ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Dense.inter_of_isOpen_right
theorem Dense.inter_of_isOpen_right (hs : Dense s) (ht : Dense t) (hto : IsOpen t) :
    Dense (s ∩ t) 
Arguments: {X : Type u} {s t : Set X} [TopologicalSpace X] (hs : Dense s) (ht : Dense t) (hto : IsOpen t)
Goal: Dense (s ∩ t)
-- Dense.inter_nhds_nonempty
theorem Dense.inter_nhds_nonempty (hs : Dense s) (ht : t ∈ 𝓝 x) :
    (s ∩ t).Nonempty 
Arguments: {X : Type u} {x : X} {s t : Set X} [TopologicalSpace X] (hs : Dense s) (ht : t ∈ 𝓝 x)
Goal: (s ∩ t).Nonempty
-- closure_diff
theorem closure_diff : closure s \ closure t ⊆ closure (s \ t) 
Arguments: {X : Type u} {s t : Set X} [TopologicalSpace X]
Goal: closure s \ closure t ⊆ closure (s \ t)
-- Filter.Frequently.mem_of_closed
theorem Filter.Frequently.mem_of_closed (h : ∃ᶠ x in 𝓝 x, x ∈ s)
    (hs : IsClosed s) : x ∈ s 
Arguments: {X : Type u} {x : X} {s : Set X} [TopologicalSpace X] (h : ∃ᶠ (x : X) in 𝓝 x, x ∈ s) (hs : IsClosed s)
Goal: x ∈ s
-- IsClosed.mem_of_frequently_of_tendsto
theorem IsClosed.mem_of_frequently_of_tendsto {f : α → X} {b : Filter α}
    (hs : IsClosed s) (h : ∃ᶠ x in b, f x ∈ s) (hf : Tendsto f b (𝓝 x)) : x ∈ s 
Arguments: {X : Type u} {α : Type u_1} {x : X} {s : Set X} [TopologicalSpace X] {f : α → X} {b : Filter α} (hs : IsClosed s) (h : ∃ᶠ (x : α) in b, f x ∈ s) (hf : Filter.Tendsto f b (𝓝 x))
Goal: x ∈ s
-- IsClosed.mem_of_tendsto
theorem IsClosed.mem_of_tendsto {f : α → X} {b : Filter α} [NeBot b]
    (hs : IsClosed s) (hf : Tendsto f b (𝓝 x)) (h : ∀ᶠ x in b, f x ∈ s) : x ∈ s 
Arguments: {X : Type u} {α : Type u_1} {x : X} {s : Set X} [TopologicalSpace X] {f : α → X} {b : Filter α} [b.NeBot] (hs : IsClosed s) (hf : Filter.Tendsto f b (𝓝 x)) (h : ∀ᶠ (x : α) in b, f x ∈ s)
Goal: x ∈ s
-- mem_closure_of_frequently_of_tendsto
theorem mem_closure_of_frequently_of_tendsto {f : α → X} {b : Filter α}
    (h : ∃ᶠ x in b, f x ∈ s) (hf : Tendsto f b (𝓝 x)) : x ∈ closure s 
Arguments: {X : Type u} {α : Type u_1} {x : X} {s : Set X} [TopologicalSpace X] {f : α → X} {b : Filter α} (h : ∃ᶠ (x : α) in b, f x ∈ s) (hf : Filter.Tendsto f b (𝓝 x))
Goal: x ∈ closure s
-- mem_closure_of_tendsto
theorem mem_closure_of_tendsto {f : α → X} {b : Filter α} [NeBot b]
    (hf : Tendsto f b (𝓝 x)) (h : ∀ᶠ x in b, f x ∈ s) : x ∈ closure s 
Arguments: {X : Type u} {α : Type u_1} {x : X} {s : Set X} [TopologicalSpace X] {f : α → X} {b : Filter α} [b.NeBot] (hf : Filter.Tendsto f b (𝓝 x)) (h : ∀ᶠ (x : α) in b, f x ∈ s)
Goal: x ∈ closure s
-- tendsto_inf_principal_nhds_iff_of_forall_eq
theorem tendsto_inf_principal_nhds_iff_of_forall_eq {f : α → X} {l : Filter α} {s : Set α}
    (h : ∀ a ∉ s, f a = x) : Tendsto f (l ⊓ 𝓟 s) (𝓝 x) ↔ Tendsto f l (𝓝 x) 
Arguments: {X : Type u} {α : Type u_1} {x : X} [TopologicalSpace X] {f : α → X} {l : Filter α} {s : Set α} (h : ∀ a ∉ s, f a = x)
Goal: Filter.Tendsto f (l ⊓ 𝓟 s) (𝓝 x) ↔ Filter.Tendsto f l (𝓝 x)
-- le_nhds_lim
theorem le_nhds_lim {f : Filter X} (h : ∃ x, f ≤ 𝓝 x) : f ≤ 𝓝 (@lim _ _ (nonempty_of_exists h) f) 
Arguments: {X : Type u} [TopologicalSpace X] {f : Filter X} (h : ∃ x, f ≤ 𝓝 x)
Goal: f ≤ 𝓝 (lim f)
-- tendsto_nhds_limUnder
theorem tendsto_nhds_limUnder {f : Filter α} {g : α → X} (h : ∃ x, Tendsto g f (𝓝 x)) :
    Tendsto g f (𝓝 (@limUnder _ _ _ (nonempty_of_exists h) f g)) 
Arguments: {X : Type u} {α : Type u_1} [TopologicalSpace X] {f : Filter α} {g : α → X} (h : ∃ x, Filter.Tendsto g f (𝓝 x))
Goal: Filter.Tendsto g f (𝓝 (limUnder f g))
-- continuous_def
theorem continuous_def {_ : TopologicalSpace X} {_ : TopologicalSpace Y} {f : X → Y} :
    Continuous f ↔ ∀ s, IsOpen s → IsOpen (f ⁻¹' s) 
Arguments: {X : Type u_1} {Y : Type u_2}
Goal: ∀ {x : TopologicalSpace X} {x_1 : TopologicalSpace Y} {f : X → Y},
  Continuous f ↔ ∀ (s : Set Y), IsOpen s → IsOpen (f ⁻¹' s)
-- IsOpen.preimage
theorem IsOpen.preimage (hf : Continuous f) {t : Set Y} (h : IsOpen t) :
    IsOpen (f ⁻¹' t) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Continuous f) {t : Set Y} (h : IsOpen t)
Goal: IsOpen (f ⁻¹' t)
-- continuous_congr
theorem continuous_congr {g : X → Y} (h : ∀ x, f x = g x) :
    Continuous f ↔ Continuous g 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f g : X → Y} (h : ∀ (x : X), f x = g x)
Goal: Continuous f ↔ Continuous g
-- Continuous.congr
theorem Continuous.congr {g : X → Y} (h : Continuous f) (h' : ∀ x, f x = g x) : Continuous g 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f g : X → Y} (h : Continuous f) (h' : ∀ (x : X), f x = g x)
Goal: Continuous g
-- ContinuousAt.tendsto
theorem ContinuousAt.tendsto (h : ContinuousAt f x) :
    Tendsto f (𝓝 x) (𝓝 (f x)) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {x : X} (h : ContinuousAt f x)
Goal: Filter.Tendsto f (𝓝 x) (𝓝 (f x))
-- continuousAt_def
theorem continuousAt_def : ContinuousAt f x ↔ ∀ A ∈ 𝓝 (f x), f ⁻¹' A ∈ 𝓝 x 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {x : X}
Goal: ContinuousAt f x ↔ ∀ A ∈ 𝓝 (f x), f ⁻¹' A ∈ 𝓝 x
-- continuousAt_congr
theorem continuousAt_congr {g : X → Y} (h : f =ᶠ[𝓝 x] g) :
    ContinuousAt f x ↔ ContinuousAt g x 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {x : X} {g : X → Y} (h : f =ᶠ[𝓝 x] g)
Goal: ContinuousAt f x ↔ ContinuousAt g x
-- ContinuousAt.congr
theorem ContinuousAt.congr {g : X → Y} (hf : ContinuousAt f x) (h : f =ᶠ[𝓝 x] g) :
    ContinuousAt g x 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {x : X} {g : X → Y} (hf : ContinuousAt f x) (h : f =ᶠ[𝓝 x] g)
Goal: ContinuousAt g x
-- ContinuousAt.preimage_mem_nhds
theorem ContinuousAt.preimage_mem_nhds {t : Set Y} (h : ContinuousAt f x)
    (ht : t ∈ 𝓝 (f x)) : f ⁻¹' t ∈ 𝓝 x 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {x : X} {t : Set Y} (h : ContinuousAt f x) (ht : t ∈ 𝓝 (f x))
Goal: f ⁻¹' t ∈ 𝓝 x
-- ContinuousAt.eventually_mem
theorem ContinuousAt.eventually_mem {f : X → Y} {x : X} (hf : ContinuousAt f x) {s : Set Y}
    (hs : s ∈ 𝓝 (f x)) : ∀ᶠ y in 𝓝 x, f y ∈ s 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {x : X} (hf : ContinuousAt f x) {s : Set Y} (hs : s ∈ 𝓝 (f x))
Goal: ∀ᶠ (y : X) in 𝓝 x, f y ∈ s
-- eventuallyEq_zero_nhds
@[deprecated (since := "2024-01-15")]
theorem eventuallyEq_zero_nhds {M₀} [Zero M₀] {f : X → M₀} :
    f =ᶠ[𝓝 x] 0 ↔ x ∉ closure (Function.support f) 
Arguments: {X : Type u_1} [TopologicalSpace X] {x : X} {M₀ : Type u_4} [Zero M₀] {f : X → M₀}
Goal: f =ᶠ[𝓝 x] 0 ↔ x ∉ closure (Function.support f)
-- ClusterPt.map
theorem ClusterPt.map {lx : Filter X} {ly : Filter Y} (H : ClusterPt x lx)
    (hfc : ContinuousAt f x) (hf : Tendsto f lx ly) : ClusterPt (f x) ly 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {x : X} {lx : Filter X} {ly : Filter Y} (H : ClusterPt x lx) (hfc : ContinuousAt f x) (hf : Filter.Tendsto f lx ly)
Goal: ClusterPt (f x) ly
-- preimage_interior_subset_interior_preimage
theorem preimage_interior_subset_interior_preimage {t : Set Y} (hf : Continuous f) :
    f ⁻¹' interior t ⊆ interior (f ⁻¹' t) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {t : Set Y} (hf : Continuous f)
Goal: f ⁻¹' interior t ⊆ interior (f ⁻¹' t)
-- continuous_id
@[continuity]
theorem continuous_id : Continuous (id : X → X) 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: Continuous id
-- continuous_id'
@[continuity, fun_prop]
theorem continuous_id' : Continuous (fun (x : X) => x) 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: Continuous fun x => x
-- Continuous.comp
theorem Continuous.comp {g : Y → Z} (hg : Continuous g) (hf : Continuous f) :
    Continuous (g ∘ f) 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → Y} {g : Y → Z} (hg : Continuous g) (hf : Continuous f)
Goal: Continuous (g ∘ f)
-- Continuous.comp'
@[continuity, fun_prop]
theorem Continuous.comp' {g : Y → Z} (hg : Continuous g) (hf : Continuous f) :
    Continuous (fun x => g (f x)) 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → Y} {g : Y → Z} (hg : Continuous g) (hf : Continuous f)
Goal: Continuous fun x => g (f x)
-- Continuous.iterate
theorem Continuous.iterate {f : X → X} (h : Continuous f) (n : ℕ) : Continuous f^[n] 
Arguments: {X : Type u_1} [TopologicalSpace X] {f : X → X} (h : Continuous f) (n : ℕ)
Goal: Continuous f^[n]
-- ContinuousAt.comp
nonrec theorem ContinuousAt.comp {g : Y → Z} (hg : ContinuousAt g (f x))
    (hf : ContinuousAt f x) : ContinuousAt (g ∘ f) x 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → Y} {x : X} {g : Y → Z} (hg : ContinuousAt g (f x)) (hf : ContinuousAt f x)
Goal: ContinuousAt (g ∘ f) x
-- ContinuousAt.comp'
@[fun_prop]
theorem ContinuousAt.comp' {g : Y → Z} {x : X} (hg : ContinuousAt g (f x))
    (hf : ContinuousAt f x) : ContinuousAt (fun x => g (f x)) x 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → Y} {g : Y → Z} {x : X} (hg : ContinuousAt g (f x)) (hf : ContinuousAt f x)
Goal: ContinuousAt (fun x => g (f x)) x
-- ContinuousAt.comp_of_eq
theorem ContinuousAt.comp_of_eq {g : Y → Z} (hg : ContinuousAt g y)
    (hf : ContinuousAt f x) (hy : f x = y) : ContinuousAt (g ∘ f) x 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → Y} {x : X} {y : Y} {g : Y → Z} (hg : ContinuousAt g y) (hf : ContinuousAt f x) (hy : f x = y)
Goal: ContinuousAt (g ∘ f) x
-- Continuous.tendsto
theorem Continuous.tendsto (hf : Continuous f) (x) : Tendsto f (𝓝 x) (𝓝 (f x)) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Continuous f) (x : X)
Goal: Filter.Tendsto f (𝓝 x) (𝓝 (f x))
-- Continuous.tendsto'
theorem Continuous.tendsto' (hf : Continuous f) (x : X) (y : Y) (h : f x = y) :
    Tendsto f (𝓝 x) (𝓝 y) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Continuous f) (x : X) (y : Y) (h : f x = y)
Goal: Filter.Tendsto f (𝓝 x) (𝓝 y)
-- Continuous.continuousAt
@[fun_prop]
theorem Continuous.continuousAt (h : Continuous f) : ContinuousAt f x 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {x : X} (h : Continuous f)
Goal: ContinuousAt f x
-- continuous_iff_continuousAt
theorem continuous_iff_continuousAt : Continuous f ↔ ∀ x, ContinuousAt f x 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y}
Goal: Continuous f ↔ ∀ (x : X), ContinuousAt f x
-- continuousAt_const
@[fun_prop]
theorem continuousAt_const : ContinuousAt (fun _ : X => y) x 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {x : X} {y : Y}
Goal: ContinuousAt (fun x => y) x
-- continuous_const
@[continuity, fun_prop]
theorem continuous_const : Continuous fun _ : X => y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {y : Y}
Goal: Continuous fun x => y
-- Filter.EventuallyEq.continuousAt
theorem Filter.EventuallyEq.continuousAt (h : f =ᶠ[𝓝 x] fun _ => y) :
    ContinuousAt f x 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {x : X} {y : Y} (h : f =ᶠ[𝓝 x] fun x => y)
Goal: ContinuousAt f x
-- continuous_of_const
theorem continuous_of_const (h : ∀ x y, f x = f y) : Continuous f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (h : ∀ (x y : X), f x = f y)
Goal: Continuous f
-- continuousAt_id
theorem continuousAt_id : ContinuousAt id x 
Arguments: {X : Type u_1} [TopologicalSpace X] {x : X}
Goal: ContinuousAt id x
-- continuousAt_id'
@[fun_prop]
theorem continuousAt_id' (y) : ContinuousAt (fun x : X => x) y 
Arguments: {X : Type u_1} [TopologicalSpace X] (y : X)
Goal: ContinuousAt (fun x => x) y
-- ContinuousAt.iterate
theorem ContinuousAt.iterate {f : X → X} (hf : ContinuousAt f x) (hx : f x = x) (n : ℕ) :
    ContinuousAt f^[n] x 
Arguments: {X : Type u_1} [TopologicalSpace X] {x : X} {f : X → X} (hf : ContinuousAt f x) (hx : f x = x) (n : ℕ)
Goal: ContinuousAt f^[n] x
-- continuous_iff_isClosed
theorem continuous_iff_isClosed : Continuous f ↔ ∀ s, IsClosed s → IsClosed (f ⁻¹' s) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y}
Goal: Continuous f ↔ ∀ (s : Set Y), IsClosed s → IsClosed (f ⁻¹' s)
-- IsClosed.preimage
theorem IsClosed.preimage (hf : Continuous f) {t : Set Y} (h : IsClosed t) :
    IsClosed (f ⁻¹' t) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Continuous f) {t : Set Y} (h : IsClosed t)
Goal: IsClosed (f ⁻¹' t)
-- mem_closure_image
theorem mem_closure_image (hf : ContinuousAt f x)
    (hx : x ∈ closure s) : f x ∈ closure (f '' s) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {s : Set X} {x : X} (hf : ContinuousAt f x) (hx : x ∈ closure s)
Goal: f x ∈ closure (f '' s)
-- continuousAt_iff_ultrafilter
theorem continuousAt_iff_ultrafilter :
    ContinuousAt f x ↔ ∀ g : Ultrafilter X, ↑g ≤ 𝓝 x → Tendsto f g (𝓝 (f x)) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {x : X}
Goal: ContinuousAt f x ↔ ∀ (g : Ultrafilter X), ↑g ≤ 𝓝 x → Filter.Tendsto f (↑g) (𝓝 (f x))
-- continuous_iff_ultrafilter
theorem continuous_iff_ultrafilter :
    Continuous f ↔ ∀ (x) (g : Ultrafilter X), ↑g ≤ 𝓝 x → Tendsto f g (𝓝 (f x)) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y}
Goal: Continuous f ↔ ∀ (x : X) (g : Ultrafilter X), ↑g ≤ 𝓝 x → Filter.Tendsto f (↑g) (𝓝 (f x))
-- Continuous.closure_preimage_subset
theorem Continuous.closure_preimage_subset (hf : Continuous f) (t : Set Y) :
    closure (f ⁻¹' t) ⊆ f ⁻¹' closure t 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Continuous f) (t : Set Y)
Goal: closure (f ⁻¹' t) ⊆ f ⁻¹' closure t
-- Continuous.frontier_preimage_subset
theorem Continuous.frontier_preimage_subset (hf : Continuous f) (t : Set Y) :
    frontier (f ⁻¹' t) ⊆ f ⁻¹' frontier t 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Continuous f) (t : Set Y)
Goal: frontier (f ⁻¹' t) ⊆ f ⁻¹' frontier t
-- Set.MapsTo.closure
protected theorem Set.MapsTo.closure {t : Set Y} (h : MapsTo f s t)
    (hc : Continuous f) : MapsTo f (closure s) (closure t) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {s : Set X} {t : Set Y} (h : Set.MapsTo f s t) (hc : Continuous f)
Goal: Set.MapsTo f (closure s) (closure t)
-- image_closure_subset_closure_image
theorem image_closure_subset_closure_image (h : Continuous f) :
    f '' closure s ⊆ closure (f '' s) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {s : Set X} (h : Continuous f)
Goal: f '' closure s ⊆ closure (f '' s)
-- closure_image_closure
theorem closure_image_closure (h : Continuous f) :
    closure (f '' closure s) = closure (f '' s) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {s : Set X} (h : Continuous f)
Goal: closure (f '' closure s) = closure (f '' s)
-- closure_subset_preimage_closure_image
theorem closure_subset_preimage_closure_image (h : Continuous f) :
    closure s ⊆ f ⁻¹' closure (f '' s) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {s : Set X} (h : Continuous f)
Goal: closure s ⊆ f ⁻¹' closure (f '' s)
-- map_mem_closure
theorem map_mem_closure {t : Set Y} (hf : Continuous f)
    (hx : x ∈ closure s) (ht : MapsTo f s t) : f x ∈ closure t 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {s : Set X} {x : X} {t : Set Y} (hf : Continuous f) (hx : x ∈ closure s) (ht : Set.MapsTo f s t)
Goal: f x ∈ closure t
-- Set.MapsTo.closure_left
theorem Set.MapsTo.closure_left {t : Set Y} (h : MapsTo f s t)
    (hc : Continuous f) (ht : IsClosed t) : MapsTo f (closure s) t 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {s : Set X} {t : Set Y} (h : Set.MapsTo f s t) (hc : Continuous f) (ht : IsClosed t)
Goal: Set.MapsTo f (closure s) t
-- Filter.Tendsto.lift'_closure
theorem Filter.Tendsto.lift'_closure (hf : Continuous f) {l l'} (h : Tendsto f l l') :
    Tendsto f (l.lift' closure) (l'.lift' closure) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Continuous f) {l : Filter X} {l' : Filter Y} (h : Filter.Tendsto f l l')
Goal: Filter.Tendsto f (l.lift' closure) (l'.lift' closure)
-- tendsto_lift'_closure_nhds
theorem tendsto_lift'_closure_nhds (hf : Continuous f) (x : X) :
    Tendsto f ((𝓝 x).lift' closure) ((𝓝 (f x)).lift' closure) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Continuous f) (x : X)
Goal: Filter.Tendsto f ((𝓝 x).lift' closure) ((𝓝 (f x)).lift' closure)
-- Function.Surjective.denseRange
theorem Function.Surjective.denseRange (hf : Function.Surjective f) : DenseRange f 
Arguments: {X : Type u_1} [TopologicalSpace X] {α : Type u_4} {f : α → X} (hf : Function.Surjective f)
Goal: DenseRange f
-- denseRange_id
theorem denseRange_id : DenseRange (id : X → X) 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: DenseRange id
-- denseRange_iff_closure_range
theorem denseRange_iff_closure_range : DenseRange f ↔ closure (range f) = univ 
Arguments: {X : Type u_1} [TopologicalSpace X] {α : Type u_4} {f : α → X}
Goal: DenseRange f ↔ closure (Set.range f) = Set.univ
-- DenseRange.closure_range
theorem DenseRange.closure_range (h : DenseRange f) : closure (range f) = univ 
Arguments: {X : Type u_1} [TopologicalSpace X] {α : Type u_4} {f : α → X} (h : DenseRange f)
Goal: closure (Set.range f) = Set.univ
-- denseRange_subtype_val
@[simp]
lemma denseRange_subtype_val {p : X → Prop} : DenseRange (@Subtype.val _ p) ↔ Dense {x | p x} 
Arguments: {X : Type u_1} [TopologicalSpace X] {p : X → Prop}
Goal: DenseRange Subtype.val ↔ Dense {x | p x}
-- Dense.denseRange_val
theorem Dense.denseRange_val (h : Dense s) : DenseRange ((↑) : s → X) 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X} (h : Dense s)
Goal: DenseRange Subtype.val
-- Continuous.range_subset_closure_image_dense
theorem Continuous.range_subset_closure_image_dense {f : X → Y} (hf : Continuous f)
    (hs : Dense s) : range f ⊆ closure (f '' s) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} {f : X → Y} (hf : Continuous f) (hs : Dense s)
Goal: Set.range f ⊆ closure (f '' s)
-- DenseRange.dense_image
theorem DenseRange.dense_image {f : X → Y} (hf' : DenseRange f) (hf : Continuous f)
    (hs : Dense s) : Dense (f '' s) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} {f : X → Y} (hf' : DenseRange f) (hf : Continuous f) (hs : Dense s)
Goal: Dense (f '' s)
-- DenseRange.subset_closure_image_preimage_of_isOpen
theorem DenseRange.subset_closure_image_preimage_of_isOpen (hf : DenseRange f) (hs : IsOpen s) :
    s ⊆ closure (f '' (f ⁻¹' s)) 
Arguments: {X : Type u_1} [TopologicalSpace X] {α : Type u_4} {f : α → X} {s : Set X} (hf : DenseRange f) (hs : IsOpen s)
Goal: s ⊆ closure (f '' (f ⁻¹' s))
-- DenseRange.dense_of_mapsTo
theorem DenseRange.dense_of_mapsTo {f : X → Y} (hf' : DenseRange f) (hf : Continuous f)
    (hs : Dense s) {t : Set Y} (ht : MapsTo f s t) : Dense t 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} {f : X → Y} (hf' : DenseRange f) (hf : Continuous f) (hs : Dense s) {t : Set Y} (ht : Set.MapsTo f s t)
Goal: Dense t
-- DenseRange.comp
theorem DenseRange.comp {g : Y → Z} {f : α → Y} (hg : DenseRange g) (hf : DenseRange f)
    (cg : Continuous g) : DenseRange (g ∘ f) 
Arguments: {Y : Type u_2} {Z : Type u_3} [TopologicalSpace Y] [TopologicalSpace Z] {α : Type u_4} {g : Y → Z} {f : α → Y} (hg : DenseRange g) (hf : DenseRange f) (cg : Continuous g)
Goal: DenseRange (g ∘ f)
-- DenseRange.nonempty_iff
nonrec theorem DenseRange.nonempty_iff (hf : DenseRange f) : Nonempty α ↔ Nonempty X 
Arguments: {X : Type u_1} [TopologicalSpace X] {α : Type u_4} {f : α → X} (hf : DenseRange f)
Goal: Nonempty α ↔ Nonempty X
-- DenseRange.nonempty
theorem DenseRange.nonempty [h : Nonempty X] (hf : DenseRange f) : Nonempty α 
Arguments: {X : Type u_1} [TopologicalSpace X] {α : Type u_4} {f : α → X} [h : Nonempty X] (hf : DenseRange f)
Goal: Nonempty α
-- DenseRange.exists_mem_open
nonrec theorem DenseRange.exists_mem_open (hf : DenseRange f) (ho : IsOpen s) (hs : s.Nonempty) :
    ∃ a, f a ∈ s 
Arguments: {X : Type u_1} [TopologicalSpace X] {α : Type u_4} {f : α → X} {s : Set X} (hf : DenseRange f) (ho : IsOpen s) (hs : s.Nonempty)
Goal: ∃ a, f a ∈ s
-- DenseRange.mem_nhds
theorem DenseRange.mem_nhds (h : DenseRange f) (hs : s ∈ 𝓝 x) :
    ∃ a, f a ∈ s 
Arguments: {X : Type u_1} [TopologicalSpace X] {x : X} {α : Type u_4} {f : α → X} {s : Set X} (h : DenseRange f) (hs : s ∈ 𝓝 x)
Goal: ∃ a, f a ∈ s
-- isProperMap_iff_isClosedMap_ultrafilter
theorem isProperMap_iff_isClosedMap_ultrafilter {X : Type u} {Y : Type v} [TopologicalSpace X]
    [TopologicalSpace Y] {f : X → Y} :
    IsProperMap f ↔ Continuous f ∧ IsClosedMap
      (Prod.map f id : X × Ultrafilter X → Y × Ultrafilter X) 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y}
Goal: IsProperMap f ↔ Continuous f ∧ IsClosedMap (Prod.map f id)
-- isProperMap_iff_universally_closed
theorem isProperMap_iff_universally_closed {X : Type u} {Y : Type v} [TopologicalSpace X]
    [TopologicalSpace Y] {f : X → Y} :
    IsProperMap f ↔ Continuous f ∧ ∀ (Z : Type u) [TopologicalSpace Z],
      IsClosedMap (Prod.map f id : X × Z → Y × Z) 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y}
Goal: IsProperMap f ↔ Continuous f ∧ ∀ (Z : Type u) [inst : TopologicalSpace Z], IsClosedMap (Prod.map f id)
-- Filter.isOpen_Iic_principal
theorem isOpen_Iic_principal {s : Set α} : IsOpen (Iic (𝓟 s)) 
Arguments: {α : Type u_2} {s : Set α}
Goal: IsOpen (Set.Iic (𝓟 s))
-- Filter.isOpen_setOf_mem
theorem isOpen_setOf_mem {s : Set α} : IsOpen { l : Filter α | s ∈ l } 
Arguments: {α : Type u_2} {s : Set α}
Goal: IsOpen {l | s ∈ l}
-- Filter.isTopologicalBasis_Iic_principal
theorem isTopologicalBasis_Iic_principal :
    IsTopologicalBasis (range (Iic ∘ 𝓟 : Set α → Set (Filter α))) 
Arguments: {α : Type u_2}
Goal: TopologicalSpace.IsTopologicalBasis (Set.range (Set.Iic ∘ 𝓟))
-- Filter.isOpen_iff
theorem isOpen_iff {s : Set (Filter α)} : IsOpen s ↔ ∃ T : Set (Set α), s = ⋃ t ∈ T, Iic (𝓟 t) 
Arguments: {α : Type u_2} {s : Set (Filter α)}
Goal: IsOpen s ↔ ∃ T, s = ⋃ t ∈ T, Set.Iic (𝓟 t)
-- Filter.nhds_eq
theorem nhds_eq (l : Filter α) : 𝓝 l = l.lift' (Iic ∘ 𝓟) 
Arguments: {α : Type u_2} (l : Filter α)
Goal: 𝓝 l = l.lift' (Set.Iic ∘ 𝓟)
-- Filter.nhds_eq'
theorem nhds_eq' (l : Filter α) : 𝓝 l = l.lift' fun s => { l' | s ∈ l' } 
Arguments: {α : Type u_2} (l : Filter α)
Goal: 𝓝 l = l.lift' fun s => {l' | s ∈ l'}
-- Filter.tendsto_nhds
protected theorem tendsto_nhds {la : Filter α} {lb : Filter β} {f : α → Filter β} :
    Tendsto f la (𝓝 lb) ↔ ∀ s ∈ lb, ∀ᶠ a in la, s ∈ f a 
Arguments: {α : Type u_2} {β : Type u_3} {la : Filter α} {lb : Filter β} {f : α → Filter β}
Goal: Filter.Tendsto f la (𝓝 lb) ↔ ∀ s ∈ lb, ∀ᶠ (a : α) in la, s ∈ f a
-- Filter.HasBasis.nhds
protected theorem HasBasis.nhds {l : Filter α} {p : ι → Prop} {s : ι → Set α} (h : HasBasis l p s) :
    HasBasis (𝓝 l) p fun i => Iic (𝓟 (s i)) 
Arguments: {ι : Sort u_1} {α : Type u_2} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (h : l.HasBasis p s)
Goal: (𝓝 l).HasBasis p fun i => Set.Iic (𝓟 (s i))