ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- HasDerivAt.lhopital_zero_right_on_Ico
theorem lhopital_zero_right_on_Ico (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x)
    (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hcf : ContinuousOn f (Ico a b))
    (hcg : ContinuousOn g (Ico a b)) (hg' : ∀ x ∈ Ioo a b, g' x ≠ 0) (hfa : f a = 0) (hga : g a = 0)
    (hdiv : Tendsto (fun x => f' x / g' x) (𝓝[>] a) l) :
    Tendsto (fun x => f x / g x) (𝓝[>] a) l 
Arguments: {a b : ℝ} (hab : a < b) {l : Filter ℝ} {f f' g g' : ℝ → ℝ} (hff' : ∀ x ∈ Set.Ioo a b, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Set.Ioo a b, HasDerivAt g (g' x) x) (hcf : ContinuousOn f (Set.Ico a b)) (hcg : ContinuousOn g (Set.Ico a b)) (hg' : ∀ x ∈ Set.Ioo a b, g' x ≠ 0) (hfa : f a = 0) (hga : g a = 0) (hdiv : Filter.Tendsto (fun x => f' x / g' x) (𝓝[>] a) l)
Goal: Filter.Tendsto (fun x => f x / g x) (𝓝[>] a) l
-- HasDerivAt.lhopital_zero_left_on_Ioo
theorem lhopital_zero_left_on_Ioo (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x)
    (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hg' : ∀ x ∈ Ioo a b, g' x ≠ 0)
    (hfb : Tendsto f (𝓝[<] b) (𝓝 0)) (hgb : Tendsto g (𝓝[<] b) (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) (𝓝[<] b) l) :
  Tendsto (fun x => f x / g x) (𝓝[<] b) l 
Arguments: {a b : ℝ} (hab : a < b) {l : Filter ℝ} {f f' g g' : ℝ → ℝ} (hff' : ∀ x ∈ Set.Ioo a b, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Set.Ioo a b, HasDerivAt g (g' x) x) (hg' : ∀ x ∈ Set.Ioo a b, g' x ≠ 0) (hfb : Filter.Tendsto f (𝓝[<] b) (𝓝 0)) (hgb : Filter.Tendsto g (𝓝[<] b) (𝓝 0)) (hdiv : Filter.Tendsto (fun x => f' x / g' x) (𝓝[<] b) l)
Goal: Filter.Tendsto (fun x => f x / g x) (𝓝[<] b) l
-- HasDerivAt.lhopital_zero_left_on_Ioc
theorem lhopital_zero_left_on_Ioc (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x)
    (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hcf : ContinuousOn f (Ioc a b))
    (hcg : ContinuousOn g (Ioc a b)) (hg' : ∀ x ∈ Ioo a b, g' x ≠ 0) (hfb : f b = 0) (hgb : g b = 0)
    (hdiv : Tendsto (fun x => f' x / g' x) (𝓝[<] b) l) :
    Tendsto (fun x => f x / g x) (𝓝[<] b) l 
Arguments: {a b : ℝ} (hab : a < b) {l : Filter ℝ} {f f' g g' : ℝ → ℝ} (hff' : ∀ x ∈ Set.Ioo a b, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Set.Ioo a b, HasDerivAt g (g' x) x) (hcf : ContinuousOn f (Set.Ioc a b)) (hcg : ContinuousOn g (Set.Ioc a b)) (hg' : ∀ x ∈ Set.Ioo a b, g' x ≠ 0) (hfb : f b = 0) (hgb : g b = 0) (hdiv : Filter.Tendsto (fun x => f' x / g' x) (𝓝[<] b) l)
Goal: Filter.Tendsto (fun x => f x / g x) (𝓝[<] b) l
-- HasDerivAt.lhopital_zero_atTop_on_Ioi
theorem lhopital_zero_atTop_on_Ioi (hff' : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x)
    (hgg' : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x) (hg' : ∀ x ∈ Ioi a, g' x ≠ 0)
    (hftop : Tendsto f atTop (𝓝 0)) (hgtop : Tendsto g atTop (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) atTop l) : Tendsto (fun x => f x / g x) atTop l 
Arguments: {a : ℝ} {l : Filter ℝ} {f f' g g' : ℝ → ℝ} (hff' : ∀ x ∈ Set.Ioi a, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Set.Ioi a, HasDerivAt g (g' x) x) (hg' : ∀ x ∈ Set.Ioi a, g' x ≠ 0) (hftop : Filter.Tendsto f Filter.atTop (𝓝 0)) (hgtop : Filter.Tendsto g Filter.atTop (𝓝 0)) (hdiv : Filter.Tendsto (fun x => f' x / g' x) Filter.atTop l)
Goal: Filter.Tendsto (fun x => f x / g x) Filter.atTop l
-- HasDerivAt.lhopital_zero_atBot_on_Iio
theorem lhopital_zero_atBot_on_Iio (hff' : ∀ x ∈ Iio a, HasDerivAt f (f' x) x)
    (hgg' : ∀ x ∈ Iio a, HasDerivAt g (g' x) x) (hg' : ∀ x ∈ Iio a, g' x ≠ 0)
    (hfbot : Tendsto f atBot (𝓝 0)) (hgbot : Tendsto g atBot (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) atBot l) : Tendsto (fun x => f x / g x) atBot l 
Arguments: {a : ℝ} {l : Filter ℝ} {f f' g g' : ℝ → ℝ} (hff' : ∀ x ∈ Set.Iio a, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Set.Iio a, HasDerivAt g (g' x) x) (hg' : ∀ x ∈ Set.Iio a, g' x ≠ 0) (hfbot : Filter.Tendsto f Filter.atBot (𝓝 0)) (hgbot : Filter.Tendsto g Filter.atBot (𝓝 0)) (hdiv : Filter.Tendsto (fun x => f' x / g' x) Filter.atBot l)
Goal: Filter.Tendsto (fun x => f x / g x) Filter.atBot l
-- deriv.lhopital_zero_right_on_Ioo
theorem lhopital_zero_right_on_Ioo (hdf : DifferentiableOn ℝ f (Ioo a b))
    (hg' : ∀ x ∈ Ioo a b, deriv g x ≠ 0) (hfa : Tendsto f (𝓝[>] a) (𝓝 0))
    (hga : Tendsto g (𝓝[>] a) (𝓝 0))
    (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) (𝓝[>] a) l) :
    Tendsto (fun x => f x / g x) (𝓝[>] a) l 
Arguments: {a b : ℝ} (hab : a < b) {l : Filter ℝ} {f g : ℝ → ℝ} (hdf : DifferentiableOn ℝ f (Set.Ioo a b)) (hg' : ∀ x ∈ Set.Ioo a b, deriv g x ≠ 0) (hfa : Filter.Tendsto f (𝓝[>] a) (𝓝 0)) (hga : Filter.Tendsto g (𝓝[>] a) (𝓝 0)) (hdiv : Filter.Tendsto (fun x => deriv f x / deriv g x) (𝓝[>] a) l)
Goal: Filter.Tendsto (fun x => f x / g x) (𝓝[>] a) l
-- deriv.lhopital_zero_right_on_Ico
theorem lhopital_zero_right_on_Ico (hdf : DifferentiableOn ℝ f (Ioo a b))
    (hcf : ContinuousOn f (Ico a b)) (hcg : ContinuousOn g (Ico a b))
    (hg' : ∀ x ∈ Ioo a b, (deriv g) x ≠ 0) (hfa : f a = 0) (hga : g a = 0)
    (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) (𝓝[>] a) l) :
    Tendsto (fun x => f x / g x) (𝓝[>] a) l 
Arguments: {a b : ℝ} (hab : a < b) {l : Filter ℝ} {f g : ℝ → ℝ} (hdf : DifferentiableOn ℝ f (Set.Ioo a b)) (hcf : ContinuousOn f (Set.Ico a b)) (hcg : ContinuousOn g (Set.Ico a b)) (hg' : ∀ x ∈ Set.Ioo a b, deriv g x ≠ 0) (hfa : f a = 0) (hga : g a = 0) (hdiv : Filter.Tendsto (fun x => deriv f x / deriv g x) (𝓝[>] a) l)
Goal: Filter.Tendsto (fun x => f x / g x) (𝓝[>] a) l
-- deriv.lhopital_zero_left_on_Ioo
theorem lhopital_zero_left_on_Ioo (hdf : DifferentiableOn ℝ f (Ioo a b))
    (hg' : ∀ x ∈ Ioo a b, (deriv g) x ≠ 0) (hfb : Tendsto f (𝓝[<] b) (𝓝 0))
    (hgb : Tendsto g (𝓝[<] b) (𝓝 0))
    (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) (𝓝[<] b) l) :
    Tendsto (fun x => f x / g x) (𝓝[<] b) l 
Arguments: {a b : ℝ} (hab : a < b) {l : Filter ℝ} {f g : ℝ → ℝ} (hdf : DifferentiableOn ℝ f (Set.Ioo a b)) (hg' : ∀ x ∈ Set.Ioo a b, deriv g x ≠ 0) (hfb : Filter.Tendsto f (𝓝[<] b) (𝓝 0)) (hgb : Filter.Tendsto g (𝓝[<] b) (𝓝 0)) (hdiv : Filter.Tendsto (fun x => deriv f x / deriv g x) (𝓝[<] b) l)
Goal: Filter.Tendsto (fun x => f x / g x) (𝓝[<] b) l
-- deriv.lhopital_zero_atTop_on_Ioi
theorem lhopital_zero_atTop_on_Ioi (hdf : DifferentiableOn ℝ f (Ioi a))
    (hg' : ∀ x ∈ Ioi a, (deriv g) x ≠ 0) (hftop : Tendsto f atTop (𝓝 0))
    (hgtop : Tendsto g atTop (𝓝 0)) (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) atTop l) :
    Tendsto (fun x => f x / g x) atTop l 
Arguments: {a : ℝ} {l : Filter ℝ} {f g : ℝ → ℝ} (hdf : DifferentiableOn ℝ f (Set.Ioi a)) (hg' : ∀ x ∈ Set.Ioi a, deriv g x ≠ 0) (hftop : Filter.Tendsto f Filter.atTop (𝓝 0)) (hgtop : Filter.Tendsto g Filter.atTop (𝓝 0)) (hdiv : Filter.Tendsto (fun x => deriv f x / deriv g x) Filter.atTop l)
Goal: Filter.Tendsto (fun x => f x / g x) Filter.atTop l
-- deriv.lhopital_zero_atBot_on_Iio
theorem lhopital_zero_atBot_on_Iio (hdf : DifferentiableOn ℝ f (Iio a))
    (hg' : ∀ x ∈ Iio a, (deriv g) x ≠ 0) (hfbot : Tendsto f atBot (𝓝 0))
    (hgbot : Tendsto g atBot (𝓝 0)) (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) atBot l) :
    Tendsto (fun x => f x / g x) atBot l 
Arguments: {a : ℝ} {l : Filter ℝ} {f g : ℝ → ℝ} (hdf : DifferentiableOn ℝ f (Set.Iio a)) (hg' : ∀ x ∈ Set.Iio a, deriv g x ≠ 0) (hfbot : Filter.Tendsto f Filter.atBot (𝓝 0)) (hgbot : Filter.Tendsto g Filter.atBot (𝓝 0)) (hdiv : Filter.Tendsto (fun x => deriv f x / deriv g x) Filter.atBot l)
Goal: Filter.Tendsto (fun x => f x / g x) Filter.atBot l
-- HasDerivAt.lhopital_zero_nhds_right
theorem lhopital_zero_nhds_right (hff' : ∀ᶠ x in 𝓝[>] a, HasDerivAt f (f' x) x)
    (hgg' : ∀ᶠ x in 𝓝[>] a, HasDerivAt g (g' x) x) (hg' : ∀ᶠ x in 𝓝[>] a, g' x ≠ 0)
    (hfa : Tendsto f (𝓝[>] a) (𝓝 0)) (hga : Tendsto g (𝓝[>] a) (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) (𝓝[>] a) l) :
    Tendsto (fun x => f x / g x) (𝓝[>] a) l 
Arguments: {a : ℝ} {l : Filter ℝ} {f f' g g' : ℝ → ℝ} (hff' : ∀ᶠ (x : ℝ) in 𝓝[>] a, HasDerivAt f (f' x) x) (hgg' : ∀ᶠ (x : ℝ) in 𝓝[>] a, HasDerivAt g (g' x) x) (hg' : ∀ᶠ (x : ℝ) in 𝓝[>] a, g' x ≠ 0) (hfa : Filter.Tendsto f (𝓝[>] a) (𝓝 0)) (hga : Filter.Tendsto g (𝓝[>] a) (𝓝 0)) (hdiv : Filter.Tendsto (fun x => f' x / g' x) (𝓝[>] a) l)
Goal: Filter.Tendsto (fun x => f x / g x) (𝓝[>] a) l
-- HasDerivAt.lhopital_zero_nhds_left
theorem lhopital_zero_nhds_left (hff' : ∀ᶠ x in 𝓝[<] a, HasDerivAt f (f' x) x)
    (hgg' : ∀ᶠ x in 𝓝[<] a, HasDerivAt g (g' x) x) (hg' : ∀ᶠ x in 𝓝[<] a, g' x ≠ 0)
    (hfa : Tendsto f (𝓝[<] a) (𝓝 0)) (hga : Tendsto g (𝓝[<] a) (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) (𝓝[<] a) l) :
    Tendsto (fun x => f x / g x) (𝓝[<] a) l 
Arguments: {a : ℝ} {l : Filter ℝ} {f f' g g' : ℝ → ℝ} (hff' : ∀ᶠ (x : ℝ) in 𝓝[<] a, HasDerivAt f (f' x) x) (hgg' : ∀ᶠ (x : ℝ) in 𝓝[<] a, HasDerivAt g (g' x) x) (hg' : ∀ᶠ (x : ℝ) in 𝓝[<] a, g' x ≠ 0) (hfa : Filter.Tendsto f (𝓝[<] a) (𝓝 0)) (hga : Filter.Tendsto g (𝓝[<] a) (𝓝 0)) (hdiv : Filter.Tendsto (fun x => f' x / g' x) (𝓝[<] a) l)
Goal: Filter.Tendsto (fun x => f x / g x) (𝓝[<] a) l
-- HasDerivAt.lhopital_zero_nhds'
theorem lhopital_zero_nhds' (hff' : ∀ᶠ x in 𝓝[≠] a, HasDerivAt f (f' x) x)
    (hgg' : ∀ᶠ x in 𝓝[≠] a, HasDerivAt g (g' x) x) (hg' : ∀ᶠ x in 𝓝[≠] a, g' x ≠ 0)
    (hfa : Tendsto f (𝓝[≠] a) (𝓝 0)) (hga : Tendsto g (𝓝[≠] a) (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) (𝓝[≠] a) l) :
    Tendsto (fun x => f x / g x) (𝓝[≠] a) l 
Arguments: {a : ℝ} {l : Filter ℝ} {f f' g g' : ℝ → ℝ} (hff' : ∀ᶠ (x : ℝ) in 𝓝[≠] a, HasDerivAt f (f' x) x) (hgg' : ∀ᶠ (x : ℝ) in 𝓝[≠] a, HasDerivAt g (g' x) x) (hg' : ∀ᶠ (x : ℝ) in 𝓝[≠] a, g' x ≠ 0) (hfa : Filter.Tendsto f (𝓝[≠] a) (𝓝 0)) (hga : Filter.Tendsto g (𝓝[≠] a) (𝓝 0)) (hdiv : Filter.Tendsto (fun x => f' x / g' x) (𝓝[≠] a) l)
Goal: Filter.Tendsto (fun x => f x / g x) (𝓝[≠] a) l
-- HasDerivAt.lhopital_zero_nhds
theorem lhopital_zero_nhds (hff' : ∀ᶠ x in 𝓝 a, HasDerivAt f (f' x) x)
    (hgg' : ∀ᶠ x in 𝓝 a, HasDerivAt g (g' x) x) (hg' : ∀ᶠ x in 𝓝 a, g' x ≠ 0)
    (hfa : Tendsto f (𝓝 a) (𝓝 0)) (hga : Tendsto g (𝓝 a) (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) (𝓝 a) l) : Tendsto (fun x => f x / g x) (𝓝[≠] a) l 
Arguments: {a : ℝ} {l : Filter ℝ} {f f' g g' : ℝ → ℝ} (hff' : ∀ᶠ (x : ℝ) in 𝓝 a, HasDerivAt f (f' x) x) (hgg' : ∀ᶠ (x : ℝ) in 𝓝 a, HasDerivAt g (g' x) x) (hg' : ∀ᶠ (x : ℝ) in 𝓝 a, g' x ≠ 0) (hfa : Filter.Tendsto f (𝓝 a) (𝓝 0)) (hga : Filter.Tendsto g (𝓝 a) (𝓝 0)) (hdiv : Filter.Tendsto (fun x => f' x / g' x) (𝓝 a) l)
Goal: Filter.Tendsto (fun x => f x / g x) (𝓝[≠] a) l
-- HasDerivAt.lhopital_zero_atTop
theorem lhopital_zero_atTop (hff' : ∀ᶠ x in atTop, HasDerivAt f (f' x) x)
    (hgg' : ∀ᶠ x in atTop, HasDerivAt g (g' x) x) (hg' : ∀ᶠ x in atTop, g' x ≠ 0)
    (hftop : Tendsto f atTop (𝓝 0)) (hgtop : Tendsto g atTop (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) atTop l) : Tendsto (fun x => f x / g x) atTop l 
Arguments: {l : Filter ℝ} {f f' g g' : ℝ → ℝ} (hff' : ∀ᶠ (x : ℝ) in Filter.atTop, HasDerivAt f (f' x) x) (hgg' : ∀ᶠ (x : ℝ) in Filter.atTop, HasDerivAt g (g' x) x) (hg' : ∀ᶠ (x : ℝ) in Filter.atTop, g' x ≠ 0) (hftop : Filter.Tendsto f Filter.atTop (𝓝 0)) (hgtop : Filter.Tendsto g Filter.atTop (𝓝 0)) (hdiv : Filter.Tendsto (fun x => f' x / g' x) Filter.atTop l)
Goal: Filter.Tendsto (fun x => f x / g x) Filter.atTop l
-- HasDerivAt.lhopital_zero_atBot
theorem lhopital_zero_atBot (hff' : ∀ᶠ x in atBot, HasDerivAt f (f' x) x)
    (hgg' : ∀ᶠ x in atBot, HasDerivAt g (g' x) x) (hg' : ∀ᶠ x in atBot, g' x ≠ 0)
    (hfbot : Tendsto f atBot (𝓝 0)) (hgbot : Tendsto g atBot (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) atBot l) : Tendsto (fun x => f x / g x) atBot l 
Arguments: {l : Filter ℝ} {f f' g g' : ℝ → ℝ} (hff' : ∀ᶠ (x : ℝ) in Filter.atBot, HasDerivAt f (f' x) x) (hgg' : ∀ᶠ (x : ℝ) in Filter.atBot, HasDerivAt g (g' x) x) (hg' : ∀ᶠ (x : ℝ) in Filter.atBot, g' x ≠ 0) (hfbot : Filter.Tendsto f Filter.atBot (𝓝 0)) (hgbot : Filter.Tendsto g Filter.atBot (𝓝 0)) (hdiv : Filter.Tendsto (fun x => f' x / g' x) Filter.atBot l)
Goal: Filter.Tendsto (fun x => f x / g x) Filter.atBot l
-- deriv.lhopital_zero_nhds_right
theorem lhopital_zero_nhds_right (hdf : ∀ᶠ x in 𝓝[>] a, DifferentiableAt ℝ f x)
    (hg' : ∀ᶠ x in 𝓝[>] a, deriv g x ≠ 0) (hfa : Tendsto f (𝓝[>] a) (𝓝 0))
    (hga : Tendsto g (𝓝[>] a) (𝓝 0))
    (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) (𝓝[>] a) l) :
    Tendsto (fun x => f x / g x) (𝓝[>] a) l 
Arguments: {a : ℝ} {l : Filter ℝ} {f g : ℝ → ℝ} (hdf : ∀ᶠ (x : ℝ) in 𝓝[>] a, DifferentiableAt ℝ f x) (hg' : ∀ᶠ (x : ℝ) in 𝓝[>] a, deriv g x ≠ 0) (hfa : Filter.Tendsto f (𝓝[>] a) (𝓝 0)) (hga : Filter.Tendsto g (𝓝[>] a) (𝓝 0)) (hdiv : Filter.Tendsto (fun x => deriv f x / deriv g x) (𝓝[>] a) l)
Goal: Filter.Tendsto (fun x => f x / g x) (𝓝[>] a) l
-- deriv.lhopital_zero_nhds_left
theorem lhopital_zero_nhds_left (hdf : ∀ᶠ x in 𝓝[<] a, DifferentiableAt ℝ f x)
    (hg' : ∀ᶠ x in 𝓝[<] a, deriv g x ≠ 0) (hfa : Tendsto f (𝓝[<] a) (𝓝 0))
    (hga : Tendsto g (𝓝[<] a) (𝓝 0))
    (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) (𝓝[<] a) l) :
    Tendsto (fun x => f x / g x) (𝓝[<] a) l 
Arguments: {a : ℝ} {l : Filter ℝ} {f g : ℝ → ℝ} (hdf : ∀ᶠ (x : ℝ) in 𝓝[<] a, DifferentiableAt ℝ f x) (hg' : ∀ᶠ (x : ℝ) in 𝓝[<] a, deriv g x ≠ 0) (hfa : Filter.Tendsto f (𝓝[<] a) (𝓝 0)) (hga : Filter.Tendsto g (𝓝[<] a) (𝓝 0)) (hdiv : Filter.Tendsto (fun x => deriv f x / deriv g x) (𝓝[<] a) l)
Goal: Filter.Tendsto (fun x => f x / g x) (𝓝[<] a) l
-- deriv.lhopital_zero_nhds'
theorem lhopital_zero_nhds' (hdf : ∀ᶠ x in 𝓝[≠] a, DifferentiableAt ℝ f x)
    (hg' : ∀ᶠ x in 𝓝[≠] a, deriv g x ≠ 0) (hfa : Tendsto f (𝓝[≠] a) (𝓝 0))
    (hga : Tendsto g (𝓝[≠] a) (𝓝 0))
    (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) (𝓝[≠] a) l) :
    Tendsto (fun x => f x / g x) (𝓝[≠] a) l 
Arguments: {a : ℝ} {l : Filter ℝ} {f g : ℝ → ℝ} (hdf : ∀ᶠ (x : ℝ) in 𝓝[≠] a, DifferentiableAt ℝ f x) (hg' : ∀ᶠ (x : ℝ) in 𝓝[≠] a, deriv g x ≠ 0) (hfa : Filter.Tendsto f (𝓝[≠] a) (𝓝 0)) (hga : Filter.Tendsto g (𝓝[≠] a) (𝓝 0)) (hdiv : Filter.Tendsto (fun x => deriv f x / deriv g x) (𝓝[≠] a) l)
Goal: Filter.Tendsto (fun x => f x / g x) (𝓝[≠] a) l
-- deriv.lhopital_zero_nhds
theorem lhopital_zero_nhds (hdf : ∀ᶠ x in 𝓝 a, DifferentiableAt ℝ f x)
    (hg' : ∀ᶠ x in 𝓝 a, deriv g x ≠ 0) (hfa : Tendsto f (𝓝 a) (𝓝 0)) (hga : Tendsto g (𝓝 a) (𝓝 0))
    (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) (𝓝 a) l) :
    Tendsto (fun x => f x / g x) (𝓝[≠] a) l 
Arguments: {a : ℝ} {l : Filter ℝ} {f g : ℝ → ℝ} (hdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x) (hg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0) (hfa : Filter.Tendsto f (𝓝 a) (𝓝 0)) (hga : Filter.Tendsto g (𝓝 a) (𝓝 0)) (hdiv : Filter.Tendsto (fun x => deriv f x / deriv g x) (𝓝 a) l)
Goal: Filter.Tendsto (fun x => f x / g x) (𝓝[≠] a) l
-- deriv.lhopital_zero_atTop
theorem lhopital_zero_atTop (hdf : ∀ᶠ x : ℝ in atTop, DifferentiableAt ℝ f x)
    (hg' : ∀ᶠ x : ℝ in atTop, deriv g x ≠ 0) (hftop : Tendsto f atTop (𝓝 0))
    (hgtop : Tendsto g atTop (𝓝 0)) (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) atTop l) :
    Tendsto (fun x => f x / g x) atTop l 
Arguments: {l : Filter ℝ} {f g : ℝ → ℝ} (hdf : ∀ᶠ (x : ℝ) in Filter.atTop, DifferentiableAt ℝ f x) (hg' : ∀ᶠ (x : ℝ) in Filter.atTop, deriv g x ≠ 0) (hftop : Filter.Tendsto f Filter.atTop (𝓝 0)) (hgtop : Filter.Tendsto g Filter.atTop (𝓝 0)) (hdiv : Filter.Tendsto (fun x => deriv f x / deriv g x) Filter.atTop l)
Goal: Filter.Tendsto (fun x => f x / g x) Filter.atTop l
-- deriv.lhopital_zero_atBot
theorem lhopital_zero_atBot (hdf : ∀ᶠ x : ℝ in atBot, DifferentiableAt ℝ f x)
    (hg' : ∀ᶠ x : ℝ in atBot, deriv g x ≠ 0) (hfbot : Tendsto f atBot (𝓝 0))
    (hgbot : Tendsto g atBot (𝓝 0)) (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) atBot l) :
    Tendsto (fun x => f x / g x) atBot l 
Arguments: {l : Filter ℝ} {f g : ℝ → ℝ} (hdf : ∀ᶠ (x : ℝ) in Filter.atBot, DifferentiableAt ℝ f x) (hg' : ∀ᶠ (x : ℝ) in Filter.atBot, deriv g x ≠ 0) (hfbot : Filter.Tendsto f Filter.atBot (𝓝 0)) (hgbot : Filter.Tendsto g Filter.atBot (𝓝 0)) (hdiv : Filter.Tendsto (fun x => deriv f x / deriv g x) Filter.atBot l)
Goal: Filter.Tendsto (fun x => f x / g x) Filter.atBot l
-- exists_smooth_tsupport_subset
theorem exists_smooth_tsupport_subset {s : Set E} {x : E} (hs : s ∈ 𝓝 x) :
    ∃ f : E → ℝ,
      tsupport f ⊆ s ∧ HasCompactSupport f ∧ ContDiff ℝ ⊤ f ∧ range f ⊆ Icc 0 1 ∧ f x = 1 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Set E} {x : E} (hs : s ∈ 𝓝 x)
Goal: ∃ f, tsupport f ⊆ s ∧ HasCompactSupport f ∧ ContDiff ℝ ⊤ f ∧ Set.range f ⊆ Set.Icc 0 1 ∧ f x = 1
-- IsOpen.exists_smooth_support_eq
theorem IsOpen.exists_smooth_support_eq {s : Set E} (hs : IsOpen s) :
    ∃ f : E → ℝ, f.support = s ∧ ContDiff ℝ ⊤ f ∧ Set.range f ⊆ Set.Icc 0 1 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Set E} (hs : IsOpen s)
Goal: ∃ f, Function.support f = s ∧ ContDiff ℝ ⊤ f ∧ Set.range f ⊆ Set.Icc 0 1
-- ExistsContDiffBumpBase.u_exists
theorem u_exists :
    ∃ u : E → ℝ,
      ContDiff ℝ ⊤ u ∧ (∀ x, u x ∈ Icc (0 : ℝ) 1) ∧ support u = ball 0 1 ∧ ∀ x, u (-x) = u x 
Arguments: (E : Type u_1) [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
Goal: ∃ u, ContDiff ℝ ⊤ u ∧ (∀ (x : E), u x ∈ Set.Icc 0 1) ∧ Function.support u = Metric.ball 0 1 ∧ ∀ (x : E), u (-x) = u x
-- ExistsContDiffBumpBase.u_smooth
theorem u_smooth : ContDiff ℝ ⊤ (u : E → ℝ) 
Arguments: (E : Type u_1) [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
Goal: ContDiff ℝ ⊤ ExistsContDiffBumpBase.u
-- ExistsContDiffBumpBase.u_continuous
theorem u_continuous : Continuous (u : E → ℝ) 
Arguments: (E : Type u_1) [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
Goal: Continuous ExistsContDiffBumpBase.u
-- ExistsContDiffBumpBase.u_support
theorem u_support : support (u : E → ℝ) = ball 0 1 
Arguments: (E : Type u_1) [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
Goal: Function.support ExistsContDiffBumpBase.u = Metric.ball 0 1
-- ExistsContDiffBumpBase.u_compact_support
theorem u_compact_support : HasCompactSupport (u : E → ℝ) 
Arguments: (E : Type u_1) [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
Goal: HasCompactSupport ExistsContDiffBumpBase.u
-- ExistsContDiffBumpBase.u_nonneg
theorem u_nonneg (x : E) : 0 ≤ u x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] (x : E)
Goal: 0 ≤ ExistsContDiffBumpBase.u x
-- ExistsContDiffBumpBase.u_le_one
theorem u_le_one (x : E) : u x ≤ 1 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] (x : E)
Goal: ExistsContDiffBumpBase.u x ≤ 1
-- ExistsContDiffBumpBase.u_neg
theorem u_neg (x : E) : u (-x) = u x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] (x : E)
Goal: ExistsContDiffBumpBase.u (-x) = ExistsContDiffBumpBase.u x
-- ExistsContDiffBumpBase.u_int_pos
theorem u_int_pos : 0 < ∫ x : E, u x ∂μ 
Arguments: (E : Type u_1) [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
Goal: 0 < ∫ (x : E), ExistsContDiffBumpBase.u x ∂MeasureTheory.Measure.addHaar
-- ExistsContDiffBumpBase.w_def
theorem w_def (D : ℝ) :
    (w D : E → ℝ) = fun x => ((∫ x : E, u x ∂μ) * |D| ^ finrank ℝ E)⁻¹ • u (D⁻¹ • x) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] (D : ℝ)
Goal: ExistsContDiffBumpBase.w D = fun x =>
  ((∫ (x : E), ExistsContDiffBumpBase.u x ∂MeasureTheory.Measure.addHaar) * |D| ^ FiniteDimensional.finrank ℝ E)⁻¹ •
    ExistsContDiffBumpBase.u (D⁻¹ • x)
-- ExistsContDiffBumpBase.w_nonneg
theorem w_nonneg (D : ℝ) (x : E) : 0 ≤ w D x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] (D : ℝ) (x : E)
Goal: 0 ≤ ExistsContDiffBumpBase.w D x
-- ExistsContDiffBumpBase.w_mul_φ_nonneg
theorem w_mul_φ_nonneg (D : ℝ) (x y : E) : 0 ≤ w D y * φ (x - y) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] (D : ℝ) (x y : E)
Goal: 0 ≤ ExistsContDiffBumpBase.w D y * ExistsContDiffBumpBase.φ (x - y)
-- ExistsContDiffBumpBase.w_integral
theorem w_integral {D : ℝ} (Dpos : 0 < D) : ∫ x : E, w D x ∂μ = 1 
Arguments: (E : Type u_1) [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {D : ℝ} (Dpos : 0 < D)
Goal: ∫ (x : E), ExistsContDiffBumpBase.w D x ∂MeasureTheory.Measure.addHaar = 1
-- ExistsContDiffBumpBase.w_support
theorem w_support {D : ℝ} (Dpos : 0 < D) : support (w D : E → ℝ) = ball 0 D 
Arguments: (E : Type u_1) [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {D : ℝ} (Dpos : 0 < D)
Goal: Function.support (ExistsContDiffBumpBase.w D) = Metric.ball 0 D
-- ExistsContDiffBumpBase.w_compact_support
theorem w_compact_support {D : ℝ} (Dpos : 0 < D) : HasCompactSupport (w D : E → ℝ) 
Arguments: (E : Type u_1) [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {D : ℝ} (Dpos : 0 < D)
Goal: HasCompactSupport (ExistsContDiffBumpBase.w D)
-- ExistsContDiffBumpBase.y_neg
theorem y_neg (D : ℝ) (x : E) : y D (-x) = y D x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] (D : ℝ) (x : E)
Goal: ExistsContDiffBumpBase.y D (-x) = ExistsContDiffBumpBase.y D x
-- ExistsContDiffBumpBase.y_eq_one_of_mem_closedBall
theorem y_eq_one_of_mem_closedBall {D : ℝ} {x : E} (Dpos : 0 < D)
    (hx : x ∈ closedBall (0 : E) (1 - D)) : y D x = 1 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {D : ℝ} {x : E} (Dpos : 0 < D) (hx : x ∈ Metric.closedBall 0 (1 - D))
Goal: ExistsContDiffBumpBase.y D x = 1
-- ExistsContDiffBumpBase.y_eq_zero_of_not_mem_ball
theorem y_eq_zero_of_not_mem_ball {D : ℝ} {x : E} (Dpos : 0 < D) (hx : x ∉ ball (0 : E) (1 + D)) :
    y D x = 0 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {D : ℝ} {x : E} (Dpos : 0 < D) (hx : x ∉ Metric.ball 0 (1 + D))
Goal: ExistsContDiffBumpBase.y D x = 0
-- ExistsContDiffBumpBase.y_nonneg
theorem y_nonneg (D : ℝ) (x : E) : 0 ≤ y D x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] (D : ℝ) (x : E)
Goal: 0 ≤ ExistsContDiffBumpBase.y D x
-- ExistsContDiffBumpBase.y_le_one
theorem y_le_one {D : ℝ} (x : E) (Dpos : 0 < D) : y D x ≤ 1 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {D : ℝ} (x : E) (Dpos : 0 < D)
Goal: ExistsContDiffBumpBase.y D x ≤ 1
-- ExistsContDiffBumpBase.y_pos_of_mem_ball
theorem y_pos_of_mem_ball {D : ℝ} {x : E} (Dpos : 0 < D) (D_lt_one : D < 1)
    (hx : x ∈ ball (0 : E) (1 + D)) : 0 < y D x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {D : ℝ} {x : E} (Dpos : 0 < D) (D_lt_one : D < 1) (hx : x ∈ Metric.ball 0 (1 + D))
Goal: 0 < ExistsContDiffBumpBase.y D x
-- ExistsContDiffBumpBase.y_smooth
theorem y_smooth : ContDiffOn ℝ ⊤ (uncurry y) (Ioo (0 : ℝ) 1 ×ˢ (univ : Set E)) 
Arguments: (E : Type u_1) [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
Goal: ContDiffOn ℝ ⊤ (Function.uncurry ExistsContDiffBumpBase.y) (Set.Ioo 0 1 ×ˢ Set.univ)
-- ExistsContDiffBumpBase.y_support
theorem y_support {D : ℝ} (Dpos : 0 < D) (D_lt_one : D < 1) :
    support (y D : E → ℝ) = ball (0 : E) (1 + D) 
Arguments: (E : Type u_1) [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {D : ℝ} (Dpos : 0 < D) (D_lt_one : D < 1)
Goal: Function.support (ExistsContDiffBumpBase.y D) = Metric.ball 0 (1 + D)