ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Filter.tendsto_top
@[simp] theorem tendsto_top {f : α → β} {l : Filter α} : Tendsto f l ⊤ 
Arguments: {α : Type u} {β : Type v} {f : α → β} {l : Filter α}
Goal: Filter.Tendsto f l ⊤
-- Filter.le_map_of_right_inverse
theorem le_map_of_right_inverse {mab : α → β} {mba : β → α} {f : Filter α} {g : Filter β}
    (h₁ : mab ∘ mba =ᶠ[g] id) (h₂ : Tendsto mba g f) : g ≤ map mab f 
Arguments: {α : Type u} {β : Type v} {mab : α → β} {mba : β → α} {f : Filter α} {g : Filter β} (h₁ : mab ∘ mba =ᶠ[g] id) (h₂ : Filter.Tendsto mba g f)
Goal: g ≤ Filter.map mab f
-- Filter.tendsto_of_isEmpty
theorem tendsto_of_isEmpty [IsEmpty α] {f : α → β} {la : Filter α} {lb : Filter β} :
    Tendsto f la lb 
Arguments: {α : Type u} {β : Type v} [IsEmpty α] {f : α → β} {la : Filter α} {lb : Filter β}
Goal: Filter.Tendsto f la lb
-- Filter.eventuallyEq_of_left_inv_of_right_inv
theorem eventuallyEq_of_left_inv_of_right_inv {f : α → β} {g₁ g₂ : β → α} {fa : Filter α}
    {fb : Filter β} (hleft : ∀ᶠ x in fa, g₁ (f x) = x) (hright : ∀ᶠ y in fb, f (g₂ y) = y)
    (htendsto : Tendsto g₂ fb fa) : g₁ =ᶠ[fb] g₂ 
Arguments: {α : Type u} {β : Type v} {f : α → β} {g₁ g₂ : β → α} {fa : Filter α} {fb : Filter β} (hleft : ∀ᶠ (x : α) in fa, g₁ (f x) = x) (hright : ∀ᶠ (y : β) in fb, f (g₂ y) = y) (htendsto : Filter.Tendsto g₂ fb fa)
Goal: g₁ =ᶠ[fb] g₂
-- Filter.tendsto_iff_comap
theorem tendsto_iff_comap {f : α → β} {l₁ : Filter α} {l₂ : Filter β} :
    Tendsto f l₁ l₂ ↔ l₁ ≤ l₂.comap f 
Arguments: {α : Type u} {β : Type v} {f : α → β} {l₁ : Filter α} {l₂ : Filter β}
Goal: Filter.Tendsto f l₁ l₂ ↔ l₁ ≤ Filter.comap f l₂
-- Filter.Tendsto.disjoint
protected theorem Tendsto.disjoint {f : α → β} {la₁ la₂ : Filter α} {lb₁ lb₂ : Filter β}
    (h₁ : Tendsto f la₁ lb₁) (hd : Disjoint lb₁ lb₂) (h₂ : Tendsto f la₂ lb₂) : Disjoint la₁ la₂ 
Arguments: {α : Type u} {β : Type v} {f : α → β} {la₁ la₂ : Filter α} {lb₁ lb₂ : Filter β} (h₁ : Filter.Tendsto f la₁ lb₁) (hd : Disjoint lb₁ lb₂) (h₂ : Filter.Tendsto f la₂ lb₂)
Goal: Disjoint la₁ la₂
-- Filter.tendsto_congr'
theorem tendsto_congr' {f₁ f₂ : α → β} {l₁ : Filter α} {l₂ : Filter β} (hl : f₁ =ᶠ[l₁] f₂) :
    Tendsto f₁ l₁ l₂ ↔ Tendsto f₂ l₁ l₂ 
Arguments: {α : Type u} {β : Type v} {f₁ f₂ : α → β} {l₁ : Filter α} {l₂ : Filter β} (hl : f₁ =ᶠ[l₁] f₂)
Goal: Filter.Tendsto f₁ l₁ l₂ ↔ Filter.Tendsto f₂ l₁ l₂
-- Filter.Tendsto.congr'
theorem Tendsto.congr' {f₁ f₂ : α → β} {l₁ : Filter α} {l₂ : Filter β} (hl : f₁ =ᶠ[l₁] f₂)
    (h : Tendsto f₁ l₁ l₂) : Tendsto f₂ l₁ l₂ 
Arguments: {α : Type u} {β : Type v} {f₁ f₂ : α → β} {l₁ : Filter α} {l₂ : Filter β} (hl : f₁ =ᶠ[l₁] f₂) (h : Filter.Tendsto f₁ l₁ l₂)
Goal: Filter.Tendsto f₂ l₁ l₂
-- Filter.tendsto_congr
theorem tendsto_congr {f₁ f₂ : α → β} {l₁ : Filter α} {l₂ : Filter β} (h : ∀ x, f₁ x = f₂ x) :
    Tendsto f₁ l₁ l₂ ↔ Tendsto f₂ l₁ l₂ 
Arguments: {α : Type u} {β : Type v} {f₁ f₂ : α → β} {l₁ : Filter α} {l₂ : Filter β} (h : ∀ (x : α), f₁ x = f₂ x)
Goal: Filter.Tendsto f₁ l₁ l₂ ↔ Filter.Tendsto f₂ l₁ l₂
-- Filter.Tendsto.congr
theorem Tendsto.congr {f₁ f₂ : α → β} {l₁ : Filter α} {l₂ : Filter β} (h : ∀ x, f₁ x = f₂ x) :
    Tendsto f₁ l₁ l₂ → Tendsto f₂ l₁ l₂ 
Arguments: {α : Type u} {β : Type v} {f₁ f₂ : α → β} {l₁ : Filter α} {l₂ : Filter β} (h : ∀ (x : α), f₁ x = f₂ x)
Goal: Filter.Tendsto f₁ l₁ l₂ → Filter.Tendsto f₂ l₁ l₂
-- Filter.tendsto_id'
theorem tendsto_id' {x y : Filter α} : Tendsto id x y ↔ x ≤ y 
Arguments: {α : Type u} {x y : Filter α}
Goal: Filter.Tendsto id x y ↔ x ≤ y
-- Filter.tendsto_id
theorem tendsto_id {x : Filter α} : Tendsto id x x 
Arguments: {α : Type u} {x : Filter α}
Goal: Filter.Tendsto id x x
-- Filter.Tendsto.comp
theorem Tendsto.comp {f : α → β} {g : β → γ} {x : Filter α} {y : Filter β} {z : Filter γ}
    (hg : Tendsto g y z) (hf : Tendsto f x y) : Tendsto (g ∘ f) x z 
Arguments: {α : Type u} {β : Type v} {γ : Type w} {f : α → β} {g : β → γ} {x : Filter α} {y : Filter β} {z : Filter γ} (hg : Filter.Tendsto g y z) (hf : Filter.Tendsto f x y)
Goal: Filter.Tendsto (g ∘ f) x z
-- Filter.Tendsto.iterate
protected theorem Tendsto.iterate {f : α → α} {l : Filter α} (h : Tendsto f l l) :
    ∀ n, Tendsto (f^[n]) l l
  
Arguments: {α : Type u} {f : α → α} {l : Filter α} (h : Filter.Tendsto f l l) (n : ℕ)
Goal: Filter.Tendsto f^[n] l l
-- Filter.Tendsto.mono_left
theorem Tendsto.mono_left {f : α → β} {x y : Filter α} {z : Filter β} (hx : Tendsto f x z)
    (h : y ≤ x) : Tendsto f y z 
Arguments: {α : Type u} {β : Type v} {f : α → β} {x y : Filter α} {z : Filter β} (hx : Filter.Tendsto f x z) (h : y ≤ x)
Goal: Filter.Tendsto f y z
-- Filter.Tendsto.mono_right
theorem Tendsto.mono_right {f : α → β} {x : Filter α} {y z : Filter β} (hy : Tendsto f x y)
    (hz : y ≤ z) : Tendsto f x z 
Arguments: {α : Type u} {β : Type v} {f : α → β} {x : Filter α} {y z : Filter β} (hy : Filter.Tendsto f x y) (hz : y ≤ z)
Goal: Filter.Tendsto f x z
-- Filter.Tendsto.neBot
theorem Tendsto.neBot {f : α → β} {x : Filter α} {y : Filter β} (h : Tendsto f x y) [hx : NeBot x] :
    NeBot y 
Arguments: {α : Type u} {β : Type v} {f : α → β} {x : Filter α} {y : Filter β} (h : Filter.Tendsto f x y) [hx : x.NeBot]
Goal: y.NeBot
-- Filter.tendsto_map
theorem tendsto_map {f : α → β} {x : Filter α} : Tendsto f x (map f x) 
Arguments: {α : Type u} {β : Type v} {f : α → β} {x : Filter α}
Goal: Filter.Tendsto f x (Filter.map f x)
-- Filter.tendsto_map'_iff
@[simp]
theorem tendsto_map'_iff {f : β → γ} {g : α → β} {x : Filter α} {y : Filter γ} :
    Tendsto f (map g x) y ↔ Tendsto (f ∘ g) x y 
Arguments: {α : Type u} {β : Type v} {γ : Type w} {f : β → γ} {g : α → β} {x : Filter α} {y : Filter γ}
Goal: Filter.Tendsto f (Filter.map g x) y ↔ Filter.Tendsto (f ∘ g) x y
-- Filter.tendsto_comap
theorem tendsto_comap {f : α → β} {x : Filter β} : Tendsto f (comap f x) x 
Arguments: {α : Type u} {β : Type v} {f : α → β} {x : Filter β}
Goal: Filter.Tendsto f (Filter.comap f x) x
-- Filter.tendsto_comap_iff
@[simp]
theorem tendsto_comap_iff {f : α → β} {g : β → γ} {a : Filter α} {c : Filter γ} :
    Tendsto f a (c.comap g) ↔ Tendsto (g ∘ f) a c 
Arguments: {α : Type u} {β : Type v} {γ : Type w} {f : α → β} {g : β → γ} {a : Filter α} {c : Filter γ}
Goal: Filter.Tendsto f a (Filter.comap g c) ↔ Filter.Tendsto (g ∘ f) a c
-- Filter.tendsto_comap'_iff
theorem tendsto_comap'_iff {m : α → β} {f : Filter α} {g : Filter β} {i : γ → α} (h : range i ∈ f) :
    Tendsto (m ∘ i) (comap i f) g ↔ Tendsto m f g 
Arguments: {α : Type u} {β : Type v} {γ : Type w} {m : α → β} {f : Filter α} {g : Filter β} {i : γ → α} (h : Set.range i ∈ f)
Goal: Filter.Tendsto (m ∘ i) (Filter.comap i f) g ↔ Filter.Tendsto m f g
-- Filter.Tendsto.of_tendsto_comp
theorem Tendsto.of_tendsto_comp {f : α → β} {g : β → γ} {a : Filter α} {b : Filter β} {c : Filter γ}
    (hfg : Tendsto (g ∘ f) a c) (hg : comap g c ≤ b) : Tendsto f a b 
Arguments: {α : Type u} {β : Type v} {γ : Type w} {f : α → β} {g : β → γ} {a : Filter α} {b : Filter β} {c : Filter γ} (hfg : Filter.Tendsto (g ∘ f) a c) (hg : Filter.comap g c ≤ b)
Goal: Filter.Tendsto f a b
-- Filter.comap_eq_of_inverse
theorem comap_eq_of_inverse {f : Filter α} {g : Filter β} {φ : α → β} (ψ : β → α) (eq : ψ ∘ φ = id)
    (hφ : Tendsto φ f g) (hψ : Tendsto ψ g f) : comap φ g = f 
Arguments: {α : Type u} {β : Type v} {f : Filter α} {g : Filter β} {φ : α → β} (ψ : β → α) (eq : ψ ∘ φ = id) (hφ : Filter.Tendsto φ f g) (hψ : Filter.Tendsto ψ g f)
Goal: Filter.comap φ g = f
-- Filter.map_eq_of_inverse
theorem map_eq_of_inverse {f : Filter α} {g : Filter β} {φ : α → β} (ψ : β → α) (eq : φ ∘ ψ = id)
    (hφ : Tendsto φ f g) (hψ : Tendsto ψ g f) : map φ f = g 
Arguments: {α : Type u} {β : Type v} {f : Filter α} {g : Filter β} {φ : α → β} (ψ : β → α) (eq : φ ∘ ψ = id) (hφ : Filter.Tendsto φ f g) (hψ : Filter.Tendsto ψ g f)
Goal: Filter.map φ f = g
-- Filter.tendsto_inf
theorem tendsto_inf {f : α → β} {x : Filter α} {y₁ y₂ : Filter β} :
    Tendsto f x (y₁ ⊓ y₂) ↔ Tendsto f x y₁ ∧ Tendsto f x y₂ 
Arguments: {α : Type u} {β : Type v} {f : α → β} {x : Filter α} {y₁ y₂ : Filter β}
Goal: Filter.Tendsto f x (y₁ ⊓ y₂) ↔ Filter.Tendsto f x y₁ ∧ Filter.Tendsto f x y₂
-- Filter.tendsto_inf_left
theorem tendsto_inf_left {f : α → β} {x₁ x₂ : Filter α} {y : Filter β} (h : Tendsto f x₁ y) :
    Tendsto f (x₁ ⊓ x₂) y 
Arguments: {α : Type u} {β : Type v} {f : α → β} {x₁ x₂ : Filter α} {y : Filter β} (h : Filter.Tendsto f x₁ y)
Goal: Filter.Tendsto f (x₁ ⊓ x₂) y
-- Filter.tendsto_inf_right
theorem tendsto_inf_right {f : α → β} {x₁ x₂ : Filter α} {y : Filter β} (h : Tendsto f x₂ y) :
    Tendsto f (x₁ ⊓ x₂) y 
Arguments: {α : Type u} {β : Type v} {f : α → β} {x₁ x₂ : Filter α} {y : Filter β} (h : Filter.Tendsto f x₂ y)
Goal: Filter.Tendsto f (x₁ ⊓ x₂) y
-- Filter.Tendsto.inf
theorem Tendsto.inf {f : α → β} {x₁ x₂ : Filter α} {y₁ y₂ : Filter β} (h₁ : Tendsto f x₁ y₁)
    (h₂ : Tendsto f x₂ y₂) : Tendsto f (x₁ ⊓ x₂) (y₁ ⊓ y₂) 
Arguments: {α : Type u} {β : Type v} {f : α → β} {x₁ x₂ : Filter α} {y₁ y₂ : Filter β} (h₁ : Filter.Tendsto f x₁ y₁) (h₂ : Filter.Tendsto f x₂ y₂)
Goal: Filter.Tendsto f (x₁ ⊓ x₂) (y₁ ⊓ y₂)
-- Filter.tendsto_iInf
@[simp]
theorem tendsto_iInf {f : α → β} {x : Filter α} {y : ι → Filter β} :
    Tendsto f x (⨅ i, y i) ↔ ∀ i, Tendsto f x (y i) 
Arguments: {α : Type u} {β : Type v} {ι : Sort x} {f : α → β} {x : Filter α} {y : ι → Filter β}
Goal: Filter.Tendsto f x (⨅ i, y i) ↔ ∀ (i : ι), Filter.Tendsto f x (y i)
-- Filter.tendsto_iInf'
theorem tendsto_iInf' {f : α → β} {x : ι → Filter α} {y : Filter β} (i : ι)
    (hi : Tendsto f (x i) y) : Tendsto f (⨅ i, x i) y 
Arguments: {α : Type u} {β : Type v} {ι : Sort x} {f : α → β} {x : ι → Filter α} {y : Filter β} (i : ι) (hi : Filter.Tendsto f (x i) y)
Goal: Filter.Tendsto f (⨅ i, x i) y
-- Filter.tendsto_iInf_iInf
theorem tendsto_iInf_iInf {f : α → β} {x : ι → Filter α} {y : ι → Filter β}
    (h : ∀ i, Tendsto f (x i) (y i)) : Tendsto f (iInf x) (iInf y) 
Arguments: {α : Type u} {β : Type v} {ι : Sort x} {f : α → β} {x : ι → Filter α} {y : ι → Filter β} (h : ∀ (i : ι), Filter.Tendsto f (x i) (y i))
Goal: Filter.Tendsto f (iInf x) (iInf y)
-- Filter.tendsto_sup
@[simp]
theorem tendsto_sup {f : α → β} {x₁ x₂ : Filter α} {y : Filter β} :
    Tendsto f (x₁ ⊔ x₂) y ↔ Tendsto f x₁ y ∧ Tendsto f x₂ y 
Arguments: {α : Type u} {β : Type v} {f : α → β} {x₁ x₂ : Filter α} {y : Filter β}
Goal: Filter.Tendsto f (x₁ ⊔ x₂) y ↔ Filter.Tendsto f x₁ y ∧ Filter.Tendsto f x₂ y
-- Filter.Tendsto.sup
theorem Tendsto.sup {f : α → β} {x₁ x₂ : Filter α} {y : Filter β} :
    Tendsto f x₁ y → Tendsto f x₂ y → Tendsto f (x₁ ⊔ x₂) y 
Arguments: {α : Type u} {β : Type v} {f : α → β} {x₁ x₂ : Filter α} {y : Filter β}
Goal: Filter.Tendsto f x₁ y → Filter.Tendsto f x₂ y → Filter.Tendsto f (x₁ ⊔ x₂) y
-- Filter.Tendsto.sup_sup
theorem Tendsto.sup_sup {f : α → β} {x₁ x₂ : Filter α} {y₁ y₂ : Filter β}
    (h₁ : Tendsto f x₁ y₁) (h₂ : Tendsto f x₂ y₂) : Tendsto f (x₁ ⊔ x₂) (y₁ ⊔ y₂) 
Arguments: {α : Type u} {β : Type v} {f : α → β} {x₁ x₂ : Filter α} {y₁ y₂ : Filter β} (h₁ : Filter.Tendsto f x₁ y₁) (h₂ : Filter.Tendsto f x₂ y₂)
Goal: Filter.Tendsto f (x₁ ⊔ x₂) (y₁ ⊔ y₂)
-- Filter.tendsto_iSup
@[simp]
theorem tendsto_iSup {f : α → β} {x : ι → Filter α} {y : Filter β} :
    Tendsto f (⨆ i, x i) y ↔ ∀ i, Tendsto f (x i) y 
Arguments: {α : Type u} {β : Type v} {ι : Sort x} {f : α → β} {x : ι → Filter α} {y : Filter β}
Goal: Filter.Tendsto f (⨆ i, x i) y ↔ ∀ (i : ι), Filter.Tendsto f (x i) y
-- Filter.tendsto_iSup_iSup
theorem tendsto_iSup_iSup {f : α → β} {x : ι → Filter α} {y : ι → Filter β}
    (h : ∀ i, Tendsto f (x i) (y i)) : Tendsto f (iSup x) (iSup y) 
Arguments: {α : Type u} {β : Type v} {ι : Sort x} {f : α → β} {x : ι → Filter α} {y : ι → Filter β} (h : ∀ (i : ι), Filter.Tendsto f (x i) (y i))
Goal: Filter.Tendsto f (iSup x) (iSup y)
-- Filter.tendsto_principal
@[simp] theorem tendsto_principal {f : α → β} {l : Filter α} {s : Set β} :
    Tendsto f l (𝓟 s) ↔ ∀ᶠ a in l, f a ∈ s 
Arguments: {α : Type u} {β : Type v} {f : α → β} {l : Filter α} {s : Set β}
Goal: Filter.Tendsto f l (𝓟 s) ↔ ∀ᶠ (a : α) in l, f a ∈ s
-- Filter.tendsto_principal_principal
theorem tendsto_principal_principal {f : α → β} {s : Set α} {t : Set β} :
    Tendsto f (𝓟 s) (𝓟 t) ↔ ∀ a ∈ s, f a ∈ t 
Arguments: {α : Type u} {β : Type v} {f : α → β} {s : Set α} {t : Set β}
Goal: Filter.Tendsto f (𝓟 s) (𝓟 t) ↔ ∀ a ∈ s, f a ∈ t
-- Filter.tendsto_pure
@[simp] theorem tendsto_pure {f : α → β} {a : Filter α} {b : β} :
    Tendsto f a (pure b) ↔ ∀ᶠ x in a, f x = b 
Arguments: {α : Type u} {β : Type v} {f : α → β} {a : Filter α} {b : β}
Goal: Filter.Tendsto f a (pure b) ↔ ∀ᶠ (x : α) in a, f x = b
-- Filter.tendsto_pure_pure
theorem tendsto_pure_pure (f : α → β) (a : α) : Tendsto f (pure a) (pure (f a)) 
Arguments: {α : Type u} {β : Type v} (f : α → β) (a : α)
Goal: Filter.Tendsto f (pure a) (pure (f a))
-- Filter.tendsto_const_pure
theorem tendsto_const_pure {a : Filter α} {b : β} : Tendsto (fun _ => b) a (pure b) 
Arguments: {α : Type u} {β : Type v} {a : Filter α} {b : β}
Goal: Filter.Tendsto (fun x => b) a (pure b)
-- Filter.pure_le_iff
theorem pure_le_iff {a : α} {l : Filter α} : pure a ≤ l ↔ ∀ s ∈ l, a ∈ s 
Arguments: {α : Type u} {a : α} {l : Filter α}
Goal: pure a ≤ l ↔ ∀ s ∈ l, a ∈ s
-- Filter.tendsto_pure_left
theorem tendsto_pure_left {f : α → β} {a : α} {l : Filter β} :
    Tendsto f (pure a) l ↔ ∀ s ∈ l, f a ∈ s 
Arguments: {α : Type u} {β : Type v} {f : α → β} {a : α} {l : Filter β}
Goal: Filter.Tendsto f (pure a) l ↔ ∀ s ∈ l, f a ∈ s
-- Filter.map_inf_principal_preimage
@[simp]
theorem map_inf_principal_preimage {f : α → β} {s : Set β} {l : Filter α} :
    map f (l ⊓ 𝓟 (f ⁻¹' s)) = map f l ⊓ 𝓟 s 
Arguments: {α : Type u} {β : Type v} {f : α → β} {s : Set β} {l : Filter α}
Goal: Filter.map f (l ⊓ 𝓟 (f ⁻¹' s)) = Filter.map f l ⊓ 𝓟 s
-- Filter.Tendsto.not_tendsto
theorem Tendsto.not_tendsto {f : α → β} {a : Filter α} {b₁ b₂ : Filter β} (hf : Tendsto f a b₁)
    [NeBot a] (hb : Disjoint b₁ b₂) : ¬Tendsto f a b₂ 
Arguments: {α : Type u} {β : Type v} {f : α → β} {a : Filter α} {b₁ b₂ : Filter β} (hf : Filter.Tendsto f a b₁) [a.NeBot] (hb : Disjoint b₁ b₂)
Goal: ¬Filter.Tendsto f a b₂
-- Filter.Tendsto.if
protected theorem Tendsto.if {l₁ : Filter α} {l₂ : Filter β} {f g : α → β} {p : α → Prop}
    [∀ x, Decidable (p x)] (h₀ : Tendsto f (l₁ ⊓ 𝓟 { x | p x }) l₂)
    (h₁ : Tendsto g (l₁ ⊓ 𝓟 { x | ¬p x }) l₂) :
    Tendsto (fun x => if p x then f x else g x) l₁ l₂ 
Arguments: {α : Type u} {β : Type v} {l₁ : Filter α} {l₂ : Filter β} {f g : α → β} {p : α → Prop} [(x : α) → Decidable (p x)] (h₀ : Filter.Tendsto f (l₁ ⊓ 𝓟 {x | p x}) l₂) (h₁ : Filter.Tendsto g (l₁ ⊓ 𝓟 {x | ¬p x}) l₂)
Goal: Filter.Tendsto (fun x => if p x then f x else g x) l₁ l₂
-- Filter.Tendsto.if'
protected theorem Tendsto.if' {α β : Type*} {l₁ : Filter α} {l₂ : Filter β} {f g : α → β}
    {p : α → Prop} [DecidablePred p] (hf : Tendsto f l₁ l₂) (hg : Tendsto g l₁ l₂) :
    Tendsto (fun a => if p a then f a else g a) l₁ l₂ 
Arguments: {α : Type u_2} {β : Type u_3} {l₁ : Filter α} {l₂ : Filter β} {f g : α → β} {p : α → Prop} [DecidablePred p] (hf : Filter.Tendsto f l₁ l₂) (hg : Filter.Tendsto g l₁ l₂)
Goal: Filter.Tendsto (fun a => if p a then f a else g a) l₁ l₂
-- Filter.Tendsto.piecewise
protected theorem Tendsto.piecewise {l₁ : Filter α} {l₂ : Filter β} {f g : α → β} {s : Set α}
    [∀ x, Decidable (x ∈ s)] (h₀ : Tendsto f (l₁ ⊓ 𝓟 s) l₂) (h₁ : Tendsto g (l₁ ⊓ 𝓟 sᶜ) l₂) :
    Tendsto (piecewise s f g) l₁ l₂ 
Arguments: {α : Type u} {β : Type v} {l₁ : Filter α} {l₂ : Filter β} {f g : α → β} {s : Set α} [(x : α) → Decidable (x ∈ s)] (h₀ : Filter.Tendsto f (l₁ ⊓ 𝓟 s) l₂) (h₁ : Filter.Tendsto g (l₁ ⊓ 𝓟 sᶜ) l₂)
Goal: Filter.Tendsto (s.piecewise f g) l₁ l₂
-- Set.EqOn.eventuallyEq
theorem Set.EqOn.eventuallyEq {α β} {s : Set α} {f g : α → β} (h : EqOn f g s) : f =ᶠ[𝓟 s] g 
Arguments: {α : Type u_1} {β : Type u_2} {s : Set α} {f g : α → β} (h : Set.EqOn f g s)
Goal: f =ᶠ[𝓟 s] g
-- Set.EqOn.eventuallyEq_of_mem
theorem Set.EqOn.eventuallyEq_of_mem {α β} {s : Set α} {l : Filter α} {f g : α → β} (h : EqOn f g s)
    (hl : s ∈ l) : f =ᶠ[l] g 
Arguments: {α : Type u_1} {β : Type u_2} {s : Set α} {l : Filter α} {f g : α → β} (h : Set.EqOn f g s) (hl : s ∈ l)
Goal: f =ᶠ[l] g
-- HasSubset.Subset.eventuallyLE
theorem HasSubset.Subset.eventuallyLE {α} {l : Filter α} {s t : Set α} (h : s ⊆ t) : s ≤ᶠ[l] t 
Arguments: {α : Type u_1} {l : Filter α} {s t : Set α} (h : s ⊆ t)
Goal: s ≤ᶠ[l] t
-- Set.MapsTo.tendsto
theorem Set.MapsTo.tendsto {α β} {s : Set α} {t : Set β} {f : α → β} (h : MapsTo f s t) :
    Filter.Tendsto f (𝓟 s) (𝓟 t) 
Arguments: {α : Type u_1} {β : Type u_2} {s : Set α} {t : Set β} {f : α → β} (h : Set.MapsTo f s t)
Goal: Filter.Tendsto f (𝓟 s) (𝓟 t)
-- Filter.EventuallyEq.comp_tendsto
theorem Filter.EventuallyEq.comp_tendsto {α β γ : Type*} {l : Filter α} {f : α → β} {f' : α → β}
    (H : f =ᶠ[l] f') {g : γ → α} {lc : Filter γ} (hg : Tendsto g lc l) :
    f ∘ g =ᶠ[lc] f' ∘ g 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {l : Filter α} {f f' : α → β} (H : f =ᶠ[l] f') {g : γ → α} {lc : Filter γ} (hg : Filter.Tendsto g lc l)
Goal: f ∘ g =ᶠ[lc] f' ∘ g
-- Filter.map_mapsTo_Iic_iff_tendsto
theorem Filter.map_mapsTo_Iic_iff_tendsto {m : α → β} :
    MapsTo (map m) (Iic F) (Iic G) ↔ Tendsto m F G 
Arguments: {α : Type u_1} {β : Type u_2} {F : Filter α} {G : Filter β} {m : α → β}
Goal: Set.MapsTo (Filter.map m) (Set.Iic F) (Set.Iic G) ↔ Filter.Tendsto m F G
-- Filter.map_mapsTo_Iic_iff_mapsTo
theorem Filter.map_mapsTo_Iic_iff_mapsTo {s : Set α} {t : Set β} {m : α → β} :
    MapsTo (map m) (Iic <| 𝓟 s) (Iic <| 𝓟 t) ↔ MapsTo m s t 
Arguments: {α : Type u_1} {β : Type u_2} {s : Set α} {t : Set β} {m : α → β}
Goal: Set.MapsTo (Filter.map m) (Set.Iic (𝓟 s)) (Set.Iic (𝓟 t)) ↔ Set.MapsTo m s t
-- Filter.map_surjOn_Iic_iff_le_map
theorem Filter.map_surjOn_Iic_iff_le_map {m : α → β} :
    SurjOn (map m) (Iic F) (Iic G) ↔ G ≤ map m F 
Arguments: {α : Type u_1} {β : Type u_2} {F : Filter α} {G : Filter β} {m : α → β}
Goal: Set.SurjOn (Filter.map m) (Set.Iic F) (Set.Iic G) ↔ G ≤ Filter.map m F
-- Filter.map_surjOn_Iic_iff_surjOn
theorem Filter.map_surjOn_Iic_iff_surjOn {s : Set α} {t : Set β} {m : α → β} :
    SurjOn (map m) (Iic <| 𝓟 s) (Iic <| 𝓟 t) ↔ SurjOn m s t 
Arguments: {α : Type u_1} {β : Type u_2} {s : Set α} {t : Set β} {m : α → β}
Goal: Set.SurjOn (Filter.map m) (Set.Iic (𝓟 s)) (Set.Iic (𝓟 t)) ↔ Set.SurjOn m s t
-- Filter.filter_injOn_Iic_iff_injOn
theorem Filter.filter_injOn_Iic_iff_injOn {s : Set α} {m : α → β} :
    InjOn (map m) (Iic <| 𝓟 s) ↔ InjOn m s 
Arguments: {α : Type u_1} {β : Type u_2} {s : Set α} {m : α → β}
Goal: Set.InjOn (Filter.map m) (Set.Iic (𝓟 s)) ↔ Set.InjOn m s
-- Filter.mem_comk
@[simp]
lemma mem_comk {p : Set α → Prop} {he hmono hunion s} :
    s ∈ comk p he hmono hunion ↔ p sᶜ 
Arguments: {α : Type u_1} {p : Set α → Prop} {he : p ∅} {hmono : ∀ (t : Set α), p t → ∀ s ⊆ t, p s} {hunion : ∀ (s : Set α), p s → ∀ (t : Set α), p t → p (s ∪ t)} {s : Set α}
Goal: s ∈ Filter.comk p he hmono hunion ↔ p sᶜ
-- Filter.compl_mem_comk
lemma compl_mem_comk {p : Set α → Prop} {he hmono hunion s} :
    sᶜ ∈ comk p he hmono hunion ↔ p s 
Arguments: {α : Type u_1} {p : Set α → Prop} {he : p ∅} {hmono : ∀ (t : Set α), p t → ∀ s ⊆ t, p s} {hunion : ∀ (s : Set α), p s → ∀ (t : Set α), p t → p (s ∪ t)} {s : Set α}
Goal: sᶜ ∈ Filter.comk p he hmono hunion ↔ p s
-- reflTransGen_of_succ_of_le
theorem reflTransGen_of_succ_of_le (r : α → α → Prop) {n m : α} (h : ∀ i ∈ Ico n m, r i (succ i))
    (hnm : n ≤ m) : ReflTransGen r n m 
Arguments: {α : Type u_1} [PartialOrder α] [SuccOrder α] [IsSuccArchimedean α] (r : α → α → Prop) {n m : α} (h : ∀ i ∈ Set.Ico n m, r i (Order.succ i)) (hnm : n ≤ m)
Goal: Relation.ReflTransGen r n m
-- reflTransGen_of_succ_of_ge
theorem reflTransGen_of_succ_of_ge (r : α → α → Prop) {n m : α} (h : ∀ i ∈ Ico m n, r (succ i) i)
    (hmn : m ≤ n) : ReflTransGen r n m 
Arguments: {α : Type u_1} [PartialOrder α] [SuccOrder α] [IsSuccArchimedean α] (r : α → α → Prop) {n m : α} (h : ∀ i ∈ Set.Ico m n, r (Order.succ i) i) (hmn : m ≤ n)
Goal: Relation.ReflTransGen r n m
-- transGen_of_succ_of_lt
theorem transGen_of_succ_of_lt (r : α → α → Prop) {n m : α} (h : ∀ i ∈ Ico n m, r i (succ i))
    (hnm : n < m) : TransGen r n m 
Arguments: {α : Type u_1} [PartialOrder α] [SuccOrder α] [IsSuccArchimedean α] (r : α → α → Prop) {n m : α} (h : ∀ i ∈ Set.Ico n m, r i (Order.succ i)) (hnm : n < m)
Goal: Relation.TransGen r n m
-- transGen_of_succ_of_gt
theorem transGen_of_succ_of_gt (r : α → α → Prop) {n m : α} (h : ∀ i ∈ Ico m n, r (succ i) i)
    (hmn : m < n) : TransGen r n m 
Arguments: {α : Type u_1} [PartialOrder α] [SuccOrder α] [IsSuccArchimedean α] (r : α → α → Prop) {n m : α} (h : ∀ i ∈ Set.Ico m n, r (Order.succ i) i) (hmn : m < n)
Goal: Relation.TransGen r n m
-- reflTransGen_of_succ
theorem reflTransGen_of_succ (r : α → α → Prop) {n m : α} (h1 : ∀ i ∈ Ico n m, r i (succ i))
    (h2 : ∀ i ∈ Ico m n, r (succ i) i) : ReflTransGen r n m 
Arguments: {α : Type u_1} [LinearOrder α] [SuccOrder α] [IsSuccArchimedean α] (r : α → α → Prop) {n m : α} (h1 : ∀ i ∈ Set.Ico n m, r i (Order.succ i)) (h2 : ∀ i ∈ Set.Ico m n, r (Order.succ i) i)
Goal: Relation.ReflTransGen r n m
-- transGen_of_succ_of_ne
theorem transGen_of_succ_of_ne (r : α → α → Prop) {n m : α} (h1 : ∀ i ∈ Ico n m, r i (succ i))
    (h2 : ∀ i ∈ Ico m n, r (succ i) i) (hnm : n ≠ m) : TransGen r n m 
Arguments: {α : Type u_1} [LinearOrder α] [SuccOrder α] [IsSuccArchimedean α] (r : α → α → Prop) {n m : α} (h1 : ∀ i ∈ Set.Ico n m, r i (Order.succ i)) (h2 : ∀ i ∈ Set.Ico m n, r (Order.succ i) i) (hnm : n ≠ m)
Goal: Relation.TransGen r n m
-- transGen_of_succ_of_reflexive
theorem transGen_of_succ_of_reflexive (r : α → α → Prop) {n m : α} (hr : Reflexive r)
    (h1 : ∀ i ∈ Ico n m, r i (succ i)) (h2 : ∀ i ∈ Ico m n, r (succ i) i) : TransGen r n m 
Arguments: {α : Type u_1} [LinearOrder α] [SuccOrder α] [IsSuccArchimedean α] (r : α → α → Prop) {n m : α} (hr : Reflexive r) (h1 : ∀ i ∈ Set.Ico n m, r i (Order.succ i)) (h2 : ∀ i ∈ Set.Ico m n, r (Order.succ i) i)
Goal: Relation.TransGen r n m
-- reflTransGen_of_pred_of_ge
theorem reflTransGen_of_pred_of_ge (r : α → α → Prop) {n m : α} (h : ∀ i ∈ Ioc m n, r i (pred i))
    (hnm : m ≤ n) : ReflTransGen r n m 
Arguments: {α : Type u_1} [PartialOrder α] [PredOrder α] [IsPredArchimedean α] (r : α → α → Prop) {n m : α} (h : ∀ i ∈ Set.Ioc m n, r i (Order.pred i)) (hnm : m ≤ n)
Goal: Relation.ReflTransGen r n m
-- reflTransGen_of_pred_of_le
theorem reflTransGen_of_pred_of_le (r : α → α → Prop) {n m : α} (h : ∀ i ∈ Ioc n m, r (pred i) i)
    (hmn : n ≤ m) : ReflTransGen r n m 
Arguments: {α : Type u_1} [PartialOrder α] [PredOrder α] [IsPredArchimedean α] (r : α → α → Prop) {n m : α} (h : ∀ i ∈ Set.Ioc n m, r (Order.pred i) i) (hmn : n ≤ m)
Goal: Relation.ReflTransGen r n m
-- transGen_of_pred_of_gt
theorem transGen_of_pred_of_gt (r : α → α → Prop) {n m : α} (h : ∀ i ∈ Ioc m n, r i (pred i))
    (hnm : m < n) : TransGen r n m 
Arguments: {α : Type u_1} [PartialOrder α] [PredOrder α] [IsPredArchimedean α] (r : α → α → Prop) {n m : α} (h : ∀ i ∈ Set.Ioc m n, r i (Order.pred i)) (hnm : m < n)
Goal: Relation.TransGen r n m
-- transGen_of_pred_of_lt
theorem transGen_of_pred_of_lt (r : α → α → Prop) {n m : α} (h : ∀ i ∈ Ioc n m, r (pred i) i)
    (hmn : n < m) : TransGen r n m 
Arguments: {α : Type u_1} [PartialOrder α] [PredOrder α] [IsPredArchimedean α] (r : α → α → Prop) {n m : α} (h : ∀ i ∈ Set.Ioc n m, r (Order.pred i) i) (hmn : n < m)
Goal: Relation.TransGen r n m
-- reflTransGen_of_pred
theorem reflTransGen_of_pred (r : α → α → Prop) {n m : α} (h1 : ∀ i ∈ Ioc m n, r i (pred i))
    (h2 : ∀ i ∈ Ioc n m, r (pred i) i) : ReflTransGen r n m 
Arguments: {α : Type u_1} [LinearOrder α] [PredOrder α] [IsPredArchimedean α] (r : α → α → Prop) {n m : α} (h1 : ∀ i ∈ Set.Ioc m n, r i (Order.pred i)) (h2 : ∀ i ∈ Set.Ioc n m, r (Order.pred i) i)
Goal: Relation.ReflTransGen r n m
-- transGen_of_pred_of_ne
theorem transGen_of_pred_of_ne (r : α → α → Prop) {n m : α} (h1 : ∀ i ∈ Ioc m n, r i (pred i))
    (h2 : ∀ i ∈ Ioc n m, r (pred i) i) (hnm : n ≠ m) : TransGen r n m 
Arguments: {α : Type u_1} [LinearOrder α] [PredOrder α] [IsPredArchimedean α] (r : α → α → Prop) {n m : α} (h1 : ∀ i ∈ Set.Ioc m n, r i (Order.pred i)) (h2 : ∀ i ∈ Set.Ioc n m, r (Order.pred i) i) (hnm : n ≠ m)
Goal: Relation.TransGen r n m