ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- uniformContinuous₂_def
theorem uniformContinuous₂_def (f : α → β → γ) :
    UniformContinuous₂ f ↔ UniformContinuous (uncurry f) 
Arguments: {α : Type ua} {β : Type ub} {γ : Type uc} [UniformSpace α] [UniformSpace β] [UniformSpace γ] (f : α → β → γ)
Goal: UniformContinuous₂ f ↔ UniformContinuous (Function.uncurry f)
-- UniformContinuous₂.uniformContinuous
theorem UniformContinuous₂.uniformContinuous {f : α → β → γ} (h : UniformContinuous₂ f) :
    UniformContinuous (uncurry f) 
Arguments: {α : Type ua} {β : Type ub} {γ : Type uc} [UniformSpace α] [UniformSpace β] [UniformSpace γ] {f : α → β → γ} (h : UniformContinuous₂ f)
Goal: UniformContinuous (Function.uncurry f)
-- uniformContinuous₂_curry
theorem uniformContinuous₂_curry (f : α × β → γ) :
    UniformContinuous₂ (Function.curry f) ↔ UniformContinuous f 
Arguments: {α : Type ua} {β : Type ub} {γ : Type uc} [UniformSpace α] [UniformSpace β] [UniformSpace γ] (f : α × β → γ)
Goal: UniformContinuous₂ (Function.curry f) ↔ UniformContinuous f
-- UniformContinuous₂.comp
theorem UniformContinuous₂.comp {f : α → β → γ} {g : γ → δ} (hg : UniformContinuous g)
    (hf : UniformContinuous₂ f) : UniformContinuous₂ (g ∘₂ f) 
Arguments: {α : Type ua} {β : Type ub} {γ : Type uc} {δ : Type ud} [UniformSpace α] [UniformSpace β] [UniformSpace γ] [UniformSpace δ] {f : α → β → γ} {g : γ → δ} (hg : UniformContinuous g) (hf : UniformContinuous₂ f)
Goal: UniformContinuous₂ (Function.bicompr g f)
-- UniformContinuous₂.bicompl
theorem UniformContinuous₂.bicompl {f : α → β → γ} {ga : δ → α} {gb : δ' → β}
    (hf : UniformContinuous₂ f) (hga : UniformContinuous ga) (hgb : UniformContinuous gb) :
    UniformContinuous₂ (bicompl f ga gb) 
Arguments: {α : Type ua} {β : Type ub} {γ : Type uc} {δ : Type ud} {δ' : Type u_2} [UniformSpace α] [UniformSpace β] [UniformSpace γ] [UniformSpace δ] [UniformSpace δ'] {f : α → β → γ} {ga : δ → α} {gb : δ' → β} (hf : UniformContinuous₂ f) (hga : UniformContinuous ga) (hgb : UniformContinuous gb)
Goal: UniformContinuous₂ (Function.bicompl f ga gb)
-- toTopologicalSpace_subtype
theorem toTopologicalSpace_subtype [u : UniformSpace α] {p : α → Prop} :
    @UniformSpace.toTopologicalSpace (Subtype p) instUniformSpaceSubtype =
      @instTopologicalSpaceSubtype α p u.toTopologicalSpace 
Arguments: {α : Type ua} [u : UniformSpace α] {p : α → Prop}
Goal: UniformSpace.toTopologicalSpace = instTopologicalSpaceSubtype
-- union_mem_uniformity_sum
theorem union_mem_uniformity_sum {a : Set (α × α)} (ha : a ∈ 𝓤 α) {b : Set (β × β)} (hb : b ∈ 𝓤 β) :
    Prod.map inl inl '' a ∪ Prod.map inr inr '' b ∈ 𝓤 (α ⊕ β) 
Arguments: {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] {a : Set (α × α)} (ha : a ∈ 𝓤 α) {b : Set (β × β)} (hb : b ∈ 𝓤 β)
Goal: Prod.map Sum.inl Sum.inl '' a ∪ Prod.map Sum.inr Sum.inr '' b ∈ 𝓤 (α ⊕ β)
-- Sum.uniformity
theorem Sum.uniformity : 𝓤 (α ⊕ β) = map (Prod.map inl inl) (𝓤 α) ⊔ map (Prod.map inr inr) (𝓤 β) 
Arguments: {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β]
Goal: 𝓤 (α ⊕ β) = Filter.map (Prod.map Sum.inl Sum.inl) (𝓤 α) ⊔ Filter.map (Prod.map Sum.inr Sum.inr) (𝓤 β)
-- uniformContinuous_inl
lemma uniformContinuous_inl : UniformContinuous (Sum.inl : α → α ⊕ β) 
Arguments: {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β]
Goal: UniformContinuous Sum.inl
-- uniformContinuous_inr
lemma uniformContinuous_inr : UniformContinuous (Sum.inr : β → α ⊕ β) 
Arguments: {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β]
Goal: UniformContinuous Sum.inr
-- lebesgue_number_lemma
theorem lebesgue_number_lemma {ι : Sort*} {U : ι → Set α} (hK : IsCompact K)
    (hopen : ∀ i, IsOpen (U i)) (hcover : K ⊆ ⋃ i, U i) :
    ∃ V ∈ 𝓤 α, ∀ x ∈ K, ∃ i, ball x V ⊆ U i 
Arguments: {α : Type ua} [UniformSpace α] {K : Set α} {ι : Sort u_2} {U : ι → Set α} (hK : IsCompact K) (hopen : ∀ (i : ι), IsOpen (U i)) (hcover : K ⊆ ⋃ i, U i)
Goal: ∃ V ∈ 𝓤 α, ∀ x ∈ K, ∃ i, UniformSpace.ball x V ⊆ U i
-- Filter.HasBasis.lebesgue_number_lemma
protected theorem Filter.HasBasis.lebesgue_number_lemma {ι' ι : Sort*} {p : ι' → Prop}
    {V : ι' → Set (α × α)} {U : ι → Set α} (hbasis : (𝓤 α).HasBasis p V) (hK : IsCompact K)
    (hopen : ∀ j, IsOpen (U j)) (hcover : K ⊆ ⋃ j, U j) :
    ∃ i, p i ∧ ∀ x ∈ K, ∃ j, ball x (V i) ⊆ U j 
Arguments: {α : Type ua} [UniformSpace α] {K : Set α} {ι' : Sort u_2} {ι : Sort u_3} {p : ι' → Prop} {V : ι' → Set (α × α)} {U : ι → Set α} (hbasis : (𝓤 α).HasBasis p V) (hK : IsCompact K) (hopen : ∀ (j : ι), IsOpen (U j)) (hcover : K ⊆ ⋃ j, U j)
Goal: ∃ i, p i ∧ ∀ x ∈ K, ∃ j, UniformSpace.ball x (V i) ⊆ U j
-- lebesgue_number_lemma_sUnion
theorem lebesgue_number_lemma_sUnion {S : Set (Set α)}
    (hK : IsCompact K) (hopen : ∀ s ∈ S, IsOpen s) (hcover : K ⊆ ⋃₀ S) :
    ∃ V ∈ 𝓤 α, ∀ x ∈ K, ∃ s ∈ S, ball x V ⊆ s 
Arguments: {α : Type ua} [UniformSpace α] {K : Set α} {S : Set (Set α)} (hK : IsCompact K) (hopen : ∀ s ∈ S, IsOpen s) (hcover : K ⊆ ⋃₀ S)
Goal: ∃ V ∈ 𝓤 α, ∀ x ∈ K, ∃ s ∈ S, UniformSpace.ball x V ⊆ s
-- IsCompact.nhdsSet_basis_uniformity
theorem IsCompact.nhdsSet_basis_uniformity {p : ι → Prop} {V : ι → Set (α × α)}
    (hbasis : (𝓤 α).HasBasis p V) (hK : IsCompact K) :
    (𝓝ˢ K).HasBasis p fun i => ⋃ x ∈ K, ball x (V i) 
Arguments: {α : Type ua} {ι : Sort u_1} [UniformSpace α] {K : Set α} {p : ι → Prop} {V : ι → Set (α × α)} (hbasis : (𝓤 α).HasBasis p V) (hK : IsCompact K)
Goal: (𝓝ˢ K).HasBasis p fun i => ⋃ x ∈ K, UniformSpace.ball x (V i)
-- Disjoint.exists_uniform_thickening
theorem Disjoint.exists_uniform_thickening {A B : Set α} (hA : IsCompact A) (hB : IsClosed B)
    (h : Disjoint A B) : ∃ V ∈ 𝓤 α, Disjoint (⋃ x ∈ A, ball x V) (⋃ x ∈ B, ball x V) 
Arguments: {α : Type ua} [UniformSpace α] {A B : Set α} (hA : IsCompact A) (hB : IsClosed B) (h : Disjoint A B)
Goal: ∃ V ∈ 𝓤 α, Disjoint (⋃ x ∈ A, UniformSpace.ball x V) (⋃ x ∈ B, UniformSpace.ball x V)
-- Disjoint.exists_uniform_thickening_of_basis
theorem Disjoint.exists_uniform_thickening_of_basis {p : ι → Prop} {s : ι → Set (α × α)}
    (hU : (𝓤 α).HasBasis p s) {A B : Set α} (hA : IsCompact A) (hB : IsClosed B)
    (h : Disjoint A B) : ∃ i, p i ∧ Disjoint (⋃ x ∈ A, ball x (s i)) (⋃ x ∈ B, ball x (s i)) 
Arguments: {α : Type ua} {ι : Sort u_1} [UniformSpace α] {p : ι → Prop} {s : ι → Set (α × α)} (hU : (𝓤 α).HasBasis p s) {A B : Set α} (hA : IsCompact A) (hB : IsClosed B) (h : Disjoint A B)
Goal: ∃ i, p i ∧ Disjoint (⋃ x ∈ A, UniformSpace.ball x (s i)) (⋃ x ∈ B, UniformSpace.ball x (s i))
-- lebesgue_number_of_compact_open
theorem lebesgue_number_of_compact_open {K U : Set α} (hK : IsCompact K)
    (hU : IsOpen U) (hKU : K ⊆ U) : ∃ V ∈ 𝓤 α, IsOpen V ∧ ∀ x ∈ K, UniformSpace.ball x V ⊆ U 
Arguments: {α : Type ua} [UniformSpace α] {K U : Set α} (hK : IsCompact K) (hU : IsOpen U) (hKU : K ⊆ U)
Goal: ∃ V ∈ 𝓤 α, IsOpen V ∧ ∀ x ∈ K, UniformSpace.ball x V ⊆ U
-- Uniform.tendsto_nhds_right
theorem tendsto_nhds_right {f : Filter β} {u : β → α} {a : α} :
    Tendsto u f (𝓝 a) ↔ Tendsto (fun x => (a, u x)) f (𝓤 α) 
Arguments: {α : Type ua} {β : Type ub} [UniformSpace α] {f : Filter β} {u : β → α} {a : α}
Goal: Filter.Tendsto u f (𝓝 a) ↔ Filter.Tendsto (fun x => (a, u x)) f (𝓤 α)
-- Uniform.tendsto_nhds_left
theorem tendsto_nhds_left {f : Filter β} {u : β → α} {a : α} :
    Tendsto u f (𝓝 a) ↔ Tendsto (fun x => (u x, a)) f (𝓤 α) 
Arguments: {α : Type ua} {β : Type ub} [UniformSpace α] {f : Filter β} {u : β → α} {a : α}
Goal: Filter.Tendsto u f (𝓝 a) ↔ Filter.Tendsto (fun x => (u x, a)) f (𝓤 α)
-- Uniform.continuousAt_iff'_right
theorem continuousAt_iff'_right [TopologicalSpace β] {f : β → α} {b : β} :
    ContinuousAt f b ↔ Tendsto (fun x => (f b, f x)) (𝓝 b) (𝓤 α) 
Arguments: {α : Type ua} {β : Type ub} [UniformSpace α] [TopologicalSpace β] {f : β → α} {b : β}
Goal: ContinuousAt f b ↔ Filter.Tendsto (fun x => (f b, f x)) (𝓝 b) (𝓤 α)
-- Uniform.continuousAt_iff'_left
theorem continuousAt_iff'_left [TopologicalSpace β] {f : β → α} {b : β} :
    ContinuousAt f b ↔ Tendsto (fun x => (f x, f b)) (𝓝 b) (𝓤 α) 
Arguments: {α : Type ua} {β : Type ub} [UniformSpace α] [TopologicalSpace β] {f : β → α} {b : β}
Goal: ContinuousAt f b ↔ Filter.Tendsto (fun x => (f x, f b)) (𝓝 b) (𝓤 α)
-- Uniform.continuousAt_iff_prod
theorem continuousAt_iff_prod [TopologicalSpace β] {f : β → α} {b : β} :
    ContinuousAt f b ↔ Tendsto (fun x : β × β => (f x.1, f x.2)) (𝓝 (b, b)) (𝓤 α) 
Arguments: {α : Type ua} {β : Type ub} [UniformSpace α] [TopologicalSpace β] {f : β → α} {b : β}
Goal: ContinuousAt f b ↔ Filter.Tendsto (fun x => (f x.1, f x.2)) (𝓝 (b, b)) (𝓤 α)
-- Uniform.continuousWithinAt_iff'_right
theorem continuousWithinAt_iff'_right [TopologicalSpace β] {f : β → α} {b : β} {s : Set β} :
    ContinuousWithinAt f s b ↔ Tendsto (fun x => (f b, f x)) (𝓝[s] b) (𝓤 α) 
Arguments: {α : Type ua} {β : Type ub} [UniformSpace α] [TopologicalSpace β] {f : β → α} {b : β} {s : Set β}
Goal: ContinuousWithinAt f s b ↔ Filter.Tendsto (fun x => (f b, f x)) (𝓝[s] b) (𝓤 α)
-- Uniform.continuousWithinAt_iff'_left
theorem continuousWithinAt_iff'_left [TopologicalSpace β] {f : β → α} {b : β} {s : Set β} :
    ContinuousWithinAt f s b ↔ Tendsto (fun x => (f x, f b)) (𝓝[s] b) (𝓤 α) 
Arguments: {α : Type ua} {β : Type ub} [UniformSpace α] [TopologicalSpace β] {f : β → α} {b : β} {s : Set β}
Goal: ContinuousWithinAt f s b ↔ Filter.Tendsto (fun x => (f x, f b)) (𝓝[s] b) (𝓤 α)
-- Uniform.continuousOn_iff'_right
theorem continuousOn_iff'_right [TopologicalSpace β] {f : β → α} {s : Set β} :
    ContinuousOn f s ↔ ∀ b ∈ s, Tendsto (fun x => (f b, f x)) (𝓝[s] b) (𝓤 α) 
Arguments: {α : Type ua} {β : Type ub} [UniformSpace α] [TopologicalSpace β] {f : β → α} {s : Set β}
Goal: ContinuousOn f s ↔ ∀ b ∈ s, Filter.Tendsto (fun x => (f b, f x)) (𝓝[s] b) (𝓤 α)
-- Uniform.continuousOn_iff'_left
theorem continuousOn_iff'_left [TopologicalSpace β] {f : β → α} {s : Set β} :
    ContinuousOn f s ↔ ∀ b ∈ s, Tendsto (fun x => (f x, f b)) (𝓝[s] b) (𝓤 α) 
Arguments: {α : Type ua} {β : Type ub} [UniformSpace α] [TopologicalSpace β] {f : β → α} {s : Set β}
Goal: ContinuousOn f s ↔ ∀ b ∈ s, Filter.Tendsto (fun x => (f x, f b)) (𝓝[s] b) (𝓤 α)
-- Uniform.continuous_iff'_right
theorem continuous_iff'_right [TopologicalSpace β] {f : β → α} :
    Continuous f ↔ ∀ b, Tendsto (fun x => (f b, f x)) (𝓝 b) (𝓤 α) 
Arguments: {α : Type ua} {β : Type ub} [UniformSpace α] [TopologicalSpace β] {f : β → α}
Goal: Continuous f ↔ ∀ (b : β), Filter.Tendsto (fun x => (f b, f x)) (𝓝 b) (𝓤 α)
-- Uniform.continuous_iff'_left
theorem continuous_iff'_left [TopologicalSpace β] {f : β → α} :
    Continuous f ↔ ∀ b, Tendsto (fun x => (f x, f b)) (𝓝 b) (𝓤 α) 
Arguments: {α : Type ua} {β : Type ub} [UniformSpace α] [TopologicalSpace β] {f : β → α}
Goal: Continuous f ↔ ∀ (b : β), Filter.Tendsto (fun x => (f x, f b)) (𝓝 b) (𝓤 α)
-- Uniform.exists_is_open_mem_uniformity_of_forall_mem_eq
lemma exists_is_open_mem_uniformity_of_forall_mem_eq
    [TopologicalSpace β] {r : Set (α × α)} {s : Set β}
    {f g : β → α} (hf : ∀ x ∈ s, ContinuousAt f x) (hg : ∀ x ∈ s, ContinuousAt g x)
    (hfg : s.EqOn f g) (hr : r ∈ 𝓤 α) :
    ∃ t, IsOpen t ∧ s ⊆ t ∧ ∀ x ∈ t, (f x, g x) ∈ r 
Arguments: {α : Type ua} {β : Type ub} [UniformSpace α] [TopologicalSpace β] {r : Set (α × α)} {s : Set β} {f g : β → α} (hf : ∀ x ∈ s, ContinuousAt f x) (hg : ∀ x ∈ s, ContinuousAt g x) (hfg : Set.EqOn f g s) (hr : r ∈ 𝓤 α)
Goal: ∃ t, IsOpen t ∧ s ⊆ t ∧ ∀ x ∈ t, (f x, g x) ∈ r
-- Filter.Tendsto.congr_uniformity
theorem Filter.Tendsto.congr_uniformity {α β} [UniformSpace β] {f g : α → β} {l : Filter α} {b : β}
    (hf : Tendsto f l (𝓝 b)) (hg : Tendsto (fun x => (f x, g x)) l (𝓤 β)) : Tendsto g l (𝓝 b) 
Arguments: {α : Type u_2} {β : Type u_3} [UniformSpace β] {f g : α → β} {l : Filter α} {b : β} (hf : Filter.Tendsto f l (𝓝 b)) (hg : Filter.Tendsto (fun x => (f x, g x)) l (𝓤 β))
Goal: Filter.Tendsto g l (𝓝 b)
-- Uniform.tendsto_congr
theorem Uniform.tendsto_congr {α β} [UniformSpace β] {f g : α → β} {l : Filter α} {b : β}
    (hfg : Tendsto (fun x => (f x, g x)) l (𝓤 β)) : Tendsto f l (𝓝 b) ↔ Tendsto g l (𝓝 b) 
Arguments: {α : Type u_2} {β : Type u_3} [UniformSpace β] {f g : α → β} {l : Filter α} {b : β} (hfg : Filter.Tendsto (fun x => (f x, g x)) l (𝓤 β))
Goal: Filter.Tendsto f l (𝓝 b) ↔ Filter.Tendsto g l (𝓝 b)
-- PolishSpace.complete
class PolishSpace (α : Type*) [h : TopologicalSpace α]
    extends SecondCountableTopology α : Prop where
  complete : ∃ m : MetricSpace α, m.toUniformSpace.toTopologicalSpace = h ∧
    @CompleteSpace α m.toUniformSpace
Arguments: {α : Type u_3} [h : TopologicalSpace α] [self : PolishSpace α]
Goal: ∃ m, UniformSpace.toTopologicalSpace = h ∧ CompleteSpace α
-- complete_polishSpaceMetric
theorem complete_polishSpaceMetric (α : Type*) [ht : TopologicalSpace α] [h : PolishSpace α] :
    @CompleteSpace α (polishSpaceMetric α).toUniformSpace 
Arguments: (α : Type u_3) [ht : TopologicalSpace α] [h : PolishSpace α]
Goal: CompleteSpace α
-- PolishSpace.t2Space
@[deprecated (since := "2024-02-23")]
theorem t2Space (α : Type*) [TopologicalSpace α] [PolishSpace α] : T2Space α 
Arguments: (α : Type u_3) [TopologicalSpace α] [PolishSpace α]
Goal: T2Space α
-- PolishSpace.exists_nat_nat_continuous_surjective
theorem exists_nat_nat_continuous_surjective (α : Type*) [TopologicalSpace α] [PolishSpace α]
    [Nonempty α] : ∃ f : (ℕ → ℕ) → α, Continuous f ∧ Surjective f 
Arguments: (α : Type u_3) [TopologicalSpace α] [PolishSpace α] [Nonempty α]
Goal: ∃ f, Continuous f ∧ Function.Surjective f
-- ClosedEmbedding.polishSpace
theorem _root_.ClosedEmbedding.polishSpace [TopologicalSpace α] [TopologicalSpace β] [PolishSpace β]
    {f : α → β} (hf : ClosedEmbedding f) : PolishSpace α 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [PolishSpace β] {f : α → β} (hf : ClosedEmbedding f)
Goal: PolishSpace α
-- Equiv.polishSpace_induced
theorem _root_.Equiv.polishSpace_induced [t : TopologicalSpace β] [PolishSpace β] (f : α ≃ β) :
    @PolishSpace α (t.induced f) 
Arguments: {α : Type u_1} {β : Type u_2} [t : TopologicalSpace β] [PolishSpace β] (f : α ≃ β)
Goal: PolishSpace α
-- IsClosed.polishSpace
theorem _root_.IsClosed.polishSpace [TopologicalSpace α] [PolishSpace α] {s : Set α}
    (hs : IsClosed s) : PolishSpace s 
Arguments: {α : Type u_1} [TopologicalSpace α] [PolishSpace α] {s : Set α} (hs : IsClosed s)
Goal: PolishSpace ↑s
-- CompletePseudometrizable.iInf
protected theorem _root_.CompletePseudometrizable.iInf {ι : Type*} [Countable ι]
    {t : ι → TopologicalSpace α} (ht₀ : ∃ t₀, @T2Space α t₀ ∧ ∀ i, t i ≤ t₀)
    (ht : ∀ i, ∃ u : UniformSpace α, CompleteSpace α ∧ 𝓤[u].IsCountablyGenerated ∧
      u.toTopologicalSpace = t i) :
    ∃ u : UniformSpace α, CompleteSpace α ∧
      𝓤[u].IsCountablyGenerated ∧ u.toTopologicalSpace = ⨅ i, t i 
Arguments: {α : Type u_1} {ι : Type u_3} [Countable ι] {t : ι → TopologicalSpace α} (ht₀ : ∃ t₀, T2Space α ∧ ∀ (i : ι), t i ≤ t₀) (ht : ∀ (i : ι), ∃ u, CompleteSpace α ∧ (𝓤 α).IsCountablyGenerated ∧ UniformSpace.toTopologicalSpace = t i)
Goal: ∃ u, CompleteSpace α ∧ (𝓤 α).IsCountablyGenerated ∧ UniformSpace.toTopologicalSpace = ⨅ i, t i
-- PolishSpace.iInf
protected theorem iInf {ι : Type*} [Countable ι] {t : ι → TopologicalSpace α}
    (ht₀ : ∃ i₀, ∀ i, t i ≤ t i₀) (ht : ∀ i, @PolishSpace α (t i)) : @PolishSpace α (⨅ i, t i) 
Arguments: {α : Type u_1} {ι : Type u_3} [Countable ι] {t : ι → TopologicalSpace α} (ht₀ : ∃ i₀, ∀ (i : ι), t i ≤ t i₀) (ht : ∀ (i : ι), PolishSpace α)
Goal: PolishSpace α
-- PolishSpace.exists_polishSpace_forall_le
theorem exists_polishSpace_forall_le {ι : Type*} [Countable ι] [t : TopologicalSpace α]
    [p : PolishSpace α] (m : ι → TopologicalSpace α) (hm : ∀ n, m n ≤ t)
    (h'm : ∀ n, @PolishSpace α (m n)) :
    ∃ t' : TopologicalSpace α, (∀ n, t' ≤ m n) ∧ t' ≤ t ∧ @PolishSpace α t' 
Arguments: {α : Type u_1} {ι : Type u_3} [Countable ι] [t : TopologicalSpace α] [p : PolishSpace α] (m : ι → TopologicalSpace α) (hm : ∀ (n : ι), m n ≤ t) (h'm : ∀ (n : ι), PolishSpace α)
Goal: ∃ t', (∀ (n : ι), t' ≤ m n) ∧ t' ≤ t ∧ PolishSpace α
-- TopologicalSpace.Opens.CompleteCopy.dist_eq
theorem dist_eq (x y : CompleteCopy s) :
    dist x y = dist x.1 y.1 + abs (1 / infDist x.1 sᶜ - 1 / infDist y.1 sᶜ) 
Arguments: {α : Type u_1} [MetricSpace α] {s : TopologicalSpace.Opens α} (x y : s.CompleteCopy)
Goal: dist x y = dist ↑x ↑y + |1 / Metric.infDist (↑x) (↑s)ᶜ - 1 / Metric.infDist (↑y) (↑s)ᶜ|
-- TopologicalSpace.Opens.CompleteCopy.dist_val_le_dist
theorem dist_val_le_dist (x y : CompleteCopy s) : dist x.1 y.1 ≤ dist x y 
Arguments: {α : Type u_1} [MetricSpace α] {s : TopologicalSpace.Opens α} (x y : s.CompleteCopy)
Goal: dist ↑x ↑y ≤ dist x y
-- IsOpen.polishSpace
theorem _root_.IsOpen.polishSpace {α : Type*} [TopologicalSpace α] [PolishSpace α] {s : Set α}
    (hs : IsOpen s) : PolishSpace s 
Arguments: {α : Type u_3} [TopologicalSpace α] [PolishSpace α] {s : Set α} (hs : IsOpen s)
Goal: PolishSpace ↑s
-- IsClosed.isClopenable
theorem _root_.IsClosed.isClopenable [TopologicalSpace α] [PolishSpace α] {s : Set α}
    (hs : IsClosed s) : IsClopenable s 
Arguments: {α : Type u_1} [TopologicalSpace α] [PolishSpace α] {s : Set α} (hs : IsClosed s)
Goal: PolishSpace.IsClopenable s
-- PolishSpace.IsClopenable.compl
theorem IsClopenable.compl [TopologicalSpace α] {s : Set α} (hs : IsClopenable s) :
    IsClopenable sᶜ 
Arguments: {α : Type u_1} [TopologicalSpace α] {s : Set α} (hs : PolishSpace.IsClopenable s)
Goal: PolishSpace.IsClopenable sᶜ
-- IsOpen.isClopenable
theorem _root_.IsOpen.isClopenable [TopologicalSpace α] [PolishSpace α] {s : Set α}
    (hs : IsOpen s) : IsClopenable s 
Arguments: {α : Type u_1} [TopologicalSpace α] [PolishSpace α] {s : Set α} (hs : IsOpen s)
Goal: PolishSpace.IsClopenable s
-- PolishSpace.IsClopenable.iUnion
theorem IsClopenable.iUnion [t : TopologicalSpace α] [PolishSpace α] {s : ℕ → Set α}
    (hs : ∀ n, IsClopenable (s n)) : IsClopenable (⋃ n, s n) 
Arguments: {α : Type u_1} [t : TopologicalSpace α] [PolishSpace α] {s : ℕ → Set α} (hs : ∀ (n : ℕ), PolishSpace.IsClopenable (s n))
Goal: PolishSpace.IsClopenable (⋃ n, s n)
-- IsMetricSeparated.symm
@[symm]
theorem symm (h : IsMetricSeparated s t) : IsMetricSeparated t s 
Arguments: {X : Type u_1} [EMetricSpace X] {s t : Set X} (h : IsMetricSeparated s t)
Goal: IsMetricSeparated t s
-- IsMetricSeparated.comm
theorem comm : IsMetricSeparated s t ↔ IsMetricSeparated t s 
Arguments: {X : Type u_1} [EMetricSpace X] {s t : Set X}
Goal: IsMetricSeparated s t ↔ IsMetricSeparated t s
-- IsMetricSeparated.empty_left
@[simp]
theorem empty_left (s : Set X) : IsMetricSeparated ∅ s 
Arguments: {X : Type u_1} [EMetricSpace X] (s : Set X)
Goal: IsMetricSeparated ∅ s
-- IsMetricSeparated.empty_right
@[simp]
theorem empty_right (s : Set X) : IsMetricSeparated s ∅ 
Arguments: {X : Type u_1} [EMetricSpace X] (s : Set X)
Goal: IsMetricSeparated s ∅
-- IsMetricSeparated.disjoint
protected theorem disjoint (h : IsMetricSeparated s t) : Disjoint s t 
Arguments: {X : Type u_1} [EMetricSpace X] {s t : Set X} (h : IsMetricSeparated s t)
Goal: Disjoint s t
-- IsMetricSeparated.subset_compl_right
theorem subset_compl_right (h : IsMetricSeparated s t) : s ⊆ tᶜ 
Arguments: {X : Type u_1} [EMetricSpace X] {s t : Set X} (h : IsMetricSeparated s t)
Goal: s ⊆ tᶜ
-- IsMetricSeparated.mono
@[mono]
theorem mono {s' t'} (hs : s ⊆ s') (ht : t ⊆ t') :
    IsMetricSeparated s' t' → IsMetricSeparated s t 
Arguments: {X : Type u_1} [EMetricSpace X] {s t s' t' : Set X} (hs : s ⊆ s') (ht : t ⊆ t')
Goal: IsMetricSeparated s' t' → IsMetricSeparated s t
-- IsMetricSeparated.mono_left
theorem mono_left {s'} (h' : IsMetricSeparated s' t) (hs : s ⊆ s') : IsMetricSeparated s t 
Arguments: {X : Type u_1} [EMetricSpace X] {s t s' : Set X} (h' : IsMetricSeparated s' t) (hs : s ⊆ s')
Goal: IsMetricSeparated s t
-- IsMetricSeparated.mono_right
theorem mono_right {t'} (h' : IsMetricSeparated s t') (ht : t ⊆ t') : IsMetricSeparated s t 
Arguments: {X : Type u_1} [EMetricSpace X] {s t t' : Set X} (h' : IsMetricSeparated s t') (ht : t ⊆ t')
Goal: IsMetricSeparated s t
-- IsMetricSeparated.union_left
theorem union_left {s'} (h : IsMetricSeparated s t) (h' : IsMetricSeparated s' t) :
    IsMetricSeparated (s ∪ s') t 
Arguments: {X : Type u_1} [EMetricSpace X] {s t s' : Set X} (h : IsMetricSeparated s t) (h' : IsMetricSeparated s' t)
Goal: IsMetricSeparated (s ∪ s') t
-- IsMetricSeparated.union_left_iff
@[simp]
theorem union_left_iff {s'} :
    IsMetricSeparated (s ∪ s') t ↔ IsMetricSeparated s t ∧ IsMetricSeparated s' t 
Arguments: {X : Type u_1} [EMetricSpace X] {s t s' : Set X}
Goal: IsMetricSeparated (s ∪ s') t ↔ IsMetricSeparated s t ∧ IsMetricSeparated s' t
-- IsMetricSeparated.union_right
theorem union_right {t'} (h : IsMetricSeparated s t) (h' : IsMetricSeparated s t') :
    IsMetricSeparated s (t ∪ t') 
Arguments: {X : Type u_1} [EMetricSpace X] {s t t' : Set X} (h : IsMetricSeparated s t) (h' : IsMetricSeparated s t')
Goal: IsMetricSeparated s (t ∪ t')
-- IsMetricSeparated.union_right_iff
@[simp]
theorem union_right_iff {t'} :
    IsMetricSeparated s (t ∪ t') ↔ IsMetricSeparated s t ∧ IsMetricSeparated s t' 
Arguments: {X : Type u_1} [EMetricSpace X] {s t t' : Set X}
Goal: IsMetricSeparated s (t ∪ t') ↔ IsMetricSeparated s t ∧ IsMetricSeparated s t'
-- IsMetricSeparated.finite_iUnion_left_iff
theorem finite_iUnion_left_iff {ι : Type*} {I : Set ι} (hI : I.Finite) {s : ι → Set X}
    {t : Set X} : IsMetricSeparated (⋃ i ∈ I, s i) t ↔ ∀ i ∈ I, IsMetricSeparated (s i) t 
Arguments: {X : Type u_1} [EMetricSpace X] {ι : Type u_2} {I : Set ι} (hI : I.Finite) {s : ι → Set X} {t : Set X}
Goal: IsMetricSeparated (⋃ i ∈ I, s i) t ↔ ∀ i ∈ I, IsMetricSeparated (s i) t
-- IsMetricSeparated.finite_iUnion_right_iff
theorem finite_iUnion_right_iff {ι : Type*} {I : Set ι} (hI : I.Finite) {s : Set X}
    {t : ι → Set X} : IsMetricSeparated s (⋃ i ∈ I, t i) ↔ ∀ i ∈ I, IsMetricSeparated s (t i) 
Arguments: {X : Type u_1} [EMetricSpace X] {ι : Type u_2} {I : Set ι} (hI : I.Finite) {s : Set X} {t : ι → Set X}
Goal: IsMetricSeparated s (⋃ i ∈ I, t i) ↔ ∀ i ∈ I, IsMetricSeparated s (t i)
-- IsMetricSeparated.finset_iUnion_left_iff
@[simp]
theorem finset_iUnion_left_iff {ι : Type*} {I : Finset ι} {s : ι → Set X} {t : Set X} :
    IsMetricSeparated (⋃ i ∈ I, s i) t ↔ ∀ i ∈ I, IsMetricSeparated (s i) t 
Arguments: {X : Type u_1} [EMetricSpace X] {ι : Type u_2} {I : Finset ι} {s : ι → Set X} {t : Set X}
Goal: IsMetricSeparated (⋃ i ∈ I, s i) t ↔ ∀ i ∈ I, IsMetricSeparated (s i) t
-- IsMetricSeparated.finset_iUnion_right_iff
@[simp]
theorem finset_iUnion_right_iff {ι : Type*} {I : Finset ι} {s : Set X} {t : ι → Set X} :
    IsMetricSeparated s (⋃ i ∈ I, t i) ↔ ∀ i ∈ I, IsMetricSeparated s (t i) 
Arguments: {X : Type u_1} [EMetricSpace X] {ι : Type u_2} {I : Finset ι} {s : Set X} {t : ι → Set X}
Goal: IsMetricSeparated s (⋃ i ∈ I, t i) ↔ ∀ i ∈ I, IsMetricSeparated s (t i)
-- holderOnWith_empty
@[simp]
theorem holderOnWith_empty (C r : ℝ≥0) (f : X → Y) : HolderOnWith C r f ∅ 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] (C r : ℝ≥0) (f : X → Y)
Goal: HolderOnWith C r f ∅
-- holderOnWith_singleton
@[simp]
theorem holderOnWith_singleton (C r : ℝ≥0) (f : X → Y) (x : X) : HolderOnWith C r f {x} 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] (C r : ℝ≥0) (f : X → Y) (x : X)
Goal: HolderOnWith C r f {x}
-- Set.Subsingleton.holderOnWith
theorem Set.Subsingleton.holderOnWith {s : Set X} (hs : s.Subsingleton) (C r : ℝ≥0) (f : X → Y) :
    HolderOnWith C r f s 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] {s : Set X} (hs : s.Subsingleton) (C r : ℝ≥0) (f : X → Y)
Goal: HolderOnWith C r f s
-- holderOnWith_univ
theorem holderOnWith_univ {C r : ℝ≥0} {f : X → Y} : HolderOnWith C r f univ ↔ HolderWith C r f 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] {C r : ℝ≥0} {f : X → Y}
Goal: HolderOnWith C r f Set.univ ↔ HolderWith C r f
-- holderOnWith_one
@[simp]
theorem holderOnWith_one {C : ℝ≥0} {f : X → Y} {s : Set X} :
    HolderOnWith C 1 f s ↔ LipschitzOnWith C f s 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] {C : ℝ≥0} {f : X → Y} {s : Set X}
Goal: HolderOnWith C 1 f s ↔ LipschitzOnWith C f s