ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- tendstoLocallyUniformlyOn_univ
theorem tendstoLocallyUniformlyOn_univ :
    TendstoLocallyUniformlyOn F f p univ ↔ TendstoLocallyUniformly F f p 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {p : Filter ι} [TopologicalSpace α]
Goal: TendstoLocallyUniformlyOn F f p Set.univ ↔ TendstoLocallyUniformly F f p
-- tendstoLocallyUniformlyOn_iff_forall_tendsto
theorem tendstoLocallyUniformlyOn_iff_forall_tendsto :
    TendstoLocallyUniformlyOn F f p s ↔
      ∀ x ∈ s, Tendsto (fun y : ι × α => (f y.2, F y.1 y.2)) (p ×ˢ 𝓝[s] x) (𝓤 β) 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {s : Set α} {p : Filter ι} [TopologicalSpace α]
Goal: TendstoLocallyUniformlyOn F f p s ↔ ∀ x ∈ s, Filter.Tendsto (fun y => (f y.2, F y.1 y.2)) (p ×ˢ 𝓝[s] x) (𝓤 β)
-- IsOpen.tendstoLocallyUniformlyOn_iff_forall_tendsto
nonrec theorem IsOpen.tendstoLocallyUniformlyOn_iff_forall_tendsto (hs : IsOpen s) :
    TendstoLocallyUniformlyOn F f p s ↔
      ∀ x ∈ s, Tendsto (fun y : ι × α => (f y.2, F y.1 y.2)) (p ×ˢ 𝓝 x) (𝓤 β) 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {s : Set α} {p : Filter ι} [TopologicalSpace α] (hs : IsOpen s)
Goal: TendstoLocallyUniformlyOn F f p s ↔ ∀ x ∈ s, Filter.Tendsto (fun y => (f y.2, F y.1 y.2)) (p ×ˢ 𝓝 x) (𝓤 β)
-- tendstoLocallyUniformly_iff_forall_tendsto
theorem tendstoLocallyUniformly_iff_forall_tendsto :
    TendstoLocallyUniformly F f p ↔
      ∀ x, Tendsto (fun y : ι × α => (f y.2, F y.1 y.2)) (p ×ˢ 𝓝 x) (𝓤 β) 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {p : Filter ι} [TopologicalSpace α]
Goal: TendstoLocallyUniformly F f p ↔ ∀ (x : α), Filter.Tendsto (fun y => (f y.2, F y.1 y.2)) (p ×ˢ 𝓝 x) (𝓤 β)
-- tendstoLocallyUniformlyOn_iff_tendstoLocallyUniformly_comp_coe
theorem tendstoLocallyUniformlyOn_iff_tendstoLocallyUniformly_comp_coe :
    TendstoLocallyUniformlyOn F f p s ↔
      TendstoLocallyUniformly (fun i (x : s) => F i x) (f ∘ (↑)) p 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {s : Set α} {p : Filter ι} [TopologicalSpace α]
Goal: TendstoLocallyUniformlyOn F f p s ↔ TendstoLocallyUniformly (fun i x => F i ↑x) (f ∘ Subtype.val) p
-- TendstoUniformlyOn.tendstoLocallyUniformlyOn
protected theorem TendstoUniformlyOn.tendstoLocallyUniformlyOn (h : TendstoUniformlyOn F f p s) :
    TendstoLocallyUniformlyOn F f p s 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {s : Set α} {p : Filter ι} [TopologicalSpace α] (h : TendstoUniformlyOn F f p s)
Goal: TendstoLocallyUniformlyOn F f p s
-- TendstoUniformly.tendstoLocallyUniformly
protected theorem TendstoUniformly.tendstoLocallyUniformly (h : TendstoUniformly F f p) :
    TendstoLocallyUniformly F f p 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {p : Filter ι} [TopologicalSpace α] (h : TendstoUniformly F f p)
Goal: TendstoLocallyUniformly F f p
-- TendstoLocallyUniformlyOn.mono
theorem TendstoLocallyUniformlyOn.mono (h : TendstoLocallyUniformlyOn F f p s) (h' : s' ⊆ s) :
    TendstoLocallyUniformlyOn F f p s' 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {s s' : Set α} {p : Filter ι} [TopologicalSpace α] (h : TendstoLocallyUniformlyOn F f p s) (h' : s' ⊆ s)
Goal: TendstoLocallyUniformlyOn F f p s'
-- tendstoLocallyUniformlyOn_iUnion
theorem tendstoLocallyUniformlyOn_iUnion {ι' : Sort*} {S : ι' → Set α} (hS : ∀ i, IsOpen (S i))
    (h : ∀ i, TendstoLocallyUniformlyOn F f p (S i)) :
    TendstoLocallyUniformlyOn F f p (⋃ i, S i) 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {p : Filter ι} [TopologicalSpace α] {ι' : Sort u_1} {S : ι' → Set α} (hS : ∀ (i : ι'), IsOpen (S i)) (h : ∀ (i : ι'), TendstoLocallyUniformlyOn F f p (S i))
Goal: TendstoLocallyUniformlyOn F f p (⋃ i, S i)
-- tendstoLocallyUniformlyOn_biUnion
theorem tendstoLocallyUniformlyOn_biUnion {s : Set γ} {S : γ → Set α} (hS : ∀ i ∈ s, IsOpen (S i))
    (h : ∀ i ∈ s, TendstoLocallyUniformlyOn F f p (S i)) :
    TendstoLocallyUniformlyOn F f p (⋃ i ∈ s, S i) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {p : Filter ι} [TopologicalSpace α] {s : Set γ} {S : γ → Set α} (hS : ∀ i ∈ s, IsOpen (S i)) (h : ∀ i ∈ s, TendstoLocallyUniformlyOn F f p (S i))
Goal: TendstoLocallyUniformlyOn F f p (⋃ i ∈ s, S i)
-- tendstoLocallyUniformlyOn_sUnion
theorem tendstoLocallyUniformlyOn_sUnion (S : Set (Set α)) (hS : ∀ s ∈ S, IsOpen s)
    (h : ∀ s ∈ S, TendstoLocallyUniformlyOn F f p s) : TendstoLocallyUniformlyOn F f p (⋃₀ S) 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {p : Filter ι} [TopologicalSpace α] (S : Set (Set α)) (hS : ∀ s ∈ S, IsOpen s) (h : ∀ s ∈ S, TendstoLocallyUniformlyOn F f p s)
Goal: TendstoLocallyUniformlyOn F f p (⋃₀ S)
-- TendstoLocallyUniformlyOn.union
theorem TendstoLocallyUniformlyOn.union {s₁ s₂ : Set α} (hs₁ : IsOpen s₁) (hs₂ : IsOpen s₂)
    (h₁ : TendstoLocallyUniformlyOn F f p s₁) (h₂ : TendstoLocallyUniformlyOn F f p s₂) :
    TendstoLocallyUniformlyOn F f p (s₁ ∪ s₂) 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {p : Filter ι} [TopologicalSpace α] {s₁ s₂ : Set α} (hs₁ : IsOpen s₁) (hs₂ : IsOpen s₂) (h₁ : TendstoLocallyUniformlyOn F f p s₁) (h₂ : TendstoLocallyUniformlyOn F f p s₂)
Goal: TendstoLocallyUniformlyOn F f p (s₁ ∪ s₂)
-- TendstoLocallyUniformly.tendstoLocallyUniformlyOn
protected theorem TendstoLocallyUniformly.tendstoLocallyUniformlyOn
    (h : TendstoLocallyUniformly F f p) : TendstoLocallyUniformlyOn F f p s 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {s : Set α} {p : Filter ι} [TopologicalSpace α] (h : TendstoLocallyUniformly F f p)
Goal: TendstoLocallyUniformlyOn F f p s
-- tendstoLocallyUniformly_iff_tendstoUniformly_of_compactSpace
theorem tendstoLocallyUniformly_iff_tendstoUniformly_of_compactSpace [CompactSpace α] :
    TendstoLocallyUniformly F f p ↔ TendstoUniformly F f p 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {p : Filter ι} [TopologicalSpace α] [CompactSpace α]
Goal: TendstoLocallyUniformly F f p ↔ TendstoUniformly F f p
-- tendstoLocallyUniformlyOn_iff_tendstoUniformlyOn_of_compact
theorem tendstoLocallyUniformlyOn_iff_tendstoUniformlyOn_of_compact (hs : IsCompact s) :
    TendstoLocallyUniformlyOn F f p s ↔ TendstoUniformlyOn F f p s 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {s : Set α} {p : Filter ι} [TopologicalSpace α] (hs : IsCompact s)
Goal: TendstoLocallyUniformlyOn F f p s ↔ TendstoUniformlyOn F f p s
-- TendstoLocallyUniformlyOn.comp
theorem TendstoLocallyUniformlyOn.comp [TopologicalSpace γ] {t : Set γ}
    (h : TendstoLocallyUniformlyOn F f p s) (g : γ → α) (hg : MapsTo g t s)
    (cg : ContinuousOn g t) : TendstoLocallyUniformlyOn (fun n => F n ∘ g) (f ∘ g) p t 
Arguments: {α : Type u} {β : Type v} {γ : Type w} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {s : Set α} {p : Filter ι} [TopologicalSpace α] [TopologicalSpace γ] {t : Set γ} (h : TendstoLocallyUniformlyOn F f p s) (g : γ → α) (hg : Set.MapsTo g t s) (cg : ContinuousOn g t)
Goal: TendstoLocallyUniformlyOn (fun n => F n ∘ g) (f ∘ g) p t
-- TendstoLocallyUniformly.comp
theorem TendstoLocallyUniformly.comp [TopologicalSpace γ] (h : TendstoLocallyUniformly F f p)
    (g : γ → α) (cg : Continuous g) : TendstoLocallyUniformly (fun n => F n ∘ g) (f ∘ g) p 
Arguments: {α : Type u} {β : Type v} {γ : Type w} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {p : Filter ι} [TopologicalSpace α] [TopologicalSpace γ] (h : TendstoLocallyUniformly F f p) (g : γ → α) (cg : Continuous g)
Goal: TendstoLocallyUniformly (fun n => F n ∘ g) (f ∘ g) p
-- tendstoLocallyUniformlyOn_TFAE
theorem tendstoLocallyUniformlyOn_TFAE [LocallyCompactSpace α] (G : ι → α → β) (g : α → β)
    (p : Filter ι) (hs : IsOpen s) :
    List.TFAE [
      TendstoLocallyUniformlyOn G g p s,
      ∀ K, K ⊆ s → IsCompact K → TendstoUniformlyOn G g p K,
      ∀ x ∈ s, ∃ v ∈ 𝓝[s] x, TendstoUniformlyOn G g p v] 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {s : Set α} [TopologicalSpace α] [LocallyCompactSpace α] (G : ι → α → β) (g : α → β) (p : Filter ι) (hs : IsOpen s)
Goal: [TendstoLocallyUniformlyOn G g p s, ∀ K ⊆ s, IsCompact K → TendstoUniformlyOn G g p K,
    ∀ x ∈ s, ∃ v ∈ 𝓝[s] x, TendstoUniformlyOn G g p v].TFAE
-- tendstoLocallyUniformlyOn_iff_forall_isCompact
theorem tendstoLocallyUniformlyOn_iff_forall_isCompact [LocallyCompactSpace α] (hs : IsOpen s) :
    TendstoLocallyUniformlyOn F f p s ↔ ∀ K, K ⊆ s → IsCompact K → TendstoUniformlyOn F f p K 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {s : Set α} {p : Filter ι} [TopologicalSpace α] [LocallyCompactSpace α] (hs : IsOpen s)
Goal: TendstoLocallyUniformlyOn F f p s ↔ ∀ K ⊆ s, IsCompact K → TendstoUniformlyOn F f p K
-- tendstoLocallyUniformly_iff_forall_isCompact
lemma tendstoLocallyUniformly_iff_forall_isCompact [LocallyCompactSpace α]  :
    TendstoLocallyUniformly F f p ↔ ∀ K : Set α, IsCompact K → TendstoUniformlyOn F f p K 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {p : Filter ι} [TopologicalSpace α] [LocallyCompactSpace α]
Goal: TendstoLocallyUniformly F f p ↔ ∀ (K : Set α), IsCompact K → TendstoUniformlyOn F f p K
-- tendstoLocallyUniformlyOn_iff_filter
theorem tendstoLocallyUniformlyOn_iff_filter :
    TendstoLocallyUniformlyOn F f p s ↔ ∀ x ∈ s, TendstoUniformlyOnFilter F f p (𝓝[s] x) 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {s : Set α} {p : Filter ι} [TopologicalSpace α]
Goal: TendstoLocallyUniformlyOn F f p s ↔ ∀ x ∈ s, TendstoUniformlyOnFilter F f p (𝓝[s] x)
-- tendstoLocallyUniformly_iff_filter
theorem tendstoLocallyUniformly_iff_filter :
    TendstoLocallyUniformly F f p ↔ ∀ x, TendstoUniformlyOnFilter F f p (𝓝 x) 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {p : Filter ι} [TopologicalSpace α]
Goal: TendstoLocallyUniformly F f p ↔ ∀ (x : α), TendstoUniformlyOnFilter F f p (𝓝 x)
-- TendstoLocallyUniformlyOn.tendsto_at
theorem TendstoLocallyUniformlyOn.tendsto_at (hf : TendstoLocallyUniformlyOn F f p s) {a : α}
    (ha : a ∈ s) : Tendsto (fun i => F i a) p (𝓝 (f a)) 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {s : Set α} {p : Filter ι} [TopologicalSpace α] (hf : TendstoLocallyUniformlyOn F f p s) {a : α} (ha : a ∈ s)
Goal: Filter.Tendsto (fun i => F i a) p (𝓝 (f a))
-- TendstoLocallyUniformlyOn.unique
theorem TendstoLocallyUniformlyOn.unique [p.NeBot] [T2Space β] {g : α → β}
    (hf : TendstoLocallyUniformlyOn F f p s) (hg : TendstoLocallyUniformlyOn F g p s) :
    s.EqOn f g 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {s : Set α} {p : Filter ι} [TopologicalSpace α] [p.NeBot] [T2Space β] {g : α → β} (hf : TendstoLocallyUniformlyOn F f p s) (hg : TendstoLocallyUniformlyOn F g p s)
Goal: Set.EqOn f g s
-- TendstoLocallyUniformlyOn.congr
theorem TendstoLocallyUniformlyOn.congr {G : ι → α → β} (hf : TendstoLocallyUniformlyOn F f p s)
    (hg : ∀ n, s.EqOn (F n) (G n)) : TendstoLocallyUniformlyOn G f p s 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {s : Set α} {p : Filter ι} [TopologicalSpace α] {G : ι → α → β} (hf : TendstoLocallyUniformlyOn F f p s) (hg : ∀ (n : ι), Set.EqOn (F n) (G n) s)
Goal: TendstoLocallyUniformlyOn G f p s
-- TendstoLocallyUniformlyOn.congr_right
theorem TendstoLocallyUniformlyOn.congr_right {g : α → β} (hf : TendstoLocallyUniformlyOn F f p s)
    (hg : s.EqOn f g) : TendstoLocallyUniformlyOn F g p s 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {s : Set α} {p : Filter ι} [TopologicalSpace α] {g : α → β} (hf : TendstoLocallyUniformlyOn F f p s) (hg : Set.EqOn f g s)
Goal: TendstoLocallyUniformlyOn F g p s
-- continuousWithinAt_of_locally_uniform_approx_of_continuousWithinAt
theorem continuousWithinAt_of_locally_uniform_approx_of_continuousWithinAt (hx : x ∈ s)
    (L : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝[s] x, ∃ F : α → β, ContinuousWithinAt F s x ∧ ∀ y ∈ t, (f y, F y) ∈ u) :
    ContinuousWithinAt f s x 
Arguments: {α : Type u} {β : Type v} [UniformSpace β] {f : α → β} {s : Set α} {x : α} [TopologicalSpace α] (hx : x ∈ s) (L : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝[s] x, ∃ F, ContinuousWithinAt F s x ∧ ∀ y ∈ t, (f y, F y) ∈ u)
Goal: ContinuousWithinAt f s x
-- continuousAt_of_locally_uniform_approx_of_continuousAt
theorem continuousAt_of_locally_uniform_approx_of_continuousAt
    (L : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝 x, ∃ F, ContinuousAt F x ∧ ∀ y ∈ t, (f y, F y) ∈ u) :
    ContinuousAt f x 
Arguments: {α : Type u} {β : Type v} [UniformSpace β] {f : α → β} {x : α} [TopologicalSpace α] (L : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝 x, ∃ F, ContinuousAt F x ∧ ∀ y ∈ t, (f y, F y) ∈ u)
Goal: ContinuousAt f x
-- continuousOn_of_locally_uniform_approx_of_continuousWithinAt
theorem continuousOn_of_locally_uniform_approx_of_continuousWithinAt
    (L : ∀ x ∈ s, ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝[s] x, ∃ F,
      ContinuousWithinAt F s x ∧ ∀ y ∈ t, (f y, F y) ∈ u) :
    ContinuousOn f s 
Arguments: {α : Type u} {β : Type v} [UniformSpace β] {f : α → β} {s : Set α} [TopologicalSpace α] (L : ∀ x ∈ s, ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝[s] x, ∃ F, ContinuousWithinAt F s x ∧ ∀ y ∈ t, (f y, F y) ∈ u)
Goal: ContinuousOn f s
-- continuousOn_of_uniform_approx_of_continuousOn
theorem continuousOn_of_uniform_approx_of_continuousOn
    (L : ∀ u ∈ 𝓤 β, ∃ F, ContinuousOn F s ∧ ∀ y ∈ s, (f y, F y) ∈ u) : ContinuousOn f s 
Arguments: {α : Type u} {β : Type v} [UniformSpace β] {f : α → β} {s : Set α} [TopologicalSpace α] (L : ∀ u ∈ 𝓤 β, ∃ F, ContinuousOn F s ∧ ∀ y ∈ s, (f y, F y) ∈ u)
Goal: ContinuousOn f s
-- continuous_of_locally_uniform_approx_of_continuousAt
theorem continuous_of_locally_uniform_approx_of_continuousAt
    (L : ∀ x : α, ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝 x, ∃ F, ContinuousAt F x ∧ ∀ y ∈ t, (f y, F y) ∈ u) :
    Continuous f 
Arguments: {α : Type u} {β : Type v} [UniformSpace β] {f : α → β} [TopologicalSpace α] (L : ∀ (x : α), ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝 x, ∃ F, ContinuousAt F x ∧ ∀ y ∈ t, (f y, F y) ∈ u)
Goal: Continuous f
-- continuous_of_uniform_approx_of_continuous
theorem continuous_of_uniform_approx_of_continuous
    (L : ∀ u ∈ 𝓤 β, ∃ F, Continuous F ∧ ∀ y, (f y, F y) ∈ u) : Continuous f 
Arguments: {α : Type u} {β : Type v} [UniformSpace β] {f : α → β} [TopologicalSpace α] (L : ∀ u ∈ 𝓤 β, ∃ F, Continuous F ∧ ∀ (y : α), (f y, F y) ∈ u)
Goal: Continuous f
-- TendstoLocallyUniformlyOn.continuousOn
protected theorem TendstoLocallyUniformlyOn.continuousOn (h : TendstoLocallyUniformlyOn F f p s)
    (hc : ∀ᶠ n in p, ContinuousOn (F n) s) [NeBot p] : ContinuousOn f s 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {s : Set α} {p : Filter ι} [TopologicalSpace α] (h : TendstoLocallyUniformlyOn F f p s) (hc : ∀ᶠ (n : ι) in p, ContinuousOn (F n) s) [p.NeBot]
Goal: ContinuousOn f s
-- TendstoUniformlyOn.continuousOn
protected theorem TendstoUniformlyOn.continuousOn (h : TendstoUniformlyOn F f p s)
    (hc : ∀ᶠ n in p, ContinuousOn (F n) s) [NeBot p] : ContinuousOn f s 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {s : Set α} {p : Filter ι} [TopologicalSpace α] (h : TendstoUniformlyOn F f p s) (hc : ∀ᶠ (n : ι) in p, ContinuousOn (F n) s) [p.NeBot]
Goal: ContinuousOn f s
-- TendstoLocallyUniformly.continuous
protected theorem TendstoLocallyUniformly.continuous (h : TendstoLocallyUniformly F f p)
    (hc : ∀ᶠ n in p, Continuous (F n)) [NeBot p] : Continuous f 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {p : Filter ι} [TopologicalSpace α] (h : TendstoLocallyUniformly F f p) (hc : ∀ᶠ (n : ι) in p, Continuous (F n)) [p.NeBot]
Goal: Continuous f
-- TendstoUniformly.continuous
protected theorem TendstoUniformly.continuous (h : TendstoUniformly F f p)
    (hc : ∀ᶠ n in p, Continuous (F n)) [NeBot p] : Continuous f 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {p : Filter ι} [TopologicalSpace α] (h : TendstoUniformly F f p) (hc : ∀ᶠ (n : ι) in p, Continuous (F n)) [p.NeBot]
Goal: Continuous f
-- tendsto_comp_of_locally_uniform_limit_within
theorem tendsto_comp_of_locally_uniform_limit_within (h : ContinuousWithinAt f s x)
    (hg : Tendsto g p (𝓝[s] x))
    (hunif : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝[s] x, ∀ᶠ n in p, ∀ y ∈ t, (f y, F n y) ∈ u) :
    Tendsto (fun n => F n (g n)) p (𝓝 (f x)) 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {s : Set α} {x : α} {p : Filter ι} {g : ι → α} [TopologicalSpace α] (h : ContinuousWithinAt f s x) (hg : Filter.Tendsto g p (𝓝[s] x)) (hunif : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝[s] x, ∀ᶠ (n : ι) in p, ∀ y ∈ t, (f y, F n y) ∈ u)
Goal: Filter.Tendsto (fun n => F n (g n)) p (𝓝 (f x))
-- tendsto_comp_of_locally_uniform_limit
theorem tendsto_comp_of_locally_uniform_limit (h : ContinuousAt f x) (hg : Tendsto g p (𝓝 x))
    (hunif : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝 x, ∀ᶠ n in p, ∀ y ∈ t, (f y, F n y) ∈ u) :
    Tendsto (fun n => F n (g n)) p (𝓝 (f x)) 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {x : α} {p : Filter ι} {g : ι → α} [TopologicalSpace α] (h : ContinuousAt f x) (hg : Filter.Tendsto g p (𝓝 x)) (hunif : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝 x, ∀ᶠ (n : ι) in p, ∀ y ∈ t, (f y, F n y) ∈ u)
Goal: Filter.Tendsto (fun n => F n (g n)) p (𝓝 (f x))
-- TendstoLocallyUniformlyOn.tendsto_comp
theorem TendstoLocallyUniformlyOn.tendsto_comp (h : TendstoLocallyUniformlyOn F f p s)
    (hf : ContinuousWithinAt f s x) (hx : x ∈ s) (hg : Tendsto g p (𝓝[s] x)) :
    Tendsto (fun n => F n (g n)) p (𝓝 (f x)) 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {s : Set α} {x : α} {p : Filter ι} {g : ι → α} [TopologicalSpace α] (h : TendstoLocallyUniformlyOn F f p s) (hf : ContinuousWithinAt f s x) (hx : x ∈ s) (hg : Filter.Tendsto g p (𝓝[s] x))
Goal: Filter.Tendsto (fun n => F n (g n)) p (𝓝 (f x))
-- TendstoUniformlyOn.tendsto_comp
theorem TendstoUniformlyOn.tendsto_comp (h : TendstoUniformlyOn F f p s)
    (hf : ContinuousWithinAt f s x) (hg : Tendsto g p (𝓝[s] x)) :
    Tendsto (fun n => F n (g n)) p (𝓝 (f x)) 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {s : Set α} {x : α} {p : Filter ι} {g : ι → α} [TopologicalSpace α] (h : TendstoUniformlyOn F f p s) (hf : ContinuousWithinAt f s x) (hg : Filter.Tendsto g p (𝓝[s] x))
Goal: Filter.Tendsto (fun n => F n (g n)) p (𝓝 (f x))
-- TendstoLocallyUniformly.tendsto_comp
theorem TendstoLocallyUniformly.tendsto_comp (h : TendstoLocallyUniformly F f p)
    (hf : ContinuousAt f x) (hg : Tendsto g p (𝓝 x)) : Tendsto (fun n => F n (g n)) p (𝓝 (f x)) 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {x : α} {p : Filter ι} {g : ι → α} [TopologicalSpace α] (h : TendstoLocallyUniformly F f p) (hf : ContinuousAt f x) (hg : Filter.Tendsto g p (𝓝 x))
Goal: Filter.Tendsto (fun n => F n (g n)) p (𝓝 (f x))
-- TendstoUniformly.tendsto_comp
theorem TendstoUniformly.tendsto_comp (h : TendstoUniformly F f p) (hf : ContinuousAt f x)
    (hg : Tendsto g p (𝓝 x)) : Tendsto (fun n => F n (g n)) p (𝓝 (f x)) 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {x : α} {p : Filter ι} {g : ι → α} [TopologicalSpace α] (h : TendstoUniformly F f p) (hf : ContinuousAt f x) (hg : Filter.Tendsto g p (𝓝 x))
Goal: Filter.Tendsto (fun n => F n (g n)) p (𝓝 (f x))
-- Metric.equicontinuousAt_iff_right
theorem equicontinuousAt_iff_right {ι : Type*} [TopologicalSpace β] {F : ι → β → α} {x₀ : β} :
    EquicontinuousAt F x₀ ↔ ∀ ε > 0, ∀ᶠ x in 𝓝 x₀, ∀ i, dist (F i x₀) (F i x) < ε 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] {ι : Type u_4} [TopologicalSpace β] {F : ι → β → α} {x₀ : β}
Goal: EquicontinuousAt F x₀ ↔ ∀ ε > 0, ∀ᶠ (x : β) in 𝓝 x₀, ∀ (i : ι), dist (F i x₀) (F i x) < ε
-- Metric.equicontinuousAt_iff
theorem equicontinuousAt_iff {ι : Type*} [PseudoMetricSpace β] {F : ι → β → α} {x₀ : β} :
    EquicontinuousAt F x₀ ↔ ∀ ε > 0, ∃ δ > 0, ∀ x, dist x x₀ < δ → ∀ i, dist (F i x₀) (F i x) < ε 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] {ι : Type u_4} [PseudoMetricSpace β] {F : ι → β → α} {x₀ : β}
Goal: EquicontinuousAt F x₀ ↔ ∀ ε > 0, ∃ δ > 0, ∀ (x : β), dist x x₀ < δ → ∀ (i : ι), dist (F i x₀) (F i x) < ε
-- Metric.equicontinuousAt_iff_pair
protected theorem equicontinuousAt_iff_pair {ι : Type*} [TopologicalSpace β] {F : ι → β → α}
    {x₀ : β} :
    EquicontinuousAt F x₀ ↔
      ∀ ε > 0, ∃ U ∈ 𝓝 x₀, ∀ x ∈ U, ∀ x' ∈ U, ∀ i, dist (F i x) (F i x') < ε 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] {ι : Type u_4} [TopologicalSpace β] {F : ι → β → α} {x₀ : β}
Goal: EquicontinuousAt F x₀ ↔ ∀ ε > 0, ∃ U ∈ 𝓝 x₀, ∀ x ∈ U, ∀ x' ∈ U, ∀ (i : ι), dist (F i x) (F i x') < ε
-- Metric.uniformEquicontinuous_iff_right
theorem uniformEquicontinuous_iff_right {ι : Type*} [UniformSpace β] {F : ι → β → α} :
    UniformEquicontinuous F ↔ ∀ ε > 0, ∀ᶠ xy : β × β in 𝓤 β, ∀ i, dist (F i xy.1) (F i xy.2) < ε 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] {ι : Type u_4} [UniformSpace β] {F : ι → β → α}
Goal: UniformEquicontinuous F ↔ ∀ ε > 0, ∀ᶠ (xy : β × β) in 𝓤 β, ∀ (i : ι), dist (F i xy.1) (F i xy.2) < ε
-- Metric.uniformEquicontinuous_iff
theorem uniformEquicontinuous_iff {ι : Type*} [PseudoMetricSpace β] {F : ι → β → α} :
    UniformEquicontinuous F ↔
      ∀ ε > 0, ∃ δ > 0, ∀ x y, dist x y < δ → ∀ i, dist (F i x) (F i y) < ε 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] {ι : Type u_4} [PseudoMetricSpace β] {F : ι → β → α}
Goal: UniformEquicontinuous F ↔ ∀ ε > 0, ∃ δ > 0, ∀ (x y : β), dist x y < δ → ∀ (i : ι), dist (F i x) (F i y) < ε
-- Metric.equicontinuousAt_of_continuity_modulus
theorem equicontinuousAt_of_continuity_modulus {ι : Type*} [TopologicalSpace β] {x₀ : β}
    (b : β → ℝ) (b_lim : Tendsto b (𝓝 x₀) (𝓝 0)) (F : ι → β → α)
    (H : ∀ᶠ x in 𝓝 x₀, ∀ i, dist (F i x₀) (F i x) ≤ b x) : EquicontinuousAt F x₀ 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] {ι : Type u_4} [TopologicalSpace β] {x₀ : β} (b : β → ℝ) (b_lim : Filter.Tendsto b (𝓝 x₀) (𝓝 0)) (F : ι → β → α) (H : ∀ᶠ (x : β) in 𝓝 x₀, ∀ (i : ι), dist (F i x₀) (F i x) ≤ b x)
Goal: EquicontinuousAt F x₀
-- Metric.uniformEquicontinuous_of_continuity_modulus
theorem uniformEquicontinuous_of_continuity_modulus {ι : Type*} [PseudoMetricSpace β] (b : ℝ → ℝ)
    (b_lim : Tendsto b (𝓝 0) (𝓝 0)) (F : ι → β → α)
    (H : ∀ (x y : β) (i), dist (F i x) (F i y) ≤ b (dist x y)) : UniformEquicontinuous F 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] {ι : Type u_4} [PseudoMetricSpace β] (b : ℝ → ℝ) (b_lim : Filter.Tendsto b (𝓝 0) (𝓝 0)) (F : ι → β → α) (H : ∀ (x y : β) (i : ι), dist (F i x) (F i y) ≤ b (dist x y))
Goal: UniformEquicontinuous F
-- Metric.equicontinuous_of_continuity_modulus
theorem equicontinuous_of_continuity_modulus {ι : Type*} [PseudoMetricSpace β] (b : ℝ → ℝ)
    (b_lim : Tendsto b (𝓝 0) (𝓝 0)) (F : ι → β → α)
    (H : ∀ (x y : β) (i), dist (F i x) (F i y) ≤ b (dist x y)) : Equicontinuous F 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] {ι : Type u_4} [PseudoMetricSpace β] (b : ℝ → ℝ) (b_lim : Filter.Tendsto b (𝓝 0) (𝓝 0)) (F : ι → β → α) (H : ∀ (x y : β) (i : ι), dist (F i x) (F i y) ≤ b (dist x y))
Goal: Equicontinuous F
-- UniformFun.toFun_ofFun
@[simp] lemma UniformFun.toFun_ofFun (f : α → β) : toFun (ofFun f) = f 
Arguments: {α : Type u_1} {β : Type u_2} (f : α → β)
Goal: UniformFun.toFun (UniformFun.ofFun f) = f
-- UniformFun.ofFun_toFun
@[simp] lemma UniformFun.ofFun_toFun (f : α →ᵤ β) : ofFun (toFun f) = f 
Arguments: {α : Type u_1} {β : Type u_2} (f : α →ᵤ β)
Goal: UniformFun.ofFun (UniformFun.toFun f) = f