f x') * (g (x n)) - (g x') * (f (x n))a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhx:∀ (n : ℕ), x n ∈ Set.Ioc a bhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)⊢ ∃ yn ∈ Set.Ioo a (x n), f (x n) / g (x n) = f' yn / g' yn
      have hdiff : DifferentiableOn ℝ h (.Icc a b) := bya:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)⊢ (∀ x ∈ Set.Ioc a b, g x ≠ 0) ∧ Filter.Tendsto (fun x => f x / g x) (nhdsWithin a (Set.Ioc a b)) (nhds L) fun_propAll goals completed! 🐙
      observe hcon : ContinuousOn h (.Icc a b)a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhx:∀ (n : ℕ), x n ∈ Set.Ioc a bhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Icc _fvar.4271 _fvar.4272) := ?_mvar.84998hcon:ContinuousOn h (Set.Icc a b)⊢ ∃ yn ∈ Set.Ioo a (x n), f (x n) / g (x n) = f' yn / g' yn
      specialize hx na:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Icc _fvar.4271 _fvar.4272) := ?_mvar.84998hcon:ContinuousOn h (Set.Icc a b)hx:x n ∈ Set.Ioc a b⊢ ∃ yn ∈ Set.Ioo a (x n), f (x n) / g (x n) = f' yn / g' yn; simp at hxa:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Icc _fvar.4271 _fvar.4272) := ?_mvar.84998hcon:ContinuousOn h (Set.Icc a b)hx:a < x n ∧ x n ≤ b⊢ ∃ yn ∈ Set.Ioo a (x n), f (x n) / g (x n) = f' yn / g' yn
      replace hcon : ContinuousOn h (.Icc a (x n)) := hcon.mono ?_refine_2a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Icc _fvar.4271 _fvar.4272) := ?_mvar.84998hx:a < x n ∧ x n ≤ bhcon:ContinuousOn _fvar.84237 (Set.Icc _fvar.4271 (@_fvar.83912 _fvar.83923)) := ContinuousOn.mono _fvar.88991 ?_mvar.91263⊢ ∃ yn ∈ Set.Ioo a (x n), f (x n) / g (x n) = f' yn / g' ynrefine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Icc _fvar.4271 _fvar.4272) := ?_mvar.84998hcon:ContinuousOn h (Set.Icc a b)hx:a < x n ∧ x n ≤ b⊢ Set.Icc a (x n) ⊆ Set.Icc a b
      replace hdiff : DifferentiableOn ℝ h (.Ioo a (x n)) := hdiff.mono ?_refine_2.refine_2a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn _fvar.84237 (Set.Icc _fvar.4271 (@_fvar.83912 _fvar.83923)) := ContinuousOn.mono _fvar.88991 ?_mvar.91263hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Ioo _fvar.4271 (@_fvar.83912 _fvar.83923)) := DifferentiableOn.mono _fvar.84999 ?_mvar.91385⊢ ∃ yn ∈ Set.Ioo a (x n), f (x n) / g (x n) = f' yn / g' ynrefine_2.refine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Icc _fvar.4271 _fvar.4272) := ?_mvar.84998hx:a < x n ∧ x n ≤ bhcon:ContinuousOn _fvar.84237 (Set.Icc _fvar.4271 (@_fvar.83912 _fvar.83923)) := ContinuousOn.mono _fvar.88991 ?_mvar.91263⊢ Set.Ioo a (x n) ⊆ Set.Icc a brefine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Icc _fvar.4271 _fvar.4272) := ?_mvar.84998hcon:ContinuousOn h (Set.Icc a b)hx:a < x n ∧ x n ≤ b⊢ Set.Icc a (x n) ⊆ Set.Icc a b
      have ha : h a = 0 := ?_refine_2.refine_2.refine_2a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn _fvar.84237 (Set.Icc _fvar.4271 (@_fvar.83912 _fvar.83923)) := ContinuousOn.mono _fvar.88991 ?_mvar.91263hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Ioo _fvar.4271 (@_fvar.83912 _fvar.83923)) := DifferentiableOn.mono _fvar.84999 ?_mvar.91385ha:@_fvar.84237 _fvar.4271 = 0 := ?_mvar.91457⊢ ∃ yn ∈ Set.Ioo a (x n), f (x n) / g (x n) = f' yn / g' ynrefine_2.refine_2.refine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn _fvar.84237 (Set.Icc _fvar.4271 (@_fvar.83912 _fvar.83923)) := ContinuousOn.mono _fvar.88991 ?_mvar.91263hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Ioo _fvar.4271 (@_fvar.83912 _fvar.83923)) := DifferentiableOn.mono _fvar.84999 ?_mvar.91385⊢ h a = 0refine_2.refine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Icc _fvar.4271 _fvar.4272) := ?_mvar.84998hx:a < x n ∧ x n ≤ bhcon:ContinuousOn _fvar.84237 (Set.Icc _fvar.4271 (@_fvar.83912 _fvar.83923)) := ContinuousOn.mono _fvar.88991 ?_mvar.91263⊢ Set.Ioo a (x n) ⊆ Set.Icc a brefine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Icc _fvar.4271 _fvar.4272) := ?_mvar.84998hcon:ContinuousOn h (Set.Icc a b)hx:a < x n ∧ x n ≤ b⊢ Set.Icc a (x n) ⊆ Set.Icc a b
      have hb : h (x n) = 0 := ?_refine_2.refine_2.refine_2.refine_2a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn _fvar.84237 (Set.Icc _fvar.4271 (@_fvar.83912 _fvar.83923)) := ContinuousOn.mono _fvar.88991 ?_mvar.91263hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Ioo _fvar.4271 (@_fvar.83912 _fvar.83923)) := DifferentiableOn.mono _fvar.84999 ?_mvar.91385ha:@_fvar.84237 _fvar.4271 = 0 := ?_mvar.91457hb:@_fvar.84237 (@_fvar.83912 _fvar.83923) = 0 := ?_mvar.91489⊢ ∃ yn ∈ Set.Ioo a (x n), f (x n) / g (x n) = f' yn / g' ynrefine_2.refine_2.refine_2.refine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn _fvar.84237 (Set.Icc _fvar.4271 (@_fvar.83912 _fvar.83923)) := ContinuousOn.mono _fvar.88991 ?_mvar.91263hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Ioo _fvar.4271 (@_fvar.83912 _fvar.83923)) := DifferentiableOn.mono _fvar.84999 ?_mvar.91385ha:@_fvar.84237 _fvar.4271 = 0 := ?_mvar.91457⊢ h (x n) = 0refine_2.refine_2.refine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn _fvar.84237 (Set.Icc _fvar.4271 (@_fvar.83912 _fvar.83923)) := ContinuousOn.mono _fvar.88991 ?_mvar.91263hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Ioo _fvar.4271 (@_fvar.83912 _fvar.83923)) := DifferentiableOn.mono _fvar.84999 ?_mvar.91385⊢ h a = 0refine_2.refine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Icc _fvar.4271 _fvar.4272) := ?_mvar.84998hx:a < x n ∧ x n ≤ bhcon:ContinuousOn _fvar.84237 (Set.Icc _fvar.4271 (@_fvar.83912 _fvar.83923)) := ContinuousOn.mono _fvar.88991 ?_mvar.91263⊢ Set.Ioo a (x n) ⊆ Set.Icc a brefine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Icc _fvar.4271 _fvar.4272) := ?_mvar.84998hcon:ContinuousOn h (Set.Icc a b)hx:a < x n ∧ x n ≤ b⊢ Set.Icc a (x n) ⊆ Set.Icc a b
      choose yn hyn hdh using HasDerivWithinAt.exist_zero hx.1 hcon hdiff (bya:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := 
  fun x hx this =>
    have this_1 :=
      HasDerivWithinAt.exist_zero (Eq.mp Set.mem_Ioc._simp_1 hx).left
        (ContinuousOn.mono _fvar.57488
          (Filter.Tendsto.of_div'._proof_5 _fvar.4286 _fvar.55943 _fvar.57488 x this (Eq.mp Set.mem_Ioc._simp_1 hx)))
        (DifferentiableOn.mono _fvar.4280
          (Filter.Tendsto.of_div'._proof_6 _fvar.4286 _fvar.55943 _fvar.57488 x this (Eq.mp Set.mem_Ioc._simp_1 hx)))
        (Filter.Tendsto.of_div'._proof_7 _fvar.4284 _fvar.4286 _fvar.55943 _fvar.57488 x this
          (Eq.mp Set.mem_Ioc._simp_1 hx));
    (fun y x_1 =>
        have this_2 :=
          Filter.Tendsto.of_div'._proof_4 _fvar.4286 _fvar.55943 _fvar.57488 x this (Eq.mp Set.mem_Ioc._simp_1 hx) y
            x_1.right (Eq.mp Set.mem_Ioo._simp_1 x_1.left);
        have hg :=
          Eq.mpr (id (congrArg (fun _a => HasDerivWithinAt _fvar.4276 (_a y) (Set.Ioo _fvar.4271 x) y) _fvar.4282))
            (HasDerivWithinAt.mono
              (DifferentiableWithinAt.hasDerivWithinAt
                (Eq.mp (congrArg (fun _a => _a) (DifferentiableOn.eq_1 ℝ _fvar.4276 (Set.Icc _fvar.4271 _fvar.4272)))
                  _fvar.4280 y this_2))
              (Filter.Tendsto.of_div'._proof_2 _fvar.4286 _fvar.55943 _fvar.57488 x this (Eq.mp Set.mem_Ioc._simp_1 hx) y
                x_1.right (Eq.mp Set.mem_Ioo._simp_1 x_1.left) this_2 (@_fvar.4285 y this_2)
                (Eq.mp (congrArg (fun _a => _a) (DifferentiableOn.eq_1 ℝ _fvar.4276 (Set.Icc _fvar.4271 _fvar.4272)))
                  _fvar.4280 y this_2)));
        have hd :=
          Chapter10.derivative_unique
            (ClusterPt.mono
              (of_eq_true
                (Eq.trans Filter.Tendsto.of_div'._simp_2
                  (Eq.trans
                    (Eq.trans
                      (congrArg (fun x => y ∈ x)
                        (closure_Ioo
                          (have this :=
                            Not.intro fun a =>
                              Mathlib.Tactic.Linarith.lt_irrefl
                                (Eq.mp
                                  (congrArg (fun _a => _a < 0)
                                    (Mathlib.Tactic.Ring.of_eq
                                      (Mathlib.Tactic.Ring.add_congr
                                        (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.4271)
                                          (Mathlib.Tactic.Ring.atom_pf y)
                                          (Mathlib.Tactic.Ring.sub_pf
                                            (Mathlib.Tactic.Ring.neg_add
                                              (Mathlib.Tactic.Ring.neg_mul y (Nat.rawCast 1)
                                                (Mathlib.Tactic.Ring.neg_one_mul
                                                  (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                                                    (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                                      (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                                                      (Mathlib.Meta.NormNum.IsNat.to_isInt
                                                        (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                                                      (Eq.refl (Int.negOfNat 1))))))
                                              Mathlib.Tactic.Ring.neg_zero)
                                            (Mathlib.Tactic.Ring.add_pf_add_lt
                                              (_fvar.4271 ^ Nat.rawCast 1 * Nat.rawCast 1)
                                              (Mathlib.Tactic.Ring.add_pf_zero_add
                                                (y ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                                        (Mathlib.Tactic.Ring.neg_congr
                                          (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.4271)
                                            (Mathlib.Tactic.Ring.atom_pf y)
                                            (Mathlib.Tactic.Ring.sub_pf
                                              (Mathlib.Tactic.Ring.neg_add
                                                (Mathlib.Tactic.Ring.neg_mul y (Nat.rawCast 1)
                                                  (Mathlib.Tactic.Ring.neg_one_mul
                                                    (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                                                      (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                                                        (Mathlib.Meta.NormNum.IsNat.to_isInt
                                                          (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                                                        (Eq.refl (Int.negOfNat 1))))))
                                                Mathlib.Tactic.Ring.neg_zero)
                                              (Mathlib.Tactic.Ring.add_pf_add_lt
                                                (_fvar.4271 ^ Nat.rawCast 1 * Nat.rawCast 1)
                                                (Mathlib.Tactic.Ring.add_pf_zero_add
                                                  (y ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                                          (Mathlib.Tactic.Ring.neg_add
                                            (Mathlib.Tactic.Ring.neg_mul _fvar.4271 (Nat.rawCast 1)
                                              (Mathlib.Tactic.Ring.neg_one_mul
                                                (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                                                  (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                                                    (Mathlib.Meta.NormNum.IsNat.to_isInt
                                                      (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                                                    (Eq.refl (Int.negOfNat 1))))))
                                            (Mathlib.Tactic.Ring.neg_add
                                              (Mathlib.Tactic.Ring.neg_mul y (Nat.rawCast 1)
                                                (Mathlib.Tactic.Ring.neg_one_mul
                                                  (Mathlib.Meta.NormNum.IsNat.to_raw_eq
                                                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                                                      (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                                                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                                                        (Eq.refl (Int.ofNat 1)))))))
                                              Mathlib.Tactic.Ring.neg_zero)))
                                        (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                                          (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.4271 (Nat.rawCast 1)
                                            (Mathlib.Meta.NormNum.IsInt.to_isNat
                                              (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                                                (Mathlib.Meta.NormNum.IsNat.to_isInt
                                                  (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                                                (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                                                (Eq.refl (Int.ofNat 0)))))
                                          (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                                            (Mathlib.Tactic.Ring.add_overlap_pf_zero y (Nat.rawCast 1)
                                              (Mathlib.Meta.NormNum.IsInt.to_isNat
                                                (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                                                  (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                                                  (Mathlib.Meta.NormNum.IsNat.to_isInt
                                                    (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                                                  (Eq.refl (Int.ofNat 0)))))
                                            (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
                                      (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
                                  (Mathlib.Tactic.Linarith.lt_of_lt_of_eq
                                    (Mathlib.Tactic.Linarith.sub_neg_of_lt (Eq.mp Set.mem_Ioo._simp_1 x_1.left).left)
                                    (neg_eq_zero.mpr (sub_eq_zero_of_eq a))));
                          this)))
                      Set.mem_Icc._simp_1)
                    (Eq.trans
                      (congr (congrArg And (eq_true (le_of_lt (Eq.mp Set.mem_Ioo._simp_1 x_1.left).left)))
                        (le_refl._simp_1 y))
                      (and_self True)))))
              (Filter.principal_mono.mpr
                (Filter.Tendsto.of_div'._proof_3 _fvar.4286 _fvar.55943 _fvar.57488 x this (Eq.mp Set.mem_Ioc._simp_1 hx)
                  y x_1.right (Eq.mp Set.mem_Ioo._simp_1 x_1.left) this_2 (@_fvar.4285 y this_2) hg)))
            hg x_1.right;
        False.elim (@_fvar.4285 y this_2 hd))
      (Classical.choose this_1) (Classical.choose_spec this_1)x:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn _fvar.84237 (Set.Icc _fvar.4271 (@_fvar.83912 _fvar.83923)) := ContinuousOn.mono _fvar.88991 ?_mvar.91263hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Ioo _fvar.4271 (@_fvar.83912 _fvar.83923)) := DifferentiableOn.mono _fvar.84999 ?_mvar.91385ha:@_fvar.84237 _fvar.4271 = 0 := ?_mvar.91457hb:@_fvar.84237 (@_fvar.83912 _fvar.83923) = 0 := ?_mvar.91489⊢ h a = h (x n) grindAll goals completed! 🐙)
      use yn, hynrighta:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn _fvar.84237 (Set.Icc _fvar.4271 (@_fvar.83912 _fvar.83923)) := ContinuousOn.mono _fvar.88991 ?_mvar.91263hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Ioo _fvar.4271 (@_fvar.83912 _fvar.83923)) := DifferentiableOn.mono _fvar.84999 ?_mvar.91385ha:@_fvar.84237 _fvar.4271 = 0 := ?_mvar.91457hb:@_fvar.84237 (@_fvar.83912 _fvar.83923) = 0 := ?_mvar.91489yn:ℝhyn:yn ∈ Set.Ioo a (x n)hdh:HasDerivWithinAt h 0 (Set.Ioo a (x n)) yn⊢ f (x n) / g (x n) = f' yn / g' ynrefine_2.refine_2.refine_2.refine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn _fvar.84237 (Set.Icc _fvar.4271 (@_fvar.83912 _fvar.83923)) := ContinuousOn.mono _fvar.88991 ?_mvar.91263hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Ioo _fvar.4271 (@_fvar.83912 _fvar.83923)) := DifferentiableOn.mono _fvar.84999 ?_mvar.91385ha:@_fvar.84237 _fvar.4271 = 0 := ?_mvar.91457⊢ h (x n) = 0refine_2.refine_2.refine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn _fvar.84237 (Set.Icc _fvar.4271 (@_fvar.83912 _fvar.83923)) := ContinuousOn.mono _fvar.88991 ?_mvar.91263hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Ioo _fvar.4271 (@_fvar.83912 _fvar.83923)) := DifferentiableOn.mono _fvar.84999 ?_mvar.91385⊢ h a = 0refine_2.refine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Icc _fvar.4271 _fvar.4272) := ?_mvar.84998hx:a < x n ∧ x n ≤ bhcon:ContinuousOn _fvar.84237 (Set.Icc _fvar.4271 (@_fvar.83912 _fvar.83923)) := ContinuousOn.mono _fvar.88991 ?_mvar.91263⊢ Set.Ioo a (x n) ⊆ Set.Icc a brefine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Icc _fvar.4271 _fvar.4272) := ?_mvar.84998hcon:ContinuousOn h (Set.Icc a b)hx:a < x n ∧ x n ≤ b⊢ Set.Icc a (x n) ⊆ Set.Icc a b
      rw [DifferentiableOn.eq_1] at hf hgrighta:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ f (Set.Icc a b) xhg:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ g (Set.Icc a b) xhf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc a b, g x ≠ 0x:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.94246 _fvar.94248) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.94246 _fvar.94248)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn h (Set.Icc a (x n))hdiff:DifferentiableOn ℝ h (Set.Ioo a (x n))ha:h a = 0hb:h (x n) = 0yn:ℝhyn:yn ∈ Set.Ioo a (x n)hdh:HasDerivWithinAt h 0 (Set.Ioo a (x n)) yn⊢ f (x n) / g (x n) = f' yn / g' ynrefine_2.refine_2.refine_2.refine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn _fvar.84237 (Set.Icc _fvar.4271 (@_fvar.83912 _fvar.83923)) := ContinuousOn.mono _fvar.88991 ?_mvar.91263hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Ioo _fvar.4271 (@_fvar.83912 _fvar.83923)) := DifferentiableOn.mono _fvar.84999 ?_mvar.91385ha:@_fvar.84237 _fvar.4271 = 0 := ?_mvar.91457⊢ h (x n) = 0refine_2.refine_2.refine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn _fvar.84237 (Set.Icc _fvar.4271 (@_fvar.83912 _fvar.83923)) := ContinuousOn.mono _fvar.88991 ?_mvar.91263hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Ioo _fvar.4271 (@_fvar.83912 _fvar.83923)) := DifferentiableOn.mono _fvar.84999 ?_mvar.91385⊢ h a = 0refine_2.refine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Icc _fvar.4271 _fvar.4272) := ?_mvar.84998hx:a < x n ∧ x n ≤ bhcon:ContinuousOn _fvar.84237 (Set.Icc _fvar.4271 (@_fvar.83912 _fvar.83923)) := ContinuousOn.mono _fvar.88991 ?_mvar.91263⊢ Set.Ioo a (x n) ⊆ Set.Icc a brefine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Icc _fvar.4271 _fvar.4272) := ?_mvar.84998hcon:ContinuousOn h (Set.Icc a b)hx:a < x n ∧ x n ≤ b⊢ Set.Icc a (x n) ⊆ Set.Icc a b
      have h1 : HasDerivWithinAt f (f' yn) (.Ioo a (x n)) yn := bya:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)⊢ (∀ x ∈ Set.Ioc a b, g x ≠ 0) ∧ Filter.Tendsto (fun x => f x / g x) (nhdsWithin a (Set.Ioc a b)) (nhds L)
        rw [hf']a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ f (Set.Icc a b) xhg:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ g (Set.Icc a b) xhf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc a b, g x ≠ 0x:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.94246 _fvar.94248) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.94246 _fvar.94248)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn h (Set.Icc a (x n))hdiff:DifferentiableOn ℝ h (Set.Ioo a (x n))ha:h a = 0hb:h (x n) = 0yn:ℝhyn:yn ∈ Set.Ioo a (x n)hdh:HasDerivWithinAt h 0 (Set.Ioo a (x n)) yn⊢ HasDerivWithinAt f (derivWithin f (Set.Icc a b) yn) (Set.Ioo a (x n)) yn; apply (hf _ _).hasDerivWithinAt.monoa:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ f (Set.Icc a b) xhg:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ g (Set.Icc a b) xhf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc a b, g x ≠ 0x:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.94246 _fvar.94248) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.94246 _fvar.94248)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn h (Set.Icc a (x n))hdiff:DifferentiableOn ℝ h (Set.Ioo a (x n))ha:h a = 0hb:h (x n) = 0yn:ℝhyn:yn ∈ Set.Ioo a (x n)hdh:HasDerivWithinAt h 0 (Set.Ioo a (x n)) yn⊢ Set.Ioo a (x n) ⊆ Set.Icc a ba:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ f (Set.Icc a b) xhg:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ g (Set.Icc a b) xhf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc a b, g x ≠ 0x:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.94246 _fvar.94248) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.94246 _fvar.94248)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn h (Set.Icc a (x n))hdiff:DifferentiableOn ℝ h (Set.Ioo a (x n))ha:h a = 0hb:h (x n) = 0yn:ℝhyn:yn ∈ Set.Ioo a (x n)hdh:HasDerivWithinAt h 0 (Set.Ioo a (x n)) yn⊢ yn ∈ Set.Icc a b <;>a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ f (Set.Icc a b) xhg:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ g (Set.Icc a b) xhf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc a b, g x ≠ 0x:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.94246 _fvar.94248) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.94246 _fvar.94248)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn h (Set.Icc a (x n))hdiff:DifferentiableOn ℝ h (Set.Ioo a (x n))ha:h a = 0hb:h (x n) = 0yn:ℝhyn:yn ∈ Set.Ioo a (x n)hdh:HasDerivWithinAt h 0 (Set.Ioo a (x n)) yn⊢ Set.Ioo a (x n) ⊆ Set.Icc a ba:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ f (Set.Icc a b) xhg:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ g (Set.Icc a b) xhf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc a b, g x ≠ 0x:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.94246 _fvar.94248) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.94246 _fvar.94248)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn h (Set.Icc a (x n))hdiff:DifferentiableOn ℝ h (Set.Ioo a (x n))ha:h a = 0hb:h (x n) = 0yn:ℝhyn:yn ∈ Set.Ioo a (x n)hdh:HasDerivWithinAt h 0 (Set.Ioo a (x n)) yn⊢ yn ∈ Set.Icc a b grindAll goals completed! 🐙
      have h2 : HasDerivWithinAt g (g' yn) (.Ioo a (x n)) yn := bya:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)⊢ (∀ x ∈ Set.Ioc a b, g x ≠ 0) ∧ Filter.Tendsto (fun x => f x / g x) (nhdsWithin a (Set.Ioc a b)) (nhds L)
        rw [hg']a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ f (Set.Icc a b) xhg:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ g (Set.Icc a b) xhf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc a b, g x ≠ 0x:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.94246 _fvar.94248) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.94246 _fvar.94248)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn h (Set.Icc a (x n))hdiff:DifferentiableOn ℝ h (Set.Ioo a (x n))ha:h a = 0hb:h (x n) = 0yn:ℝhyn:yn ∈ Set.Ioo a (x n)hdh:HasDerivWithinAt h 0 (Set.Ioo a (x n)) ynh1:HasDerivWithinAt _fvar.4275 (@_fvar.4277 _fvar.94255) (Set.Ioo _fvar.4271 (@_fvar.94246 _fvar.94248)) _fvar.94255 := ?_mvar.95497⊢ HasDerivWithinAt g (derivWithin g (Set.Icc a b) yn) (Set.Ioo a (x n)) yn; apply (hg _ _).hasDerivWithinAt.monoa:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ f (Set.Icc a b) xhg:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ g (Set.Icc a b) xhf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc a b, g x ≠ 0x:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.94246 _fvar.94248) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.94246 _fvar.94248)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn h (Set.Icc a (x n))hdiff:DifferentiableOn ℝ h (Set.Ioo a (x n))ha:h a = 0hb:h (x n) = 0yn:ℝhyn:yn ∈ Set.Ioo a (x n)hdh:HasDerivWithinAt h 0 (Set.Ioo a (x n)) ynh1:HasDerivWithinAt _fvar.4275 (@_fvar.4277 _fvar.94255) (Set.Ioo _fvar.4271 (@_fvar.94246 _fvar.94248)) _fvar.94255 := ?_mvar.95497⊢ Set.Ioo a (x n) ⊆ Set.Icc a ba:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ f (Set.Icc a b) xhg:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ g (Set.Icc a b) xhf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc a b, g x ≠ 0x:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.94246 _fvar.94248) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.94246 _fvar.94248)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn h (Set.Icc a (x n))hdiff:DifferentiableOn ℝ h (Set.Ioo a (x n))ha:h a = 0hb:h (x n) = 0yn:ℝhyn:yn ∈ Set.Ioo a (x n)hdh:HasDerivWithinAt h 0 (Set.Ioo a (x n)) ynh1:HasDerivWithinAt _fvar.4275 (@_fvar.4277 _fvar.94255) (Set.Ioo _fvar.4271 (@_fvar.94246 _fvar.94248)) _fvar.94255 := ?_mvar.95497⊢ yn ∈ Set.Icc a b <;>a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ f (Set.Icc a b) xhg:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ g (Set.Icc a b) xhf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc a b, g x ≠ 0x:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.94246 _fvar.94248) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.94246 _fvar.94248)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn h (Set.Icc a (x n))hdiff:DifferentiableOn ℝ h (Set.Ioo a (x n))ha:h a = 0hb:h (x n) = 0yn:ℝhyn:yn ∈ Set.Ioo a (x n)hdh:HasDerivWithinAt h 0 (Set.Ioo a (x n)) ynh1:HasDerivWithinAt _fvar.4275 (@_fvar.4277 _fvar.94255) (Set.Ioo _fvar.4271 (@_fvar.94246 _fvar.94248)) _fvar.94255 := ?_mvar.95497⊢ Set.Ioo a (x n) ⊆ Set.Icc a ba:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ f (Set.Icc a b) xhg:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ g (Set.Icc a b) xhf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc a b, g x ≠ 0x:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.94246 _fvar.94248) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.94246 _fvar.94248)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn h (Set.Icc a (x n))hdiff:DifferentiableOn ℝ h (Set.Ioo a (x n))ha:h a = 0hb:h (x n) = 0yn:ℝhyn:yn ∈ Set.Ioo a (x n)hdh:HasDerivWithinAt h 0 (Set.Ioo a (x n)) ynh1:HasDerivWithinAt _fvar.4275 (@_fvar.4277 _fvar.94255) (Set.Ioo _fvar.4271 (@_fvar.94246 _fvar.94248)) _fvar.94255 := ?_mvar.95497⊢ yn ∈ Set.Icc a b grindAll goals completed! 🐙
      observe h3 : HasDerivWithinAt (fun x' ↦ (f x') * (g (x n))) ((f' yn)*(g (x n))) (.Ioo a (x n)) ynrighta:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ f (Set.Icc a b) xhg:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ g (Set.Icc a b) xhf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc a b, g x ≠ 0x:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.94246 _fvar.94248) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.94246 _fvar.94248)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn h (Set.Icc a (x n))hdiff:DifferentiableOn ℝ h (Set.Ioo a (x n))ha:h a = 0hb:h (x n) = 0yn:ℝhyn:yn ∈ Set.Ioo a (x n)hdh:HasDerivWithinAt h 0 (Set.Ioo a (x n)) ynh1:HasDerivWithinAt _fvar.4275 (@_fvar.4277 _fvar.94255) (Set.Ioo _fvar.4271 (@_fvar.94246 _fvar.94248)) _fvar.94255 := ?_mvar.95497h2:HasDerivWithinAt _fvar.4276 (@_fvar.4278 _fvar.94255) (Set.Ioo _fvar.4271 (@_fvar.94246 _fvar.94248)) _fvar.94255 := ?_mvar.103370h3:HasDerivWithinAt (fun y => f y * g (x n)) (f' yn * g (x n)) (Set.Ioo a (x n)) yn⊢ f (x n) / g (x n) = f' yn / g' ynrefine_2.refine_2.refine_2.refine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn _fvar.84237 (Set.Icc _fvar.4271 (@_fvar.83912 _fvar.83923)) := ContinuousOn.mono _fvar.88991 ?_mvar.91263hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Ioo _fvar.4271 (@_fvar.83912 _fvar.83923)) := DifferentiableOn.mono _fvar.84999 ?_mvar.91385ha:@_fvar.84237 _fvar.4271 = 0 := ?_mvar.91457⊢ h (x n) = 0refine_2.refine_2.refine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn _fvar.84237 (Set.Icc _fvar.4271 (@_fvar.83912 _fvar.83923)) := ContinuousOn.mono _fvar.88991 ?_mvar.91263hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Ioo _fvar.4271 (@_fvar.83912 _fvar.83923)) := DifferentiableOn.mono _fvar.84999 ?_mvar.91385⊢ h a = 0refine_2.refine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Icc _fvar.4271 _fvar.4272) := ?_mvar.84998hx:a < x n ∧ x n ≤ bhcon:ContinuousOn _fvar.84237 (Set.Icc _fvar.4271 (@_fvar.83912 _fvar.83923)) := ContinuousOn.mono _fvar.88991 ?_mvar.91263⊢ Set.Ioo a (x n) ⊆ Set.Icc a brefine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Icc _fvar.4271 _fvar.4272) := ?_mvar.84998hcon:ContinuousOn h (Set.Icc a b)hx:a < x n ∧ x n ≤ b⊢ Set.Icc a (x n) ⊆ Set.Icc a b
      observe h4 : HasDerivWithinAt (fun x' ↦ (g x') * (f (x n))) ((g' yn)*(f (x n))) (.Ioo a (x n)) ynrighta:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ f (Set.Icc a b) xhg:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ g (Set.Icc a b) xhf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc a b, g x ≠ 0x:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.94246 _fvar.94248) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.94246 _fvar.94248)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn h (Set.Icc a (x n))hdiff:DifferentiableOn ℝ h (Set.Ioo a (x n))ha:h a = 0hb:h (x n) = 0yn:ℝhyn:yn ∈ Set.Ioo a (x n)hdh:HasDerivWithinAt h 0 (Set.Ioo a (x n)) ynh1:HasDerivWithinAt _fvar.4275 (@_fvar.4277 _fvar.94255) (Set.Ioo _fvar.4271 (@_fvar.94246 _fvar.94248)) _fvar.94255 := ?_mvar.95497h2:HasDerivWithinAt _fvar.4276 (@_fvar.4278 _fvar.94255) (Set.Ioo _fvar.4271 (@_fvar.94246 _fvar.94248)) _fvar.94255 := ?_mvar.103370h3:HasDerivWithinAt (fun y => f y * g (x n)) (f' yn * g (x n)) (Set.Ioo a (x n)) ynh4:HasDerivWithinAt (fun y => g y * f (x n)) (g' yn * f (x n)) (Set.Ioo a (x n)) yn⊢ f (x n) / g (x n) = f' yn / g' ynrefine_2.refine_2.refine_2.refine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn _fvar.84237 (Set.Icc _fvar.4271 (@_fvar.83912 _fvar.83923)) := ContinuousOn.mono _fvar.88991 ?_mvar.91263hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Ioo _fvar.4271 (@_fvar.83912 _fvar.83923)) := DifferentiableOn.mono _fvar.84999 ?_mvar.91385ha:@_fvar.84237 _fvar.4271 = 0 := ?_mvar.91457⊢ h (x n) = 0refine_2.refine_2.refine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn _fvar.84237 (Set.Icc _fvar.4271 (@_fvar.83912 _fvar.83923)) := ContinuousOn.mono _fvar.88991 ?_mvar.91263hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Ioo _fvar.4271 (@_fvar.83912 _fvar.83923)) := DifferentiableOn.mono _fvar.84999 ?_mvar.91385⊢ h a = 0refine_2.refine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Icc _fvar.4271 _fvar.4272) := ?_mvar.84998hx:a < x n ∧ x n ≤ bhcon:ContinuousOn _fvar.84237 (Set.Icc _fvar.4271 (@_fvar.83912 _fvar.83923)) := ContinuousOn.mono _fvar.88991 ?_mvar.91263⊢ Set.Ioo a (x n) ⊆ Set.Icc a brefine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Icc _fvar.4271 _fvar.4272) := ?_mvar.84998hcon:ContinuousOn h (Set.Icc a b)hx:a < x n ∧ x n ≤ b⊢ Set.Icc a (x n) ⊆ Set.Icc a b
      have h5 : HasDerivWithinAt h (f' yn * g (x n) - g' yn * f (x n)) (.Ioo a (x n)) yn := bya:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)⊢ (∀ x ∈ Set.Ioc a b, g x ≠ 0) ∧ Filter.Tendsto (fun x => f x / g x) (nhdsWithin a (Set.Ioc a b)) (nhds L)
        simp [h]a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ f (Set.Icc a b) xhg:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ g (Set.Icc a b) xhf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc a b, g x ≠ 0x:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.94246 _fvar.94248) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.94246 _fvar.94248)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn h (Set.Icc a (x n))hdiff:DifferentiableOn ℝ h (Set.Ioo a (x n))ha:h a = 0hb:h (x n) = 0yn:ℝhyn:yn ∈ Set.Ioo a (x n)hdh:HasDerivWithinAt h 0 (Set.Ioo a (x n)) ynh1:HasDerivWithinAt _fvar.4275 (@_fvar.4277 _fvar.94255) (Set.Ioo _fvar.4271 (@_fvar.94246 _fvar.94248)) _fvar.94255 := ?_mvar.95497h2:HasDerivWithinAt _fvar.4276 (@_fvar.4278 _fvar.94255) (Set.Ioo _fvar.4271 (@_fvar.94246 _fvar.94248)) _fvar.94255 := ?_mvar.103370h3:HasDerivWithinAt (fun y => f y * g (x n)) (f' yn * g (x n)) (Set.Ioo a (x n)) ynh4:HasDerivWithinAt (fun y => g y * f (x n)) (g' yn * f (x n)) (Set.Ioo a (x n)) yn⊢ HasDerivWithinAt (fun x' => f x' * g (x n) - g x' * f (x n)) (f' yn * g (x n) - g' yn * f (x n)) (Set.Ioo a (x n)) yn; exact h3.sub h4All goals completed! 🐙
      have h6 : f' yn * g (x n) - g' yn * f (x n) = 0 := bya:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)⊢ (∀ x ∈ Set.Ioc a b, g x ≠ 0) ∧ Filter.Tendsto (fun x => f x / g x) (nhdsWithin a (Set.Ioc a b)) (nhds L)
        apply derivative_unique _ h5 hdha:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ f (Set.Icc a b) xhg:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ g (Set.Icc a b) xhf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc a b, g x ≠ 0x:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.94246 _fvar.94248) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.94246 _fvar.94248)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn h (Set.Icc a (x n))hdiff:DifferentiableOn ℝ h (Set.Ioo a (x n))ha:h a = 0hb:h (x n) = 0yn:ℝhyn:yn ∈ Set.Ioo a (x n)hdh:HasDerivWithinAt h 0 (Set.Ioo a (x n)) ynh1:HasDerivWithinAt _fvar.4275 (@_fvar.4277 _fvar.94255) (Set.Ioo _fvar.4271 (@_fvar.94246 _fvar.94248)) _fvar.94255 := ?_mvar.95497h2:HasDerivWithinAt _fvar.4276 (@_fvar.4278 _fvar.94255) (Set.Ioo _fvar.4271 (@_fvar.94246 _fvar.94248)) _fvar.94255 := ?_mvar.103370h3:HasDerivWithinAt (fun y => f y * g (x n)) (f' yn * g (x n)) (Set.Ioo a (x n)) ynh4:HasDerivWithinAt (fun y => g y * f (x n)) (g' yn * f (x n)) (Set.Ioo a (x n)) ynh5:HasDerivWithinAt _fvar.94249
  (@_fvar.4277 _fvar.94255 * @_fvar.4276 (@_fvar.94246 _fvar.94248) -
    @_fvar.4278 _fvar.94255 * @_fvar.4275 (@_fvar.94246 _fvar.94248))
  (Set.Ioo _fvar.4271 (@_fvar.94246 _fvar.94248)) _fvar.94255 := 
  ?_mvar.124802⊢ ClusterPt yn (Filter.principal (Set.Ioo a (x n) \ {yn}))
        apply ClusterPt.mono _ ((Filter.principal_mono (s := .Ioo a yn)).mpr  _)a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ f (Set.Icc a b) xhg:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ g (Set.Icc a b) xhf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc a b, g x ≠ 0x:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.94246 _fvar.94248) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.94246 _fvar.94248)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn h (Set.Icc a (x n))hdiff:DifferentiableOn ℝ h (Set.Ioo a (x n))ha:h a = 0hb:h (x n) = 0yn:ℝhyn:yn ∈ Set.Ioo a (x n)hdh:HasDerivWithinAt h 0 (Set.Ioo a (x n)) ynh1:HasDerivWithinAt _fvar.4275 (@_fvar.4277 _fvar.94255) (Set.Ioo _fvar.4271 (@_fvar.94246 _fvar.94248)) _fvar.94255 := ?_mvar.95497h2:HasDerivWithinAt _fvar.4276 (@_fvar.4278 _fvar.94255) (Set.Ioo _fvar.4271 (@_fvar.94246 _fvar.94248)) _fvar.94255 := ?_mvar.103370h3:HasDerivWithinAt (fun y => f y * g (x n)) (f' yn * g (x n)) (Set.Ioo a (x n)) ynh4:HasDerivWithinAt (fun y => g y * f (x n)) (g' yn * f (x n)) (Set.Ioo a (x n)) ynh5:HasDerivWithinAt _fvar.94249
  (@_fvar.4277 _fvar.94255 * @_fvar.4276 (@_fvar.94246 _fvar.94248) -
    @_fvar.4278 _fvar.94255 * @_fvar.4275 (@_fvar.94246 _fvar.94248))
  (Set.Ioo _fvar.4271 (@_fvar.94246 _fvar.94248)) _fvar.94255 := 
  ?_mvar.124802⊢ ClusterPt yn (Filter.principal (Set.Ioo a yn))a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ f (Set.Icc a b) xhg:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ g (Set.Icc a b) xhf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc a b, g x ≠ 0x:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.94246 _fvar.94248) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.94246 _fvar.94248)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn h (Set.Icc a (x n))hdiff:DifferentiableOn ℝ h (Set.Ioo a (x n))ha:h a = 0hb:h (x n) = 0yn:ℝhyn:yn ∈ Set.Ioo a (x n)hdh:HasDerivWithinAt h 0 (Set.Ioo a (x n)) ynh1:HasDerivWithinAt _fvar.4275 (@_fvar.4277 _fvar.94255) (Set.Ioo _fvar.4271 (@_fvar.94246 _fvar.94248)) _fvar.94255 := ?_mvar.95497h2:HasDerivWithinAt _fvar.4276 (@_fvar.4278 _fvar.94255) (Set.Ioo _fvar.4271 (@_fvar.94246 _fvar.94248)) _fvar.94255 := ?_mvar.103370h3:HasDerivWithinAt (fun y => f y * g (x n)) (f' yn * g (x n)) (Set.Ioo a (x n)) ynh4:HasDerivWithinAt (fun y => g y * f (x n)) (g' yn * f (x n)) (Set.Ioo a (x n)) ynh5:HasDerivWithinAt _fvar.94249
  (@_fvar.4277 _fvar.94255 * @_fvar.4276 (@_fvar.94246 _fvar.94248) -
    @_fvar.4278 _fvar.94255 * @_fvar.4275 (@_fvar.94246 _fvar.94248))
  (Set.Ioo _fvar.4271 (@_fvar.94246 _fvar.94248)) _fvar.94255 := 
  ?_mvar.124802⊢ Set.Ioo a yn ⊆ Set.Ioo a (x n) \ {yn}
        .a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ f (Set.Icc a b) xhg:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ g (Set.Icc a b) xhf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc a b, g x ≠ 0x:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.94246 _fvar.94248) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.94246 _fvar.94248)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn h (Set.Icc a (x n))hdiff:DifferentiableOn ℝ h (Set.Ioo a (x n))ha:h a = 0hb:h (x n) = 0yn:ℝhyn:yn ∈ Set.Ioo a (x n)hdh:HasDerivWithinAt h 0 (Set.Ioo a (x n)) ynh1:HasDerivWithinAt _fvar.4275 (@_fvar.4277 _fvar.94255) (Set.Ioo _fvar.4271 (@_fvar.94246 _fvar.94248)) _fvar.94255 := ?_mvar.95497h2:HasDerivWithinAt _fvar.4276 (@_fvar.4278 _fvar.94255) (Set.Ioo _fvar.4271 (@_fvar.94246 _fvar.94248)) _fvar.94255 := ?_mvar.103370h3:HasDerivWithinAt (fun y => f y * g (x n)) (f' yn * g (x n)) (Set.Ioo a (x n)) ynh4:HasDerivWithinAt (fun y => g y * f (x n)) (g' yn * f (x n)) (Set.Ioo a (x n)) ynh5:HasDerivWithinAt _fvar.94249
  (@_fvar.4277 _fvar.94255 * @_fvar.4276 (@_fvar.94246 _fvar.94248) -
    @_fvar.4278 _fvar.94255 * @_fvar.4275 (@_fvar.94246 _fvar.94248))
  (Set.Ioo _fvar.4271 (@_fvar.94246 _fvar.94248)) _fvar.94255 := 
  ?_mvar.124802⊢ ClusterPt yn (Filter.principal (Set.Ioo a yn)) simp [←mem_closure_iff_clusterPt, closure_Ioo (show a ≠ yn by grind)]a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ f (Set.Icc a b) xhg:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ g (Set.Icc a b) xhf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc a b, g x ≠ 0x:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.94246 _fvar.94248) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.94246 _fvar.94248)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn h (Set.Icc a (x n))hdiff:DifferentiableOn ℝ h (Set.Ioo a (x n))ha:h a = 0hb:h (x n) = 0yn:ℝhyn:yn ∈ Set.Ioo a (x n)hdh:HasDerivWithinAt h 0 (Set.Ioo a (x n)) ynh1:HasDerivWithinAt _fvar.4275 (@_fvar.4277 _fvar.94255) (Set.Ioo _fvar.4271 (@_fvar.94246 _fvar.94248)) _fvar.94255 := ?_mvar.95497h2:HasDerivWithinAt _fvar.4276 (@_fvar.4278 _fvar.94255) (Set.Ioo _fvar.4271 (@_fvar.94246 _fvar.94248)) _fvar.94255 := ?_mvar.103370h3:HasDerivWithinAt (fun y => f y * g (x n)) (f' yn * g (x n)) (Set.Ioo a (x n)) ynh4:HasDerivWithinAt (fun y => g y * f (x n)) (g' yn * f (x n)) (Set.Ioo a (x n)) ynh5:HasDerivWithinAt _fvar.94249
  (@_fvar.4277 _fvar.94255 * @_fvar.4276 (@_fvar.94246 _fvar.94248) -
    @_fvar.4278 _fvar.94255 * @_fvar.4275 (@_fvar.94246 _fvar.94248))
  (Set.Ioo _fvar.4271 (@_fvar.94246 _fvar.94248)) _fvar.94255 := 
  ?_mvar.124802⊢ a ≤ yn; grindAll goals completed! 🐙
        grindAll goals completed! 🐙
      have h7 : g (x n) ≠ 0 := ?_right.refine_2a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ f (Set.Icc a b) xhg:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ g (Set.Icc a b) xhf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc a b, g x ≠ 0x:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.94246 _fvar.94248) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.94246 _fvar.94248)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn h (Set.Icc a (x n))hdiff:DifferentiableOn ℝ h (Set.Ioo a (x n))ha:h a = 0hb:h (x n) = 0yn:ℝhyn:yn ∈ Set.Ioo a (x n)hdh:HasDerivWithinAt h 0 (Set.Ioo a (x n)) ynh1:HasDerivWithinAt _fvar.4275 (@_fvar.4277 _fvar.94255) (Set.Ioo _fvar.4271 (@_fvar.94246 _fvar.94248)) _fvar.94255 := ?_mvar.95497h2:HasDerivWithinAt _fvar.4276 (@_fvar.4278 _fvar.94255) (Set.Ioo _fvar.4271 (@_fvar.94246 _fvar.94248)) _fvar.94255 := ?_mvar.103370h3:HasDerivWithinAt (fun y => f y * g (x n)) (f' yn * g (x n)) (Set.Ioo a (x n)) ynh4:HasDerivWithinAt (fun y => g y * f (x n)) (g' yn * f (x n)) (Set.Ioo a (x n)) ynh5:HasDerivWithinAt _fvar.94249
  (@_fvar.4277 _fvar.94255 * @_fvar.4276 (@_fvar.94246 _fvar.94248) -
    @_fvar.4278 _fvar.94255 * @_fvar.4275 (@_fvar.94246 _fvar.94248))
  (Set.Ioo _fvar.4271 (@_fvar.94246 _fvar.94248)) _fvar.94255 := 
  ?_mvar.124802h6:@_fvar.4277 _fvar.94255 * @_fvar.4276 (@_fvar.94246 _fvar.94248) -
    @_fvar.4278 _fvar.94255 * @_fvar.4275 (@_fvar.94246 _fvar.94248) =
  0 := 
  ?_mvar.127142h7:@_fvar.4276 (@_fvar.94246 _fvar.94248) ≠ 0 := ?_mvar.142089⊢ f (x n) / g (x n) = f' yn / g' ynright.refine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ f (Set.Icc a b) xhg:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ g (Set.Icc a b) xhf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc a b, g x ≠ 0x:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.94246 _fvar.94248) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.94246 _fvar.94248)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn h (Set.Icc a (x n))hdiff:DifferentiableOn ℝ h (Set.Ioo a (x n))ha:h a = 0hb:h (x n) = 0yn:ℝhyn:yn ∈ Set.Ioo a (x n)hdh:HasDerivWithinAt h 0 (Set.Ioo a (x n)) ynh1:HasDerivWithinAt _fvar.4275 (@_fvar.4277 _fvar.94255) (Set.Ioo _fvar.4271 (@_fvar.94246 _fvar.94248)) _fvar.94255 := ?_mvar.95497h2:HasDerivWithinAt _fvar.4276 (@_fvar.4278 _fvar.94255) (Set.Ioo _fvar.4271 (@_fvar.94246 _fvar.94248)) _fvar.94255 := ?_mvar.103370h3:HasDerivWithinAt (fun y => f y * g (x n)) (f' yn * g (x n)) (Set.Ioo a (x n)) ynh4:HasDerivWithinAt (fun y => g y * f (x n)) (g' yn * f (x n)) (Set.Ioo a (x n)) ynh5:HasDerivWithinAt _fvar.94249
  (@_fvar.4277 _fvar.94255 * @_fvar.4276 (@_fvar.94246 _fvar.94248) -
    @_fvar.4278 _fvar.94255 * @_fvar.4275 (@_fvar.94246 _fvar.94248))
  (Set.Ioo _fvar.4271 (@_fvar.94246 _fvar.94248)) _fvar.94255 := 
  ?_mvar.124802h6:@_fvar.4277 _fvar.94255 * @_fvar.4276 (@_fvar.94246 _fvar.94248) -
    @_fvar.4278 _fvar.94255 * @_fvar.4275 (@_fvar.94246 _fvar.94248) =
  0 := 
  ?_mvar.127142⊢ g (x n) ≠ 0refine_2.refine_2.refine_2.refine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn _fvar.84237 (Set.Icc _fvar.4271 (@_fvar.83912 _fvar.83923)) := ContinuousOn.mono _fvar.88991 ?_mvar.91263hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Ioo _fvar.4271 (@_fvar.83912 _fvar.83923)) := DifferentiableOn.mono _fvar.84999 ?_mvar.91385ha:@_fvar.84237 _fvar.4271 = 0 := ?_mvar.91457⊢ h (x n) = 0refine_2.refine_2.refine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn _fvar.84237 (Set.Icc _fvar.4271 (@_fvar.83912 _fvar.83923)) := ContinuousOn.mono _fvar.88991 ?_mvar.91263hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Ioo _fvar.4271 (@_fvar.83912 _fvar.83923)) := DifferentiableOn.mono _fvar.84999 ?_mvar.91385⊢ h a = 0refine_2.refine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Icc _fvar.4271 _fvar.4272) := ?_mvar.84998hx:a < x n ∧ x n ≤ bhcon:ContinuousOn _fvar.84237 (Set.Icc _fvar.4271 (@_fvar.83912 _fvar.83923)) := ContinuousOn.mono _fvar.88991 ?_mvar.91263⊢ Set.Ioo a (x n) ⊆ Set.Icc a brefine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Icc _fvar.4271 _fvar.4272) := ?_mvar.84998hcon:ContinuousOn h (Set.Icc a b)hx:a < x n ∧ x n ≤ b⊢ Set.Icc a (x n) ⊆ Set.Icc a b
      have h8 : g' (yn) ≠ 0 := ?_right.refine_2.refine_2a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ f (Set.Icc a b) xhg:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ g (Set.Icc a b) xhf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc a b, g x ≠ 0x:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.94246 _fvar.94248) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.94246 _fvar.94248)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn h (Set.Icc a (x n))hdiff:DifferentiableOn ℝ h (Set.Ioo a (x n))ha:h a = 0hb:h (x n) = 0yn:ℝhyn:yn ∈ Set.Ioo a (x n)hdh:HasDerivWithinAt h 0 (Set.Ioo a (x n)) ynh1:HasDerivWithinAt _fvar.4275 (@_fvar.4277 _fvar.94255) (Set.Ioo _fvar.4271 (@_fvar.94246 _fvar.94248)) _fvar.94255 := ?_mvar.95497h2:HasDerivWithinAt _fvar.4276 (@_fvar.4278 _fvar.94255) (Set.Ioo _fvar.4271 (@_fvar.94246 _fvar.94248)) _fvar.94255 := ?_mvar.103370h3:HasDerivWithinAt (fun y => f y * g (x n)) (f' yn * g (x n)) (Set.Ioo a (x n)) ynh4:HasDerivWithinAt (fun y => g y * f (x n)) (g' yn * f (x n)) (Set.Ioo a (x n)) ynh5:HasDerivWithinAt _fvar.94249
  (@_fvar.4277 _fvar.94255 * @_fvar.4276 (@_fvar.94246 _fvar.94248) -
    @_fvar.4278 _fvar.94255 * @_fvar.4275 (@_fvar.94246 _fvar.94248))
  (Set.Ioo _fvar.4271 (@_fvar.94246 _fvar.94248)) _fvar.94255 := 
  ?_mvar.124802h6:@_fvar.4277 _fvar.94255 * @_fvar.4276 (@_fvar.94246 _fvar.94248) -
    @_fvar.4278 _fvar.94255 * @_fvar.4275 (@_fvar.94246 _fvar.94248) =
  0 := 
  ?_mvar.127142h7:@_fvar.4276 (@_fvar.94246 _fvar.94248) ≠ 0 := ?_mvar.142089h8:@_fvar.4278 _fvar.94255 ≠ 0 := ?_mvar.142121⊢ f (x n) / g (x n) = f' yn / g' ynright.refine_2.refine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ f (Set.Icc a b) xhg:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ g (Set.Icc a b) xhf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc a b, g x ≠ 0x:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.94246 _fvar.94248) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.94246 _fvar.94248)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn h (Set.Icc a (x n))hdiff:DifferentiableOn ℝ h (Set.Ioo a (x n))ha:h a = 0hb:h (x n) = 0yn:ℝhyn:yn ∈ Set.Ioo a (x n)hdh:HasDerivWithinAt h 0 (Set.Ioo a (x n)) ynh1:HasDerivWithinAt _fvar.4275 (@_fvar.4277 _fvar.94255) (Set.Ioo _fvar.4271 (@_fvar.94246 _fvar.94248)) _fvar.94255 := ?_mvar.95497h2:HasDerivWithinAt _fvar.4276 (@_fvar.4278 _fvar.94255) (Set.Ioo _fvar.4271 (@_fvar.94246 _fvar.94248)) _fvar.94255 := ?_mvar.103370h3:HasDerivWithinAt (fun y => f y * g (x n)) (f' yn * g (x n)) (Set.Ioo a (x n)) ynh4:HasDerivWithinAt (fun y => g y * f (x n)) (g' yn * f (x n)) (Set.Ioo a (x n)) ynh5:HasDerivWithinAt _fvar.94249
  (@_fvar.4277 _fvar.94255 * @_fvar.4276 (@_fvar.94246 _fvar.94248) -
    @_fvar.4278 _fvar.94255 * @_fvar.4275 (@_fvar.94246 _fvar.94248))
  (Set.Ioo _fvar.4271 (@_fvar.94246 _fvar.94248)) _fvar.94255 := 
  ?_mvar.124802h6:@_fvar.4277 _fvar.94255 * @_fvar.4276 (@_fvar.94246 _fvar.94248) -
    @_fvar.4278 _fvar.94255 * @_fvar.4275 (@_fvar.94246 _fvar.94248) =
  0 := 
  ?_mvar.127142h7:@_fvar.4276 (@_fvar.94246 _fvar.94248) ≠ 0 := ?_mvar.142089⊢ g' yn ≠ 0right.refine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ f (Set.Icc a b) xhg:∀ x ∈ Set.Icc a b, DifferentiableWithinAt ℝ g (Set.Icc a b) xhf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc a b, g x ≠ 0x:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.94246 _fvar.94248) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.94246 _fvar.94248)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn h (Set.Icc a (x n))hdiff:DifferentiableOn ℝ h (Set.Ioo a (x n))ha:h a = 0hb:h (x n) = 0yn:ℝhyn:yn ∈ Set.Ioo a (x n)hdh:HasDerivWithinAt h 0 (Set.Ioo a (x n)) ynh1:HasDerivWithinAt _fvar.4275 (@_fvar.4277 _fvar.94255) (Set.Ioo _fvar.4271 (@_fvar.94246 _fvar.94248)) _fvar.94255 := ?_mvar.95497h2:HasDerivWithinAt _fvar.4276 (@_fvar.4278 _fvar.94255) (Set.Ioo _fvar.4271 (@_fvar.94246 _fvar.94248)) _fvar.94255 := ?_mvar.103370h3:HasDerivWithinAt (fun y => f y * g (x n)) (f' yn * g (x n)) (Set.Ioo a (x n)) ynh4:HasDerivWithinAt (fun y => g y * f (x n)) (g' yn * f (x n)) (Set.Ioo a (x n)) ynh5:HasDerivWithinAt _fvar.94249
  (@_fvar.4277 _fvar.94255 * @_fvar.4276 (@_fvar.94246 _fvar.94248) -
    @_fvar.4278 _fvar.94255 * @_fvar.4275 (@_fvar.94246 _fvar.94248))
  (Set.Ioo _fvar.4271 (@_fvar.94246 _fvar.94248)) _fvar.94255 := 
  ?_mvar.124802h6:@_fvar.4277 _fvar.94255 * @_fvar.4276 (@_fvar.94246 _fvar.94248) -
    @_fvar.4278 _fvar.94255 * @_fvar.4275 (@_fvar.94246 _fvar.94248) =
  0 := 
  ?_mvar.127142⊢ g (x n) ≠ 0refine_2.refine_2.refine_2.refine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn _fvar.84237 (Set.Icc _fvar.4271 (@_fvar.83912 _fvar.83923)) := ContinuousOn.mono _fvar.88991 ?_mvar.91263hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Ioo _fvar.4271 (@_fvar.83912 _fvar.83923)) := DifferentiableOn.mono _fvar.84999 ?_mvar.91385ha:@_fvar.84237 _fvar.4271 = 0 := ?_mvar.91457⊢ h (x n) = 0refine_2.refine_2.refine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hx:a < x n ∧ x n ≤ bhcon:ContinuousOn _fvar.84237 (Set.Icc _fvar.4271 (@_fvar.83912 _fvar.83923)) := ContinuousOn.mono _fvar.88991 ?_mvar.91263hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Ioo _fvar.4271 (@_fvar.83912 _fvar.83923)) := DifferentiableOn.mono _fvar.84999 ?_mvar.91385⊢ h a = 0refine_2.refine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Icc _fvar.4271 _fvar.4272) := ?_mvar.84998hx:a < x n ∧ x n ≤ bhcon:ContinuousOn _fvar.84237 (Set.Icc _fvar.4271 (@_fvar.83912 _fvar.83923)) := ContinuousOn.mono _fvar.88991 ?_mvar.91263⊢ Set.Ioo a (x n) ⊆ Set.Icc a brefine_1a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhconv:Filter.Tendsto x Filter.atTop (nhds a)n:ℕh:ℝ → ℝ := 
  fun x' =>
    @_fvar.4275 x' * @_fvar.4276 (@_fvar.83912 _fvar.83923) - @_fvar.4276 x' * @_fvar.4275 (@_fvar.83912 _fvar.83923)hdiff:DifferentiableOn ℝ _fvar.84237 (Set.Icc _fvar.4271 _fvar.4272) := ?_mvar.84998hcon:ContinuousOn h (Set.Icc a b)hx:a < x n ∧ x n ≤ b⊢ Set.Icc a (x n) ⊆ Set.Icc a b
      all_goals grindAll goals completed! 🐙
    choose y hy hy' using hxya:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhx:∀ (n : ℕ), x n ∈ Set.Ioc a bhconv:Filter.Tendsto x Filter.atTop (nhds a)y:ℕ → ℝhy:∀ (n : ℕ), y n ∈ Set.Ioo a (x n)hy':∀ (n : ℕ), f (x n) / g (x n) = f' (y n) / g' (y n)⊢ Filter.Tendsto (fun n => f (x n) / g (x n)) Filter.atTop (nhds L)
    have hyconv : Filter.atTop.Tendsto y (nhds a) := bya:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)⊢ (∀ x ∈ Set.Ioc a b, g x ≠ 0) ∧ Filter.Tendsto (fun x => f x / g x) (nhdsWithin a (Set.Ioc a b)) (nhds L)
      apply tendsto_const_nhds.squeeze hconvhgfa:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhx:∀ (n : ℕ), x n ∈ Set.Ioc a bhconv:Filter.Tendsto x Filter.atTop (nhds a)y:ℕ → ℝhy:∀ (n : ℕ), y n ∈ Set.Ioo a (x n)hy':∀ (n : ℕ), f (x n) / g (x n) = f' (y n) / g' (y n)⊢ (fun x => a) ≤ yhfha:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhx:∀ (n : ℕ), x n ∈ Set.Ioc a bhconv:Filter.Tendsto x Filter.atTop (nhds a)y:ℕ → ℝhy:∀ (n : ℕ), y n ∈ Set.Ioo a (x n)hy':∀ (n : ℕ), f (x n) / g (x n) = f' (y n) / g' (y n)⊢ y ≤ x <;>hgfa:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhx:∀ (n : ℕ), x n ∈ Set.Ioc a bhconv:Filter.Tendsto x Filter.atTop (nhds a)y:ℕ → ℝhy:∀ (n : ℕ), y n ∈ Set.Ioo a (x n)hy':∀ (n : ℕ), f (x n) / g (x n) = f' (y n) / g' (y n)⊢ (fun x => a) ≤ yhfha:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhx:∀ (n : ℕ), x n ∈ Set.Ioc a bhconv:Filter.Tendsto x Filter.atTop (nhds a)y:ℕ → ℝhy:∀ (n : ℕ), y n ∈ Set.Ioo a (x n)hy':∀ (n : ℕ), f (x n) / g (x n) = f' (y n) / g' (y n)⊢ y ≤ x (intro _hfha:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhx:∀ (n : ℕ), x n ∈ Set.Ioc a bhconv:Filter.Tendsto x Filter.atTop (nhds a)y:ℕ → ℝhy:∀ (n : ℕ), y n ∈ Set.Ioo a (x n)hy':∀ (n : ℕ), f (x n) / g (x n) = f' (y n) / g' (y n)i✝:ℕ⊢ y i✝ ≤ x i✝; grindAll goals completed! 🐙)
    replace hy : ∀ n, y n ∈ Set.Icc a b := bya:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)⊢ (∀ x ∈ Set.Ioc a b, g x ≠ 0) ∧ Filter.Tendsto (fun x => f x / g x) (nhdsWithin a (Set.Ioc a b)) (nhds L) grindAll goals completed! 🐙
    simp_rw [hy']a:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhx:∀ (n : ℕ), x n ∈ Set.Ioc a bhconv:Filter.Tendsto x Filter.atTop (nhds a)y:ℕ → ℝhy':∀ (n : ℕ), f (x n) / g (x n) = f' (y n) / g' (y n)hyconv:Filter.Tendsto _fvar.162520 Filter.atTop (nhds _fvar.4271) := ?_mvar.162712hy:∀ (n : ℕ), @_fvar.162520 n ∈ Set.Icc _fvar.4271 _fvar.4272 := ?_mvar.168811⊢ Filter.Tendsto (fun n => f' (y n) / g' (y n)) Filter.atTop (nhds L); apply hderiv.compa:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hxx:ℕ → ℝhx:∀ (n : ℕ), x n ∈ Set.Ioc a bhconv:Filter.Tendsto x Filter.atTop (nhds a)y:ℕ → ℝhy':∀ (n : ℕ), f (x n) / g (x n) = f' (y n) / g' (y n)hyconv:Filter.Tendsto _fvar.162520 Filter.atTop (nhds _fvar.4271) := ?_mvar.162712hy:∀ (n : ℕ), @_fvar.162520 n ∈ Set.Icc _fvar.4271 _fvar.4272 := ?_mvar.168811⊢ Filter.Tendsto y Filter.atTop (nhdsWithin a (Set.Icc a b))
    solve_by_elim [tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within,
    Filter.Eventually.of_forall]All goals completed! 🐙
  simp [←closure_def', closure_Ioc (show a ≠ b by grind)]ha:ℝb:ℝL:ℝhab:a < bf:ℝ → ℝg:ℝ → ℝf':ℝ → ℝg':ℝ → ℝhf:DifferentiableOn ℝ f (Set.Icc a b)hg:DifferentiableOn ℝ g (Set.Icc a b)hf':f' = derivWithin f (Set.Icc a b)hg':g' = derivWithin g (Set.Icc a b)hfa:f a = 0hga:g a = 0hgnon:∀ x ∈ Set.Icc a b, g' x ≠ 0hderiv:Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Icc a b)) (nhds L)hfcon:ContinuousOn f (Set.Icc a b)hgcon:ContinuousOn g (Set.Icc a b)this:∀ x ∈ Set.Ioc _fvar.4271 _fvar.4272, @_fvar.4276 x ≠ 0 := fun x hx => @?_mvar.57675 x hx⊢ a ≤ b; grindAll goals completed! 🐙

end Chapter10