ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- nhdsWithin_prod
theorem nhdsWithin_prod {α : Type*} [TopologicalSpace α] {β : Type*} [TopologicalSpace β]
    {s u : Set α} {t v : Set β} {a : α} {b : β} (hu : u ∈ 𝓝[s] a) (hv : v ∈ 𝓝[t] b) :
    u ×ˢ v ∈ 𝓝[s ×ˢ t] (a, b) 
Arguments: {α : Type u_5} [TopologicalSpace α] {β : Type u_6} [TopologicalSpace β] {s u : Set α} {t v : Set β} {a : α} {b : β} (hu : u ∈ 𝓝[s] a) (hv : v ∈ 𝓝[t] b)
Goal: u ×ˢ v ∈ 𝓝[s ×ˢ t] (a, b)
-- nhdsWithin_pi_eq'
theorem nhdsWithin_pi_eq' {I : Set ι} (hI : I.Finite) (s : ∀ i, Set (π i)) (x : ∀ i, π i) :
    𝓝[pi I s] x = ⨅ i, comap (fun x => x i) (𝓝 (x i) ⊓ ⨅ (_ : i ∈ I), 𝓟 (s i)) 
Arguments: {ι : Type u_5} {π : ι → Type u_6} [(i : ι) → TopologicalSpace (π i)] {I : Set ι} (hI : I.Finite) (s : (i : ι) → Set (π i)) (x : (i : ι) → π i)
Goal: 𝓝[I.pi s] x = ⨅ i, Filter.comap (fun x => x i) (𝓝 (x i) ⊓ ⨅ (_ : i ∈ I), 𝓟 (s i))
-- nhdsWithin_pi_eq
theorem nhdsWithin_pi_eq {I : Set ι} (hI : I.Finite) (s : ∀ i, Set (π i)) (x : ∀ i, π i) :
    𝓝[pi I s] x =
      (⨅ i ∈ I, comap (fun x => x i) (𝓝[s i] x i)) ⊓
        ⨅ (i) (_ : i ∉ I), comap (fun x => x i) (𝓝 (x i)) 
Arguments: {ι : Type u_5} {π : ι → Type u_6} [(i : ι) → TopologicalSpace (π i)] {I : Set ι} (hI : I.Finite) (s : (i : ι) → Set (π i)) (x : (i : ι) → π i)
Goal: 𝓝[I.pi s] x =
  (⨅ i ∈ I, Filter.comap (fun x => x i) (𝓝[s i] x i)) ⊓ ⨅ i, ⨅ (_ : i ∉ I), Filter.comap (fun x => x i) (𝓝 (x i))
-- nhdsWithin_pi_univ_eq
theorem nhdsWithin_pi_univ_eq [Finite ι] (s : ∀ i, Set (π i)) (x : ∀ i, π i) :
    𝓝[pi univ s] x = ⨅ i, comap (fun x => x i) (𝓝[s i] x i) 
Arguments: {ι : Type u_5} {π : ι → Type u_6} [(i : ι) → TopologicalSpace (π i)] [Finite ι] (s : (i : ι) → Set (π i)) (x : (i : ι) → π i)
Goal: 𝓝[Set.univ.pi s] x = ⨅ i, Filter.comap (fun x => x i) (𝓝[s i] x i)
-- nhdsWithin_pi_eq_bot
theorem nhdsWithin_pi_eq_bot {I : Set ι} {s : ∀ i, Set (π i)} {x : ∀ i, π i} :
    𝓝[pi I s] x = ⊥ ↔ ∃ i ∈ I, 𝓝[s i] x i = ⊥ 
Arguments: {ι : Type u_5} {π : ι → Type u_6} [(i : ι) → TopologicalSpace (π i)] {I : Set ι} {s : (i : ι) → Set (π i)} {x : (i : ι) → π i}
Goal: 𝓝[I.pi s] x = ⊥ ↔ ∃ i ∈ I, 𝓝[s i] x i = ⊥
-- nhdsWithin_pi_neBot
theorem nhdsWithin_pi_neBot {I : Set ι} {s : ∀ i, Set (π i)} {x : ∀ i, π i} :
    (𝓝[pi I s] x).NeBot ↔ ∀ i ∈ I, (𝓝[s i] x i).NeBot 
Arguments: {ι : Type u_5} {π : ι → Type u_6} [(i : ι) → TopologicalSpace (π i)] {I : Set ι} {s : (i : ι) → Set (π i)} {x : (i : ι) → π i}
Goal: (𝓝[I.pi s] x).NeBot ↔ ∀ i ∈ I, (𝓝[s i] x i).NeBot
-- Filter.Tendsto.piecewise_nhdsWithin
theorem Filter.Tendsto.piecewise_nhdsWithin {f g : α → β} {t : Set α} [∀ x, Decidable (x ∈ t)]
    {a : α} {s : Set α} {l : Filter β} (h₀ : Tendsto f (𝓝[s ∩ t] a) l)
    (h₁ : Tendsto g (𝓝[s ∩ tᶜ] a) l) : Tendsto (piecewise t f g) (𝓝[s] a) l 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] {f g : α → β} {t : Set α} [(x : α) → Decidable (x ∈ t)] {a : α} {s : Set α} {l : Filter β} (h₀ : Filter.Tendsto f (𝓝[s ∩ t] a) l) (h₁ : Filter.Tendsto g (𝓝[s ∩ tᶜ] a) l)
Goal: Filter.Tendsto (t.piecewise f g) (𝓝[s] a) l
-- Filter.Tendsto.if_nhdsWithin
theorem Filter.Tendsto.if_nhdsWithin {f g : α → β} {p : α → Prop} [DecidablePred p] {a : α}
    {s : Set α} {l : Filter β} (h₀ : Tendsto f (𝓝[s ∩ { x | p x }] a) l)
    (h₁ : Tendsto g (𝓝[s ∩ { x | ¬p x }] a) l) :
    Tendsto (fun x => if p x then f x else g x) (𝓝[s] a) l 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] {f g : α → β} {p : α → Prop} [DecidablePred p] {a : α} {s : Set α} {l : Filter β} (h₀ : Filter.Tendsto f (𝓝[s ∩ {x | p x}] a) l) (h₁ : Filter.Tendsto g (𝓝[s ∩ {x | ¬p x}] a) l)
Goal: Filter.Tendsto (fun x => if p x then f x else g x) (𝓝[s] a) l
-- map_nhdsWithin
theorem map_nhdsWithin (f : α → β) (a : α) (s : Set α) :
    map f (𝓝[s] a) = ⨅ t ∈ { t : Set α | a ∈ t ∧ IsOpen t }, 𝓟 (f '' (t ∩ s)) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] (f : α → β) (a : α) (s : Set α)
Goal: Filter.map f (𝓝[s] a) = ⨅ t ∈ {t | a ∈ t ∧ IsOpen t}, 𝓟 (f '' (t ∩ s))
-- tendsto_nhdsWithin_mono_left
theorem tendsto_nhdsWithin_mono_left {f : α → β} {a : α} {s t : Set α} {l : Filter β} (hst : s ⊆ t)
    (h : Tendsto f (𝓝[t] a) l) : Tendsto f (𝓝[s] a) l 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] {f : α → β} {a : α} {s t : Set α} {l : Filter β} (hst : s ⊆ t) (h : Filter.Tendsto f (𝓝[t] a) l)
Goal: Filter.Tendsto f (𝓝[s] a) l
-- tendsto_nhdsWithin_mono_right
theorem tendsto_nhdsWithin_mono_right {f : β → α} {l : Filter β} {a : α} {s t : Set α} (hst : s ⊆ t)
    (h : Tendsto f l (𝓝[s] a)) : Tendsto f l (𝓝[t] a) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] {f : β → α} {l : Filter β} {a : α} {s t : Set α} (hst : s ⊆ t) (h : Filter.Tendsto f l (𝓝[s] a))
Goal: Filter.Tendsto f l (𝓝[t] a)
-- tendsto_nhdsWithin_of_tendsto_nhds
theorem tendsto_nhdsWithin_of_tendsto_nhds {f : α → β} {a : α} {s : Set α} {l : Filter β}
    (h : Tendsto f (𝓝 a) l) : Tendsto f (𝓝[s] a) l 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] {f : α → β} {a : α} {s : Set α} {l : Filter β} (h : Filter.Tendsto f (𝓝 a) l)
Goal: Filter.Tendsto f (𝓝[s] a) l
-- eventually_mem_of_tendsto_nhdsWithin
theorem eventually_mem_of_tendsto_nhdsWithin {f : β → α} {a : α} {s : Set α} {l : Filter β}
    (h : Tendsto f l (𝓝[s] a)) : ∀ᶠ i in l, f i ∈ s 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] {f : β → α} {a : α} {s : Set α} {l : Filter β} (h : Filter.Tendsto f l (𝓝[s] a))
Goal: ∀ᶠ (i : β) in l, f i ∈ s
-- tendsto_nhds_of_tendsto_nhdsWithin
theorem tendsto_nhds_of_tendsto_nhdsWithin {f : β → α} {a : α} {s : Set α} {l : Filter β}
    (h : Tendsto f l (𝓝[s] a)) : Tendsto f l (𝓝 a) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] {f : β → α} {a : α} {s : Set α} {l : Filter β} (h : Filter.Tendsto f l (𝓝[s] a))
Goal: Filter.Tendsto f l (𝓝 a)
-- nhdsWithin_neBot_of_mem
theorem nhdsWithin_neBot_of_mem {s : Set α} {x : α} (hx : x ∈ s) : NeBot (𝓝[s] x) 
Arguments: {α : Type u_1} [TopologicalSpace α] {s : Set α} {x : α} (hx : x ∈ s)
Goal: (𝓝[s] x).NeBot
-- IsClosed.mem_of_nhdsWithin_neBot
theorem IsClosed.mem_of_nhdsWithin_neBot {s : Set α} (hs : IsClosed s) {x : α}
    (hx : NeBot <| 𝓝[s] x) : x ∈ s 
Arguments: {α : Type u_1} [TopologicalSpace α] {s : Set α} (hs : IsClosed s) {x : α} (hx : (𝓝[s] x).NeBot)
Goal: x ∈ s
-- DenseRange.nhdsWithin_neBot
theorem DenseRange.nhdsWithin_neBot {ι : Type*} {f : ι → α} (h : DenseRange f) (x : α) :
    NeBot (𝓝[range f] x) 
Arguments: {α : Type u_1} [TopologicalSpace α] {ι : Type u_7} {f : ι → α} (h : DenseRange f) (x : α)
Goal: (𝓝[Set.range f] x).NeBot
-- mem_closure_pi
theorem mem_closure_pi {ι : Type*} {α : ι → Type*} [∀ i, TopologicalSpace (α i)] {I : Set ι}
    {s : ∀ i, Set (α i)} {x : ∀ i, α i} : x ∈ closure (pi I s) ↔ ∀ i ∈ I, x i ∈ closure (s i) 
Arguments: {ι : Type u_7} {α : ι → Type u_8} [(i : ι) → TopologicalSpace (α i)] {I : Set ι} {s : (i : ι) → Set (α i)} {x : (i : ι) → α i}
Goal: x ∈ closure (I.pi s) ↔ ∀ i ∈ I, x i ∈ closure (s i)
-- closure_pi_set
theorem closure_pi_set {ι : Type*} {α : ι → Type*} [∀ i, TopologicalSpace (α i)] (I : Set ι)
    (s : ∀ i, Set (α i)) : closure (pi I s) = pi I fun i => closure (s i) 
Arguments: {ι : Type u_7} {α : ι → Type u_8} [(i : ι) → TopologicalSpace (α i)] (I : Set ι) (s : (i : ι) → Set (α i))
Goal: closure (I.pi s) = I.pi fun i => closure (s i)
-- dense_pi
theorem dense_pi {ι : Type*} {α : ι → Type*} [∀ i, TopologicalSpace (α i)] {s : ∀ i, Set (α i)}
    (I : Set ι) (hs : ∀ i ∈ I, Dense (s i)) : Dense (pi I s) 
Arguments: {ι : Type u_7} {α : ι → Type u_8} [(i : ι) → TopologicalSpace (α i)] {s : (i : ι) → Set (α i)} (I : Set ι) (hs : ∀ i ∈ I, Dense (s i))
Goal: Dense (I.pi s)
-- eventuallyEq_nhdsWithin_iff
theorem eventuallyEq_nhdsWithin_iff {f g : α → β} {s : Set α} {a : α} :
    f =ᶠ[𝓝[s] a] g ↔ ∀ᶠ x in 𝓝 a, x ∈ s → f x = g x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] {f g : α → β} {s : Set α} {a : α}
Goal: f =ᶠ[𝓝[s] a] g ↔ ∀ᶠ (x : α) in 𝓝 a, x ∈ s → f x = g x
-- eventuallyEq_nhdsWithin_of_eqOn
theorem eventuallyEq_nhdsWithin_of_eqOn {f g : α → β} {s : Set α} {a : α} (h : EqOn f g s) :
    f =ᶠ[𝓝[s] a] g 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] {f g : α → β} {s : Set α} {a : α} (h : Set.EqOn f g s)
Goal: f =ᶠ[𝓝[s] a] g
-- Set.EqOn.eventuallyEq_nhdsWithin
theorem Set.EqOn.eventuallyEq_nhdsWithin {f g : α → β} {s : Set α} {a : α} (h : EqOn f g s) :
    f =ᶠ[𝓝[s] a] g 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] {f g : α → β} {s : Set α} {a : α} (h : Set.EqOn f g s)
Goal: f =ᶠ[𝓝[s] a] g
-- tendsto_nhdsWithin_congr
theorem tendsto_nhdsWithin_congr {f g : α → β} {s : Set α} {a : α} {l : Filter β}
    (hfg : ∀ x ∈ s, f x = g x) (hf : Tendsto f (𝓝[s] a) l) : Tendsto g (𝓝[s] a) l 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] {f g : α → β} {s : Set α} {a : α} {l : Filter β} (hfg : ∀ x ∈ s, f x = g x) (hf : Filter.Tendsto f (𝓝[s] a) l)
Goal: Filter.Tendsto g (𝓝[s] a) l
-- eventually_nhdsWithin_of_forall
theorem eventually_nhdsWithin_of_forall {s : Set α} {a : α} {p : α → Prop} (h : ∀ x ∈ s, p x) :
    ∀ᶠ x in 𝓝[s] a, p x 
Arguments: {α : Type u_1} [TopologicalSpace α] {s : Set α} {a : α} {p : α → Prop} (h : ∀ x ∈ s, p x)
Goal: ∀ᶠ (x : α) in 𝓝[s] a, p x
-- tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within
theorem tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within {a : α} {l : Filter β} {s : Set α}
    (f : β → α) (h1 : Tendsto f l (𝓝 a)) (h2 : ∀ᶠ x in l, f x ∈ s) : Tendsto f l (𝓝[s] a) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] {a : α} {l : Filter β} {s : Set α} (f : β → α) (h1 : Filter.Tendsto f l (𝓝 a)) (h2 : ∀ᶠ (x : β) in l, f x ∈ s)
Goal: Filter.Tendsto f l (𝓝[s] a)
-- tendsto_nhdsWithin_iff
theorem tendsto_nhdsWithin_iff {a : α} {l : Filter β} {s : Set α} {f : β → α} :
    Tendsto f l (𝓝[s] a) ↔ Tendsto f l (𝓝 a) ∧ ∀ᶠ n in l, f n ∈ s 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] {a : α} {l : Filter β} {s : Set α} {f : β → α}
Goal: Filter.Tendsto f l (𝓝[s] a) ↔ Filter.Tendsto f l (𝓝 a) ∧ ∀ᶠ (n : β) in l, f n ∈ s
-- tendsto_nhdsWithin_range
@[simp]
theorem tendsto_nhdsWithin_range {a : α} {l : Filter β} {f : β → α} :
    Tendsto f l (𝓝[range f] a) ↔ Tendsto f l (𝓝 a) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] {a : α} {l : Filter β} {f : β → α}
Goal: Filter.Tendsto f l (𝓝[Set.range f] a) ↔ Filter.Tendsto f l (𝓝 a)
-- Filter.EventuallyEq.eq_of_nhdsWithin
theorem Filter.EventuallyEq.eq_of_nhdsWithin {s : Set α} {f g : α → β} {a : α} (h : f =ᶠ[𝓝[s] a] g)
    (hmem : a ∈ s) : f a = g a 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] {s : Set α} {f g : α → β} {a : α} (h : f =ᶠ[𝓝[s] a] g) (hmem : a ∈ s)
Goal: f a = g a
-- eventually_nhdsWithin_of_eventually_nhds
theorem eventually_nhdsWithin_of_eventually_nhds {α : Type*} [TopologicalSpace α] {s : Set α}
    {a : α} {p : α → Prop} (h : ∀ᶠ x in 𝓝 a, p x) : ∀ᶠ x in 𝓝[s] a, p x 
Arguments: {α : Type u_7} [TopologicalSpace α] {s : Set α} {a : α} {p : α → Prop} (h : ∀ᶠ (x : α) in 𝓝 a, p x)
Goal: ∀ᶠ (x : α) in 𝓝[s] a, p x
-- mem_nhdsWithin_subtype
theorem mem_nhdsWithin_subtype {s : Set α} {a : { x // x ∈ s }} {t u : Set { x // x ∈ s }} :
    t ∈ 𝓝[u] a ↔ t ∈ comap ((↑) : s → α) (𝓝[(↑) '' u] a) 
Arguments: {α : Type u_1} [TopologicalSpace α] {s : Set α} {a : { x // x ∈ s }} {t u : Set { x // x ∈ s }}
Goal: t ∈ 𝓝[u] a ↔ t ∈ Filter.comap Subtype.val (𝓝[Subtype.val '' u] ↑a)
-- nhdsWithin_subtype
theorem nhdsWithin_subtype (s : Set α) (a : { x // x ∈ s }) (t : Set { x // x ∈ s }) :
    𝓝[t] a = comap ((↑) : s → α) (𝓝[(↑) '' t] a) 
Arguments: {α : Type u_1} [TopologicalSpace α] (s : Set α) (a : { x // x ∈ s }) (t : Set { x // x ∈ s })
Goal: 𝓝[t] a = Filter.comap Subtype.val (𝓝[Subtype.val '' t] ↑a)
-- nhdsWithin_eq_map_subtype_coe
theorem nhdsWithin_eq_map_subtype_coe {s : Set α} {a : α} (h : a ∈ s) :
    𝓝[s] a = map ((↑) : s → α) (𝓝 ⟨a, h⟩) 
Arguments: {α : Type u_1} [TopologicalSpace α] {s : Set α} {a : α} (h : a ∈ s)
Goal: 𝓝[s] a = Filter.map Subtype.val (𝓝 ⟨a, h⟩)
-- mem_nhds_subtype_iff_nhdsWithin
theorem mem_nhds_subtype_iff_nhdsWithin {s : Set α} {a : s} {t : Set s} :
    t ∈ 𝓝 a ↔ (↑) '' t ∈ 𝓝[s] (a : α) 
Arguments: {α : Type u_1} [TopologicalSpace α] {s : Set α} {a : ↑s} {t : Set ↑s}
Goal: t ∈ 𝓝 a ↔ Subtype.val '' t ∈ 𝓝[s] ↑a
-- preimage_coe_mem_nhds_subtype
theorem preimage_coe_mem_nhds_subtype {s t : Set α} {a : s} : (↑) ⁻¹' t ∈ 𝓝 a ↔ t ∈ 𝓝[s] ↑a 
Arguments: {α : Type u_1} [TopologicalSpace α] {s t : Set α} {a : ↑s}
Goal: Subtype.val ⁻¹' t ∈ 𝓝 a ↔ t ∈ 𝓝[s] ↑a
-- eventually_nhds_subtype_iff
theorem eventually_nhds_subtype_iff (s : Set α) (a : s) (P : α → Prop) :
    (∀ᶠ x : s in 𝓝 a, P x) ↔ ∀ᶠ x in 𝓝[s] a, P x 
Arguments: {α : Type u_1} [TopologicalSpace α] (s : Set α) (a : ↑s) (P : α → Prop)
Goal: (∀ᶠ (x : ↑s) in 𝓝 a, P ↑x) ↔ ∀ᶠ (x : α) in 𝓝[s] ↑a, P x
-- frequently_nhds_subtype_iff
theorem frequently_nhds_subtype_iff (s : Set α) (a : s) (P : α → Prop) :
    (∃ᶠ x : s in 𝓝 a, P x) ↔ ∃ᶠ x in 𝓝[s] a, P x 
Arguments: {α : Type u_1} [TopologicalSpace α] (s : Set α) (a : ↑s) (P : α → Prop)
Goal: (∃ᶠ (x : ↑s) in 𝓝 a, P ↑x) ↔ ∃ᶠ (x : α) in 𝓝[s] ↑a, P x
-- tendsto_nhdsWithin_iff_subtype
theorem tendsto_nhdsWithin_iff_subtype {s : Set α} {a : α} (h : a ∈ s) (f : α → β) (l : Filter β) :
    Tendsto f (𝓝[s] a) l ↔ Tendsto (s.restrict f) (𝓝 ⟨a, h⟩) l 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] {s : Set α} {a : α} (h : a ∈ s) (f : α → β) (l : Filter β)
Goal: Filter.Tendsto f (𝓝[s] a) l ↔ Filter.Tendsto (s.restrict f) (𝓝 ⟨a, h⟩) l
-- ContinuousWithinAt.tendsto
theorem ContinuousWithinAt.tendsto {f : α → β} {s : Set α} {x : α} (h : ContinuousWithinAt f s x) :
    Tendsto f (𝓝[s] x) (𝓝 (f x)) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s : Set α} {x : α} (h : ContinuousWithinAt f s x)
Goal: Filter.Tendsto f (𝓝[s] x) (𝓝 (f x))
-- ContinuousOn.continuousWithinAt
theorem ContinuousOn.continuousWithinAt {f : α → β} {s : Set α} {x : α} (hf : ContinuousOn f s)
    (hx : x ∈ s) : ContinuousWithinAt f s x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s : Set α} {x : α} (hf : ContinuousOn f s) (hx : x ∈ s)
Goal: ContinuousWithinAt f s x
-- continuousWithinAt_univ
theorem continuousWithinAt_univ (f : α → β) (x : α) :
    ContinuousWithinAt f Set.univ x ↔ ContinuousAt f x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : α → β) (x : α)
Goal: ContinuousWithinAt f Set.univ x ↔ ContinuousAt f x
-- continuous_iff_continuousOn_univ
theorem continuous_iff_continuousOn_univ {f : α → β} : Continuous f ↔ ContinuousOn f univ 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β}
Goal: Continuous f ↔ ContinuousOn f Set.univ
-- continuousWithinAt_iff_continuousAt_restrict
theorem continuousWithinAt_iff_continuousAt_restrict (f : α → β) {x : α} {s : Set α} (h : x ∈ s) :
    ContinuousWithinAt f s x ↔ ContinuousAt (s.restrict f) ⟨x, h⟩ 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : α → β) {x : α} {s : Set α} (h : x ∈ s)
Goal: ContinuousWithinAt f s x ↔ ContinuousAt (s.restrict f) ⟨x, h⟩
-- ContinuousWithinAt.tendsto_nhdsWithin
theorem ContinuousWithinAt.tendsto_nhdsWithin {f : α → β} {x : α} {s : Set α} {t : Set β}
    (h : ContinuousWithinAt f s x) (ht : MapsTo f s t) : Tendsto f (𝓝[s] x) (𝓝[t] f x) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {x : α} {s : Set α} {t : Set β} (h : ContinuousWithinAt f s x) (ht : Set.MapsTo f s t)
Goal: Filter.Tendsto f (𝓝[s] x) (𝓝[t] f x)
-- ContinuousWithinAt.tendsto_nhdsWithin_image
theorem ContinuousWithinAt.tendsto_nhdsWithin_image {f : α → β} {x : α} {s : Set α}
    (h : ContinuousWithinAt f s x) : Tendsto f (𝓝[s] x) (𝓝[f '' s] f x) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {x : α} {s : Set α} (h : ContinuousWithinAt f s x)
Goal: Filter.Tendsto f (𝓝[s] x) (𝓝[f '' s] f x)
-- ContinuousWithinAt.prod_map
theorem ContinuousWithinAt.prod_map {f : α → γ} {g : β → δ} {s : Set α} {t : Set β} {x : α} {y : β}
    (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g t y) :
    ContinuousWithinAt (Prod.map f g) (s ×ˢ t) (x, y) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] {f : α → γ} {g : β → δ} {s : Set α} {t : Set β} {x : α} {y : β} (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g t y)
Goal: ContinuousWithinAt (Prod.map f g) (s ×ˢ t) (x, y)
-- continuousWithinAt_prod_of_discrete_left
theorem continuousWithinAt_prod_of_discrete_left [DiscreteTopology α]
    {f : α × β → γ} {s : Set (α × β)} {x : α × β} :
    ContinuousWithinAt f s x ↔ ContinuousWithinAt (f ⟨x.1, ·⟩) {b | (x.1, b) ∈ s} x.2 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [DiscreteTopology α] {f : α × β → γ} {s : Set (α × β)} {x : α × β}
Goal: ContinuousWithinAt f s x ↔ ContinuousWithinAt (fun x_1 => f (x.1, x_1)) {b | (x.1, b) ∈ s} x.2
-- continuousWithinAt_prod_of_discrete_right
theorem continuousWithinAt_prod_of_discrete_right [DiscreteTopology β]
    {f : α × β → γ} {s : Set (α × β)} {x : α × β} :
    ContinuousWithinAt f s x ↔ ContinuousWithinAt (f ⟨·, x.2⟩) {a | (a, x.2) ∈ s} x.1 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [DiscreteTopology β] {f : α × β → γ} {s : Set (α × β)} {x : α × β}
Goal: ContinuousWithinAt f s x ↔ ContinuousWithinAt (fun x_1 => f (x_1, x.2)) {a | (a, x.2) ∈ s} x.1
-- continuousAt_prod_of_discrete_left
theorem continuousAt_prod_of_discrete_left [DiscreteTopology α] {f : α × β → γ} {x : α × β} :
    ContinuousAt f x ↔ ContinuousAt (f ⟨x.1, ·⟩) x.2 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [DiscreteTopology α] {f : α × β → γ} {x : α × β}
Goal: ContinuousAt f x ↔ ContinuousAt (fun x_1 => f (x.1, x_1)) x.2
-- continuousAt_prod_of_discrete_right
theorem continuousAt_prod_of_discrete_right [DiscreteTopology β] {f : α × β → γ} {x : α × β} :
    ContinuousAt f x ↔ ContinuousAt (f ⟨·, x.2⟩) x.1 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [DiscreteTopology β] {f : α × β → γ} {x : α × β}
Goal: ContinuousAt f x ↔ ContinuousAt (fun x_1 => f (x_1, x.2)) x.1
-- continuousOn_prod_of_discrete_left
theorem continuousOn_prod_of_discrete_left [DiscreteTopology α] {f : α × β → γ} {s : Set (α × β)} :
    ContinuousOn f s ↔ ∀ a, ContinuousOn (f ⟨a, ·⟩) {b | (a, b) ∈ s} 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [DiscreteTopology α] {f : α × β → γ} {s : Set (α × β)}
Goal: ContinuousOn f s ↔ ∀ (a : α), ContinuousOn (fun x => f (a, x)) {b | (a, b) ∈ s}
-- continuousOn_prod_of_discrete_right
theorem continuousOn_prod_of_discrete_right [DiscreteTopology β] {f : α × β → γ} {s : Set (α × β)} :
    ContinuousOn f s ↔ ∀ b, ContinuousOn (f ⟨·, b⟩) {a | (a, b) ∈ s} 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [DiscreteTopology β] {f : α × β → γ} {s : Set (α × β)}
Goal: ContinuousOn f s ↔ ∀ (b : β), ContinuousOn (fun x => f (x, b)) {a | (a, b) ∈ s}
-- continuous_prod_of_discrete_left
theorem continuous_prod_of_discrete_left [DiscreteTopology α] {f : α × β → γ} :
    Continuous f ↔ ∀ a, Continuous (f ⟨a, ·⟩) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [DiscreteTopology α] {f : α × β → γ}
Goal: Continuous f ↔ ∀ (a : α), Continuous fun x => f (a, x)
-- continuous_prod_of_discrete_right
theorem continuous_prod_of_discrete_right [DiscreteTopology β] {f : α × β → γ} :
    Continuous f ↔ ∀ b, Continuous (f ⟨·, b⟩) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [DiscreteTopology β] {f : α × β → γ}
Goal: Continuous f ↔ ∀ (b : β), Continuous fun x => f (x, b)
-- isOpenMap_prod_of_discrete_left
theorem isOpenMap_prod_of_discrete_left [DiscreteTopology α] {f : α × β → γ} :
    IsOpenMap f ↔ ∀ a, IsOpenMap (f ⟨a, ·⟩) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [DiscreteTopology α] {f : α × β → γ}
Goal: IsOpenMap f ↔ ∀ (a : α), IsOpenMap fun x => f (a, x)
-- isOpenMap_prod_of_discrete_right
theorem isOpenMap_prod_of_discrete_right [DiscreteTopology β] {f : α × β → γ} :
    IsOpenMap f ↔ ∀ b, IsOpenMap (f ⟨·, b⟩) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [DiscreteTopology β] {f : α × β → γ}
Goal: IsOpenMap f ↔ ∀ (b : β), IsOpenMap fun x => f (x, b)
-- continuousWithinAt_pi
theorem continuousWithinAt_pi {ι : Type*} {π : ι → Type*} [∀ i, TopologicalSpace (π i)]
    {f : α → ∀ i, π i} {s : Set α} {x : α} :
    ContinuousWithinAt f s x ↔ ∀ i, ContinuousWithinAt (fun y => f y i) s x 
Arguments: {α : Type u_1} [TopologicalSpace α] {ι : Type u_7} {π : ι → Type u_8} [(i : ι) → TopologicalSpace (π i)] {f : α → (i : ι) → π i} {s : Set α} {x : α}
Goal: ContinuousWithinAt f s x ↔ ∀ (i : ι), ContinuousWithinAt (fun y => f y i) s x
-- continuousOn_pi
theorem continuousOn_pi {ι : Type*} {π : ι → Type*} [∀ i, TopologicalSpace (π i)]
    {f : α → ∀ i, π i} {s : Set α} : ContinuousOn f s ↔ ∀ i, ContinuousOn (fun y => f y i) s 
Arguments: {α : Type u_1} [TopologicalSpace α] {ι : Type u_7} {π : ι → Type u_8} [(i : ι) → TopologicalSpace (π i)] {f : α → (i : ι) → π i} {s : Set α}
Goal: ContinuousOn f s ↔ ∀ (i : ι), ContinuousOn (fun y => f y i) s
-- continuousOn_pi'
@[fun_prop]
theorem continuousOn_pi' {ι : Type*} {π : ι → Type*} [∀ i, TopologicalSpace (π i)]
    {f : α → ∀ i, π i} {s : Set α} (hf : ∀ i, ContinuousOn (fun y => f y i) s) :
    ContinuousOn f s 
Arguments: {α : Type u_1} [TopologicalSpace α] {ι : Type u_7} {π : ι → Type u_8} [(i : ι) → TopologicalSpace (π i)] {f : α → (i : ι) → π i} {s : Set α} (hf : ∀ (i : ι), ContinuousOn (fun y => f y i) s)
Goal: ContinuousOn f s
-- ContinuousWithinAt.fin_insertNth
theorem ContinuousWithinAt.fin_insertNth {n} {π : Fin (n + 1) → Type*}
    [∀ i, TopologicalSpace (π i)] (i : Fin (n + 1)) {f : α → π i} {a : α} {s : Set α}
    (hf : ContinuousWithinAt f s a) {g : α → ∀ j : Fin n, π (i.succAbove j)}
    (hg : ContinuousWithinAt g s a) : ContinuousWithinAt (fun a => i.insertNth (f a) (g a)) s a 
Arguments: {α : Type u_1} [TopologicalSpace α] {n : ℕ} {π : Fin (n + 1) → Type u_7} [(i : Fin (n + 1)) → TopologicalSpace (π i)] (i : Fin (n + 1)) {f : α → π i} {a : α} {s : Set α} (hf : ContinuousWithinAt f s a) {g : α → (j : Fin n) → π (i.succAbove j)} (hg : ContinuousWithinAt g s a)
Goal: ContinuousWithinAt (fun a => i.insertNth (f a) (g a)) s a
-- ContinuousOn.fin_insertNth
nonrec theorem ContinuousOn.fin_insertNth {n} {π : Fin (n + 1) → Type*}
    [∀ i, TopologicalSpace (π i)] (i : Fin (n + 1)) {f : α → π i} {s : Set α}
    (hf : ContinuousOn f s) {g : α → ∀ j : Fin n, π (i.succAbove j)} (hg : ContinuousOn g s) :
    ContinuousOn (fun a => i.insertNth (f a) (g a)) s 
Arguments: {α : Type u_1} [TopologicalSpace α] {n : ℕ} {π : Fin (n + 1) → Type u_7} [(i : Fin (n + 1)) → TopologicalSpace (π i)] (i : Fin (n + 1)) {f : α → π i} {s : Set α} (hf : ContinuousOn f s) {g : α → (j : Fin n) → π (i.succAbove j)} (hg : ContinuousOn g s)
Goal: ContinuousOn (fun a => i.insertNth (f a) (g a)) s
-- continuousOn_iff
theorem continuousOn_iff {f : α → β} {s : Set α} :
    ContinuousOn f s ↔
      ∀ x ∈ s, ∀ t : Set β, IsOpen t → f x ∈ t → ∃ u, IsOpen u ∧ x ∈ u ∧ u ∩ s ⊆ f ⁻¹' t 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s : Set α}
Goal: ContinuousOn f s ↔ ∀ x ∈ s, ∀ (t : Set β), IsOpen t → f x ∈ t → ∃ u, IsOpen u ∧ x ∈ u ∧ u ∩ s ⊆ f ⁻¹' t
-- continuousOn_iff_continuous_restrict
theorem continuousOn_iff_continuous_restrict {f : α → β} {s : Set α} :
    ContinuousOn f s ↔ Continuous (s.restrict f) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s : Set α}
Goal: ContinuousOn f s ↔ Continuous (s.restrict f)