ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Filter.Tendsto.apply_nhds
theorem Filter.Tendsto.apply_nhds {l : Filter Y} {f : Y → ∀ i, π i} {x : ∀ i, π i}
    (h : Tendsto f l (𝓝 x)) (i : ι) : Tendsto (fun a => f a i) l (𝓝 <| x i) 
Arguments: {Y : Type v} {ι : Type u_5} {π : ι → Type u_6} [T : (i : ι) → TopologicalSpace (π i)] {l : Filter Y} {f : Y → (i : ι) → π i} {x : (i : ι) → π i} (h : Filter.Tendsto f l (𝓝 x)) (i : ι)
Goal: Filter.Tendsto (fun a => f a i) l (𝓝 (x i))
-- nhds_pi
theorem nhds_pi {a : ∀ i, π i} : 𝓝 a = pi fun i => 𝓝 (a i) 
Arguments: {ι : Type u_5} {π : ι → Type u_6} [T : (i : ι) → TopologicalSpace (π i)] {a : (i : ι) → π i}
Goal: 𝓝 a = Filter.pi fun i => 𝓝 (a i)
-- tendsto_pi_nhds
theorem tendsto_pi_nhds {f : Y → ∀ i, π i} {g : ∀ i, π i} {u : Filter Y} :
    Tendsto f u (𝓝 g) ↔ ∀ x, Tendsto (fun i => f i x) u (𝓝 (g x)) 
Arguments: {Y : Type v} {ι : Type u_5} {π : ι → Type u_6} [T : (i : ι) → TopologicalSpace (π i)] {f : Y → (i : ι) → π i} {g : (i : ι) → π i} {u : Filter Y}
Goal: Filter.Tendsto f u (𝓝 g) ↔ ∀ (x : ι), Filter.Tendsto (fun i => f i x) u (𝓝 (g x))
-- continuousAt_pi
theorem continuousAt_pi {f : X → ∀ i, π i} {x : X} :
    ContinuousAt f x ↔ ∀ i, ContinuousAt (fun y => f y i) x 
Arguments: {X : Type u} {ι : Type u_5} {π : ι → Type u_6} [TopologicalSpace X] [T : (i : ι) → TopologicalSpace (π i)] {f : X → (i : ι) → π i} {x : X}
Goal: ContinuousAt f x ↔ ∀ (i : ι), ContinuousAt (fun y => f y i) x
-- continuousAt_pi'
@[fun_prop]
theorem continuousAt_pi' {f : X → ∀ i, π i} {x : X} (hf : ∀ i, ContinuousAt (fun y => f y i) x) :
    ContinuousAt f x 
Arguments: {X : Type u} {ι : Type u_5} {π : ι → Type u_6} [TopologicalSpace X] [T : (i : ι) → TopologicalSpace (π i)] {f : X → (i : ι) → π i} {x : X} (hf : ∀ (i : ι), ContinuousAt (fun y => f y i) x)
Goal: ContinuousAt f x
-- Pi.continuous_precomp'
theorem Pi.continuous_precomp' {ι' : Type*} (φ : ι' → ι) :
    Continuous (fun (f : (∀ i, π i)) (j : ι') ↦ f (φ j)) 
Arguments: {ι : Type u_5} {π : ι → Type u_6} [T : (i : ι) → TopologicalSpace (π i)] {ι' : Type u_8} (φ : ι' → ι)
Goal: Continuous fun f j => f (φ j)
-- Pi.continuous_precomp
theorem Pi.continuous_precomp {ι' : Type*} (φ : ι' → ι) :
    Continuous (· ∘ φ : (ι → X) → (ι' → X)) 
Arguments: {X : Type u} {ι : Type u_5} [TopologicalSpace X] {ι' : Type u_8} (φ : ι' → ι)
Goal: Continuous fun x => x ∘ φ
-- Pi.continuous_postcomp'
theorem Pi.continuous_postcomp' {X : ι → Type*} [∀ i, TopologicalSpace (X i)]
    {g : ∀ i, π i → X i} (hg : ∀ i, Continuous (g i)) :
    Continuous (fun (f : (∀ i, π i)) (i : ι) ↦ g i (f i)) 
Arguments: {ι : Type u_5} {π : ι → Type u_6} [T : (i : ι) → TopologicalSpace (π i)] {X : ι → Type u_8} [(i : ι) → TopologicalSpace (X i)] {g : (i : ι) → π i → X i} (hg : ∀ (i : ι), Continuous (g i))
Goal: Continuous fun f i => g i (f i)
-- Pi.continuous_postcomp
theorem Pi.continuous_postcomp [TopologicalSpace Y] {g : X → Y} (hg : Continuous g) :
    Continuous (g ∘ · : (ι → X) → (ι → Y)) 
Arguments: {X : Type u} {Y : Type v} {ι : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] {g : X → Y} (hg : Continuous g)
Goal: Continuous fun x => g ∘ x
-- Pi.induced_precomp'
lemma Pi.induced_precomp' {ι' : Type*} (φ : ι' → ι) :
    induced (fun (f : (∀ i, π i)) (j : ι') ↦ f (φ j)) Pi.topologicalSpace =
    ⨅ i', induced (eval (φ i')) (T (φ i')) 
Arguments: {ι : Type u_5} {π : ι → Type u_6} [T : (i : ι) → TopologicalSpace (π i)] {ι' : Type u_8} (φ : ι' → ι)
Goal: TopologicalSpace.induced (fun f j => f (φ j)) Pi.topologicalSpace =
  ⨅ i', TopologicalSpace.induced (Function.eval (φ i')) (T (φ i'))
-- Pi.induced_precomp
lemma Pi.induced_precomp [TopologicalSpace Y] {ι' : Type*} (φ : ι' → ι) :
    induced (· ∘ φ) Pi.topologicalSpace =
    ⨅ i', induced (eval (φ i')) ‹TopologicalSpace Y› 
Arguments: {Y : Type v} {ι : Type u_5} [TopologicalSpace Y] {ι' : Type u_8} (φ : ι' → ι)
Goal: TopologicalSpace.induced (fun x => x ∘ φ) Pi.topologicalSpace =
  ⨅ i', TopologicalSpace.induced (Function.eval (φ i')) inst✝
-- Pi.continuous_restrict
lemma Pi.continuous_restrict (S : Set ι) :
    Continuous (S.restrict : (∀ i : ι, π i) → (∀ i : S, π i)) 
Arguments: {ι : Type u_5} {π : ι → Type u_6} [T : (i : ι) → TopologicalSpace (π i)] (S : Set ι)
Goal: Continuous S.restrict
-- Pi.induced_restrict
lemma Pi.induced_restrict (S : Set ι) :
    induced (S.restrict) Pi.topologicalSpace =
    ⨅ i ∈ S, induced (eval i) (T i) 
Arguments: {ι : Type u_5} {π : ι → Type u_6} [T : (i : ι) → TopologicalSpace (π i)] (S : Set ι)
Goal: TopologicalSpace.induced S.restrict Pi.topologicalSpace = ⨅ i ∈ S, TopologicalSpace.induced (Function.eval i) (T i)
-- Pi.induced_restrict_sUnion
lemma Pi.induced_restrict_sUnion (𝔖 : Set (Set ι)) :
    induced (⋃₀ 𝔖).restrict (Pi.topologicalSpace (Y := fun i : (⋃₀ 𝔖) ↦ π i)) =
    ⨅ S ∈ 𝔖, induced S.restrict Pi.topologicalSpace 
Arguments: {ι : Type u_5} {π : ι → Type u_6} [T : (i : ι) → TopologicalSpace (π i)] (𝔖 : Set (Set ι))
Goal: TopologicalSpace.induced (⋃₀ 𝔖).restrict Pi.topologicalSpace =
  ⨅ S ∈ 𝔖, TopologicalSpace.induced S.restrict Pi.topologicalSpace
-- Filter.Tendsto.update
theorem Filter.Tendsto.update [DecidableEq ι] {l : Filter Y} {f : Y → ∀ i, π i} {x : ∀ i, π i}
    (hf : Tendsto f l (𝓝 x)) (i : ι) {g : Y → π i} {xi : π i} (hg : Tendsto g l (𝓝 xi)) :
    Tendsto (fun a => update (f a) i (g a)) l (𝓝 <| update x i xi) 
Arguments: {Y : Type v} {ι : Type u_5} {π : ι → Type u_6} [T : (i : ι) → TopologicalSpace (π i)] [DecidableEq ι] {l : Filter Y} {f : Y → (i : ι) → π i} {x : (i : ι) → π i} (hf : Filter.Tendsto f l (𝓝 x)) (i : ι) {g : Y → π i} {xi : π i} (hg : Filter.Tendsto g l (𝓝 xi))
Goal: Filter.Tendsto (fun a => Function.update (f a) i (g a)) l (𝓝 (Function.update x i xi))
-- ContinuousAt.update
theorem ContinuousAt.update [DecidableEq ι] {x : X} (hf : ContinuousAt f x) (i : ι) {g : X → π i}
    (hg : ContinuousAt g x) : ContinuousAt (fun a => update (f a) i (g a)) x 
Arguments: {X : Type u} {ι : Type u_5} {π : ι → Type u_6} [TopologicalSpace X] [T : (i : ι) → TopologicalSpace (π i)] {f : X → (i : ι) → π i} [DecidableEq ι] {x : X} (hf : ContinuousAt f x) (i : ι) {g : X → π i} (hg : ContinuousAt g x)
Goal: ContinuousAt (fun a => Function.update (f a) i (g a)) x
-- Continuous.update
theorem Continuous.update [DecidableEq ι] (hf : Continuous f) (i : ι) {g : X → π i}
    (hg : Continuous g) : Continuous fun a => update (f a) i (g a) 
Arguments: {X : Type u} {ι : Type u_5} {π : ι → Type u_6} [TopologicalSpace X] [T : (i : ι) → TopologicalSpace (π i)] {f : X → (i : ι) → π i} [DecidableEq ι] (hf : Continuous f) (i : ι) {g : X → π i} (hg : Continuous g)
Goal: Continuous fun a => Function.update (f a) i (g a)
-- continuous_update
@[continuity, fun_prop]
theorem continuous_update [DecidableEq ι] (i : ι) :
    Continuous fun f : (∀ j, π j) × π i => update f.1 i f.2 
Arguments: {ι : Type u_5} {π : ι → Type u_6} [T : (i : ι) → TopologicalSpace (π i)] [DecidableEq ι] (i : ι)
Goal: Continuous fun f => Function.update f.1 i f.2
-- continuous_single
-- Porting note (#11215): TODO: restore @[continuity]
@[to_additive "`Pi.single i x` is continuous in `x`."]
theorem continuous_mulSingle [∀ i, One (π i)] [DecidableEq ι] (i : ι) :
    Continuous fun x => (Pi.mulSingle i x : ∀ i, π i) 
Arguments: {ι : Type u_5} {π : ι → Type u_6} [T : (i : ι) → TopologicalSpace (π i)] [(i : ι) → Zero (π i)] [DecidableEq ι] (i : ι)
Goal: Continuous fun x => Pi.single i x
-- continuous_mulSingle
-- Porting note (#11215): TODO: restore @[continuity]
@[to_additive "`Pi.single i x` is continuous in `x`."]
theorem continuous_mulSingle [∀ i, One (π i)] [DecidableEq ι] (i : ι) :
    Continuous fun x => (Pi.mulSingle i x : ∀ i, π i) 
Arguments: {ι : Type u_5} {π : ι → Type u_6} [T : (i : ι) → TopologicalSpace (π i)] [(i : ι) → One (π i)] [DecidableEq ι] (i : ι)
Goal: Continuous fun x => Pi.mulSingle i x
-- Filter.Tendsto.fin_insertNth
theorem Filter.Tendsto.fin_insertNth {n} {π : Fin (n + 1) → Type*} [∀ i, TopologicalSpace (π i)]
    (i : Fin (n + 1)) {f : Y → π i} {l : Filter Y} {x : π i} (hf : Tendsto f l (𝓝 x))
    {g : Y → ∀ j : Fin n, π (i.succAbove j)} {y : ∀ j, π (i.succAbove j)} (hg : Tendsto g l (𝓝 y)) :
    Tendsto (fun a => i.insertNth (f a) (g a)) l (𝓝 <| i.insertNth x y) 
Arguments: {Y : Type v} {n : ℕ} {π : Fin (n + 1) → Type u_8} [(i : Fin (n + 1)) → TopologicalSpace (π i)] (i : Fin (n + 1)) {f : Y → π i} {l : Filter Y} {x : π i} (hf : Filter.Tendsto f l (𝓝 x)) {g : Y → (j : Fin n) → π (i.succAbove j)} {y : (j : Fin n) → π (i.succAbove j)} (hg : Filter.Tendsto g l (𝓝 y))
Goal: Filter.Tendsto (fun a => i.insertNth (f a) (g a)) l (𝓝 (i.insertNth x y))
-- ContinuousAt.fin_insertNth
theorem ContinuousAt.fin_insertNth {n} {π : Fin (n + 1) → Type*} [∀ i, TopologicalSpace (π i)]
    (i : Fin (n + 1)) {f : X → π i} {x : X} (hf : ContinuousAt f x)
    {g : X → ∀ j : Fin n, π (i.succAbove j)} (hg : ContinuousAt g x) :
    ContinuousAt (fun a => i.insertNth (f a) (g a)) x 
Arguments: {X : Type u} [TopologicalSpace X] {n : ℕ} {π : Fin (n + 1) → Type u_8} [(i : Fin (n + 1)) → TopologicalSpace (π i)] (i : Fin (n + 1)) {f : X → π i} {x : X} (hf : ContinuousAt f x) {g : X → (j : Fin n) → π (i.succAbove j)} (hg : ContinuousAt g x)
Goal: ContinuousAt (fun a => i.insertNth (f a) (g a)) x
-- Continuous.fin_insertNth
theorem Continuous.fin_insertNth {n} {π : Fin (n + 1) → Type*} [∀ i, TopologicalSpace (π i)]
    (i : Fin (n + 1)) {f : X → π i} (hf : Continuous f) {g : X → ∀ j : Fin n, π (i.succAbove j)}
    (hg : Continuous g) : Continuous fun a => i.insertNth (f a) (g a) 
Arguments: {X : Type u} [TopologicalSpace X] {n : ℕ} {π : Fin (n + 1) → Type u_8} [(i : Fin (n + 1)) → TopologicalSpace (π i)] (i : Fin (n + 1)) {f : X → π i} (hf : Continuous f) {g : X → (j : Fin n) → π (i.succAbove j)} (hg : Continuous g)
Goal: Continuous fun a => i.insertNth (f a) (g a)
-- isOpen_set_pi
theorem isOpen_set_pi {i : Set ι} {s : ∀ a, Set (π a)} (hi : i.Finite)
    (hs : ∀ a ∈ i, IsOpen (s a)) : IsOpen (pi i s) 
Arguments: {ι : Type u_5} {π : ι → Type u_6} [T : (i : ι) → TopologicalSpace (π i)] {i : Set ι} {s : (a : ι) → Set (π a)} (hi : i.Finite) (hs : ∀ a ∈ i, IsOpen (s a))
Goal: IsOpen (i.pi s)
-- isOpen_pi_iff
theorem isOpen_pi_iff {s : Set (∀ a, π a)} :
    IsOpen s ↔
      ∀ f, f ∈ s → ∃ (I : Finset ι) (u : ∀ a, Set (π a)),
        (∀ a, a ∈ I → IsOpen (u a) ∧ f a ∈ u a) ∧ (I : Set ι).pi u ⊆ s 
Arguments: {ι : Type u_5} {π : ι → Type u_6} [T : (i : ι) → TopologicalSpace (π i)] {s : Set ((a : ι) → π a)}
Goal: IsOpen s ↔ ∀ f ∈ s, ∃ I u, (∀ a ∈ I, IsOpen (u a) ∧ f a ∈ u a) ∧ (↑I).pi u ⊆ s
-- isOpen_pi_iff'
theorem isOpen_pi_iff' [Finite ι] {s : Set (∀ a, π a)} :
    IsOpen s ↔
      ∀ f, f ∈ s → ∃ u : ∀ a, Set (π a), (∀ a, IsOpen (u a) ∧ f a ∈ u a) ∧ univ.pi u ⊆ s 
Arguments: {ι : Type u_5} {π : ι → Type u_6} [T : (i : ι) → TopologicalSpace (π i)] [Finite ι] {s : Set ((a : ι) → π a)}
Goal: IsOpen s ↔ ∀ f ∈ s, ∃ u, (∀ (a : ι), IsOpen (u a) ∧ f a ∈ u a) ∧ Set.univ.pi u ⊆ s
-- isClosed_set_pi
theorem isClosed_set_pi {i : Set ι} {s : ∀ a, Set (π a)} (hs : ∀ a ∈ i, IsClosed (s a)) :
    IsClosed (pi i s) 
Arguments: {ι : Type u_5} {π : ι → Type u_6} [T : (i : ι) → TopologicalSpace (π i)] {i : Set ι} {s : (a : ι) → Set (π a)} (hs : ∀ a ∈ i, IsClosed (s a))
Goal: IsClosed (i.pi s)
-- mem_nhds_of_pi_mem_nhds
theorem mem_nhds_of_pi_mem_nhds {I : Set ι} {s : ∀ i, Set (π i)} (a : ∀ i, π i) (hs : I.pi s ∈ 𝓝 a)
    {i : ι} (hi : i ∈ I) : s i ∈ 𝓝 (a i) 
Arguments: {ι : Type u_5} {π : ι → Type u_6} [T : (i : ι) → TopologicalSpace (π i)] {I : Set ι} {s : (i : ι) → Set (π i)} (a : (i : ι) → π i) (hs : I.pi s ∈ 𝓝 a) {i : ι} (hi : i ∈ I)
Goal: s i ∈ 𝓝 (a i)
-- set_pi_mem_nhds
theorem set_pi_mem_nhds {i : Set ι} {s : ∀ a, Set (π a)} {x : ∀ a, π a} (hi : i.Finite)
    (hs : ∀ a ∈ i, s a ∈ 𝓝 (x a)) : pi i s ∈ 𝓝 x 
Arguments: {ι : Type u_5} {π : ι → Type u_6} [T : (i : ι) → TopologicalSpace (π i)] {i : Set ι} {s : (a : ι) → Set (π a)} {x : (a : ι) → π a} (hi : i.Finite) (hs : ∀ a ∈ i, s a ∈ 𝓝 (x a))
Goal: i.pi s ∈ 𝓝 x
-- set_pi_mem_nhds_iff
theorem set_pi_mem_nhds_iff {I : Set ι} (hI : I.Finite) {s : ∀ i, Set (π i)} (a : ∀ i, π i) :
    I.pi s ∈ 𝓝 a ↔ ∀ i : ι, i ∈ I → s i ∈ 𝓝 (a i) 
Arguments: {ι : Type u_5} {π : ι → Type u_6} [T : (i : ι) → TopologicalSpace (π i)] {I : Set ι} (hI : I.Finite) {s : (i : ι) → Set (π i)} (a : (i : ι) → π i)
Goal: I.pi s ∈ 𝓝 a ↔ ∀ i ∈ I, s i ∈ 𝓝 (a i)
-- interior_pi_set
theorem interior_pi_set {I : Set ι} (hI : I.Finite) {s : ∀ i, Set (π i)} :
    interior (pi I s) = I.pi fun i => interior (s i) 
Arguments: {ι : Type u_5} {π : ι → Type u_6} [T : (i : ι) → TopologicalSpace (π i)] {I : Set ι} (hI : I.Finite) {s : (i : ι) → Set (π i)}
Goal: interior (I.pi s) = I.pi fun i => interior (s i)
-- exists_finset_piecewise_mem_of_mem_nhds
theorem exists_finset_piecewise_mem_of_mem_nhds [DecidableEq ι] {s : Set (∀ a, π a)} {x : ∀ a, π a}
    (hs : s ∈ 𝓝 x) (y : ∀ a, π a) : ∃ I : Finset ι, I.piecewise x y ∈ s 
Arguments: {ι : Type u_5} {π : ι → Type u_6} [T : (i : ι) → TopologicalSpace (π i)] [DecidableEq ι] {s : Set ((a : ι) → π a)} {x : (a : ι) → π a} (hs : s ∈ 𝓝 x) (y : (a : ι) → π a)
Goal: ∃ I, I.piecewise x y ∈ s
-- pi_generateFrom_eq
theorem pi_generateFrom_eq {π : ι → Type*} {g : ∀ a, Set (Set (π a))} :
    (@Pi.topologicalSpace ι π fun a => generateFrom (g a)) =
      generateFrom
        { t | ∃ (s : ∀ a, Set (π a)) (i : Finset ι), (∀ a ∈ i, s a ∈ g a) ∧ t = pi (↑i) s } 
Arguments: {ι : Type u_5} {π : ι → Type u_8} {g : (a : ι) → Set (Set (π a))}
Goal: Pi.topologicalSpace = TopologicalSpace.generateFrom {t | ∃ s i, (∀ a ∈ i, s a ∈ g a) ∧ t = (↑i).pi s}
-- pi_eq_generateFrom
theorem pi_eq_generateFrom :
    Pi.topologicalSpace =
      generateFrom
        { g | ∃ (s : ∀ a, Set (π a)) (i : Finset ι), (∀ a ∈ i, IsOpen (s a)) ∧ g = pi (↑i) s } 
Arguments: {ι : Type u_5} {π : ι → Type u_6} [T : (i : ι) → TopologicalSpace (π i)]
Goal: Pi.topologicalSpace = TopologicalSpace.generateFrom {g | ∃ s i, (∀ a ∈ i, IsOpen (s a)) ∧ g = (↑i).pi s}
-- pi_generateFrom_eq_finite
theorem pi_generateFrom_eq_finite {π : ι → Type*} {g : ∀ a, Set (Set (π a))} [Finite ι]
    (hg : ∀ a, ⋃₀ g a = univ) :
    (@Pi.topologicalSpace ι π fun a => generateFrom (g a)) =
      generateFrom { t | ∃ s : ∀ a, Set (π a), (∀ a, s a ∈ g a) ∧ t = pi univ s } 
Arguments: {ι : Type u_5} {π : ι → Type u_8} {g : (a : ι) → Set (Set (π a))} [Finite ι] (hg : ∀ (a : ι), ⋃₀ g a = Set.univ)
Goal: Pi.topologicalSpace = TopologicalSpace.generateFrom {t | ∃ s, (∀ (a : ι), s a ∈ g a) ∧ t = Set.univ.pi s}
-- induced_to_pi
theorem induced_to_pi {X : Type*} (f : X → ∀ i, π i) :
    induced f Pi.topologicalSpace = ⨅ i, induced (f · i) inferInstance 
Arguments: {ι : Type u_5} {π : ι → Type u_6} [T : (i : ι) → TopologicalSpace (π i)] {X : Type u_8} (f : X → (i : ι) → π i)
Goal: TopologicalSpace.induced f Pi.topologicalSpace = ⨅ i, TopologicalSpace.induced (fun x => f x i) inferInstance
-- inducing_iInf_to_pi
theorem inducing_iInf_to_pi {X : Type*} (f : ∀ i, X → π i) :
    @Inducing X (∀ i, π i) (⨅ i, induced (f i) inferInstance) _ fun x i => f i x 
Arguments: {ι : Type u_5} {π : ι → Type u_6} [T : (i : ι) → TopologicalSpace (π i)] {X : Type u_8} (f : (i : ι) → X → π i)
Goal: Inducing fun x i => f i x
-- continuous_sigmaMk
@[continuity, fun_prop]
theorem continuous_sigmaMk {i : ι} : Continuous (@Sigma.mk ι σ i) 
Arguments: {ι : Type u_5} {σ : ι → Type u_7} [(i : ι) → TopologicalSpace (σ i)] {i : ι}
Goal: Continuous (Sigma.mk i)
-- isOpen_sigma_iff
theorem isOpen_sigma_iff {s : Set (Sigma σ)} : IsOpen s ↔ ∀ i, IsOpen (Sigma.mk i ⁻¹' s) 
Arguments: {ι : Type u_5} {σ : ι → Type u_7} [(i : ι) → TopologicalSpace (σ i)] {s : Set (Sigma σ)}
Goal: IsOpen s ↔ ∀ (i : ι), IsOpen (Sigma.mk i ⁻¹' s)
-- isClosed_sigma_iff
theorem isClosed_sigma_iff {s : Set (Sigma σ)} : IsClosed s ↔ ∀ i, IsClosed (Sigma.mk i ⁻¹' s) 
Arguments: {ι : Type u_5} {σ : ι → Type u_7} [(i : ι) → TopologicalSpace (σ i)] {s : Set (Sigma σ)}
Goal: IsClosed s ↔ ∀ (i : ι), IsClosed (Sigma.mk i ⁻¹' s)
-- isOpenMap_sigmaMk
theorem isOpenMap_sigmaMk {i : ι} : IsOpenMap (@Sigma.mk ι σ i) 
Arguments: {ι : Type u_5} {σ : ι → Type u_7} [(i : ι) → TopologicalSpace (σ i)] {i : ι}
Goal: IsOpenMap (Sigma.mk i)
-- isOpen_range_sigmaMk
theorem isOpen_range_sigmaMk {i : ι} : IsOpen (range (@Sigma.mk ι σ i)) 
Arguments: {ι : Type u_5} {σ : ι → Type u_7} [(i : ι) → TopologicalSpace (σ i)] {i : ι}
Goal: IsOpen (Set.range (Sigma.mk i))
-- isClosedMap_sigmaMk
theorem isClosedMap_sigmaMk {i : ι} : IsClosedMap (@Sigma.mk ι σ i) 
Arguments: {ι : Type u_5} {σ : ι → Type u_7} [(i : ι) → TopologicalSpace (σ i)] {i : ι}
Goal: IsClosedMap (Sigma.mk i)
-- isClosed_range_sigmaMk
theorem isClosed_range_sigmaMk {i : ι} : IsClosed (range (@Sigma.mk ι σ i)) 
Arguments: {ι : Type u_5} {σ : ι → Type u_7} [(i : ι) → TopologicalSpace (σ i)] {i : ι}
Goal: IsClosed (Set.range (Sigma.mk i))
-- openEmbedding_sigmaMk
theorem openEmbedding_sigmaMk {i : ι} : OpenEmbedding (@Sigma.mk ι σ i) 
Arguments: {ι : Type u_5} {σ : ι → Type u_7} [(i : ι) → TopologicalSpace (σ i)] {i : ι}
Goal: OpenEmbedding (Sigma.mk i)
-- closedEmbedding_sigmaMk
theorem closedEmbedding_sigmaMk {i : ι} : ClosedEmbedding (@Sigma.mk ι σ i) 
Arguments: {ι : Type u_5} {σ : ι → Type u_7} [(i : ι) → TopologicalSpace (σ i)] {i : ι}
Goal: ClosedEmbedding (Sigma.mk i)
-- embedding_sigmaMk
theorem embedding_sigmaMk {i : ι} : Embedding (@Sigma.mk ι σ i) 
Arguments: {ι : Type u_5} {σ : ι → Type u_7} [(i : ι) → TopologicalSpace (σ i)] {i : ι}
Goal: Embedding (Sigma.mk i)
-- Sigma.nhds_mk
theorem Sigma.nhds_mk (i : ι) (x : σ i) : 𝓝 (⟨i, x⟩ : Sigma σ) = Filter.map (Sigma.mk i) (𝓝 x) 
Arguments: {ι : Type u_5} {σ : ι → Type u_7} [(i : ι) → TopologicalSpace (σ i)] (i : ι) (x : σ i)
Goal: 𝓝 ⟨i, x⟩ = Filter.map (Sigma.mk i) (𝓝 x)
-- Sigma.nhds_eq
theorem Sigma.nhds_eq (x : Sigma σ) : 𝓝 x = Filter.map (Sigma.mk x.1) (𝓝 x.2) 
Arguments: {ι : Type u_5} {σ : ι → Type u_7} [(i : ι) → TopologicalSpace (σ i)] (x : Sigma σ)
Goal: 𝓝 x = Filter.map (Sigma.mk x.fst) (𝓝 x.snd)
-- comap_sigmaMk_nhds
theorem comap_sigmaMk_nhds (i : ι) (x : σ i) : comap (Sigma.mk i) (𝓝 ⟨i, x⟩) = 𝓝 x 
Arguments: {ι : Type u_5} {σ : ι → Type u_7} [(i : ι) → TopologicalSpace (σ i)] (i : ι) (x : σ i)
Goal: Filter.comap (Sigma.mk i) (𝓝 ⟨i, x⟩) = 𝓝 x
-- isOpen_sigma_fst_preimage
theorem isOpen_sigma_fst_preimage (s : Set ι) : IsOpen (Sigma.fst ⁻¹' s : Set (Σ a, σ a)) 
Arguments: {ι : Type u_5} {σ : ι → Type u_7} [(i : ι) → TopologicalSpace (σ i)] (s : Set ι)
Goal: IsOpen (Sigma.fst ⁻¹' s)
-- continuous_sigma_iff
@[simp]
theorem continuous_sigma_iff {f : Sigma σ → X} :
    Continuous f ↔ ∀ i, Continuous fun a => f ⟨i, a⟩ 
Arguments: {X : Type u} {ι : Type u_5} {σ : ι → Type u_7} [(i : ι) → TopologicalSpace (σ i)] [TopologicalSpace X] {f : Sigma σ → X}
Goal: Continuous f ↔ ∀ (i : ι), Continuous fun a => f ⟨i, a⟩
-- continuous_sigma
@[continuity, fun_prop]
theorem continuous_sigma {f : Sigma σ → X} (hf : ∀ i, Continuous fun a => f ⟨i, a⟩) :
    Continuous f 
Arguments: {X : Type u} {ι : Type u_5} {σ : ι → Type u_7} [(i : ι) → TopologicalSpace (σ i)] [TopologicalSpace X] {f : Sigma σ → X} (hf : ∀ (i : ι), Continuous fun a => f ⟨i, a⟩)
Goal: Continuous f
-- inducing_sigma
theorem inducing_sigma {f : Sigma σ → X} :
    Inducing f ↔ (∀ i, Inducing (f ∘ Sigma.mk i)) ∧
      (∀ i, ∃ U, IsOpen U ∧ ∀ x, f x ∈ U ↔ x.1 = i) 
Arguments: {X : Type u} {ι : Type u_5} {σ : ι → Type u_7} [(i : ι) → TopologicalSpace (σ i)] [TopologicalSpace X] {f : Sigma σ → X}
Goal: Inducing f ↔ (∀ (i : ι), Inducing (f ∘ Sigma.mk i)) ∧ ∀ (i : ι), ∃ U, IsOpen U ∧ ∀ (x : Sigma σ), f x ∈ U ↔ x.fst = i
-- continuous_sigma_map
@[simp 1100]
theorem continuous_sigma_map {f₁ : ι → κ} {f₂ : ∀ i, σ i → τ (f₁ i)} :
    Continuous (Sigma.map f₁ f₂) ↔ ∀ i, Continuous (f₂ i) 
Arguments: {ι : Type u_5} {κ : Type u_6} {σ : ι → Type u_7} {τ : κ → Type u_8} [(i : ι) → TopologicalSpace (σ i)] [(k : κ) → TopologicalSpace (τ k)] {f₁ : ι → κ} {f₂ : (i : ι) → σ i → τ (f₁ i)}
Goal: Continuous (Sigma.map f₁ f₂) ↔ ∀ (i : ι), Continuous (f₂ i)
-- Continuous.sigma_map
@[continuity, fun_prop]
theorem Continuous.sigma_map {f₁ : ι → κ} {f₂ : ∀ i, σ i → τ (f₁ i)} (hf : ∀ i, Continuous (f₂ i)) :
    Continuous (Sigma.map f₁ f₂) 
Arguments: {ι : Type u_5} {κ : Type u_6} {σ : ι → Type u_7} {τ : κ → Type u_8} [(i : ι) → TopologicalSpace (σ i)] [(k : κ) → TopologicalSpace (τ k)] {f₁ : ι → κ} {f₂ : (i : ι) → σ i → τ (f₁ i)} (hf : ∀ (i : ι), Continuous (f₂ i))
Goal: Continuous (Sigma.map f₁ f₂)
-- isOpenMap_sigma
theorem isOpenMap_sigma {f : Sigma σ → X} : IsOpenMap f ↔ ∀ i, IsOpenMap fun a => f ⟨i, a⟩ 
Arguments: {X : Type u} {ι : Type u_5} {σ : ι → Type u_7} [(i : ι) → TopologicalSpace (σ i)] [TopologicalSpace X] {f : Sigma σ → X}
Goal: IsOpenMap f ↔ ∀ (i : ι), IsOpenMap fun a => f ⟨i, a⟩
-- isOpenMap_sigma_map
theorem isOpenMap_sigma_map {f₁ : ι → κ} {f₂ : ∀ i, σ i → τ (f₁ i)} :
    IsOpenMap (Sigma.map f₁ f₂) ↔ ∀ i, IsOpenMap (f₂ i) 
Arguments: {ι : Type u_5} {κ : Type u_6} {σ : ι → Type u_7} {τ : κ → Type u_8} [(i : ι) → TopologicalSpace (σ i)] [(k : κ) → TopologicalSpace (τ k)] {f₁ : ι → κ} {f₂ : (i : ι) → σ i → τ (f₁ i)}
Goal: IsOpenMap (Sigma.map f₁ f₂) ↔ ∀ (i : ι), IsOpenMap (f₂ i)
-- inducing_sigma_map
theorem inducing_sigma_map {f₁ : ι → κ} {f₂ : ∀ i, σ i → τ (f₁ i)} (h₁ : Injective f₁) :
    Inducing (Sigma.map f₁ f₂) ↔ ∀ i, Inducing (f₂ i) 
Arguments: {ι : Type u_5} {κ : Type u_6} {σ : ι → Type u_7} {τ : κ → Type u_8} [(i : ι) → TopologicalSpace (σ i)] [(k : κ) → TopologicalSpace (τ k)] {f₁ : ι → κ} {f₂ : (i : ι) → σ i → τ (f₁ i)} (h₁ : Function.Injective f₁)
Goal: Inducing (Sigma.map f₁ f₂) ↔ ∀ (i : ι), Inducing (f₂ i)
-- embedding_sigma_map
theorem embedding_sigma_map {f₁ : ι → κ} {f₂ : ∀ i, σ i → τ (f₁ i)} (h : Injective f₁) :
    Embedding (Sigma.map f₁ f₂) ↔ ∀ i, Embedding (f₂ i) 
Arguments: {ι : Type u_5} {κ : Type u_6} {σ : ι → Type u_7} {τ : κ → Type u_8} [(i : ι) → TopologicalSpace (σ i)] [(k : κ) → TopologicalSpace (τ k)] {f₁ : ι → κ} {f₂ : (i : ι) → σ i → τ (f₁ i)} (h : Function.Injective f₁)
Goal: Embedding (Sigma.map f₁ f₂) ↔ ∀ (i : ι), Embedding (f₂ i)
-- openEmbedding_sigma_map
theorem openEmbedding_sigma_map {f₁ : ι → κ} {f₂ : ∀ i, σ i → τ (f₁ i)} (h : Injective f₁) :
    OpenEmbedding (Sigma.map f₁ f₂) ↔ ∀ i, OpenEmbedding (f₂ i) 
Arguments: {ι : Type u_5} {κ : Type u_6} {σ : ι → Type u_7} {τ : κ → Type u_8} [(i : ι) → TopologicalSpace (σ i)] [(k : κ) → TopologicalSpace (τ k)] {f₁ : ι → κ} {f₂ : (i : ι) → σ i → τ (f₁ i)} (h : Function.Injective f₁)
Goal: OpenEmbedding (Sigma.map f₁ f₂) ↔ ∀ (i : ι), OpenEmbedding (f₂ i)
-- ULift.isOpen_iff
theorem ULift.isOpen_iff [TopologicalSpace X] {s : Set (ULift.{v} X)} :
    IsOpen s ↔ IsOpen (ULift.up ⁻¹' s) 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set (ULift.{v, u} X)}
Goal: IsOpen s ↔ IsOpen (ULift.up ⁻¹' s)
-- ULift.isClosed_iff
theorem ULift.isClosed_iff [TopologicalSpace X] {s : Set (ULift.{v} X)} :
    IsClosed s ↔ IsClosed (ULift.up ⁻¹' s) 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set (ULift.{v, u} X)}
Goal: IsClosed s ↔ IsClosed (ULift.up ⁻¹' s)
-- continuous_uLift_down
@[continuity]
theorem continuous_uLift_down [TopologicalSpace X] : Continuous (ULift.down : ULift.{v, u} X → X) 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: Continuous ULift.down
-- continuous_uLift_up
@[continuity]
theorem continuous_uLift_up [TopologicalSpace X] : Continuous (ULift.up : X → ULift.{v, u} X) 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: Continuous ULift.up
-- embedding_uLift_down
theorem embedding_uLift_down [TopologicalSpace X] : Embedding (ULift.down : ULift.{v, u} X → X) 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: Embedding ULift.down
-- ULift.closedEmbedding_down
theorem ULift.closedEmbedding_down [TopologicalSpace X] :
    ClosedEmbedding (ULift.down : ULift.{v, u} X → X) 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: ClosedEmbedding ULift.down
-- IsOpen.trans
theorem IsOpen.trans (ht : IsOpen t) (hs : IsOpen s) : IsOpen (t : Set X) 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} {t : Set ↑s} (ht : IsOpen t) (hs : IsOpen s)
Goal: IsOpen (Subtype.val '' t)
-- IsClosed.trans
theorem IsClosed.trans (ht : IsClosed t) (hs : IsClosed s) : IsClosed (t : Set X) 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} {t : Set ↑s} (ht : IsClosed t) (hs : IsClosed s)
Goal: IsClosed (Subtype.val '' t)