ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- TendstoUniformly.tendsto_at
theorem TendstoUniformly.tendsto_at (h : TendstoUniformly F f p) (x : α) :
    Tendsto (fun n => F n x) p <| 𝓝 (f x) 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {p : Filter ι} (h : TendstoUniformly F f p) (x : α)
Goal: Filter.Tendsto (fun n => F n x) p (𝓝 (f x))
-- TendstoUniformlyOnFilter.mono_left
theorem TendstoUniformlyOnFilter.mono_left {p'' : Filter ι} (h : TendstoUniformlyOnFilter F f p p')
    (hp : p'' ≤ p) : TendstoUniformlyOnFilter F f p'' p' 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {p : Filter ι} {p' : Filter α} {p'' : Filter ι} (h : TendstoUniformlyOnFilter F f p p') (hp : p'' ≤ p)
Goal: TendstoUniformlyOnFilter F f p'' p'
-- TendstoUniformlyOnFilter.mono_right
theorem TendstoUniformlyOnFilter.mono_right {p'' : Filter α} (h : TendstoUniformlyOnFilter F f p p')
    (hp : p'' ≤ p') : TendstoUniformlyOnFilter F f p p'' 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {p : Filter ι} {p' p'' : Filter α} (h : TendstoUniformlyOnFilter F f p p') (hp : p'' ≤ p')
Goal: TendstoUniformlyOnFilter F f p p''
-- TendstoUniformlyOn.mono
theorem TendstoUniformlyOn.mono {s' : Set α} (h : TendstoUniformlyOn F f p s) (h' : s' ⊆ 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} {s' : Set α} (h : TendstoUniformlyOn F f p s) (h' : s' ⊆ s)
Goal: TendstoUniformlyOn F f p s'
-- TendstoUniformlyOnFilter.congr
theorem TendstoUniformlyOnFilter.congr {F' : ι → α → β} (hf : TendstoUniformlyOnFilter F f p p')
    (hff' : ∀ᶠ n : ι × α in p ×ˢ p', F n.fst n.snd = F' n.fst n.snd) :
    TendstoUniformlyOnFilter F' f p p' 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {p : Filter ι} {p' : Filter α} {F' : ι → α → β} (hf : TendstoUniformlyOnFilter F f p p') (hff' : ∀ᶠ (n : ι × α) in p ×ˢ p', F n.1 n.2 = F' n.1 n.2)
Goal: TendstoUniformlyOnFilter F' f p p'
-- TendstoUniformlyOn.congr
theorem TendstoUniformlyOn.congr {F' : ι → α → β} (hf : TendstoUniformlyOn F f p s)
    (hff' : ∀ᶠ n in p, Set.EqOn (F n) (F' n) 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} {F' : ι → α → β} (hf : TendstoUniformlyOn F f p s) (hff' : ∀ᶠ (n : ι) in p, Set.EqOn (F n) (F' n) s)
Goal: TendstoUniformlyOn F' f p s
-- TendstoUniformlyOn.congr_right
theorem TendstoUniformlyOn.congr_right {g : α → β} (hf : TendstoUniformlyOn F f p s)
    (hfg : EqOn f g s) : TendstoUniformlyOn 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} {g : α → β} (hf : TendstoUniformlyOn F f p s) (hfg : Set.EqOn f g s)
Goal: TendstoUniformlyOn F g p s
-- TendstoUniformly.tendstoUniformlyOn
protected theorem TendstoUniformly.tendstoUniformlyOn (h : TendstoUniformly F f p) :
    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} (h : TendstoUniformly F f p)
Goal: TendstoUniformlyOn F f p s
-- TendstoUniformlyOnFilter.comp
theorem TendstoUniformlyOnFilter.comp (h : TendstoUniformlyOnFilter F f p p') (g : γ → α) :
    TendstoUniformlyOnFilter (fun n => F n ∘ g) (f ∘ g) p (p'.comap g) 
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} {p' : Filter α} (h : TendstoUniformlyOnFilter F f p p') (g : γ → α)
Goal: TendstoUniformlyOnFilter (fun n => F n ∘ g) (f ∘ g) p (Filter.comap g p')
-- TendstoUniformlyOn.comp
theorem TendstoUniformlyOn.comp (h : TendstoUniformlyOn F f p s) (g : γ → α) :
    TendstoUniformlyOn (fun n => F n ∘ g) (f ∘ g) p (g ⁻¹' s) 
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} (h : TendstoUniformlyOn F f p s) (g : γ → α)
Goal: TendstoUniformlyOn (fun n => F n ∘ g) (f ∘ g) p (g ⁻¹' s)
-- TendstoUniformly.comp
theorem TendstoUniformly.comp (h : TendstoUniformly F f p) (g : γ → α) :
    TendstoUniformly (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} (h : TendstoUniformly F f p) (g : γ → α)
Goal: TendstoUniformly (fun n => F n ∘ g) (f ∘ g) p
-- UniformContinuous.comp_tendstoUniformlyOnFilter
theorem UniformContinuous.comp_tendstoUniformlyOnFilter [UniformSpace γ] {g : β → γ}
    (hg : UniformContinuous g) (h : TendstoUniformlyOnFilter F f p p') :
    TendstoUniformlyOnFilter (fun i => g ∘ F i) (g ∘ f) p 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} {p' : Filter α} [UniformSpace γ] {g : β → γ} (hg : UniformContinuous g) (h : TendstoUniformlyOnFilter F f p p')
Goal: TendstoUniformlyOnFilter (fun i => g ∘ F i) (g ∘ f) p p'
-- UniformContinuous.comp_tendstoUniformlyOn
theorem UniformContinuous.comp_tendstoUniformlyOn [UniformSpace γ] {g : β → γ}
    (hg : UniformContinuous g) (h : TendstoUniformlyOn F f p s) :
    TendstoUniformlyOn (fun i => g ∘ F i) (g ∘ f) p s 
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} [UniformSpace γ] {g : β → γ} (hg : UniformContinuous g) (h : TendstoUniformlyOn F f p s)
Goal: TendstoUniformlyOn (fun i => g ∘ F i) (g ∘ f) p s
-- UniformContinuous.comp_tendstoUniformly
theorem UniformContinuous.comp_tendstoUniformly [UniformSpace γ] {g : β → γ}
    (hg : UniformContinuous g) (h : TendstoUniformly F f p) :
    TendstoUniformly (fun i => g ∘ F i) (g ∘ f) 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} [UniformSpace γ] {g : β → γ} (hg : UniformContinuous g) (h : TendstoUniformly F f p)
Goal: TendstoUniformly (fun i => g ∘ F i) (g ∘ f) p
-- TendstoUniformlyOnFilter.prod_map
theorem TendstoUniformlyOnFilter.prod_map {ι' α' β' : Type*} [UniformSpace β'] {F' : ι' → α' → β'}
    {f' : α' → β'} {q : Filter ι'} {q' : Filter α'} (h : TendstoUniformlyOnFilter F f p p')
    (h' : TendstoUniformlyOnFilter F' f' q q') :
    TendstoUniformlyOnFilter (fun i : ι × ι' => Prod.map (F i.1) (F' i.2)) (Prod.map f f')
      (p ×ˢ q) (p' ×ˢ q') 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {p : Filter ι} {p' : Filter α} {ι' : Type u_1} {α' : Type u_2} {β' : Type u_3} [UniformSpace β'] {F' : ι' → α' → β'} {f' : α' → β'} {q : Filter ι'} {q' : Filter α'} (h : TendstoUniformlyOnFilter F f p p') (h' : TendstoUniformlyOnFilter F' f' q q')
Goal: TendstoUniformlyOnFilter (fun i => Prod.map (F i.1) (F' i.2)) (Prod.map f f') (p ×ˢ q) (p' ×ˢ q')
-- TendstoUniformlyOn.prod_map
theorem TendstoUniformlyOn.prod_map {ι' α' β' : Type*} [UniformSpace β'] {F' : ι' → α' → β'}
    {f' : α' → β'} {p' : Filter ι'} {s' : Set α'} (h : TendstoUniformlyOn F f p s)
    (h' : TendstoUniformlyOn F' f' p' s') :
    TendstoUniformlyOn (fun i : ι × ι' => Prod.map (F i.1) (F' i.2)) (Prod.map f f') (p ×ˢ p')
      (s ×ˢ 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} {ι' : Type u_1} {α' : Type u_2} {β' : Type u_3} [UniformSpace β'] {F' : ι' → α' → β'} {f' : α' → β'} {p' : Filter ι'} {s' : Set α'} (h : TendstoUniformlyOn F f p s) (h' : TendstoUniformlyOn F' f' p' s')
Goal: TendstoUniformlyOn (fun i => Prod.map (F i.1) (F' i.2)) (Prod.map f f') (p ×ˢ p') (s ×ˢ s')
-- TendstoUniformly.prod_map
theorem TendstoUniformly.prod_map {ι' α' β' : Type*} [UniformSpace β'] {F' : ι' → α' → β'}
    {f' : α' → β'} {p' : Filter ι'} (h : TendstoUniformly F f p) (h' : TendstoUniformly F' f' p') :
    TendstoUniformly (fun i : ι × ι' => Prod.map (F i.1) (F' i.2)) (Prod.map f f') (p ×ˢ p') 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {p : Filter ι} {ι' : Type u_1} {α' : Type u_2} {β' : Type u_3} [UniformSpace β'] {F' : ι' → α' → β'} {f' : α' → β'} {p' : Filter ι'} (h : TendstoUniformly F f p) (h' : TendstoUniformly F' f' p')
Goal: TendstoUniformly (fun i => Prod.map (F i.1) (F' i.2)) (Prod.map f f') (p ×ˢ p')
-- TendstoUniformlyOnFilter.prod
theorem TendstoUniformlyOnFilter.prod {ι' β' : Type*} [UniformSpace β'] {F' : ι' → α → β'}
    {f' : α → β'} {q : Filter ι'} (h : TendstoUniformlyOnFilter F f p p')
    (h' : TendstoUniformlyOnFilter F' f' q p') :
    TendstoUniformlyOnFilter (fun (i : ι × ι') a => (F i.1 a, F' i.2 a)) (fun a => (f a, f' a))
      (p ×ˢ q) p' 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {p : Filter ι} {p' : Filter α} {ι' : Type u_1} {β' : Type u_2} [UniformSpace β'] {F' : ι' → α → β'} {f' : α → β'} {q : Filter ι'} (h : TendstoUniformlyOnFilter F f p p') (h' : TendstoUniformlyOnFilter F' f' q p')
Goal: TendstoUniformlyOnFilter (fun i a => (F i.1 a, F' i.2 a)) (fun a => (f a, f' a)) (p ×ˢ q) p'
-- TendstoUniformlyOn.prod
theorem TendstoUniformlyOn.prod {ι' β' : Type*} [UniformSpace β'] {F' : ι' → α → β'} {f' : α → β'}
    {p' : Filter ι'} (h : TendstoUniformlyOn F f p s) (h' : TendstoUniformlyOn F' f' p' s) :
    TendstoUniformlyOn (fun (i : ι × ι') a => (F i.1 a, F' i.2 a)) (fun a => (f a, f' a))
      (p.prod 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} {ι' : Type u_1} {β' : Type u_2} [UniformSpace β'] {F' : ι' → α → β'} {f' : α → β'} {p' : Filter ι'} (h : TendstoUniformlyOn F f p s) (h' : TendstoUniformlyOn F' f' p' s)
Goal: TendstoUniformlyOn (fun i a => (F i.1 a, F' i.2 a)) (fun a => (f a, f' a)) (p.prod p') s
-- TendstoUniformly.prod
theorem TendstoUniformly.prod {ι' β' : Type*} [UniformSpace β'] {F' : ι' → α → β'} {f' : α → β'}
    {p' : Filter ι'} (h : TendstoUniformly F f p) (h' : TendstoUniformly F' f' p') :
    TendstoUniformly (fun (i : ι × ι') a => (F i.1 a, F' i.2 a)) (fun a => (f a, f' a))
      (p ×ˢ p') 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {p : Filter ι} {ι' : Type u_1} {β' : Type u_2} [UniformSpace β'] {F' : ι' → α → β'} {f' : α → β'} {p' : Filter ι'} (h : TendstoUniformly F f p) (h' : TendstoUniformly F' f' p')
Goal: TendstoUniformly (fun i a => (F i.1 a, F' i.2 a)) (fun a => (f a, f' a)) (p ×ˢ p')
-- tendsto_prod_filter_iff
theorem tendsto_prod_filter_iff {c : β} :
    Tendsto (↿F) (p ×ˢ p') (𝓝 c) ↔ TendstoUniformlyOnFilter F (fun _ => c) p p' 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {p : Filter ι} {p' : Filter α} {c : β}
Goal: Filter.Tendsto (↿F) (p ×ˢ p') (𝓝 c) ↔ TendstoUniformlyOnFilter F (fun x => c) p p'
-- tendsto_prod_principal_iff
theorem tendsto_prod_principal_iff {c : β} :
    Tendsto (↿F) (p ×ˢ 𝓟 s) (𝓝 c) ↔ TendstoUniformlyOn F (fun _ => c) p s 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {s : Set α} {p : Filter ι} {c : β}
Goal: Filter.Tendsto (↿F) (p ×ˢ 𝓟 s) (𝓝 c) ↔ TendstoUniformlyOn F (fun x => c) p s
-- tendsto_prod_top_iff
theorem tendsto_prod_top_iff {c : β} :
    Tendsto (↿F) (p ×ˢ ⊤) (𝓝 c) ↔ TendstoUniformly F (fun _ => c) p 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {p : Filter ι} {c : β}
Goal: Filter.Tendsto (↿F) (p ×ˢ ⊤) (𝓝 c) ↔ TendstoUniformly F (fun x => c) p
-- tendstoUniformlyOn_empty
theorem tendstoUniformlyOn_empty : TendstoUniformlyOn 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}
Goal: TendstoUniformlyOn F f p ∅
-- tendstoUniformlyOn_singleton_iff_tendsto
theorem tendstoUniformlyOn_singleton_iff_tendsto :
    TendstoUniformlyOn F f p {x} ↔ Tendsto (fun n : ι => F n x) 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}
Goal: TendstoUniformlyOn F f p {x} ↔ Filter.Tendsto (fun n => F n x) p (𝓝 (f x))
-- Filter.Tendsto.tendstoUniformlyOnFilter_const
theorem Filter.Tendsto.tendstoUniformlyOnFilter_const {g : ι → β} {b : β} (hg : Tendsto g p (𝓝 b))
    (p' : Filter α) :
    TendstoUniformlyOnFilter (fun n : ι => fun _ : α => g n) (fun _ : α => b) p p' 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {p : Filter ι} {g : ι → β} {b : β} (hg : Filter.Tendsto g p (𝓝 b)) (p' : Filter α)
Goal: TendstoUniformlyOnFilter (fun n x => g n) (fun x => b) p p'
-- Filter.Tendsto.tendstoUniformlyOn_const
theorem Filter.Tendsto.tendstoUniformlyOn_const {g : ι → β} {b : β} (hg : Tendsto g p (𝓝 b))
    (s : Set α) : TendstoUniformlyOn (fun n : ι => fun _ : α => g n) (fun _ : α => b) p s 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {p : Filter ι} {g : ι → β} {b : β} (hg : Filter.Tendsto g p (𝓝 b)) (s : Set α)
Goal: TendstoUniformlyOn (fun n x => g n) (fun x => b) p s
-- UniformContinuousOn.tendstoUniformlyOn
theorem UniformContinuousOn.tendstoUniformlyOn [UniformSpace α] [UniformSpace γ] {x : α} {U : Set α}
    {V : Set β} {F : α → β → γ} (hF : UniformContinuousOn (↿F) (U ×ˢ V)) (hU : x ∈ U) :
    TendstoUniformlyOn F (F x) (𝓝[U] x) V 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [UniformSpace β] [UniformSpace α] [UniformSpace γ] {x : α} {U : Set α} {V : Set β} {F : α → β → γ} (hF : UniformContinuousOn (↿F) (U ×ˢ V)) (hU : x ∈ U)
Goal: TendstoUniformlyOn F (F x) (𝓝[U] x) V
-- UniformContinuousOn.tendstoUniformly
theorem UniformContinuousOn.tendstoUniformly [UniformSpace α] [UniformSpace γ] {x : α} {U : Set α}
    (hU : U ∈ 𝓝 x) {F : α → β → γ} (hF : UniformContinuousOn (↿F) (U ×ˢ (univ : Set β))) :
    TendstoUniformly F (F x) (𝓝 x) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [UniformSpace β] [UniformSpace α] [UniformSpace γ] {x : α} {U : Set α} (hU : U ∈ 𝓝 x) {F : α → β → γ} (hF : UniformContinuousOn (↿F) (U ×ˢ Set.univ))
Goal: TendstoUniformly F (F x) (𝓝 x)
-- UniformContinuous₂.tendstoUniformly
theorem UniformContinuous₂.tendstoUniformly [UniformSpace α] [UniformSpace γ] {f : α → β → γ}
    (h : UniformContinuous₂ f) {x : α} : TendstoUniformly f (f x) (𝓝 x) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [UniformSpace β] [UniformSpace α] [UniformSpace γ] {f : α → β → γ} (h : UniformContinuous₂ f) {x : α}
Goal: TendstoUniformly f (f x) (𝓝 x)
-- uniformCauchySeqOn_iff_uniformCauchySeqOnFilter
theorem uniformCauchySeqOn_iff_uniformCauchySeqOnFilter :
    UniformCauchySeqOn F p s ↔ UniformCauchySeqOnFilter F p (𝓟 s) 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {s : Set α} {p : Filter ι}
Goal: UniformCauchySeqOn F p s ↔ UniformCauchySeqOnFilter F p (𝓟 s)
-- UniformCauchySeqOn.uniformCauchySeqOnFilter
theorem UniformCauchySeqOn.uniformCauchySeqOnFilter (hF : UniformCauchySeqOn F p s) :
    UniformCauchySeqOnFilter F p (𝓟 s) 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {s : Set α} {p : Filter ι} (hF : UniformCauchySeqOn F p s)
Goal: UniformCauchySeqOnFilter F p (𝓟 s)
-- TendstoUniformlyOnFilter.uniformCauchySeqOnFilter
theorem TendstoUniformlyOnFilter.uniformCauchySeqOnFilter (hF : TendstoUniformlyOnFilter F f p p') :
    UniformCauchySeqOnFilter F p p' 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {p : Filter ι} {p' : Filter α} (hF : TendstoUniformlyOnFilter F f p p')
Goal: UniformCauchySeqOnFilter F p p'
-- TendstoUniformlyOn.uniformCauchySeqOn
theorem TendstoUniformlyOn.uniformCauchySeqOn (hF : TendstoUniformlyOn F f p s) :
    UniformCauchySeqOn 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} (hF : TendstoUniformlyOn F f p s)
Goal: UniformCauchySeqOn F p s
-- UniformCauchySeqOnFilter.tendstoUniformlyOnFilter_of_tendsto
theorem UniformCauchySeqOnFilter.tendstoUniformlyOnFilter_of_tendsto [NeBot p]
    (hF : UniformCauchySeqOnFilter F p p')
    (hF' : ∀ᶠ x : α in p', Tendsto (fun n => F n x) p (𝓝 (f x))) :
    TendstoUniformlyOnFilter F f p p' 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {p : Filter ι} {p' : Filter α} [p.NeBot] (hF : UniformCauchySeqOnFilter F p p') (hF' : ∀ᶠ (x : α) in p', Filter.Tendsto (fun n => F n x) p (𝓝 (f x)))
Goal: TendstoUniformlyOnFilter F f p p'
-- UniformCauchySeqOn.tendstoUniformlyOn_of_tendsto
theorem UniformCauchySeqOn.tendstoUniformlyOn_of_tendsto [NeBot p] (hF : UniformCauchySeqOn F p s)
    (hF' : ∀ x : α, x ∈ s → Tendsto (fun n => F n x) p (𝓝 (f x))) : 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} [p.NeBot] (hF : UniformCauchySeqOn F p s) (hF' : ∀ x ∈ s, Filter.Tendsto (fun n => F n x) p (𝓝 (f x)))
Goal: TendstoUniformlyOn F f p s
-- UniformCauchySeqOnFilter.mono_left
theorem UniformCauchySeqOnFilter.mono_left {p'' : Filter ι} (hf : UniformCauchySeqOnFilter F p p')
    (hp : p'' ≤ p) : UniformCauchySeqOnFilter F p'' p' 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {p : Filter ι} {p' : Filter α} {p'' : Filter ι} (hf : UniformCauchySeqOnFilter F p p') (hp : p'' ≤ p)
Goal: UniformCauchySeqOnFilter F p'' p'
-- UniformCauchySeqOnFilter.mono_right
theorem UniformCauchySeqOnFilter.mono_right {p'' : Filter α} (hf : UniformCauchySeqOnFilter F p p')
    (hp : p'' ≤ p') : UniformCauchySeqOnFilter F p p'' 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {p : Filter ι} {p' p'' : Filter α} (hf : UniformCauchySeqOnFilter F p p') (hp : p'' ≤ p')
Goal: UniformCauchySeqOnFilter F p p''
-- UniformCauchySeqOn.mono
theorem UniformCauchySeqOn.mono {s' : Set α} (hf : UniformCauchySeqOn F p s) (hss' : s' ⊆ s) :
    UniformCauchySeqOn F p s' 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {s : Set α} {p : Filter ι} {s' : Set α} (hf : UniformCauchySeqOn F p s) (hss' : s' ⊆ s)
Goal: UniformCauchySeqOn F p s'
-- UniformCauchySeqOnFilter.comp
theorem UniformCauchySeqOnFilter.comp {γ : Type*} (hf : UniformCauchySeqOnFilter F p p')
    (g : γ → α) : UniformCauchySeqOnFilter (fun n => F n ∘ g) p (p'.comap g) 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {p : Filter ι} {p' : Filter α} {γ : Type u_1} (hf : UniformCauchySeqOnFilter F p p') (g : γ → α)
Goal: UniformCauchySeqOnFilter (fun n => F n ∘ g) p (Filter.comap g p')
-- UniformCauchySeqOn.comp
theorem UniformCauchySeqOn.comp {γ : Type*} (hf : UniformCauchySeqOn F p s) (g : γ → α) :
    UniformCauchySeqOn (fun n => F n ∘ g) p (g ⁻¹' s) 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {s : Set α} {p : Filter ι} {γ : Type u_1} (hf : UniformCauchySeqOn F p s) (g : γ → α)
Goal: UniformCauchySeqOn (fun n => F n ∘ g) p (g ⁻¹' s)
-- UniformContinuous.comp_uniformCauchySeqOn
theorem UniformContinuous.comp_uniformCauchySeqOn [UniformSpace γ] {g : β → γ}
    (hg : UniformContinuous g) (hf : UniformCauchySeqOn F p s) :
    UniformCauchySeqOn (fun n => g ∘ F n) p s 
Arguments: {α : Type u} {β : Type v} {γ : Type w} {ι : Type x} [UniformSpace β] {F : ι → α → β} {s : Set α} {p : Filter ι} [UniformSpace γ] {g : β → γ} (hg : UniformContinuous g) (hf : UniformCauchySeqOn F p s)
Goal: UniformCauchySeqOn (fun n => g ∘ F n) p s
-- UniformCauchySeqOn.prod_map
theorem UniformCauchySeqOn.prod_map {ι' α' β' : Type*} [UniformSpace β'] {F' : ι' → α' → β'}
    {p' : Filter ι'} {s' : Set α'} (h : UniformCauchySeqOn F p s)
    (h' : UniformCauchySeqOn F' p' s') :
    UniformCauchySeqOn (fun i : ι × ι' => Prod.map (F i.1) (F' i.2)) (p ×ˢ p') (s ×ˢ s') 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {s : Set α} {p : Filter ι} {ι' : Type u_1} {α' : Type u_2} {β' : Type u_3} [UniformSpace β'] {F' : ι' → α' → β'} {p' : Filter ι'} {s' : Set α'} (h : UniformCauchySeqOn F p s) (h' : UniformCauchySeqOn F' p' s')
Goal: UniformCauchySeqOn (fun i => Prod.map (F i.1) (F' i.2)) (p ×ˢ p') (s ×ˢ s')
-- UniformCauchySeqOn.prod
theorem UniformCauchySeqOn.prod {ι' β' : Type*} [UniformSpace β'] {F' : ι' → α → β'}
    {p' : Filter ι'} (h : UniformCauchySeqOn F p s) (h' : UniformCauchySeqOn F' p' s) :
    UniformCauchySeqOn (fun (i : ι × ι') a => (F i.fst a, F' i.snd a)) (p ×ˢ p') s 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {s : Set α} {p : Filter ι} {ι' : Type u_1} {β' : Type u_2} [UniformSpace β'] {F' : ι' → α → β'} {p' : Filter ι'} (h : UniformCauchySeqOn F p s) (h' : UniformCauchySeqOn F' p' s)
Goal: UniformCauchySeqOn (fun i a => (F i.1 a, F' i.2 a)) (p ×ˢ p') s
-- UniformCauchySeqOn.prod'
theorem UniformCauchySeqOn.prod' {β' : Type*} [UniformSpace β'] {F' : ι → α → β'}
    (h : UniformCauchySeqOn F p s) (h' : UniformCauchySeqOn F' p s) :
    UniformCauchySeqOn (fun (i : ι) a => (F i a, F' i a)) p s 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {s : Set α} {p : Filter ι} {β' : Type u_1} [UniformSpace β'] {F' : ι → α → β'} (h : UniformCauchySeqOn F p s) (h' : UniformCauchySeqOn F' p s)
Goal: UniformCauchySeqOn (fun i a => (F i a, F' i a)) p s
-- UniformCauchySeqOn.cauchy_map
theorem UniformCauchySeqOn.cauchy_map [hp : NeBot p] (hf : UniformCauchySeqOn F p s) (hx : x ∈ s) :
    Cauchy (map (fun i => F i x) p) 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {s : Set α} {x : α} {p : Filter ι} [hp : p.NeBot] (hf : UniformCauchySeqOn F p s) (hx : x ∈ s)
Goal: Cauchy (Filter.map (fun i => F i x) p)
-- UniformCauchySeqOn.cauchySeq
theorem UniformCauchySeqOn.cauchySeq [Nonempty ι] [SemilatticeSup ι]
    (hf : UniformCauchySeqOn F atTop s) (hx : x ∈ s) :
    CauchySeq fun i ↦ F i x 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {s : Set α} {x : α} [Nonempty ι] [SemilatticeSup ι] (hf : UniformCauchySeqOn F Filter.atTop s) (hx : x ∈ s)
Goal: CauchySeq fun i => F i x
-- tendstoUniformlyOn_of_seq_tendstoUniformlyOn
theorem tendstoUniformlyOn_of_seq_tendstoUniformlyOn {l : Filter ι} [l.IsCountablyGenerated]
    (h : ∀ u : ℕ → ι, Tendsto u atTop l → TendstoUniformlyOn (fun n => F (u n)) f atTop s) :
    TendstoUniformlyOn F f l s 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {s : Set α} {l : Filter ι} [l.IsCountablyGenerated] (h : ∀ (u : ℕ → ι), Filter.Tendsto u Filter.atTop l → TendstoUniformlyOn (fun n => F (u n)) f Filter.atTop s)
Goal: TendstoUniformlyOn F f l s
-- TendstoUniformlyOn.seq_tendstoUniformlyOn
theorem TendstoUniformlyOn.seq_tendstoUniformlyOn {l : Filter ι} (h : TendstoUniformlyOn F f l s)
    (u : ℕ → ι) (hu : Tendsto u atTop l) : TendstoUniformlyOn (fun n => F (u n)) f atTop s 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {s : Set α} {l : Filter ι} (h : TendstoUniformlyOn F f l s) (u : ℕ → ι) (hu : Filter.Tendsto u Filter.atTop l)
Goal: TendstoUniformlyOn (fun n => F (u n)) f Filter.atTop s
-- tendstoUniformlyOn_iff_seq_tendstoUniformlyOn
theorem tendstoUniformlyOn_iff_seq_tendstoUniformlyOn {l : Filter ι} [l.IsCountablyGenerated] :
    TendstoUniformlyOn F f l s ↔
      ∀ u : ℕ → ι, Tendsto u atTop l → TendstoUniformlyOn (fun n => F (u n)) f atTop s 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {s : Set α} {l : Filter ι} [l.IsCountablyGenerated]
Goal: TendstoUniformlyOn F f l s ↔
  ∀ (u : ℕ → ι), Filter.Tendsto u Filter.atTop l → TendstoUniformlyOn (fun n => F (u n)) f Filter.atTop s
-- tendstoUniformly_iff_seq_tendstoUniformly
theorem tendstoUniformly_iff_seq_tendstoUniformly {l : Filter ι} [l.IsCountablyGenerated] :
    TendstoUniformly F f l ↔
      ∀ u : ℕ → ι, Tendsto u atTop l → TendstoUniformly (fun n => F (u n)) f atTop 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {l : Filter ι} [l.IsCountablyGenerated]
Goal: TendstoUniformly F f l ↔
  ∀ (u : ℕ → ι), Filter.Tendsto u Filter.atTop l → TendstoUniformly (fun n => F (u n)) f Filter.atTop