ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Set.EquicontinuousOn.mono
protected theorem Set.EquicontinuousOn.mono {H H' : Set <| X → α} {S : Set X}
    (h : H.EquicontinuousOn S) (hH : H' ⊆ H) : H'.EquicontinuousOn S 
Arguments: {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {H H' : Set (X → α)} {S : Set X} (h : H.EquicontinuousOn S) (hH : H' ⊆ H)
Goal: H'.EquicontinuousOn S
-- UniformEquicontinuous.comp
theorem UniformEquicontinuous.comp {F : ι → β → α} (h : UniformEquicontinuous F) (u : κ → ι) :
    UniformEquicontinuous (F ∘ u) 
Arguments: {ι : Type u_1} {κ : Type u_2} {α : Type u_7} {β : Type u_9} [uα : UniformSpace α] [uβ : UniformSpace β] {F : ι → β → α} (h : UniformEquicontinuous F) (u : κ → ι)
Goal: UniformEquicontinuous (F ∘ u)
-- UniformEquicontinuousOn.comp
theorem UniformEquicontinuousOn.comp {F : ι → β → α} {S : Set β} (h : UniformEquicontinuousOn F S)
    (u : κ → ι) : UniformEquicontinuousOn (F ∘ u) S 
Arguments: {ι : Type u_1} {κ : Type u_2} {α : Type u_7} {β : Type u_9} [uα : UniformSpace α] [uβ : UniformSpace β] {F : ι → β → α} {S : Set β} (h : UniformEquicontinuousOn F S) (u : κ → ι)
Goal: UniformEquicontinuousOn (F ∘ u) S
-- Set.UniformEquicontinuous.mono
protected theorem Set.UniformEquicontinuous.mono {H H' : Set <| β → α} (h : H.UniformEquicontinuous)
    (hH : H' ⊆ H) : H'.UniformEquicontinuous 
Arguments: {α : Type u_7} {β : Type u_9} [uα : UniformSpace α] [uβ : UniformSpace β] {H H' : Set (β → α)} (h : H.UniformEquicontinuous) (hH : H' ⊆ H)
Goal: H'.UniformEquicontinuous
-- Set.UniformEquicontinuousOn.mono
protected theorem Set.UniformEquicontinuousOn.mono {H H' : Set <| β → α} {S : Set β}
    (h : H.UniformEquicontinuousOn S) (hH : H' ⊆ H) : H'.UniformEquicontinuousOn S 
Arguments: {α : Type u_7} {β : Type u_9} [uα : UniformSpace α] [uβ : UniformSpace β] {H H' : Set (β → α)} {S : Set β} (h : H.UniformEquicontinuousOn S) (hH : H' ⊆ H)
Goal: H'.UniformEquicontinuousOn S
-- equicontinuousAt_iff_range
theorem equicontinuousAt_iff_range {F : ι → X → α} {x₀ : X} :
    EquicontinuousAt F x₀ ↔ EquicontinuousAt ((↑) : range F → X → α) x₀ 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {F : ι → X → α} {x₀ : X}
Goal: EquicontinuousAt F x₀ ↔ EquicontinuousAt Subtype.val x₀
-- equicontinuousWithinAt_iff_range
theorem equicontinuousWithinAt_iff_range {F : ι → X → α} {S : Set X} {x₀ : X} :
    EquicontinuousWithinAt F S x₀ ↔ EquicontinuousWithinAt ((↑) : range F → X → α) S x₀ 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {F : ι → X → α} {S : Set X} {x₀ : X}
Goal: EquicontinuousWithinAt F S x₀ ↔ EquicontinuousWithinAt Subtype.val S x₀
-- equicontinuous_iff_range
theorem equicontinuous_iff_range {F : ι → X → α} :
    Equicontinuous F ↔ Equicontinuous ((↑) : range F → X → α) 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {F : ι → X → α}
Goal: Equicontinuous F ↔ Equicontinuous Subtype.val
-- equicontinuousOn_iff_range
theorem equicontinuousOn_iff_range {F : ι → X → α} {S : Set X} :
    EquicontinuousOn F S ↔ EquicontinuousOn ((↑) : range F → X → α) S 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {F : ι → X → α} {S : Set X}
Goal: EquicontinuousOn F S ↔ EquicontinuousOn Subtype.val S
-- uniformEquicontinuous_iff_range
theorem uniformEquicontinuous_iff_range {F : ι → β → α} :
    UniformEquicontinuous F ↔ UniformEquicontinuous ((↑) : range F → β → α) 
Arguments: {ι : Type u_1} {α : Type u_7} {β : Type u_9} [uα : UniformSpace α] [uβ : UniformSpace β] {F : ι → β → α}
Goal: UniformEquicontinuous F ↔ UniformEquicontinuous Subtype.val
-- uniformEquicontinuousOn_iff_range
theorem uniformEquicontinuousOn_iff_range {F : ι → β → α} {S : Set β} :
    UniformEquicontinuousOn F S ↔ UniformEquicontinuousOn ((↑) : range F → β → α) S 
Arguments: {ι : Type u_1} {α : Type u_7} {β : Type u_9} [uα : UniformSpace α] [uβ : UniformSpace β] {F : ι → β → α} {S : Set β}
Goal: UniformEquicontinuousOn F S ↔ UniformEquicontinuousOn Subtype.val S
-- equicontinuousAt_iff_continuousAt
theorem equicontinuousAt_iff_continuousAt {F : ι → X → α} {x₀ : X} :
    EquicontinuousAt F x₀ ↔ ContinuousAt (ofFun ∘ Function.swap F : X → ι →ᵤ α) x₀ 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {F : ι → X → α} {x₀ : X}
Goal: EquicontinuousAt F x₀ ↔ ContinuousAt (⇑UniformFun.ofFun ∘ Function.swap F) x₀
-- equicontinuousWithinAt_iff_continuousWithinAt
theorem equicontinuousWithinAt_iff_continuousWithinAt {F : ι → X → α} {S : Set X} {x₀ : X} :
    EquicontinuousWithinAt F S x₀ ↔
    ContinuousWithinAt (ofFun ∘ Function.swap F : X → ι →ᵤ α) S x₀ 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {F : ι → X → α} {S : Set X} {x₀ : X}
Goal: EquicontinuousWithinAt F S x₀ ↔ ContinuousWithinAt (⇑UniformFun.ofFun ∘ Function.swap F) S x₀
-- equicontinuous_iff_continuous
theorem equicontinuous_iff_continuous {F : ι → X → α} :
    Equicontinuous F ↔ Continuous (ofFun ∘ Function.swap F : X → ι →ᵤ α) 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {F : ι → X → α}
Goal: Equicontinuous F ↔ Continuous (⇑UniformFun.ofFun ∘ Function.swap F)
-- equicontinuousOn_iff_continuousOn
theorem equicontinuousOn_iff_continuousOn {F : ι → X → α} {S : Set X} :
    EquicontinuousOn F S ↔ ContinuousOn (ofFun ∘ Function.swap F : X → ι →ᵤ α) S 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {F : ι → X → α} {S : Set X}
Goal: EquicontinuousOn F S ↔ ContinuousOn (⇑UniformFun.ofFun ∘ Function.swap F) S
-- uniformEquicontinuous_iff_uniformContinuous
theorem uniformEquicontinuous_iff_uniformContinuous {F : ι → β → α} :
    UniformEquicontinuous F ↔ UniformContinuous (ofFun ∘ Function.swap F : β → ι →ᵤ α) 
Arguments: {ι : Type u_1} {α : Type u_7} {β : Type u_9} [uα : UniformSpace α] [uβ : UniformSpace β] {F : ι → β → α}
Goal: UniformEquicontinuous F ↔ UniformContinuous (⇑UniformFun.ofFun ∘ Function.swap F)
-- uniformEquicontinuousOn_iff_uniformContinuousOn
theorem uniformEquicontinuousOn_iff_uniformContinuousOn {F : ι → β → α} {S : Set β} :
    UniformEquicontinuousOn F S ↔ UniformContinuousOn (ofFun ∘ Function.swap F : β → ι →ᵤ α) S 
Arguments: {ι : Type u_1} {α : Type u_7} {β : Type u_9} [uα : UniformSpace α] [uβ : UniformSpace β] {F : ι → β → α} {S : Set β}
Goal: UniformEquicontinuousOn F S ↔ UniformContinuousOn (⇑UniformFun.ofFun ∘ Function.swap F) S
-- equicontinuousWithinAt_iInf_rng
theorem equicontinuousWithinAt_iInf_rng {u : κ → UniformSpace α'} {F : ι → X → α'}
    {S : Set X} {x₀ : X} : EquicontinuousWithinAt (uα :=  ⨅ k, u k) F S x₀ ↔
      ∀ k, EquicontinuousWithinAt (uα :=  u k) F S x₀ 
Arguments: {ι : Type u_1} {κ : Type u_2} {X : Type u_3} {α' : Type u_8} [tX : TopologicalSpace X] {u : κ → UniformSpace α'} {F : ι → X → α'} {S : Set X} {x₀ : X}
Goal: EquicontinuousWithinAt F S x₀ ↔ ∀ (k : κ), EquicontinuousWithinAt F S x₀
-- equicontinuousAt_iInf_rng
theorem equicontinuousAt_iInf_rng {u : κ → UniformSpace α'} {F : ι → X → α'}
    {x₀ : X} :
    EquicontinuousAt (uα := ⨅ k, u k) F x₀ ↔ ∀ k, EquicontinuousAt (uα := u k) F x₀ 
Arguments: {ι : Type u_1} {κ : Type u_2} {X : Type u_3} {α' : Type u_8} [tX : TopologicalSpace X] {u : κ → UniformSpace α'} {F : ι → X → α'} {x₀ : X}
Goal: EquicontinuousAt F x₀ ↔ ∀ (k : κ), EquicontinuousAt F x₀
-- equicontinuous_iInf_rng
theorem equicontinuous_iInf_rng {u : κ → UniformSpace α'} {F : ι → X → α'} :
    Equicontinuous (uα := ⨅ k, u k) F ↔ ∀ k, Equicontinuous (uα := u k) F 
Arguments: {ι : Type u_1} {κ : Type u_2} {X : Type u_3} {α' : Type u_8} [tX : TopologicalSpace X] {u : κ → UniformSpace α'} {F : ι → X → α'}
Goal: Equicontinuous F ↔ ∀ (k : κ), Equicontinuous F
-- equicontinuousOn_iInf_rng
theorem equicontinuousOn_iInf_rng {u : κ → UniformSpace α'} {F : ι → X → α'}
    {S : Set X} :
    EquicontinuousOn (uα := ⨅ k, u k) F S ↔ ∀ k, EquicontinuousOn (uα := u k) F S 
Arguments: {ι : Type u_1} {κ : Type u_2} {X : Type u_3} {α' : Type u_8} [tX : TopologicalSpace X] {u : κ → UniformSpace α'} {F : ι → X → α'} {S : Set X}
Goal: EquicontinuousOn F S ↔ ∀ (k : κ), EquicontinuousOn F S
-- uniformEquicontinuous_iInf_rng
theorem uniformEquicontinuous_iInf_rng {u : κ → UniformSpace α'} {F : ι → β → α'} :
    UniformEquicontinuous (uα := ⨅ k, u k) F ↔ ∀ k, UniformEquicontinuous (uα := u k) F 
Arguments: {ι : Type u_1} {κ : Type u_2} {α' : Type u_8} {β : Type u_9} [uβ : UniformSpace β] {u : κ → UniformSpace α'} {F : ι → β → α'}
Goal: UniformEquicontinuous F ↔ ∀ (k : κ), UniformEquicontinuous F
-- uniformEquicontinuousOn_iInf_rng
theorem uniformEquicontinuousOn_iInf_rng {u : κ → UniformSpace α'} {F : ι → β → α'}
    {S : Set β} : UniformEquicontinuousOn (uα := ⨅ k, u k) F S ↔
      ∀ k, UniformEquicontinuousOn (uα := u k) F S 
Arguments: {ι : Type u_1} {κ : Type u_2} {α' : Type u_8} {β : Type u_9} [uβ : UniformSpace β] {u : κ → UniformSpace α'} {F : ι → β → α'} {S : Set β}
Goal: UniformEquicontinuousOn F S ↔ ∀ (k : κ), UniformEquicontinuousOn F S
-- equicontinuousWithinAt_iInf_dom
theorem equicontinuousWithinAt_iInf_dom {t : κ → TopologicalSpace X'} {F : ι → X' → α}
    {S : Set X'} {x₀ : X'} {k : κ} (hk : EquicontinuousWithinAt (tX := t k) F S x₀) :
    EquicontinuousWithinAt (tX := ⨅ k, t k) F S x₀ 
Arguments: {ι : Type u_1} {κ : Type u_2} {X' : Type u_4} {α : Type u_7} [uα : UniformSpace α] {t : κ → TopologicalSpace X'} {F : ι → X' → α} {S : Set X'} {x₀ : X'} {k : κ} (hk : EquicontinuousWithinAt F S x₀)
Goal: EquicontinuousWithinAt F S x₀
-- equicontinuousAt_iInf_dom
theorem equicontinuousAt_iInf_dom {t : κ → TopologicalSpace X'} {F : ι → X' → α}
    {x₀ : X'} {k : κ} (hk : EquicontinuousAt (tX := t k) F x₀) :
    EquicontinuousAt (tX := ⨅ k, t k) F x₀ 
Arguments: {ι : Type u_1} {κ : Type u_2} {X' : Type u_4} {α : Type u_7} [uα : UniformSpace α] {t : κ → TopologicalSpace X'} {F : ι → X' → α} {x₀ : X'} {k : κ} (hk : EquicontinuousAt F x₀)
Goal: EquicontinuousAt F x₀
-- equicontinuous_iInf_dom
theorem equicontinuous_iInf_dom {t : κ → TopologicalSpace X'} {F : ι → X' → α}
    {k : κ} (hk : Equicontinuous (tX := t k) F) :
    Equicontinuous (tX := ⨅ k, t k) F 
Arguments: {ι : Type u_1} {κ : Type u_2} {X' : Type u_4} {α : Type u_7} [uα : UniformSpace α] {t : κ → TopologicalSpace X'} {F : ι → X' → α} {k : κ} (hk : Equicontinuous F)
Goal: Equicontinuous F
-- equicontinuousOn_iInf_dom
theorem equicontinuousOn_iInf_dom {t : κ → TopologicalSpace X'} {F : ι → X' → α}
    {S : Set X'} {k : κ} (hk : EquicontinuousOn (tX := t k) F S) :
    EquicontinuousOn (tX := ⨅ k, t k) F S 
Arguments: {ι : Type u_1} {κ : Type u_2} {X' : Type u_4} {α : Type u_7} [uα : UniformSpace α] {t : κ → TopologicalSpace X'} {F : ι → X' → α} {S : Set X'} {k : κ} (hk : EquicontinuousOn F S)
Goal: EquicontinuousOn F S
-- uniformEquicontinuous_iInf_dom
theorem uniformEquicontinuous_iInf_dom {u : κ → UniformSpace β'} {F : ι → β' → α}
    {k : κ} (hk : UniformEquicontinuous (uβ := u k) F) :
    UniformEquicontinuous (uβ := ⨅ k, u k) F 
Arguments: {ι : Type u_1} {κ : Type u_2} {α : Type u_7} {β' : Type u_10} [uα : UniformSpace α] {u : κ → UniformSpace β'} {F : ι → β' → α} {k : κ} (hk : UniformEquicontinuous F)
Goal: UniformEquicontinuous F
-- uniformEquicontinuousOn_iInf_dom
theorem uniformEquicontinuousOn_iInf_dom {u : κ → UniformSpace β'} {F : ι → β' → α}
    {S : Set β'} {k : κ} (hk : UniformEquicontinuousOn (uβ := u k) F S) :
    UniformEquicontinuousOn (uβ := ⨅ k, u k) F S 
Arguments: {ι : Type u_1} {κ : Type u_2} {α : Type u_7} {β' : Type u_10} [uα : UniformSpace α] {u : κ → UniformSpace β'} {F : ι → β' → α} {S : Set β'} {k : κ} (hk : UniformEquicontinuousOn F S)
Goal: UniformEquicontinuousOn F S
-- Filter.HasBasis.equicontinuousAt_iff_left
theorem Filter.HasBasis.equicontinuousAt_iff_left {p : κ → Prop} {s : κ → Set X}
    {F : ι → X → α} {x₀ : X} (hX : (𝓝 x₀).HasBasis p s) :
    EquicontinuousAt F x₀ ↔ ∀ U ∈ 𝓤 α, ∃ k, p k ∧ ∀ x ∈ s k, ∀ i, (F i x₀, F i x) ∈ U 
Arguments: {ι : Type u_1} {κ : Type u_2} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {p : κ → Prop} {s : κ → Set X} {F : ι → X → α} {x₀ : X} (hX : (nhds x₀).HasBasis p s)
Goal: EquicontinuousAt F x₀ ↔ ∀ U ∈ uniformity α, ∃ k, p k ∧ ∀ x ∈ s k, ∀ (i : ι), (F i x₀, F i x) ∈ U
-- Filter.HasBasis.equicontinuousWithinAt_iff_left
theorem Filter.HasBasis.equicontinuousWithinAt_iff_left {p : κ → Prop} {s : κ → Set X}
    {F : ι → X → α} {S : Set X} {x₀ : X} (hX : (𝓝[S] x₀).HasBasis p s) :
    EquicontinuousWithinAt F S x₀ ↔ ∀ U ∈ 𝓤 α, ∃ k, p k ∧ ∀ x ∈ s k, ∀ i, (F i x₀, F i x) ∈ U 
Arguments: {ι : Type u_1} {κ : Type u_2} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {p : κ → Prop} {s : κ → Set X} {F : ι → X → α} {S : Set X} {x₀ : X} (hX : (nhdsWithin x₀ S).HasBasis p s)
Goal: EquicontinuousWithinAt F S x₀ ↔ ∀ U ∈ uniformity α, ∃ k, p k ∧ ∀ x ∈ s k, ∀ (i : ι), (F i x₀, F i x) ∈ U
-- Filter.HasBasis.equicontinuousAt_iff_right
theorem Filter.HasBasis.equicontinuousAt_iff_right {p : κ → Prop} {s : κ → Set (α × α)}
    {F : ι → X → α} {x₀ : X} (hα : (𝓤 α).HasBasis p s) :
    EquicontinuousAt F x₀ ↔ ∀ k, p k → ∀ᶠ x in 𝓝 x₀, ∀ i, (F i x₀, F i x) ∈ s k 
Arguments: {ι : Type u_1} {κ : Type u_2} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {p : κ → Prop} {s : κ → Set (α × α)} {F : ι → X → α} {x₀ : X} (hα : (uniformity α).HasBasis p s)
Goal: EquicontinuousAt F x₀ ↔ ∀ (k : κ), p k → ∀ᶠ (x : X) in nhds x₀, ∀ (i : ι), (F i x₀, F i x) ∈ s k
-- Filter.HasBasis.equicontinuousWithinAt_iff_right
theorem Filter.HasBasis.equicontinuousWithinAt_iff_right {p : κ → Prop}
    {s : κ → Set (α × α)} {F : ι → X → α} {S : Set X} {x₀ : X} (hα : (𝓤 α).HasBasis p s) :
    EquicontinuousWithinAt F S x₀ ↔ ∀ k, p k → ∀ᶠ x in 𝓝[S] x₀, ∀ i, (F i x₀, F i x) ∈ s k 
Arguments: {ι : Type u_1} {κ : Type u_2} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {p : κ → Prop} {s : κ → Set (α × α)} {F : ι → X → α} {S : Set X} {x₀ : X} (hα : (uniformity α).HasBasis p s)
Goal: EquicontinuousWithinAt F S x₀ ↔ ∀ (k : κ), p k → ∀ᶠ (x : X) in nhdsWithin x₀ S, ∀ (i : ι), (F i x₀, F i x) ∈ s k
-- Filter.HasBasis.equicontinuousAt_iff
theorem Filter.HasBasis.equicontinuousAt_iff {κ₁ κ₂ : Type*} {p₁ : κ₁ → Prop} {s₁ : κ₁ → Set X}
    {p₂ : κ₂ → Prop} {s₂ : κ₂ → Set (α × α)} {F : ι → X → α} {x₀ : X} (hX : (𝓝 x₀).HasBasis p₁ s₁)
    (hα : (𝓤 α).HasBasis p₂ s₂) :
    EquicontinuousAt F x₀ ↔
      ∀ k₂, p₂ k₂ → ∃ k₁, p₁ k₁ ∧ ∀ x ∈ s₁ k₁, ∀ i, (F i x₀, F i x) ∈ s₂ k₂ 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {κ₁ : Type u_13} {κ₂ : Type u_14} {p₁ : κ₁ → Prop} {s₁ : κ₁ → Set X} {p₂ : κ₂ → Prop} {s₂ : κ₂ → Set (α × α)} {F : ι → X → α} {x₀ : X} (hX : (nhds x₀).HasBasis p₁ s₁) (hα : (uniformity α).HasBasis p₂ s₂)
Goal: EquicontinuousAt F x₀ ↔ ∀ (k₂ : κ₂), p₂ k₂ → ∃ k₁, p₁ k₁ ∧ ∀ x ∈ s₁ k₁, ∀ (i : ι), (F i x₀, F i x) ∈ s₂ k₂
-- Filter.HasBasis.equicontinuousWithinAt_iff
theorem Filter.HasBasis.equicontinuousWithinAt_iff {κ₁ κ₂ : Type*} {p₁ : κ₁ → Prop}
    {s₁ : κ₁ → Set X} {p₂ : κ₂ → Prop} {s₂ : κ₂ → Set (α × α)} {F : ι → X → α} {S : Set X} {x₀ : X}
    (hX : (𝓝[S] x₀).HasBasis p₁ s₁) (hα : (𝓤 α).HasBasis p₂ s₂) :
    EquicontinuousWithinAt F S x₀ ↔
      ∀ k₂, p₂ k₂ → ∃ k₁, p₁ k₁ ∧ ∀ x ∈ s₁ k₁, ∀ i, (F i x₀, F i x) ∈ s₂ k₂ 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {κ₁ : Type u_13} {κ₂ : Type u_14} {p₁ : κ₁ → Prop} {s₁ : κ₁ → Set X} {p₂ : κ₂ → Prop} {s₂ : κ₂ → Set (α × α)} {F : ι → X → α} {S : Set X} {x₀ : X} (hX : (nhdsWithin x₀ S).HasBasis p₁ s₁) (hα : (uniformity α).HasBasis p₂ s₂)
Goal: EquicontinuousWithinAt F S x₀ ↔ ∀ (k₂ : κ₂), p₂ k₂ → ∃ k₁, p₁ k₁ ∧ ∀ x ∈ s₁ k₁, ∀ (i : ι), (F i x₀, F i x) ∈ s₂ k₂
-- Filter.HasBasis.uniformEquicontinuous_iff_left
theorem Filter.HasBasis.uniformEquicontinuous_iff_left {p : κ → Prop}
    {s : κ → Set (β × β)} {F : ι → β → α} (hβ : (𝓤 β).HasBasis p s) :
    UniformEquicontinuous F ↔
      ∀ U ∈ 𝓤 α, ∃ k, p k ∧ ∀ x y, (x, y) ∈ s k → ∀ i, (F i x, F i y) ∈ U 
Arguments: {ι : Type u_1} {κ : Type u_2} {α : Type u_7} {β : Type u_9} [uα : UniformSpace α] [uβ : UniformSpace β] {p : κ → Prop} {s : κ → Set (β × β)} {F : ι → β → α} (hβ : (uniformity β).HasBasis p s)
Goal: UniformEquicontinuous F ↔ ∀ U ∈ uniformity α, ∃ k, p k ∧ ∀ (x y : β), (x, y) ∈ s k → ∀ (i : ι), (F i x, F i y) ∈ U
-- Filter.HasBasis.uniformEquicontinuousOn_iff_left
theorem Filter.HasBasis.uniformEquicontinuousOn_iff_left {p : κ → Prop}
    {s : κ → Set (β × β)} {F : ι → β → α} {S : Set β} (hβ : (𝓤 β ⊓ 𝓟 (S ×ˢ S)).HasBasis p s) :
    UniformEquicontinuousOn F S ↔
      ∀ U ∈ 𝓤 α, ∃ k, p k ∧ ∀ x y, (x, y) ∈ s k → ∀ i, (F i x, F i y) ∈ U 
Arguments: {ι : Type u_1} {κ : Type u_2} {α : Type u_7} {β : Type u_9} [uα : UniformSpace α] [uβ : UniformSpace β] {p : κ → Prop} {s : κ → Set (β × β)} {F : ι → β → α} {S : Set β} (hβ : (uniformity β ⊓ Filter.principal (S ×ˢ S)).HasBasis p s)
Goal: UniformEquicontinuousOn F S ↔ ∀ U ∈ uniformity α, ∃ k, p k ∧ ∀ (x y : β), (x, y) ∈ s k → ∀ (i : ι), (F i x, F i y) ∈ U
-- Filter.HasBasis.uniformEquicontinuous_iff_right
theorem Filter.HasBasis.uniformEquicontinuous_iff_right {p : κ → Prop}
    {s : κ → Set (α × α)} {F : ι → β → α} (hα : (𝓤 α).HasBasis p s) :
    UniformEquicontinuous F ↔ ∀ k, p k → ∀ᶠ xy : β × β in 𝓤 β, ∀ i, (F i xy.1, F i xy.2) ∈ s k 
Arguments: {ι : Type u_1} {κ : Type u_2} {α : Type u_7} {β : Type u_9} [uα : UniformSpace α] [uβ : UniformSpace β] {p : κ → Prop} {s : κ → Set (α × α)} {F : ι → β → α} (hα : (uniformity α).HasBasis p s)
Goal: UniformEquicontinuous F ↔ ∀ (k : κ), p k → ∀ᶠ (xy : β × β) in uniformity β, ∀ (i : ι), (F i xy.1, F i xy.2) ∈ s k
-- Filter.HasBasis.uniformEquicontinuousOn_iff_right
theorem Filter.HasBasis.uniformEquicontinuousOn_iff_right {p : κ → Prop}
    {s : κ → Set (α × α)} {F : ι → β → α} {S : Set β} (hα : (𝓤 α).HasBasis p s) :
    UniformEquicontinuousOn F S ↔
      ∀ k, p k → ∀ᶠ xy : β × β in 𝓤 β ⊓ 𝓟 (S ×ˢ S), ∀ i, (F i xy.1, F i xy.2) ∈ s k 
Arguments: {ι : Type u_1} {κ : Type u_2} {α : Type u_7} {β : Type u_9} [uα : UniformSpace α] [uβ : UniformSpace β] {p : κ → Prop} {s : κ → Set (α × α)} {F : ι → β → α} {S : Set β} (hα : (uniformity α).HasBasis p s)
Goal: UniformEquicontinuousOn F S ↔
  ∀ (k : κ), p k → ∀ᶠ (xy : β × β) in uniformity β ⊓ Filter.principal (S ×ˢ S), ∀ (i : ι), (F i xy.1, F i xy.2) ∈ s k
-- Filter.HasBasis.uniformEquicontinuous_iff
theorem Filter.HasBasis.uniformEquicontinuous_iff {κ₁ κ₂ : Type*} {p₁ : κ₁ → Prop}
    {s₁ : κ₁ → Set (β × β)} {p₂ : κ₂ → Prop} {s₂ : κ₂ → Set (α × α)} {F : ι → β → α}
    (hβ : (𝓤 β).HasBasis p₁ s₁) (hα : (𝓤 α).HasBasis p₂ s₂) :
    UniformEquicontinuous F ↔
      ∀ k₂, p₂ k₂ → ∃ k₁, p₁ k₁ ∧ ∀ x y, (x, y) ∈ s₁ k₁ → ∀ i, (F i x, F i y) ∈ s₂ k₂ 
Arguments: {ι : Type u_1} {α : Type u_7} {β : Type u_9} [uα : UniformSpace α] [uβ : UniformSpace β] {κ₁ : Type u_13} {κ₂ : Type u_14} {p₁ : κ₁ → Prop} {s₁ : κ₁ → Set (β × β)} {p₂ : κ₂ → Prop} {s₂ : κ₂ → Set (α × α)} {F : ι → β → α} (hβ : (uniformity β).HasBasis p₁ s₁) (hα : (uniformity α).HasBasis p₂ s₂)
Goal: UniformEquicontinuous F ↔
  ∀ (k₂ : κ₂), p₂ k₂ → ∃ k₁, p₁ k₁ ∧ ∀ (x y : β), (x, y) ∈ s₁ k₁ → ∀ (i : ι), (F i x, F i y) ∈ s₂ k₂
-- Filter.HasBasis.uniformEquicontinuousOn_iff
theorem Filter.HasBasis.uniformEquicontinuousOn_iff {κ₁ κ₂ : Type*} {p₁ : κ₁ → Prop}
    {s₁ : κ₁ → Set (β × β)} {p₂ : κ₂ → Prop} {s₂ : κ₂ → Set (α × α)} {F : ι → β → α}
    {S : Set β} (hβ : (𝓤 β ⊓ 𝓟 (S ×ˢ S)).HasBasis p₁ s₁) (hα : (𝓤 α).HasBasis p₂ s₂) :
    UniformEquicontinuousOn F S ↔
      ∀ k₂, p₂ k₂ → ∃ k₁, p₁ k₁ ∧ ∀ x y, (x, y) ∈ s₁ k₁ → ∀ i, (F i x, F i y) ∈ s₂ k₂ 
Arguments: {ι : Type u_1} {α : Type u_7} {β : Type u_9} [uα : UniformSpace α] [uβ : UniformSpace β] {κ₁ : Type u_13} {κ₂ : Type u_14} {p₁ : κ₁ → Prop} {s₁ : κ₁ → Set (β × β)} {p₂ : κ₂ → Prop} {s₂ : κ₂ → Set (α × α)} {F : ι → β → α} {S : Set β} (hβ : (uniformity β ⊓ Filter.principal (S ×ˢ S)).HasBasis p₁ s₁) (hα : (uniformity α).HasBasis p₂ s₂)
Goal: UniformEquicontinuousOn F S ↔
  ∀ (k₂ : κ₂), p₂ k₂ → ∃ k₁, p₁ k₁ ∧ ∀ (x y : β), (x, y) ∈ s₁ k₁ → ∀ (i : ι), (F i x, F i y) ∈ s₂ k₂
-- UniformInducing.equicontinuousAt_iff
theorem UniformInducing.equicontinuousAt_iff {F : ι → X → α} {x₀ : X} {u : α → β}
    (hu : UniformInducing u) : EquicontinuousAt F x₀ ↔ EquicontinuousAt ((u ∘ ·) ∘ F) x₀ 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} {β : Type u_9} [tX : TopologicalSpace X] [uα : UniformSpace α] [uβ : UniformSpace β] {F : ι → X → α} {x₀ : X} {u : α → β} (hu : UniformInducing u)
Goal: EquicontinuousAt F x₀ ↔ EquicontinuousAt ((fun x => u ∘ x) ∘ F) x₀
-- UniformInducing.equicontinuousWithinAt_iff
theorem UniformInducing.equicontinuousWithinAt_iff {F : ι → X → α} {S : Set X} {x₀ : X} {u : α → β}
    (hu : UniformInducing u) : EquicontinuousWithinAt F S x₀ ↔
      EquicontinuousWithinAt ((u ∘ ·) ∘ F) S x₀ 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} {β : Type u_9} [tX : TopologicalSpace X] [uα : UniformSpace α] [uβ : UniformSpace β] {F : ι → X → α} {S : Set X} {x₀ : X} {u : α → β} (hu : UniformInducing u)
Goal: EquicontinuousWithinAt F S x₀ ↔ EquicontinuousWithinAt ((fun x => u ∘ x) ∘ F) S x₀
-- UniformInducing.equicontinuous_iff
theorem UniformInducing.equicontinuous_iff {F : ι → X → α} {u : α → β} (hu : UniformInducing u) :
    Equicontinuous F ↔ Equicontinuous ((u ∘ ·) ∘ F) 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} {β : Type u_9} [tX : TopologicalSpace X] [uα : UniformSpace α] [uβ : UniformSpace β] {F : ι → X → α} {u : α → β} (hu : UniformInducing u)
Goal: Equicontinuous F ↔ Equicontinuous ((fun x => u ∘ x) ∘ F)
-- UniformInducing.equicontinuousOn_iff
theorem UniformInducing.equicontinuousOn_iff {F : ι → X → α} {S : Set X} {u : α → β}
    (hu : UniformInducing u) : EquicontinuousOn F S ↔ EquicontinuousOn ((u ∘ ·) ∘ F) S 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} {β : Type u_9} [tX : TopologicalSpace X] [uα : UniformSpace α] [uβ : UniformSpace β] {F : ι → X → α} {S : Set X} {u : α → β} (hu : UniformInducing u)
Goal: EquicontinuousOn F S ↔ EquicontinuousOn ((fun x => u ∘ x) ∘ F) S
-- UniformInducing.uniformEquicontinuous_iff
theorem UniformInducing.uniformEquicontinuous_iff {F : ι → β → α} {u : α → γ}
    (hu : UniformInducing u) : UniformEquicontinuous F ↔ UniformEquicontinuous ((u ∘ ·) ∘ F) 
Arguments: {ι : Type u_1} {α : Type u_7} {β : Type u_9} {γ : Type u_11} [uα : UniformSpace α] [uβ : UniformSpace β] [uγ : UniformSpace γ] {F : ι → β → α} {u : α → γ} (hu : UniformInducing u)
Goal: UniformEquicontinuous F ↔ UniformEquicontinuous ((fun x => u ∘ x) ∘ F)
-- UniformInducing.uniformEquicontinuousOn_iff
theorem UniformInducing.uniformEquicontinuousOn_iff {F : ι → β → α} {S : Set β} {u : α → γ}
    (hu : UniformInducing u) :
    UniformEquicontinuousOn F S ↔ UniformEquicontinuousOn ((u ∘ ·) ∘ F) S 
Arguments: {ι : Type u_1} {α : Type u_7} {β : Type u_9} {γ : Type u_11} [uα : UniformSpace α] [uβ : UniformSpace β] [uγ : UniformSpace γ] {F : ι → β → α} {S : Set β} {u : α → γ} (hu : UniformInducing u)
Goal: UniformEquicontinuousOn F S ↔ UniformEquicontinuousOn ((fun x => u ∘ x) ∘ F) S
-- EquicontinuousWithinAt.closure'
theorem EquicontinuousWithinAt.closure' {A : Set Y} {u : Y → X → α} {S : Set X} {x₀ : X}
    (hA : EquicontinuousWithinAt (u ∘ (↑) : A → X → α) S x₀) (hu₁ : Continuous (S.restrict ∘ u))
    (hu₂ : Continuous (eval x₀ ∘ u)) :
    EquicontinuousWithinAt (u ∘ (↑) : closure A → X → α) S x₀ 
Arguments: {X : Type u_3} {Y : Type u_5} {α : Type u_7} [tX : TopologicalSpace X] [tY : TopologicalSpace Y] [uα : UniformSpace α] {A : Set Y} {u : Y → X → α} {S : Set X} {x₀ : X} (hA : EquicontinuousWithinAt (u ∘ Subtype.val) S x₀) (hu₁ : Continuous (S.restrict ∘ u)) (hu₂ : Continuous (Function.eval x₀ ∘ u))
Goal: EquicontinuousWithinAt (u ∘ Subtype.val) S x₀