ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Filter.tendsto_div_const_atBot_iff
lemma tendsto_div_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
-- Filter.tendsto_const_mul_atTop_iff_neg
theorem tendsto_const_mul_atTop_iff_neg [NeBot l] (h : Tendsto f l atBot) :
    Tendsto (fun x => r * f x) l atTop ↔ r < 0 
Arguments: {α : Type u_3} {β : Type u_4} [LinearOrderedField α] {l : Filter β} {f : β → α} {r : α} [l.NeBot] (h : Filter.Tendsto f l Filter.atBot)
Goal: Filter.Tendsto (fun x => r * f x) l Filter.atTop ↔ r < 0
-- Filter.tendsto_mul_const_atTop_iff_neg
theorem tendsto_mul_const_atTop_iff_neg [NeBot l] (h : Tendsto f l atBot) :
    Tendsto (fun x => f x * r) l atTop ↔ r < 0 
Arguments: {α : Type u_3} {β : Type u_4} [LinearOrderedField α] {l : Filter β} {f : β → α} {r : α} [l.NeBot] (h : Filter.Tendsto f l Filter.atBot)
Goal: Filter.Tendsto (fun x => f x * r) l Filter.atTop ↔ r < 0
-- Filter.tendsto_div_const_atTop_iff_neg
lemma tendsto_div_const_atTop_iff_neg [NeBot l] (h : Tendsto f l atBot) :
    Tendsto (fun x ↦ f x / r) l atTop ↔ r < 0 
Arguments: {α : Type u_3} {β : Type u_4} [LinearOrderedField α] {l : Filter β} {f : β → α} {r : α} [l.NeBot] (h : Filter.Tendsto f l Filter.atBot)
Goal: Filter.Tendsto (fun x => f x / r) l Filter.atTop ↔ r < 0
-- Filter.tendsto_const_mul_atBot_iff_pos
theorem tendsto_const_mul_atBot_iff_pos [NeBot l] (h : Tendsto f l atBot) :
    Tendsto (fun x => r * f x) l atBot ↔ 0 < r 
Arguments: {α : Type u_3} {β : Type u_4} [LinearOrderedField α] {l : Filter β} {f : β → α} {r : α} [l.NeBot] (h : Filter.Tendsto f l Filter.atBot)
Goal: Filter.Tendsto (fun x => r * f x) l Filter.atBot ↔ 0 < r
-- Filter.tendsto_mul_const_atBot_iff_pos
theorem tendsto_mul_const_atBot_iff_pos [NeBot l] (h : Tendsto f l atBot) :
    Tendsto (fun x => f x * r) l atBot ↔ 0 < r 
Arguments: {α : Type u_3} {β : Type u_4} [LinearOrderedField α] {l : Filter β} {f : β → α} {r : α} [l.NeBot] (h : Filter.Tendsto f l Filter.atBot)
Goal: Filter.Tendsto (fun x => f x * r) l Filter.atBot ↔ 0 < r
-- Filter.tendsto_div_const_atBot_iff_pos
lemma tendsto_div_const_atBot_iff_pos [NeBot l] (h : Tendsto f l atBot) :
    Tendsto (fun x ↦ f x / r) l atBot ↔ 0 < r 
Arguments: {α : Type u_3} {β : Type u_4} [LinearOrderedField α] {l : Filter β} {f : β → α} {r : α} [l.NeBot] (h : Filter.Tendsto f l Filter.atBot)
Goal: Filter.Tendsto (fun x => f x / r) l Filter.atBot ↔ 0 < r
-- Filter.tendsto_const_mul_atBot_iff_neg
theorem tendsto_const_mul_atBot_iff_neg [NeBot l] (h : Tendsto f l atTop) :
    Tendsto (fun x => r * f x) l atBot ↔ r < 0 
Arguments: {α : Type u_3} {β : Type u_4} [LinearOrderedField 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.atBot ↔ r < 0
-- Filter.tendsto_mul_const_atBot_iff_neg
theorem tendsto_mul_const_atBot_iff_neg [NeBot l] (h : Tendsto f l atTop) :
    Tendsto (fun x => f x * r) l atBot ↔ r < 0 
Arguments: {α : Type u_3} {β : Type u_4} [LinearOrderedField 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.atBot ↔ r < 0
-- Filter.tendsto_div_const_atBot_iff_neg
lemma tendsto_div_const_atBot_iff_neg [NeBot l] (h : Tendsto f l atTop) :
    Tendsto (fun x ↦ f x / r) l atBot ↔ r < 0 
Arguments: {α : Type u_3} {β : Type u_4} [LinearOrderedField 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.atBot ↔ r < 0
-- Filter.Tendsto.const_mul_atTop_of_neg
theorem Tendsto.const_mul_atTop_of_neg (hr : r < 0) (hf : Tendsto f l atTop) :
    Tendsto (fun x => r * f x) l atBot 
Arguments: {α : Type u_3} {β : Type u_4} [LinearOrderedField α] {l : Filter β} {f : β → α} {r : α} (hr : r < 0) (hf : Filter.Tendsto f l Filter.atTop)
Goal: Filter.Tendsto (fun x => r * f x) l Filter.atBot
-- Filter.Tendsto.atTop_mul_const_of_neg
theorem Tendsto.atTop_mul_const_of_neg (hr : r < 0) (hf : Tendsto f l atTop) :
    Tendsto (fun x => f x * r) l atBot 
Arguments: {α : Type u_3} {β : Type u_4} [LinearOrderedField α] {l : Filter β} {f : β → α} {r : α} (hr : r < 0) (hf : Filter.Tendsto f l Filter.atTop)
Goal: Filter.Tendsto (fun x => f x * r) l Filter.atBot
-- Filter.Tendsto.atTop_div_const_of_neg
lemma Tendsto.atTop_div_const_of_neg (hr : r < 0) (hf : Tendsto f l atTop) :
    Tendsto (fun x ↦ f x / r) l atBot 
Arguments: {α : Type u_3} {β : Type u_4} [LinearOrderedField α] {l : Filter β} {f : β → α} {r : α} (hr : r < 0) (hf : Filter.Tendsto f l Filter.atTop)
Goal: Filter.Tendsto (fun x => f x / r) l Filter.atBot
-- Filter.Tendsto.const_mul_atBot
theorem Tendsto.const_mul_atBot (hr : 0 < r) (hf : Tendsto f l atBot) :
    Tendsto (fun x => r * f x) l atBot 
Arguments: {α : Type u_3} {β : Type u_4} [LinearOrderedField α] {l : Filter β} {f : β → α} {r : α} (hr : 0 < r) (hf : Filter.Tendsto f l Filter.atBot)
Goal: Filter.Tendsto (fun x => r * f x) l Filter.atBot
-- Filter.Tendsto.atBot_mul_const
theorem Tendsto.atBot_mul_const (hr : 0 < r) (hf : Tendsto f l atBot) :
    Tendsto (fun x => f x * r) l atBot 
Arguments: {α : Type u_3} {β : Type u_4} [LinearOrderedField α] {l : Filter β} {f : β → α} {r : α} (hr : 0 < r) (hf : Filter.Tendsto f l Filter.atBot)
Goal: Filter.Tendsto (fun x => f x * r) l Filter.atBot
-- Filter.Tendsto.atBot_div_const
theorem Tendsto.atBot_div_const (hr : 0 < r) (hf : Tendsto f l atBot) :
    Tendsto (fun x => f x / r) l atBot 
Arguments: {α : Type u_3} {β : Type u_4} [LinearOrderedField α] {l : Filter β} {f : β → α} {r : α} (hr : 0 < r) (hf : Filter.Tendsto f l Filter.atBot)
Goal: Filter.Tendsto (fun x => f x / r) l Filter.atBot
-- Filter.Tendsto.const_mul_atBot_of_neg
theorem Tendsto.const_mul_atBot_of_neg (hr : r < 0) (hf : Tendsto f l atBot) :
    Tendsto (fun x => r * f x) l atTop 
Arguments: {α : Type u_3} {β : Type u_4} [LinearOrderedField α] {l : Filter β} {f : β → α} {r : α} (hr : r < 0) (hf : Filter.Tendsto f l Filter.atBot)
Goal: Filter.Tendsto (fun x => r * f x) l Filter.atTop
-- Filter.Tendsto.atBot_mul_const_of_neg
theorem Tendsto.atBot_mul_const_of_neg (hr : r < 0) (hf : Tendsto f l atBot) :
    Tendsto (fun x => f x * r) l atTop 
Arguments: {α : Type u_3} {β : Type u_4} [LinearOrderedField α] {l : Filter β} {f : β → α} {r : α} (hr : r < 0) (hf : Filter.Tendsto f l Filter.atBot)
Goal: Filter.Tendsto (fun x => f x * r) l Filter.atTop
-- Filter.tendsto_neg_const_mul_pow_atTop
theorem tendsto_neg_const_mul_pow_atTop {c : α} {n : ℕ} (hn : n ≠ 0) (hc : c < 0) :
    Tendsto (fun x => c * x ^ n) atTop atBot 
Arguments: {α : Type u_3} [LinearOrderedField α] {c : α} {n : ℕ} (hn : n ≠ 0) (hc : c < 0)
Goal: Filter.Tendsto (fun x => c * x ^ n) Filter.atTop Filter.atBot
-- Filter.tendsto_const_mul_pow_atBot_iff
theorem tendsto_const_mul_pow_atBot_iff {c : α} {n : ℕ} :
    Tendsto (fun x => c * x ^ n) atTop atBot ↔ n ≠ 0 ∧ c < 0 
Arguments: {α : Type u_3} [LinearOrderedField α] {c : α} {n : ℕ}
Goal: Filter.Tendsto (fun x => c * x ^ n) Filter.atTop Filter.atBot ↔ n ≠ 0 ∧ c < 0
-- Filter.tendsto_atTop'
theorem tendsto_atTop' [Nonempty α] [SemilatticeSup α] {f : α → β} {l : Filter β} :
    Tendsto f atTop l ↔ ∀ s ∈ l, ∃ a, ∀ b ≥ a, f b ∈ s 
Arguments: {α : Type u_3} {β : Type u_4} [Nonempty α] [SemilatticeSup α] {f : α → β} {l : Filter β}
Goal: Filter.Tendsto f Filter.atTop l ↔ ∀ s ∈ l, ∃ a, ∀ b ≥ a, f b ∈ s
-- Filter.tendsto_atBot'
theorem tendsto_atBot' [Nonempty α] [SemilatticeInf α] {f : α → β} {l : Filter β} :
    Tendsto f atBot l ↔ ∀ s ∈ l, ∃ a, ∀ b ≤ a, f b ∈ s 
Arguments: {α : Type u_3} {β : Type u_4} [Nonempty α] [SemilatticeInf α] {f : α → β} {l : Filter β}
Goal: Filter.Tendsto f Filter.atBot l ↔ ∀ s ∈ l, ∃ a, ∀ b ≤ a, f b ∈ s
-- Filter.tendsto_atTop_principal
theorem tendsto_atTop_principal [Nonempty β] [SemilatticeSup β] {f : β → α} {s : Set α} :
    Tendsto f atTop (𝓟 s) ↔ ∃ N, ∀ n ≥ N, f n ∈ s 
Arguments: {α : Type u_3} {β : Type u_4} [Nonempty β] [SemilatticeSup β] {f : β → α} {s : Set α}
Goal: Filter.Tendsto f Filter.atTop (𝓟 s) ↔ ∃ N, ∀ n ≥ N, f n ∈ s
-- Filter.tendsto_atBot_principal
theorem tendsto_atBot_principal [Nonempty β] [SemilatticeInf β] {f : β → α} {s : Set α} :
    Tendsto f atBot (𝓟 s) ↔ ∃ N, ∀ n ≤ N, f n ∈ s 
Arguments: {α : Type u_3} {β : Type u_4} [Nonempty β] [SemilatticeInf β] {f : β → α} {s : Set α}
Goal: Filter.Tendsto f Filter.atBot (𝓟 s) ↔ ∃ N, ∀ n ≤ N, f n ∈ s
-- Filter.tendsto_atTop_atTop
theorem tendsto_atTop_atTop [Nonempty α] [SemilatticeSup α] [Preorder β] {f : α → β} :
    Tendsto f atTop atTop ↔ ∀ b : β, ∃ i : α, ∀ a : α, i ≤ a → b ≤ f a 
Arguments: {α : Type u_3} {β : Type u_4} [Nonempty α] [SemilatticeSup α] [Preorder β] {f : α → β}
Goal: Filter.Tendsto f Filter.atTop Filter.atTop ↔ ∀ (b : β), ∃ i, ∀ (a : α), i ≤ a → b ≤ f a
-- Filter.tendsto_atTop_atBot
theorem tendsto_atTop_atBot [Nonempty α] [SemilatticeSup α] [Preorder β] {f : α → β} :
    Tendsto f atTop atBot ↔ ∀ b : β, ∃ i : α, ∀ a : α, i ≤ a → f a ≤ b 
Arguments: {α : Type u_3} {β : Type u_4} [Nonempty α] [SemilatticeSup α] [Preorder β] {f : α → β}
Goal: Filter.Tendsto f Filter.atTop Filter.atBot ↔ ∀ (b : β), ∃ i, ∀ (a : α), i ≤ a → f a ≤ b
-- Filter.tendsto_atBot_atTop
theorem tendsto_atBot_atTop [Nonempty α] [SemilatticeInf α] [Preorder β] {f : α → β} :
    Tendsto f atBot atTop ↔ ∀ b : β, ∃ i : α, ∀ a : α, a ≤ i → b ≤ f a 
Arguments: {α : Type u_3} {β : Type u_4} [Nonempty α] [SemilatticeInf α] [Preorder β] {f : α → β}
Goal: Filter.Tendsto f Filter.atBot Filter.atTop ↔ ∀ (b : β), ∃ i, ∀ a ≤ i, b ≤ f a
-- Filter.tendsto_atBot_atBot
theorem tendsto_atBot_atBot [Nonempty α] [SemilatticeInf α] [Preorder β] {f : α → β} :
    Tendsto f atBot atBot ↔ ∀ b : β, ∃ i : α, ∀ a : α, a ≤ i → f a ≤ b 
Arguments: {α : Type u_3} {β : Type u_4} [Nonempty α] [SemilatticeInf α] [Preorder β] {f : α → β}
Goal: Filter.Tendsto f Filter.atBot Filter.atBot ↔ ∀ (b : β), ∃ i, ∀ a ≤ i, f a ≤ b
-- Filter.tendsto_atTop_atTop_of_monotone
theorem tendsto_atTop_atTop_of_monotone [Preorder α] [Preorder β] {f : α → β} (hf : Monotone f)
    (h : ∀ b, ∃ a, b ≤ f a) : Tendsto f atTop atTop 
Arguments: {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : α → β} (hf : Monotone f) (h : ∀ (b : β), ∃ a, b ≤ f a)
Goal: Filter.Tendsto f Filter.atTop Filter.atTop
-- Filter.tendsto_atTop_atBot_of_antitone
theorem tendsto_atTop_atBot_of_antitone [Preorder α] [Preorder β] {f : α → β} (hf : Antitone f)
    (h : ∀ b, ∃ a, f a ≤ b) : Tendsto f atTop atBot 
Arguments: {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : α → β} (hf : Antitone f) (h : ∀ (b : β), ∃ a, f a ≤ b)
Goal: Filter.Tendsto f Filter.atTop Filter.atBot
-- Filter.tendsto_atBot_atBot_of_monotone
theorem tendsto_atBot_atBot_of_monotone [Preorder α] [Preorder β] {f : α → β} (hf : Monotone f)
    (h : ∀ b, ∃ a, f a ≤ b) : Tendsto f atBot atBot 
Arguments: {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : α → β} (hf : Monotone f) (h : ∀ (b : β), ∃ a, f a ≤ b)
Goal: Filter.Tendsto f Filter.atBot Filter.atBot
-- Filter.tendsto_atBot_atTop_of_antitone
theorem tendsto_atBot_atTop_of_antitone [Preorder α] [Preorder β] {f : α → β} (hf : Antitone f)
    (h : ∀ b, ∃ a, b ≤ f a) : Tendsto f atBot atTop 
Arguments: {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : α → β} (hf : Antitone f) (h : ∀ (b : β), ∃ a, b ≤ f a)
Goal: Filter.Tendsto f Filter.atBot Filter.atTop
-- Filter.tendsto_atTop_atTop_iff_of_monotone
theorem tendsto_atTop_atTop_iff_of_monotone [Nonempty α] [SemilatticeSup α] [Preorder β] {f : α → β}
    (hf : Monotone f) : Tendsto f atTop atTop ↔ ∀ b : β, ∃ a : α, b ≤ f a 
Arguments: {α : Type u_3} {β : Type u_4} [Nonempty α] [SemilatticeSup α] [Preorder β] {f : α → β} (hf : Monotone f)
Goal: Filter.Tendsto f Filter.atTop Filter.atTop ↔ ∀ (b : β), ∃ a, b ≤ f a
-- Filter.tendsto_atTop_atBot_iff_of_antitone
theorem tendsto_atTop_atBot_iff_of_antitone [Nonempty α] [SemilatticeSup α] [Preorder β] {f : α → β}
    (hf : Antitone f) : Tendsto f atTop atBot ↔ ∀ b : β, ∃ a : α, f a ≤ b 
Arguments: {α : Type u_3} {β : Type u_4} [Nonempty α] [SemilatticeSup α] [Preorder β] {f : α → β} (hf : Antitone f)
Goal: Filter.Tendsto f Filter.atTop Filter.atBot ↔ ∀ (b : β), ∃ a, f a ≤ b
-- Filter.tendsto_atBot_atBot_iff_of_monotone
theorem tendsto_atBot_atBot_iff_of_monotone [Nonempty α] [SemilatticeInf α] [Preorder β] {f : α → β}
    (hf : Monotone f) : Tendsto f atBot atBot ↔ ∀ b : β, ∃ a : α, f a ≤ b 
Arguments: {α : Type u_3} {β : Type u_4} [Nonempty α] [SemilatticeInf α] [Preorder β] {f : α → β} (hf : Monotone f)
Goal: Filter.Tendsto f Filter.atBot Filter.atBot ↔ ∀ (b : β), ∃ a, f a ≤ b
-- Filter.tendsto_atBot_atTop_iff_of_antitone
theorem tendsto_atBot_atTop_iff_of_antitone [Nonempty α] [SemilatticeInf α] [Preorder β] {f : α → β}
    (hf : Antitone f) : Tendsto f atBot atTop ↔ ∀ b : β, ∃ a : α, b ≤ f a 
Arguments: {α : Type u_3} {β : Type u_4} [Nonempty α] [SemilatticeInf α] [Preorder β] {f : α → β} (hf : Antitone f)
Goal: Filter.Tendsto f Filter.atBot Filter.atTop ↔ ∀ (b : β), ∃ a, b ≤ f a
-- Filter.comap_embedding_atTop
theorem comap_embedding_atTop [Preorder β] [Preorder γ] {e : β → γ}
    (hm : ∀ b₁ b₂, e b₁ ≤ e b₂ ↔ b₁ ≤ b₂) (hu : ∀ c, ∃ b, c ≤ e b) : comap e atTop = atTop 
Arguments: {β : Type u_4} {γ : Type u_5} [Preorder β] [Preorder γ] {e : β → γ} (hm : ∀ (b₁ b₂ : β), e b₁ ≤ e b₂ ↔ b₁ ≤ b₂) (hu : ∀ (c : γ), ∃ b, c ≤ e b)
Goal: Filter.comap e Filter.atTop = Filter.atTop
-- Filter.comap_embedding_atBot
theorem comap_embedding_atBot [Preorder β] [Preorder γ] {e : β → γ}
    (hm : ∀ b₁ b₂, e b₁ ≤ e b₂ ↔ b₁ ≤ b₂) (hu : ∀ c, ∃ b, e b ≤ c) : comap e atBot = atBot 
Arguments: {β : Type u_4} {γ : Type u_5} [Preorder β] [Preorder γ] {e : β → γ} (hm : ∀ (b₁ b₂ : β), e b₁ ≤ e b₂ ↔ b₁ ≤ b₂) (hu : ∀ (c : γ), ∃ b, e b ≤ c)
Goal: Filter.comap e Filter.atBot = Filter.atBot
-- Filter.tendsto_atTop_embedding
theorem tendsto_atTop_embedding [Preorder β] [Preorder γ] {f : α → β} {e : β → γ} {l : Filter α}
    (hm : ∀ b₁ b₂, e b₁ ≤ e b₂ ↔ b₁ ≤ b₂) (hu : ∀ c, ∃ b, c ≤ e b) :
    Tendsto (e ∘ f) l atTop ↔ Tendsto f l atTop 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Preorder β] [Preorder γ] {f : α → β} {e : β → γ} {l : Filter α} (hm : ∀ (b₁ b₂ : β), e b₁ ≤ e b₂ ↔ b₁ ≤ b₂) (hu : ∀ (c : γ), ∃ b, c ≤ e b)
Goal: Filter.Tendsto (e ∘ f) l Filter.atTop ↔ Filter.Tendsto f l Filter.atTop
-- Filter.tendsto_atBot_embedding
theorem tendsto_atBot_embedding [Preorder β] [Preorder γ] {f : α → β} {e : β → γ} {l : Filter α}
    (hm : ∀ b₁ b₂, e b₁ ≤ e b₂ ↔ b₁ ≤ b₂) (hu : ∀ c, ∃ b, e b ≤ c) :
    Tendsto (e ∘ f) l atBot ↔ Tendsto f l atBot 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Preorder β] [Preorder γ] {f : α → β} {e : β → γ} {l : Filter α} (hm : ∀ (b₁ b₂ : β), e b₁ ≤ e b₂ ↔ b₁ ≤ b₂) (hu : ∀ (c : γ), ∃ b, e b ≤ c)
Goal: Filter.Tendsto (e ∘ f) l Filter.atBot ↔ Filter.Tendsto f l Filter.atBot
-- Filter.tendsto_finset_range
theorem tendsto_finset_range : Tendsto Finset.range atTop atTop 
Goal: Filter.Tendsto Finset.range Filter.atTop Filter.atTop
-- Filter.atTop_finset_eq_iInf
theorem atTop_finset_eq_iInf : (atTop : Filter (Finset α)) = ⨅ x : α, 𝓟 (Ici {x}) 
Arguments: {α : Type u_3}
Goal: Filter.atTop = ⨅ x, 𝓟 (Set.Ici {x})
-- Filter.tendsto_atTop_finset_of_monotone
theorem tendsto_atTop_finset_of_monotone [Preorder β] {f : β → Finset α} (h : Monotone f)
    (h' : ∀ x : α, ∃ n, x ∈ f n) : Tendsto f atTop atTop 
Arguments: {α : Type u_3} {β : Type u_4} [Preorder β] {f : β → Finset α} (h : Monotone f) (h' : ∀ (x : α), ∃ n, x ∈ f n)
Goal: Filter.Tendsto f Filter.atTop Filter.atTop
-- Filter.tendsto_finset_image_atTop_atTop
theorem tendsto_finset_image_atTop_atTop [DecidableEq β] {i : β → γ} {j : γ → β}
    (h : Function.LeftInverse j i) : Tendsto (Finset.image j) atTop atTop 
Arguments: {β : Type u_4} {γ : Type u_5} [DecidableEq β] {i : β → γ} {j : γ → β} (h : Function.LeftInverse j i)
Goal: Filter.Tendsto (Finset.image j) Filter.atTop Filter.atTop
-- Filter.tendsto_finset_preimage_atTop_atTop
theorem tendsto_finset_preimage_atTop_atTop {f : α → β} (hf : Function.Injective f) :
    Tendsto (fun s : Finset β => s.preimage f (hf.injOn)) atTop atTop 
Arguments: {α : Type u_3} {β : Type u_4} {f : α → β} (hf : Function.Injective f)
Goal: Filter.Tendsto (fun s => s.preimage f ⋯) Filter.atTop Filter.atTop
-- Filter.prod_atTop_atTop_eq
theorem prod_atTop_atTop_eq [Preorder α] [Preorder β] :
    (atTop : Filter α) ×ˢ (atTop : Filter β) = (atTop : Filter (α × β)) 
Arguments: {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β]
Goal: Filter.atTop ×ˢ Filter.atTop = Filter.atTop
-- Filter.prod_atBot_atBot_eq
theorem prod_atBot_atBot_eq [Preorder α] [Preorder β] :
    (atBot : Filter α) ×ˢ (atBot : Filter β) = (atBot : Filter (α × β)) 
Arguments: {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β]
Goal: Filter.atBot ×ˢ Filter.atBot = Filter.atBot
-- Filter.prod_map_atTop_eq
theorem prod_map_atTop_eq {α₁ α₂ β₁ β₂ : Type*} [Preorder β₁] [Preorder β₂]
    (u₁ : β₁ → α₁) (u₂ : β₂ → α₂) : map u₁ atTop ×ˢ map u₂ atTop = map (Prod.map u₁ u₂) atTop 
Arguments: {α₁ : Type u_6} {α₂ : Type u_7} {β₁ : Type u_8} {β₂ : Type u_9} [Preorder β₁] [Preorder β₂] (u₁ : β₁ → α₁) (u₂ : β₂ → α₂)
Goal: Filter.map u₁ Filter.atTop ×ˢ Filter.map u₂ Filter.atTop = Filter.map (Prod.map u₁ u₂) Filter.atTop
-- Filter.prod_map_atBot_eq
theorem prod_map_atBot_eq {α₁ α₂ β₁ β₂ : Type*} [Preorder β₁] [Preorder β₂]
    (u₁ : β₁ → α₁) (u₂ : β₂ → α₂) : map u₁ atBot ×ˢ map u₂ atBot = map (Prod.map u₁ u₂) atBot 
Arguments: {α₁ : Type u_6} {α₂ : Type u_7} {β₁ : Type u_8} {β₂ : Type u_9} [Preorder β₁] [Preorder β₂] (u₁ : β₁ → α₁) (u₂ : β₂ → α₂)
Goal: Filter.map u₁ Filter.atBot ×ˢ Filter.map u₂ Filter.atBot = Filter.map (Prod.map u₁ u₂) Filter.atBot
-- Filter.Tendsto.subseq_mem
theorem Tendsto.subseq_mem {F : Filter α} {V : ℕ → Set α} (h : ∀ n, V n ∈ F) {u : ℕ → α}
    (hu : Tendsto u atTop F) : ∃ φ : ℕ → ℕ, StrictMono φ ∧ ∀ n, u (φ n) ∈ V n 
Arguments: {α : Type u_3} {F : Filter α} {V : ℕ → Set α} (h : ∀ (n : ℕ), V n ∈ F) {u : ℕ → α} (hu : Filter.Tendsto u Filter.atTop F)
Goal: ∃ φ, StrictMono φ ∧ ∀ (n : ℕ), u (φ n) ∈ V n
-- Filter.tendsto_atBot_diagonal
theorem tendsto_atBot_diagonal [SemilatticeInf α] : Tendsto (fun a : α => (a, a)) atBot atBot 
Arguments: {α : Type u_3} [SemilatticeInf α]
Goal: Filter.Tendsto (fun a => (a, a)) Filter.atBot Filter.atBot
-- Filter.tendsto_atTop_diagonal
theorem tendsto_atTop_diagonal [SemilatticeSup α] : Tendsto (fun a : α => (a, a)) atTop atTop 
Arguments: {α : Type u_3} [SemilatticeSup α]
Goal: Filter.Tendsto (fun a => (a, a)) Filter.atTop Filter.atTop
-- Filter.Tendsto.prod_map_prod_atBot
theorem Tendsto.prod_map_prod_atBot [SemilatticeInf γ] {F : Filter α} {G : Filter β} {f : α → γ}
    {g : β → γ} (hf : Tendsto f F atBot) (hg : Tendsto g G atBot) :
    Tendsto (Prod.map f g) (F ×ˢ G) atBot 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} [SemilatticeInf γ] {F : Filter α} {G : Filter β} {f : α → γ} {g : β → γ} (hf : Filter.Tendsto f F Filter.atBot) (hg : Filter.Tendsto g G Filter.atBot)
Goal: Filter.Tendsto (Prod.map f g) (F ×ˢ G) Filter.atBot
-- Filter.Tendsto.prod_map_prod_atTop
theorem Tendsto.prod_map_prod_atTop [SemilatticeSup γ] {F : Filter α} {G : Filter β} {f : α → γ}
    {g : β → γ} (hf : Tendsto f F atTop) (hg : Tendsto g G atTop) :
    Tendsto (Prod.map f g) (F ×ˢ G) atTop 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} [SemilatticeSup γ] {F : Filter α} {G : Filter β} {f : α → γ} {g : β → γ} (hf : Filter.Tendsto f F Filter.atTop) (hg : Filter.Tendsto g G Filter.atTop)
Goal: Filter.Tendsto (Prod.map f g) (F ×ˢ G) Filter.atTop
-- Filter.Tendsto.prod_atBot
theorem Tendsto.prod_atBot [SemilatticeInf α] [SemilatticeInf γ] {f g : α → γ}
    (hf : Tendsto f atBot atBot) (hg : Tendsto g atBot atBot) :
    Tendsto (Prod.map f g) atBot atBot 
Arguments: {α : Type u_3} {γ : Type u_5} [SemilatticeInf α] [SemilatticeInf γ] {f g : α → γ} (hf : Filter.Tendsto f Filter.atBot Filter.atBot) (hg : Filter.Tendsto g Filter.atBot Filter.atBot)
Goal: Filter.Tendsto (Prod.map f g) Filter.atBot Filter.atBot
-- Filter.Tendsto.prod_atTop
theorem Tendsto.prod_atTop [SemilatticeSup α] [SemilatticeSup γ] {f g : α → γ}
    (hf : Tendsto f atTop atTop) (hg : Tendsto g atTop atTop) :
    Tendsto (Prod.map f g) atTop atTop 
Arguments: {α : Type u_3} {γ : Type u_5} [SemilatticeSup α] [SemilatticeSup γ] {f g : α → γ} (hf : Filter.Tendsto f Filter.atTop Filter.atTop) (hg : Filter.Tendsto g Filter.atTop Filter.atTop)
Goal: Filter.Tendsto (Prod.map f g) Filter.atTop Filter.atTop
-- Filter.eventually_atBot_prod_self
theorem eventually_atBot_prod_self [SemilatticeInf α] [Nonempty α] {p : α × α → Prop} :
    (∀ᶠ x in atBot, p x) ↔ ∃ a, ∀ k l, k ≤ a → l ≤ a → p (k, l) 
Arguments: {α : Type u_3} [SemilatticeInf α] [Nonempty α] {p : α × α → Prop}
Goal: (∀ᶠ (x : α × α) in Filter.atBot, p x) ↔ ∃ a, ∀ (k l : α), k ≤ a → l ≤ a → p (k, l)
-- Filter.eventually_atTop_prod_self
theorem eventually_atTop_prod_self [SemilatticeSup α] [Nonempty α] {p : α × α → Prop} :
    (∀ᶠ x in atTop, p x) ↔ ∃ a, ∀ k l, a ≤ k → a ≤ l → p (k, l) 
Arguments: {α : Type u_3} [SemilatticeSup α] [Nonempty α] {p : α × α → Prop}
Goal: (∀ᶠ (x : α × α) in Filter.atTop, p x) ↔ ∃ a, ∀ (k l : α), a ≤ k → a ≤ l → p (k, l)
-- Filter.eventually_atBot_prod_self'
theorem eventually_atBot_prod_self' [SemilatticeInf α] [Nonempty α] {p : α × α → Prop} :
    (∀ᶠ x in atBot, p x) ↔ ∃ a, ∀ k ≤ a, ∀ l ≤ a, p (k, l) 
Arguments: {α : Type u_3} [SemilatticeInf α] [Nonempty α] {p : α × α → Prop}
Goal: (∀ᶠ (x : α × α) in Filter.atBot, p x) ↔ ∃ a, ∀ k ≤ a, ∀ l ≤ a, p (k, l)
-- Filter.eventually_atTop_prod_self'
theorem eventually_atTop_prod_self' [SemilatticeSup α] [Nonempty α] {p : α × α → Prop} :
    (∀ᶠ x in atTop, p x) ↔ ∃ a, ∀ k ≥ a, ∀ l ≥ a, p (k, l) 
Arguments: {α : Type u_3} [SemilatticeSup α] [Nonempty α] {p : α × α → Prop}
Goal: (∀ᶠ (x : α × α) in Filter.atTop, p x) ↔ ∃ a, ∀ k ≥ a, ∀ l ≥ a, p (k, l)
-- Filter.eventually_atTop_curry
theorem eventually_atTop_curry [SemilatticeSup α] [SemilatticeSup β] {p : α × β → Prop}
    (hp : ∀ᶠ x : α × β in Filter.atTop, p x) : ∀ᶠ k in atTop, ∀ᶠ l in atTop, p (k, l) 
Arguments: {α : Type u_3} {β : Type u_4} [SemilatticeSup α] [SemilatticeSup β] {p : α × β → Prop} (hp : ∀ᶠ (x : α × β) in Filter.atTop, p x)
Goal: ∀ᶠ (k : α) in Filter.atTop, ∀ᶠ (l : β) in Filter.atTop, p (k, l)
-- Filter.eventually_atBot_curry
theorem eventually_atBot_curry [SemilatticeInf α] [SemilatticeInf β] {p : α × β → Prop}
    (hp : ∀ᶠ x : α × β in Filter.atBot, p x) : ∀ᶠ k in atBot, ∀ᶠ l in atBot, p (k, l) 
Arguments: {α : Type u_3} {β : Type u_4} [SemilatticeInf α] [SemilatticeInf β] {p : α × β → Prop} (hp : ∀ᶠ (x : α × β) in Filter.atBot, p x)
Goal: ∀ᶠ (k : α) in Filter.atBot, ∀ᶠ (l : β) in Filter.atBot, p (k, l)
-- Filter.map_atTop_eq_of_gc
theorem map_atTop_eq_of_gc [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (g : β → α) (b' : β)
    (hf : Monotone f) (gc : ∀ a, ∀ b ≥ b', f a ≤ b ↔ a ≤ g b) (hgi : ∀ b ≥ b', b ≤ f (g b)) :
    map f atTop = atTop 
Arguments: {α : Type u_3} {β : Type u_4} [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (g : β → α) (b' : β) (hf : Monotone f) (gc : ∀ (a : α), ∀ b ≥ b', f a ≤ b ↔ a ≤ g b) (hgi : ∀ b ≥ b', b ≤ f (g b))
Goal: Filter.map f Filter.atTop = Filter.atTop
-- Filter.map_atBot_eq_of_gc
theorem map_atBot_eq_of_gc [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (g : β → α) (b' : β)
    (hf : Monotone f) (gc : ∀ a, ∀ b ≤ b', b ≤ f a ↔ g b ≤ a) (hgi : ∀ b ≤ b', f (g b) ≤ b) :
    map f atBot = atBot 
Arguments: {α : Type u_3} {β : Type u_4} [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (g : β → α) (b' : β) (hf : Monotone f) (gc : ∀ (a : α), ∀ b ≤ b', b ≤ f a ↔ g b ≤ a) (hgi : ∀ b ≤ b', f (g b) ≤ b)
Goal: Filter.map f Filter.atBot = Filter.atBot
-- Filter.map_val_atTop_of_Ici_subset
theorem map_val_atTop_of_Ici_subset [SemilatticeSup α] {a : α} {s : Set α} (h : Ici a ⊆ s) :
    map ((↑) : s → α) atTop = atTop 
Arguments: {α : Type u_3} [SemilatticeSup α] {a : α} {s : Set α} (h : Set.Ici a ⊆ s)
Goal: Filter.map Subtype.val Filter.atTop = Filter.atTop