ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Filter.tendsto_atTop_of_add_const_right
theorem tendsto_atTop_of_add_const_right (C : β) (hf : Tendsto (fun x => f x + C) l atTop) :
    Tendsto f l atTop 
Arguments: {α : Type u_3} {β : Type u_4} [OrderedCancelAddCommMonoid β] {l : Filter α} {f : α → β} (C : β) (hf : Filter.Tendsto (fun x => f x + C) l Filter.atTop)
Goal: Filter.Tendsto f l Filter.atTop
-- Filter.tendsto_atBot_of_add_const_right
theorem tendsto_atBot_of_add_const_right (C : β) (hf : Tendsto (fun x => f x + C) l atBot) :
    Tendsto f l atBot 
Arguments: {α : Type u_3} {β : Type u_4} [OrderedCancelAddCommMonoid β] {l : Filter α} {f : α → β} (C : β) (hf : Filter.Tendsto (fun x => f x + C) l Filter.atBot)
Goal: Filter.Tendsto f l Filter.atBot
-- Filter.tendsto_atTop_of_add_bdd_above_left'
theorem tendsto_atTop_of_add_bdd_above_left' (C) (hC : ∀ᶠ x in l, f x ≤ C)
    (h : Tendsto (fun x => f x + g x) l atTop) : Tendsto g l atTop 
Arguments: {α : Type u_3} {β : Type u_4} [OrderedCancelAddCommMonoid β] {l : Filter α} {f g : α → β} (C : β) (hC : ∀ᶠ (x : α) in l, f x ≤ C) (h : Filter.Tendsto (fun x => f x + g x) l Filter.atTop)
Goal: Filter.Tendsto g l Filter.atTop
-- Filter.tendsto_atBot_of_add_bdd_below_left'
theorem tendsto_atBot_of_add_bdd_below_left' (C) (hC : ∀ᶠ x in l, C ≤ f x)
    (h : Tendsto (fun x => f x + g x) l atBot) : Tendsto g l atBot 
Arguments: {α : Type u_3} {β : Type u_4} [OrderedCancelAddCommMonoid β] {l : Filter α} {f g : α → β} (C : β) (hC : ∀ᶠ (x : α) in l, C ≤ f x) (h : Filter.Tendsto (fun x => f x + g x) l Filter.atBot)
Goal: Filter.Tendsto g l Filter.atBot
-- Filter.tendsto_atTop_of_add_bdd_above_left
theorem tendsto_atTop_of_add_bdd_above_left (C) (hC : ∀ x, f x ≤ C) :
    Tendsto (fun x => f x + g x) l atTop → Tendsto g l atTop 
Arguments: {α : Type u_3} {β : Type u_4} [OrderedCancelAddCommMonoid β] {l : Filter α} {f g : α → β} (C : β) (hC : ∀ (x : α), f x ≤ C)
Goal: Filter.Tendsto (fun x => f x + g x) l Filter.atTop → Filter.Tendsto g l Filter.atTop
-- Filter.tendsto_atBot_of_add_bdd_below_left
theorem tendsto_atBot_of_add_bdd_below_left (C) (hC : ∀ x, C ≤ f x) :
    Tendsto (fun x => f x + g x) l atBot → Tendsto g l atBot 
Arguments: {α : Type u_3} {β : Type u_4} [OrderedCancelAddCommMonoid β] {l : Filter α} {f g : α → β} (C : β) (hC : ∀ (x : α), C ≤ f x)
Goal: Filter.Tendsto (fun x => f x + g x) l Filter.atBot → Filter.Tendsto g l Filter.atBot
-- Filter.tendsto_atTop_of_add_bdd_above_right'
theorem tendsto_atTop_of_add_bdd_above_right' (C) (hC : ∀ᶠ x in l, g x ≤ C)
    (h : Tendsto (fun x => f x + g x) l atTop) : Tendsto f l atTop 
Arguments: {α : Type u_3} {β : Type u_4} [OrderedCancelAddCommMonoid β] {l : Filter α} {f g : α → β} (C : β) (hC : ∀ᶠ (x : α) in l, g x ≤ C) (h : Filter.Tendsto (fun x => f x + g x) l Filter.atTop)
Goal: Filter.Tendsto f l Filter.atTop
-- Filter.tendsto_atBot_of_add_bdd_below_right'
theorem tendsto_atBot_of_add_bdd_below_right' (C) (hC : ∀ᶠ x in l, C ≤ g x)
    (h : Tendsto (fun x => f x + g x) l atBot) : Tendsto f l atBot 
Arguments: {α : Type u_3} {β : Type u_4} [OrderedCancelAddCommMonoid β] {l : Filter α} {f g : α → β} (C : β) (hC : ∀ᶠ (x : α) in l, C ≤ g x) (h : Filter.Tendsto (fun x => f x + g x) l Filter.atBot)
Goal: Filter.Tendsto f l Filter.atBot
-- Filter.tendsto_atTop_of_add_bdd_above_right
theorem tendsto_atTop_of_add_bdd_above_right (C) (hC : ∀ x, g x ≤ C) :
    Tendsto (fun x => f x + g x) l atTop → Tendsto f l atTop 
Arguments: {α : Type u_3} {β : Type u_4} [OrderedCancelAddCommMonoid β] {l : Filter α} {f g : α → β} (C : β) (hC : ∀ (x : α), g x ≤ C)
Goal: Filter.Tendsto (fun x => f x + g x) l Filter.atTop → Filter.Tendsto f l Filter.atTop
-- Filter.tendsto_atBot_of_add_bdd_below_right
theorem tendsto_atBot_of_add_bdd_below_right (C) (hC : ∀ x, C ≤ g x) :
    Tendsto (fun x => f x + g x) l atBot → Tendsto f l atBot 
Arguments: {α : Type u_3} {β : Type u_4} [OrderedCancelAddCommMonoid β] {l : Filter α} {f g : α → β} (C : β) (hC : ∀ (x : α), C ≤ g x)
Goal: Filter.Tendsto (fun x => f x + g x) l Filter.atBot → Filter.Tendsto f l Filter.atBot
-- Filter.tendsto_atTop_add_left_of_le'
theorem tendsto_atTop_add_left_of_le' (C : β) (hf : ∀ᶠ x in l, C ≤ f x) (hg : Tendsto g l atTop) :
    Tendsto (fun x => f x + g x) l atTop 
Arguments: {α : Type u_3} {β : Type u_4} [OrderedAddCommGroup β] (l : Filter α) {f g : α → β} (C : β) (hf : ∀ᶠ (x : α) in l, C ≤ f x) (hg : Filter.Tendsto g l Filter.atTop)
Goal: Filter.Tendsto (fun x => f x + g x) l Filter.atTop
-- Filter.tendsto_atBot_add_left_of_ge'
theorem tendsto_atBot_add_left_of_ge' (C : β) (hf : ∀ᶠ x in l, f x ≤ C) (hg : Tendsto g l atBot) :
    Tendsto (fun x => f x + g x) l atBot 
Arguments: {α : Type u_3} {β : Type u_4} [OrderedAddCommGroup β] (l : Filter α) {f g : α → β} (C : β) (hf : ∀ᶠ (x : α) in l, f x ≤ C) (hg : Filter.Tendsto g l Filter.atBot)
Goal: Filter.Tendsto (fun x => f x + g x) l Filter.atBot
-- Filter.tendsto_atTop_add_left_of_le
theorem tendsto_atTop_add_left_of_le (C : β) (hf : ∀ x, C ≤ f x) (hg : Tendsto g l atTop) :
    Tendsto (fun x => f x + g x) l atTop 
Arguments: {α : Type u_3} {β : Type u_4} [OrderedAddCommGroup β] (l : Filter α) {f g : α → β} (C : β) (hf : ∀ (x : α), C ≤ f x) (hg : Filter.Tendsto g l Filter.atTop)
Goal: Filter.Tendsto (fun x => f x + g x) l Filter.atTop
-- Filter.tendsto_atBot_add_left_of_ge
theorem tendsto_atBot_add_left_of_ge (C : β) (hf : ∀ x, f x ≤ C) (hg : Tendsto g l atBot) :
    Tendsto (fun x => f x + g x) l atBot 
Arguments: {α : Type u_3} {β : Type u_4} [OrderedAddCommGroup β] (l : Filter α) {f g : α → β} (C : β) (hf : ∀ (x : α), f x ≤ C) (hg : Filter.Tendsto g l Filter.atBot)
Goal: Filter.Tendsto (fun x => f x + g x) l Filter.atBot
-- Filter.tendsto_atTop_add_right_of_le'
theorem tendsto_atTop_add_right_of_le' (C : β) (hf : Tendsto f l atTop) (hg : ∀ᶠ x in l, C ≤ g x) :
    Tendsto (fun x => f x + g x) l atTop 
Arguments: {α : Type u_3} {β : Type u_4} [OrderedAddCommGroup β] (l : Filter α) {f g : α → β} (C : β) (hf : Filter.Tendsto f l Filter.atTop) (hg : ∀ᶠ (x : α) in l, C ≤ g x)
Goal: Filter.Tendsto (fun x => f x + g x) l Filter.atTop
-- Filter.tendsto_atBot_add_right_of_ge'
theorem tendsto_atBot_add_right_of_ge' (C : β) (hf : Tendsto f l atBot) (hg : ∀ᶠ x in l, g x ≤ C) :
    Tendsto (fun x => f x + g x) l atBot 
Arguments: {α : Type u_3} {β : Type u_4} [OrderedAddCommGroup β] (l : Filter α) {f g : α → β} (C : β) (hf : Filter.Tendsto f l Filter.atBot) (hg : ∀ᶠ (x : α) in l, g x ≤ C)
Goal: Filter.Tendsto (fun x => f x + g x) l Filter.atBot
-- Filter.tendsto_atTop_add_right_of_le
theorem tendsto_atTop_add_right_of_le (C : β) (hf : Tendsto f l atTop) (hg : ∀ x, C ≤ g x) :
    Tendsto (fun x => f x + g x) l atTop 
Arguments: {α : Type u_3} {β : Type u_4} [OrderedAddCommGroup β] (l : Filter α) {f g : α → β} (C : β) (hf : Filter.Tendsto f l Filter.atTop) (hg : ∀ (x : α), C ≤ g x)
Goal: Filter.Tendsto (fun x => f x + g x) l Filter.atTop
-- Filter.tendsto_atBot_add_right_of_ge
theorem tendsto_atBot_add_right_of_ge (C : β) (hf : Tendsto f l atBot) (hg : ∀ x, g x ≤ C) :
    Tendsto (fun x => f x + g x) l atBot 
Arguments: {α : Type u_3} {β : Type u_4} [OrderedAddCommGroup β] (l : Filter α) {f g : α → β} (C : β) (hf : Filter.Tendsto f l Filter.atBot) (hg : ∀ (x : α), g x ≤ C)
Goal: Filter.Tendsto (fun x => f x + g x) l Filter.atBot
-- Filter.tendsto_atTop_add_const_left
theorem tendsto_atTop_add_const_left (C : β) (hf : Tendsto f l atTop) :
    Tendsto (fun x => C + f x) l atTop 
Arguments: {α : Type u_3} {β : Type u_4} [OrderedAddCommGroup β] (l : Filter α) {f : α → β} (C : β) (hf : Filter.Tendsto f l Filter.atTop)
Goal: Filter.Tendsto (fun x => C + f x) l Filter.atTop
-- Filter.tendsto_atBot_add_const_left
theorem tendsto_atBot_add_const_left (C : β) (hf : Tendsto f l atBot) :
    Tendsto (fun x => C + f x) l atBot 
Arguments: {α : Type u_3} {β : Type u_4} [OrderedAddCommGroup β] (l : Filter α) {f : α → β} (C : β) (hf : Filter.Tendsto f l Filter.atBot)
Goal: Filter.Tendsto (fun x => C + f x) l Filter.atBot
-- Filter.tendsto_atTop_add_const_right
theorem tendsto_atTop_add_const_right (C : β) (hf : Tendsto f l atTop) :
    Tendsto (fun x => f x + C) l atTop 
Arguments: {α : Type u_3} {β : Type u_4} [OrderedAddCommGroup β] (l : Filter α) {f : α → β} (C : β) (hf : Filter.Tendsto f l Filter.atTop)
Goal: Filter.Tendsto (fun x => f x + C) l Filter.atTop
-- Filter.tendsto_atBot_add_const_right
theorem tendsto_atBot_add_const_right (C : β) (hf : Tendsto f l atBot) :
    Tendsto (fun x => f x + C) l atBot 
Arguments: {α : Type u_3} {β : Type u_4} [OrderedAddCommGroup β] (l : Filter α) {f : α → β} (C : β) (hf : Filter.Tendsto f l Filter.atBot)
Goal: Filter.Tendsto (fun x => f x + C) l Filter.atBot
-- Filter.map_neg_atBot
theorem map_neg_atBot : map (Neg.neg : β → β) atBot = atTop 
Arguments: {β : Type u_4} [OrderedAddCommGroup β]
Goal: Filter.map Neg.neg Filter.atBot = Filter.atTop
-- Filter.map_neg_atTop
theorem map_neg_atTop : map (Neg.neg : β → β) atTop = atBot 
Arguments: {β : Type u_4} [OrderedAddCommGroup β]
Goal: Filter.map Neg.neg Filter.atTop = Filter.atBot
-- Filter.comap_neg_atBot
theorem comap_neg_atBot : comap (Neg.neg : β → β) atBot = atTop 
Arguments: {β : Type u_4} [OrderedAddCommGroup β]
Goal: Filter.comap Neg.neg Filter.atBot = Filter.atTop
-- Filter.comap_neg_atTop
theorem comap_neg_atTop : comap (Neg.neg : β → β) atTop = atBot 
Arguments: {β : Type u_4} [OrderedAddCommGroup β]
Goal: Filter.comap Neg.neg Filter.atTop = Filter.atBot
-- Filter.tendsto_neg_atTop_atBot
theorem tendsto_neg_atTop_atBot : Tendsto (Neg.neg : β → β) atTop atBot 
Arguments: {β : Type u_4} [OrderedAddCommGroup β]
Goal: Filter.Tendsto Neg.neg Filter.atTop Filter.atBot
-- Filter.tendsto_neg_atBot_atTop
theorem tendsto_neg_atBot_atTop : Tendsto (Neg.neg : β → β) atBot atTop 
Arguments: {β : Type u_4} [OrderedAddCommGroup β]
Goal: Filter.Tendsto Neg.neg Filter.atBot Filter.atTop
-- Filter.tendsto_neg_atTop_iff
@[simp]
theorem tendsto_neg_atTop_iff : Tendsto (fun x => -f x) l atTop ↔ Tendsto f l atBot 
Arguments: {α : Type u_3} {β : Type u_4} [OrderedAddCommGroup β] {l : Filter α} {f : α → β}
Goal: Filter.Tendsto (fun x => -f x) l Filter.atTop ↔ Filter.Tendsto f l Filter.atBot
-- Filter.tendsto_neg_atBot_iff
@[simp]
theorem tendsto_neg_atBot_iff : Tendsto (fun x => -f x) l atBot ↔ Tendsto f l atTop 
Arguments: {α : Type u_3} {β : Type u_4} [OrderedAddCommGroup β] {l : Filter α} {f : α → β}
Goal: Filter.Tendsto (fun x => -f x) l Filter.atBot ↔ Filter.Tendsto f l Filter.atTop
-- Filter.Tendsto.atTop_mul_atTop
theorem Tendsto.atTop_mul_atTop (hf : Tendsto f l atTop) (hg : Tendsto g l atTop) :
    Tendsto (fun x => f x * g x) l atTop 
Arguments: {α : Type u_3} {β : Type u_4} [OrderedSemiring α] {l : Filter β} {f g : β → α} (hf : Filter.Tendsto f l Filter.atTop) (hg : Filter.Tendsto g l Filter.atTop)
Goal: Filter.Tendsto (fun x => f x * g x) l Filter.atTop
-- Filter.tendsto_mul_self_atTop
theorem tendsto_mul_self_atTop : Tendsto (fun x : α => x * x) atTop atTop 
Arguments: {α : Type u_3} [OrderedSemiring α]
Goal: Filter.Tendsto (fun x => x * x) Filter.atTop Filter.atTop
-- Filter.tendsto_pow_atTop
theorem tendsto_pow_atTop {n : ℕ} (hn : n ≠ 0) : Tendsto (fun x : α => x ^ n) atTop atTop 
Arguments: {α : Type u_3} [OrderedSemiring α] {n : ℕ} (hn : n ≠ 0)
Goal: Filter.Tendsto (fun x => x ^ n) Filter.atTop Filter.atTop
-- Filter.zero_pow_eventuallyEq
theorem zero_pow_eventuallyEq [MonoidWithZero α] :
    (fun n : ℕ => (0 : α) ^ n) =ᶠ[atTop] fun _ => 0 
Arguments: {α : Type u_3} [MonoidWithZero α]
Goal: (fun n => 0 ^ n) =ᶠ[Filter.atTop] fun x => 0
-- Filter.Tendsto.atTop_mul_atBot
theorem Tendsto.atTop_mul_atBot (hf : Tendsto f l atTop) (hg : Tendsto g l atBot) :
    Tendsto (fun x => f x * g x) l atBot 
Arguments: {α : Type u_3} {β : Type u_4} [OrderedRing α] {l : Filter β} {f g : β → α} (hf : Filter.Tendsto f l Filter.atTop) (hg : Filter.Tendsto g l Filter.atBot)
Goal: Filter.Tendsto (fun x => f x * g x) l Filter.atBot
-- Filter.Tendsto.atBot_mul_atTop
theorem Tendsto.atBot_mul_atTop (hf : Tendsto f l atBot) (hg : Tendsto g l atTop) :
    Tendsto (fun x => f x * g x) l atBot 
Arguments: {α : Type u_3} {β : Type u_4} [OrderedRing α] {l : Filter β} {f g : β → α} (hf : Filter.Tendsto f l Filter.atBot) (hg : Filter.Tendsto g l Filter.atTop)
Goal: Filter.Tendsto (fun x => f x * g x) l Filter.atBot
-- Filter.Tendsto.atBot_mul_atBot
theorem Tendsto.atBot_mul_atBot (hf : Tendsto f l atBot) (hg : Tendsto g l atBot) :
    Tendsto (fun x => f x * g x) l atTop 
Arguments: {α : Type u_3} {β : Type u_4} [OrderedRing α] {l : Filter β} {f g : β → α} (hf : Filter.Tendsto f l Filter.atBot) (hg : Filter.Tendsto g l Filter.atBot)
Goal: Filter.Tendsto (fun x => f x * g x) l Filter.atTop
-- Filter.tendsto_abs_atTop_atTop
theorem tendsto_abs_atTop_atTop : Tendsto (abs : α → α) atTop atTop 
Arguments: {α : Type u_3} [LinearOrderedAddCommGroup α]
Goal: Filter.Tendsto abs Filter.atTop Filter.atTop
-- Filter.tendsto_abs_atBot_atTop
theorem tendsto_abs_atBot_atTop : Tendsto (abs : α → α) atBot atTop 
Arguments: {α : Type u_3} [LinearOrderedAddCommGroup α]
Goal: Filter.Tendsto abs Filter.atBot Filter.atTop
-- Filter.comap_abs_atTop
@[simp]
theorem comap_abs_atTop : comap (abs : α → α) atTop = atBot ⊔ atTop 
Arguments: {α : Type u_3} [LinearOrderedAddCommGroup α]
Goal: Filter.comap abs Filter.atTop = Filter.atBot ⊔ Filter.atTop
-- Filter.Tendsto.atTop_of_const_mul
theorem Tendsto.atTop_of_const_mul {c : α} (hc : 0 < c) (hf : Tendsto (fun x => c * f x) l atTop) :
    Tendsto f l atTop 
Arguments: {α : Type u_3} {β : Type u_4} [LinearOrderedSemiring α] {l : Filter β} {f : β → α} {c : α} (hc : 0 < c) (hf : Filter.Tendsto (fun x => c * f x) l Filter.atTop)
Goal: Filter.Tendsto f l Filter.atTop
-- Filter.Tendsto.atTop_of_mul_const
theorem Tendsto.atTop_of_mul_const {c : α} (hc : 0 < c) (hf : Tendsto (fun x => f x * c) l atTop) :
    Tendsto f l atTop 
Arguments: {α : Type u_3} {β : Type u_4} [LinearOrderedSemiring α] {l : Filter β} {f : β → α} {c : α} (hc : 0 < c) (hf : Filter.Tendsto (fun x => f x * c) l Filter.atTop)
Goal: Filter.Tendsto f l Filter.atTop
-- Filter.tendsto_pow_atTop_iff
@[simp]
theorem tendsto_pow_atTop_iff {n : ℕ} : Tendsto (fun x : α => x ^ n) atTop atTop ↔ n ≠ 0 
Arguments: {α : Type u_3} [LinearOrderedSemiring α] {n : ℕ}
Goal: Filter.Tendsto (fun x => x ^ n) Filter.atTop Filter.atTop ↔ n ≠ 0
-- Filter.not_tendsto_pow_atTop_atBot
theorem not_tendsto_pow_atTop_atBot [LinearOrderedRing α] :
    ∀ {n : ℕ}, ¬Tendsto (fun x : α => x ^ n) atTop atBot
  
Arguments: {α : Type u_3} [LinearOrderedRing α] {n : ℕ}
Goal: ¬Filter.Tendsto (fun x => x ^ n) Filter.atTop Filter.atBot
-- Filter.tendsto_const_mul_atTop_of_pos
theorem tendsto_const_mul_atTop_of_pos (hr : 0 < r) :
    Tendsto (fun x => r * f x) l atTop ↔ Tendsto f l atTop 
Arguments: {α : Type u_3} {β : Type u_4} [LinearOrderedSemifield α] {l : Filter β} {f : β → α} {r : α} (hr : 0 < r)
Goal: Filter.Tendsto (fun x => r * f x) l Filter.atTop ↔ Filter.Tendsto f l Filter.atTop
-- Filter.tendsto_mul_const_atTop_of_pos
theorem tendsto_mul_const_atTop_of_pos (hr : 0 < r) :
    Tendsto (fun x => f x * r) l atTop ↔ Tendsto f l atTop 
Arguments: {α : Type u_3} {β : Type u_4} [LinearOrderedSemifield α] {l : Filter β} {f : β → α} {r : α} (hr : 0 < r)
Goal: Filter.Tendsto (fun x => f x * r) l Filter.atTop ↔ Filter.Tendsto f l Filter.atTop
-- Filter.tendsto_div_const_atTop_of_pos
lemma tendsto_div_const_atTop_of_pos (hr : 0 < r) :
    Tendsto (fun x ↦ f x / r) l atTop ↔ Tendsto f l atTop 
Arguments: {α : Type u_3} {β : Type u_4} [LinearOrderedSemifield α] {l : Filter β} {f : β → α} {r : α} (hr : 0 < r)
Goal: Filter.Tendsto (fun x => f x / r) l Filter.atTop ↔ Filter.Tendsto f l Filter.atTop
-- Filter.tendsto_const_mul_atTop_iff_pos
theorem tendsto_const_mul_atTop_iff_pos [NeBot l] (h : Tendsto f l atTop) :
    Tendsto (fun x => r * f x) l atTop ↔ 0 < r 
Arguments: {α : Type u_3} {β : Type u_4} [LinearOrderedSemifield α] {l : Filter β} {f : β → α} {r : α} [l.NeBot] (h : Filter.Tendsto f l Filter.atTop)
Goal: Filter.Tendsto (fun x => r * f x) l Filter.atTop ↔ 0 < r
-- Filter.tendsto_mul_const_atTop_iff_pos
theorem tendsto_mul_const_atTop_iff_pos [NeBot l] (h : Tendsto f l atTop) :
    Tendsto (fun x => f x * r) l atTop ↔ 0 < r 
Arguments: {α : Type u_3} {β : Type u_4} [LinearOrderedSemifield α] {l : Filter β} {f : β → α} {r : α} [l.NeBot] (h : Filter.Tendsto f l Filter.atTop)
Goal: Filter.Tendsto (fun x => f x * r) l Filter.atTop ↔ 0 < r
-- Filter.tendsto_div_const_atTop_iff_pos
lemma tendsto_div_const_atTop_iff_pos [NeBot l] (h : Tendsto f l atTop) :
    Tendsto (fun x ↦ f x / r) l atTop ↔ 0 < r 
Arguments: {α : Type u_3} {β : Type u_4} [LinearOrderedSemifield α] {l : Filter β} {f : β → α} {r : α} [l.NeBot] (h : Filter.Tendsto f l Filter.atTop)
Goal: Filter.Tendsto (fun x => f x / r) l Filter.atTop ↔ 0 < r
-- Filter.Tendsto.const_mul_atTop
theorem Tendsto.const_mul_atTop (hr : 0 < r) (hf : Tendsto f l atTop) :
    Tendsto (fun x => r * f x) l atTop 
Arguments: {α : Type u_3} {β : Type u_4} [LinearOrderedSemifield α] {l : Filter β} {f : β → α} {r : α} (hr : 0 < r) (hf : Filter.Tendsto f l Filter.atTop)
Goal: Filter.Tendsto (fun x => r * f x) l Filter.atTop
-- Filter.Tendsto.atTop_mul_const
theorem Tendsto.atTop_mul_const (hr : 0 < r) (hf : Tendsto f l atTop) :
    Tendsto (fun x => f x * r) l atTop 
Arguments: {α : Type u_3} {β : Type u_4} [LinearOrderedSemifield α] {l : Filter β} {f : β → α} {r : α} (hr : 0 < r) (hf : Filter.Tendsto f l Filter.atTop)
Goal: Filter.Tendsto (fun x => f x * r) l Filter.atTop
-- Filter.Tendsto.atTop_div_const
theorem Tendsto.atTop_div_const (hr : 0 < r) (hf : Tendsto f l atTop) :
    Tendsto (fun x => f x / r) l atTop 
Arguments: {α : Type u_3} {β : Type u_4} [LinearOrderedSemifield α] {l : Filter β} {f : β → α} {r : α} (hr : 0 < r) (hf : Filter.Tendsto f l Filter.atTop)
Goal: Filter.Tendsto (fun x => f x / r) l Filter.atTop
-- Filter.tendsto_const_mul_pow_atTop
theorem tendsto_const_mul_pow_atTop (hn : n ≠ 0) (hc : 0 < c) :
    Tendsto (fun x => c * x ^ n) atTop atTop 
Arguments: {α : Type u_3} [LinearOrderedSemifield α] {c : α} {n : ℕ} (hn : n ≠ 0) (hc : 0 < c)
Goal: Filter.Tendsto (fun x => c * x ^ n) Filter.atTop Filter.atTop
-- Filter.tendsto_const_mul_pow_atTop_iff
theorem tendsto_const_mul_pow_atTop_iff :
    Tendsto (fun x => c * x ^ n) atTop atTop ↔ n ≠ 0 ∧ 0 < c 
Arguments: {α : Type u_3} [LinearOrderedSemifield α] {c : α} {n : ℕ}
Goal: Filter.Tendsto (fun x => c * x ^ n) Filter.atTop Filter.atTop ↔ n ≠ 0 ∧ 0 < c
-- Filter.tendsto_zpow_atTop_atTop
lemma tendsto_zpow_atTop_atTop {n : ℤ} (hn : 0 < n) : Tendsto (fun x : α ↦ x ^ n) atTop atTop 
Arguments: {α : Type u_3} [LinearOrderedSemifield α] {n : ℤ} (hn : 0 < n)
Goal: Filter.Tendsto (fun x => x ^ n) Filter.atTop Filter.atTop
-- Filter.tendsto_const_mul_atBot_of_pos
theorem tendsto_const_mul_atBot_of_pos (hr : 0 < r) :
    Tendsto (fun x => r * f x) l atBot ↔ Tendsto f l atBot 
Arguments: {α : Type u_3} {β : Type u_4} [LinearOrderedField α] {l : Filter β} {f : β → α} {r : α} (hr : 0 < r)
Goal: Filter.Tendsto (fun x => r * f x) l Filter.atBot ↔ Filter.Tendsto f l Filter.atBot
-- Filter.tendsto_mul_const_atBot_of_pos
theorem tendsto_mul_const_atBot_of_pos (hr : 0 < r) :
    Tendsto (fun x => f x * r) l atBot ↔ Tendsto f l atBot 
Arguments: {α : Type u_3} {β : Type u_4} [LinearOrderedField α] {l : Filter β} {f : β → α} {r : α} (hr : 0 < r)
Goal: Filter.Tendsto (fun x => f x * r) l Filter.atBot ↔ Filter.Tendsto f l Filter.atBot
-- Filter.tendsto_div_const_atBot_of_pos
lemma tendsto_div_const_atBot_of_pos (hr : 0 < r) :
    Tendsto (fun x ↦ f x / r) l atBot ↔ Tendsto f l atBot 
Arguments: {α : Type u_3} {β : Type u_4} [LinearOrderedField α] {l : Filter β} {f : β → α} {r : α} (hr : 0 < r)
Goal: Filter.Tendsto (fun x => f x / r) l Filter.atBot ↔ Filter.Tendsto f l Filter.atBot
-- Filter.tendsto_const_mul_atTop_of_neg
theorem tendsto_const_mul_atTop_of_neg (hr : r < 0) :
    Tendsto (fun x => r * f x) l atTop ↔ Tendsto f l atBot 
Arguments: {α : Type u_3} {β : Type u_4} [LinearOrderedField α] {l : Filter β} {f : β → α} {r : α} (hr : r < 0)
Goal: Filter.Tendsto (fun x => r * f x) l Filter.atTop ↔ Filter.Tendsto f l Filter.atBot
-- Filter.tendsto_mul_const_atTop_of_neg
theorem tendsto_mul_const_atTop_of_neg (hr : r < 0) :
    Tendsto (fun x => f x * r) l atTop ↔ Tendsto f l atBot 
Arguments: {α : Type u_3} {β : Type u_4} [LinearOrderedField α] {l : Filter β} {f : β → α} {r : α} (hr : r < 0)
Goal: Filter.Tendsto (fun x => f x * r) l Filter.atTop ↔ Filter.Tendsto f l Filter.atBot
-- Filter.tendsto_div_const_atTop_of_neg
lemma tendsto_div_const_atTop_of_neg (hr : r < 0) :
    Tendsto (fun x ↦ f x / r) l atTop ↔ Tendsto f l atBot 
Arguments: {α : Type u_3} {β : Type u_4} [LinearOrderedField α] {l : Filter β} {f : β → α} {r : α} (hr : r < 0)
Goal: Filter.Tendsto (fun x => f x / r) l Filter.atTop ↔ Filter.Tendsto f l Filter.atBot
-- Filter.tendsto_const_mul_atBot_of_neg
theorem tendsto_const_mul_atBot_of_neg (hr : r < 0) :
    Tendsto (fun x => r * f x) l atBot ↔ Tendsto f l atTop 
Arguments: {α : Type u_3} {β : Type u_4} [LinearOrderedField α] {l : Filter β} {f : β → α} {r : α} (hr : r < 0)
Goal: Filter.Tendsto (fun x => r * f x) l Filter.atBot ↔ Filter.Tendsto f l Filter.atTop
-- Filter.tendsto_mul_const_atBot_of_neg
theorem tendsto_mul_const_atBot_of_neg (hr : r < 0) :
    Tendsto (fun x => f x * r) l atBot ↔ Tendsto f l atTop 
Arguments: {α : Type u_3} {β : Type u_4} [LinearOrderedField α] {l : Filter β} {f : β → α} {r : α} (hr : r < 0)
Goal: Filter.Tendsto (fun x => f x * r) l Filter.atBot ↔ Filter.Tendsto f l Filter.atTop
-- Filter.tendsto_div_const_atBot_of_neg
lemma tendsto_div_const_atBot_of_neg (hr : r < 0) :
    Tendsto (fun x ↦ f x / r) l atBot ↔ Tendsto f l atTop 
Arguments: {α : Type u_3} {β : Type u_4} [LinearOrderedField α] {l : Filter β} {f : β → α} {r : α} (hr : r < 0)
Goal: Filter.Tendsto (fun x => f x / r) l Filter.atBot ↔ Filter.Tendsto f l Filter.atTop
-- Filter.tendsto_const_mul_atTop_iff
theorem tendsto_const_mul_atTop_iff [NeBot l] :
    Tendsto (fun x => r * f x) l atTop ↔ 0 < r ∧ Tendsto f l atTop ∨ r < 0 ∧ Tendsto f l atBot 
Arguments: {α : Type u_3} {β : Type u_4} [LinearOrderedField α] {l : Filter β} {f : β → α} {r : α} [l.NeBot]
Goal: Filter.Tendsto (fun x => r * f x) l Filter.atTop ↔
  0 < r ∧ Filter.Tendsto f l Filter.atTop ∨ r < 0 ∧ Filter.Tendsto f l Filter.atBot
-- Filter.tendsto_mul_const_atTop_iff
theorem tendsto_mul_const_atTop_iff [NeBot l] :
    Tendsto (fun x => f x * r) l atTop ↔ 0 < r ∧ Tendsto f l atTop ∨ r < 0 ∧ Tendsto f l atBot 
Arguments: {α : Type u_3} {β : Type u_4} [LinearOrderedField α] {l : Filter β} {f : β → α} {r : α} [l.NeBot]
Goal: Filter.Tendsto (fun x => f x * r) l Filter.atTop ↔
  0 < r ∧ Filter.Tendsto f l Filter.atTop ∨ r < 0 ∧ Filter.Tendsto f l Filter.atBot
-- Filter.tendsto_div_const_atTop_iff
lemma tendsto_div_const_atTop_iff [NeBot l] :
    Tendsto (fun x ↦ f x / r) l atTop ↔ 0 < r ∧ Tendsto f l atTop ∨ r < 0 ∧ Tendsto f l atBot 
Arguments: {α : Type u_3} {β : Type u_4} [LinearOrderedField α] {l : Filter β} {f : β → α} {r : α} [l.NeBot]
Goal: Filter.Tendsto (fun x => f x / r) l Filter.atTop ↔
  0 < r ∧ Filter.Tendsto f l Filter.atTop ∨ r < 0 ∧ Filter.Tendsto f l Filter.atBot
-- Filter.tendsto_const_mul_atBot_iff
theorem tendsto_const_mul_atBot_iff [NeBot l] :
    Tendsto (fun x => r * f x) l atBot ↔ 0 < r ∧ Tendsto f l atBot ∨ r < 0 ∧ Tendsto f l atTop 
Arguments: {α : Type u_3} {β : Type u_4} [LinearOrderedField α] {l : Filter β} {f : β → α} {r : α} [l.NeBot]
Goal: Filter.Tendsto (fun x => r * f x) l Filter.atBot ↔
  0 < r ∧ Filter.Tendsto f l Filter.atBot ∨ r < 0 ∧ Filter.Tendsto f l Filter.atTop
-- Filter.tendsto_mul_const_atBot_iff
theorem tendsto_mul_const_atBot_iff [NeBot l] :
    Tendsto (fun x => f x * r) l atBot ↔ 0 < r ∧ Tendsto f l atBot ∨ r < 0 ∧ Tendsto f l atTop 
Arguments: {α : Type u_3} {β : Type u_4} [LinearOrderedField α] {l : Filter β} {f : β → α} {r : α} [l.NeBot]
Goal: Filter.Tendsto (fun x => f x * r) l Filter.atBot ↔
  0 < r ∧ Filter.Tendsto f l Filter.atBot ∨ r < 0 ∧ Filter.Tendsto f l Filter.atTop