ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Urysohns.CU.approx_of_nmem_U
theorem approx_of_nmem_U (c : CU P) (n : ℕ) {x : X} (hx : x ∉ c.U) : c.approx n x = 1 
Arguments: {X : Type u_1} [TopologicalSpace X] {P : Set X → Prop} (c : Urysohns.CU P) (n : ℕ) {x : X} (hx : x ∉ c.U)
Goal: Urysohns.CU.approx n c x = 1
-- Urysohns.CU.approx_nonneg
theorem approx_nonneg (c : CU P) (n : ℕ) (x : X) : 0 ≤ c.approx n x 
Arguments: {X : Type u_1} [TopologicalSpace X] {P : Set X → Prop} (c : Urysohns.CU P) (n : ℕ) (x : X)
Goal: 0 ≤ Urysohns.CU.approx n c x
-- Urysohns.CU.approx_le_one
theorem approx_le_one (c : CU P) (n : ℕ) (x : X) : c.approx n x ≤ 1 
Arguments: {X : Type u_1} [TopologicalSpace X] {P : Set X → Prop} (c : Urysohns.CU P) (n : ℕ) (x : X)
Goal: Urysohns.CU.approx n c x ≤ 1
-- Urysohns.CU.bddAbove_range_approx
theorem bddAbove_range_approx (c : CU P) (x : X) : BddAbove (range fun n => c.approx n x) 
Arguments: {X : Type u_1} [TopologicalSpace X] {P : Set X → Prop} (c : Urysohns.CU P) (x : X)
Goal: BddAbove (Set.range fun n => Urysohns.CU.approx n c x)
-- Urysohns.CU.approx_le_approx_of_U_sub_C
theorem approx_le_approx_of_U_sub_C {c₁ c₂ : CU P} (h : c₁.U ⊆ c₂.C) (n₁ n₂ : ℕ) (x : X) :
    c₂.approx n₂ x ≤ c₁.approx n₁ x 
Arguments: {X : Type u_1} [TopologicalSpace X] {P : Set X → Prop} {c₁ c₂ : Urysohns.CU P} (h : c₁.U ⊆ c₂.C) (n₁ n₂ : ℕ) (x : X)
Goal: Urysohns.CU.approx n₂ c₂ x ≤ Urysohns.CU.approx n₁ c₁ x
-- Urysohns.CU.approx_mem_Icc_right_left
theorem approx_mem_Icc_right_left (c : CU P) (n : ℕ) (x : X) :
    c.approx n x ∈ Icc (c.right.approx n x) (c.left.approx n x) 
Arguments: {X : Type u_1} [TopologicalSpace X] {P : Set X → Prop} (c : Urysohns.CU P) (n : ℕ) (x : X)
Goal: Urysohns.CU.approx n c x ∈ Set.Icc (Urysohns.CU.approx n c.right x) (Urysohns.CU.approx n c.left x)
-- Urysohns.CU.approx_le_succ
theorem approx_le_succ (c : CU P) (n : ℕ) (x : X) : c.approx n x ≤ c.approx (n + 1) x 
Arguments: {X : Type u_1} [TopologicalSpace X] {P : Set X → Prop} (c : Urysohns.CU P) (n : ℕ) (x : X)
Goal: Urysohns.CU.approx n c x ≤ Urysohns.CU.approx (n + 1) c x
-- Urysohns.CU.approx_mono
theorem approx_mono (c : CU P) (x : X) : Monotone fun n => c.approx n x 
Arguments: {X : Type u_1} [TopologicalSpace X] {P : Set X → Prop} (c : Urysohns.CU P) (x : X)
Goal: Monotone fun n => Urysohns.CU.approx n c x
-- Urysohns.CU.tendsto_approx_atTop
theorem tendsto_approx_atTop (c : CU P) (x : X) :
    Tendsto (fun n => c.approx n x) atTop (𝓝 <| c.lim x) 
Arguments: {X : Type u_1} [TopologicalSpace X] {P : Set X → Prop} (c : Urysohns.CU P) (x : X)
Goal: Filter.Tendsto (fun n => Urysohns.CU.approx n c x) Filter.atTop (𝓝 (c.lim x))
-- Urysohns.CU.lim_of_mem_C
theorem lim_of_mem_C (c : CU P) (x : X) (h : x ∈ c.C) : c.lim x = 0 
Arguments: {X : Type u_1} [TopologicalSpace X] {P : Set X → Prop} (c : Urysohns.CU P) (x : X) (h : x ∈ c.C)
Goal: c.lim x = 0
-- Urysohns.CU.lim_of_nmem_U
theorem lim_of_nmem_U (c : CU P) (x : X) (h : x ∉ c.U) : c.lim x = 1 
Arguments: {X : Type u_1} [TopologicalSpace X] {P : Set X → Prop} (c : Urysohns.CU P) (x : X) (h : x ∉ c.U)
Goal: c.lim x = 1
-- Urysohns.CU.lim_eq_midpoint
theorem lim_eq_midpoint (c : CU P) (x : X) :
    c.lim x = midpoint ℝ (c.left.lim x) (c.right.lim x) 
Arguments: {X : Type u_1} [TopologicalSpace X] {P : Set X → Prop} (c : Urysohns.CU P) (x : X)
Goal: c.lim x = midpoint ℝ (c.left.lim x) (c.right.lim x)
-- Urysohns.CU.approx_le_lim
theorem approx_le_lim (c : CU P) (x : X) (n : ℕ) : c.approx n x ≤ c.lim x 
Arguments: {X : Type u_1} [TopologicalSpace X] {P : Set X → Prop} (c : Urysohns.CU P) (x : X) (n : ℕ)
Goal: Urysohns.CU.approx n c x ≤ c.lim x
-- Urysohns.CU.lim_nonneg
theorem lim_nonneg (c : CU P) (x : X) : 0 ≤ c.lim x 
Arguments: {X : Type u_1} [TopologicalSpace X] {P : Set X → Prop} (c : Urysohns.CU P) (x : X)
Goal: 0 ≤ c.lim x
-- Urysohns.CU.lim_le_one
theorem lim_le_one (c : CU P) (x : X) : c.lim x ≤ 1 
Arguments: {X : Type u_1} [TopologicalSpace X] {P : Set X → Prop} (c : Urysohns.CU P) (x : X)
Goal: c.lim x ≤ 1
-- Urysohns.CU.lim_mem_Icc
theorem lim_mem_Icc (c : CU P) (x : X) : c.lim x ∈ Icc (0 : ℝ) 1 
Arguments: {X : Type u_1} [TopologicalSpace X] {P : Set X → Prop} (c : Urysohns.CU P) (x : X)
Goal: c.lim x ∈ Set.Icc 0 1
-- Urysohns.CU.continuous_lim
theorem continuous_lim (c : CU P) : Continuous c.lim 
Arguments: {X : Type u_1} [TopologicalSpace X] {P : Set X → Prop} (c : Urysohns.CU P)
Goal: Continuous c.lim
-- exists_continuous_zero_one_of_isClosed
theorem exists_continuous_zero_one_of_isClosed [NormalSpace X]
    {s t : Set X} (hs : IsClosed s) (ht : IsClosed t)
    (hd : Disjoint s t) : ∃ f : C(X, ℝ), EqOn f 0 s ∧ EqOn f 1 t ∧ ∀ x, f x ∈ Icc (0 : ℝ) 1 
Arguments: {X : Type u_1} [TopologicalSpace X] [NormalSpace X] {s t : Set X} (hs : IsClosed s) (ht : IsClosed t) (hd : Disjoint s t)
Goal: ∃ f, Set.EqOn (⇑f) 0 s ∧ Set.EqOn (⇑f) 1 t ∧ ∀ (x : X), f x ∈ Set.Icc 0 1
-- exists_continuous_zero_one_of_isCompact
theorem exists_continuous_zero_one_of_isCompact [RegularSpace X] [LocallyCompactSpace X]
    {s t : Set X} (hs : IsCompact s) (ht : IsClosed t) (hd : Disjoint s t) :
    ∃ f : C(X, ℝ), EqOn f 0 s ∧ EqOn f 1 t ∧ ∀ x, f x ∈ Icc (0 : ℝ) 1 
Arguments: {X : Type u_1} [TopologicalSpace X] [RegularSpace X] [LocallyCompactSpace X] {s t : Set X} (hs : IsCompact s) (ht : IsClosed t) (hd : Disjoint s t)
Goal: ∃ f, Set.EqOn (⇑f) 0 s ∧ Set.EqOn (⇑f) 1 t ∧ ∀ (x : X), f x ∈ Set.Icc 0 1
-- exists_continuous_one_zero_of_isCompact
theorem exists_continuous_one_zero_of_isCompact [RegularSpace X] [LocallyCompactSpace X]
    {s t : Set X} (hs : IsCompact s) (ht : IsClosed t) (hd : Disjoint s t) :
    ∃ f : C(X, ℝ), EqOn f 1 s ∧ EqOn f 0 t ∧ HasCompactSupport f ∧ ∀ x, f x ∈ Icc (0 : ℝ) 1 
Arguments: {X : Type u_1} [TopologicalSpace X] [RegularSpace X] [LocallyCompactSpace X] {s t : Set X} (hs : IsCompact s) (ht : IsClosed t) (hd : Disjoint s t)
Goal: ∃ f, Set.EqOn (⇑f) 1 s ∧ Set.EqOn (⇑f) 0 t ∧ HasCompactSupport ⇑f ∧ ∀ (x : X), f x ∈ Set.Icc 0 1
-- exists_continuous_one_zero_of_isCompact_of_isGδ
theorem exists_continuous_one_zero_of_isCompact_of_isGδ [RegularSpace X] [LocallyCompactSpace X]
    {s t : Set X} (hs : IsCompact s) (h's : IsGδ s) (ht : IsClosed t) (hd : Disjoint s t) :
    ∃ f : C(X, ℝ), s = f ⁻¹' {1} ∧ EqOn f 0 t ∧ HasCompactSupport f
      ∧ ∀ x, f x ∈ Icc (0 : ℝ) 1 
Arguments: {X : Type u_1} [TopologicalSpace X] [RegularSpace X] [LocallyCompactSpace X] {s t : Set X} (hs : IsCompact s) (h's : IsGδ s) (ht : IsClosed t) (hd : Disjoint s t)
Goal: ∃ f, s = ⇑f ⁻¹' {1} ∧ Set.EqOn (⇑f) 0 t ∧ HasCompactSupport ⇑f ∧ ∀ (x : X), f x ∈ Set.Icc 0 1
-- exists_continuous_nonneg_pos
theorem exists_continuous_nonneg_pos [RegularSpace X] [LocallyCompactSpace X] (x : X) :
    ∃ f : C(X, ℝ), HasCompactSupport f ∧ 0 ≤ (f : X → ℝ) ∧ f x ≠ 0 
Arguments: {X : Type u_1} [TopologicalSpace X] [RegularSpace X] [LocallyCompactSpace X] (x : X)
Goal: ∃ f, HasCompactSupport ⇑f ∧ 0 ≤ ⇑f ∧ f x ≠ 0
-- SeqCompact.lebesgue_number_lemma_of_metric
@[deprecated lebesgue_number_lemma_of_metric (since := "2024-02-24")]
nonrec theorem SeqCompact.lebesgue_number_lemma_of_metric {ι : Sort*} {c : ι → Set X} {s : Set X}
    (hs : IsSeqCompact s) (hc₁ : ∀ i, IsOpen (c i)) (hc₂ : s ⊆ ⋃ i, c i) :
    ∃ δ > 0, ∀ a ∈ s, ∃ i, ball a δ ⊆ c i 
Arguments: {X : Type u_1} [PseudoMetricSpace X] {ι : Sort u_2} {c : ι → Set X} {s : Set X} (hs : IsSeqCompact s) (hc₁ : ∀ (i : ι), IsOpen (c i)) (hc₂ : s ⊆ ⋃ i, c i)
Goal: ∃ δ > 0, ∀ a ∈ s, ∃ i, Metric.ball a δ ⊆ c i
-- tendsto_subseq_of_frequently_bounded
theorem tendsto_subseq_of_frequently_bounded (hs : IsBounded s) {x : ℕ → X}
    (hx : ∃ᶠ n in atTop, x n ∈ s) :
    ∃ a ∈ closure s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (x ∘ φ) atTop (𝓝 a) 
Arguments: {X : Type u_1} [PseudoMetricSpace X] [ProperSpace X] {s : Set X} (hs : Bornology.IsBounded s) {x : ℕ → X} (hx : ∃ᶠ (n : ℕ) in Filter.atTop, x n ∈ s)
Goal: ∃ a ∈ closure s, ∃ φ, StrictMono φ ∧ Filter.Tendsto (x ∘ φ) Filter.atTop (𝓝 a)
-- tendsto_subseq_of_bounded
theorem tendsto_subseq_of_bounded (hs : IsBounded s) {x : ℕ → X} (hx : ∀ n, x n ∈ s) :
    ∃ a ∈ closure s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (x ∘ φ) atTop (𝓝 a) 
Arguments: {X : Type u_1} [PseudoMetricSpace X] [ProperSpace X] {s : Set X} (hs : Bornology.IsBounded s) {x : ℕ → X} (hx : ∀ (n : ℕ), x n ∈ s)
Goal: ∃ a ∈ closure s, ∃ φ, StrictMono φ ∧ Filter.Tendsto (x ∘ φ) Filter.atTop (𝓝 a)
-- UniformInducing.comap_uniformity
@[mk_iff]
structure UniformInducing (f : α → β) : Prop where
    comap_uniformity : comap (fun x : α × α => (f x.1, f x.2)) (𝓤 β) = 𝓤 α
Arguments: {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] {f : α → β} (self : UniformInducing f)
Goal: Filter.comap (fun x => (f x.1, f x.2)) (𝓤 β) = 𝓤 α
-- uniformInducing_iff
@[mk_iff]
structure UniformInducing (f : α → β) : Prop where
    comap_uniformity : comap (fun x : α × α => (f x.1, f x.2)) (𝓤 β) = 𝓤 α
Arguments: {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] (f : α → β)
Goal: UniformInducing f ↔ Filter.comap (fun x => (f x.1, f x.2)) (𝓤 β) = 𝓤 α
-- uniformInducing_iff_uniformSpace
lemma uniformInducing_iff_uniformSpace {f : α → β} :
    UniformInducing f ↔ ‹UniformSpace β›.comap f = ‹UniformSpace α› 
Arguments: {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] {f : α → β}
Goal: UniformInducing f ↔ UniformSpace.comap f inst✝ = inst✝¹
-- uniformInducing_iff'
lemma uniformInducing_iff' {f : α → β} :
    UniformInducing f ↔ UniformContinuous f ∧ comap (Prod.map f f) (𝓤 β) ≤ 𝓤 α 
Arguments: {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] {f : α → β}
Goal: UniformInducing f ↔ UniformContinuous f ∧ Filter.comap (Prod.map f f) (𝓤 β) ≤ 𝓤 α
-- Filter.HasBasis.uniformInducing_iff
protected lemma Filter.HasBasis.uniformInducing_iff {ι ι'} {p : ι → Prop} {p' : ι' → Prop} {s s'}
    (h : (𝓤 α).HasBasis p s) (h' : (𝓤 β).HasBasis p' s') {f : α → β} :
    UniformInducing f ↔
      (∀ i, p' i → ∃ j, p j ∧ ∀ x y, (x, y) ∈ s j → (f x, f y) ∈ s' i) ∧
        (∀ j, p j → ∃ i, p' i ∧ ∀ x y, (f x, f y) ∈ s' i → (x, y) ∈ s j) 
Arguments: {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] {ι : Sort u_1} {ι' : Sort u_2} {p : ι → Prop} {p' : ι' → Prop} {s : ι → Set (α × α)} {s' : ι' → Set (β × β)} (h : (𝓤 α).HasBasis p s) (h' : (𝓤 β).HasBasis p' s') {f : α → β}
Goal: UniformInducing f ↔
  (∀ (i : ι'), p' i → ∃ j, p j ∧ ∀ (x y : α), (x, y) ∈ s j → (f x, f y) ∈ s' i) ∧
    ∀ (j : ι), p j → ∃ i, p' i ∧ ∀ (x y : α), (f x, f y) ∈ s' i → (x, y) ∈ s j
-- UniformInducing.mk'
theorem UniformInducing.mk' {f : α → β}
    (h : ∀ s, s ∈ 𝓤 α ↔ ∃ t ∈ 𝓤 β, ∀ x y : α, (f x, f y) ∈ t → (x, y) ∈ s) : UniformInducing f 
Arguments: {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] {f : α → β} (h : ∀ (s : Set (α × α)), s ∈ 𝓤 α ↔ ∃ t ∈ 𝓤 β, ∀ (x y : α), (f x, f y) ∈ t → (x, y) ∈ s)
Goal: UniformInducing f
-- uniformInducing_id
theorem uniformInducing_id : UniformInducing (@id α) 
Arguments: {α : Type u} [UniformSpace α]
Goal: UniformInducing id
-- UniformInducing.comp
theorem UniformInducing.comp {g : β → γ} (hg : UniformInducing g) {f : α → β}
    (hf : UniformInducing f) : UniformInducing (g ∘ f) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [UniformSpace α] [UniformSpace β] [UniformSpace γ] {g : β → γ} (hg : UniformInducing g) {f : α → β} (hf : UniformInducing f)
Goal: UniformInducing (g ∘ f)
-- UniformInducing.of_comp_iff
theorem UniformInducing.of_comp_iff {g : β → γ} (hg : UniformInducing g) {f : α → β} :
    UniformInducing (g ∘ f) ↔ UniformInducing f 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [UniformSpace α] [UniformSpace β] [UniformSpace γ] {g : β → γ} (hg : UniformInducing g) {f : α → β}
Goal: UniformInducing (g ∘ f) ↔ UniformInducing f
-- UniformInducing.basis_uniformity
theorem UniformInducing.basis_uniformity {f : α → β} (hf : UniformInducing f) {ι : Sort*}
    {p : ι → Prop} {s : ι → Set (β × β)} (H : (𝓤 β).HasBasis p s) :
    (𝓤 α).HasBasis p fun i => Prod.map f f ⁻¹' s i 
Arguments: {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] {f : α → β} (hf : UniformInducing f) {ι : Sort u_1} {p : ι → Prop} {s : ι → Set (β × β)} (H : (𝓤 β).HasBasis p s)
Goal: (𝓤 α).HasBasis p fun i => Prod.map f f ⁻¹' s i
-- UniformInducing.cauchy_map_iff
theorem UniformInducing.cauchy_map_iff {f : α → β} (hf : UniformInducing f) {F : Filter α} :
    Cauchy (map f F) ↔ Cauchy F 
Arguments: {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] {f : α → β} (hf : UniformInducing f) {F : Filter α}
Goal: Cauchy (Filter.map f F) ↔ Cauchy F
-- uniformInducing_of_compose
theorem uniformInducing_of_compose {f : α → β} {g : β → γ} (hf : UniformContinuous f)
    (hg : UniformContinuous g) (hgf : UniformInducing (g ∘ f)) : UniformInducing f 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [UniformSpace α] [UniformSpace β] [UniformSpace γ] {f : α → β} {g : β → γ} (hf : UniformContinuous f) (hg : UniformContinuous g) (hgf : UniformInducing (g ∘ f))
Goal: UniformInducing f
-- UniformInducing.uniformContinuous
theorem UniformInducing.uniformContinuous {f : α → β} (hf : UniformInducing f) :
    UniformContinuous f 
Arguments: {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] {f : α → β} (hf : UniformInducing f)
Goal: UniformContinuous f
-- UniformInducing.uniformContinuous_iff
theorem UniformInducing.uniformContinuous_iff {f : α → β} {g : β → γ} (hg : UniformInducing g) :
    UniformContinuous f ↔ UniformContinuous (g ∘ f) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [UniformSpace α] [UniformSpace β] [UniformSpace γ] {f : α → β} {g : β → γ} (hg : UniformInducing g)
Goal: UniformContinuous f ↔ UniformContinuous (g ∘ f)
-- UniformInducing.uniformInducing_comp_iff
protected theorem UniformInducing.uniformInducing_comp_iff {f : α → β} {g : β → γ}
    (hg : UniformInducing g) : UniformInducing (g ∘ f) ↔ UniformInducing f 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [UniformSpace α] [UniformSpace β] [UniformSpace γ] {f : α → β} {g : β → γ} (hg : UniformInducing g)
Goal: UniformInducing (g ∘ f) ↔ UniformInducing f
-- UniformInducing.uniformContinuousOn_iff
theorem UniformInducing.uniformContinuousOn_iff {f : α → β} {g : β → γ} {S : Set α}
    (hg : UniformInducing g) :
    UniformContinuousOn f S ↔ UniformContinuousOn (g ∘ f) S 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [UniformSpace α] [UniformSpace β] [UniformSpace γ] {f : α → β} {g : β → γ} {S : Set α} (hg : UniformInducing g)
Goal: UniformContinuousOn f S ↔ UniformContinuousOn (g ∘ f) S
-- UniformInducing.inducing
theorem UniformInducing.inducing {f : α → β} (h : UniformInducing f) : Inducing f 
Arguments: {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] {f : α → β} (h : UniformInducing f)
Goal: Inducing f
-- UniformInducing.prod
theorem UniformInducing.prod {α' : Type*} {β' : Type*} [UniformSpace α'] [UniformSpace β']
    {e₁ : α → α'} {e₂ : β → β'} (h₁ : UniformInducing e₁) (h₂ : UniformInducing e₂) :
    UniformInducing fun p : α × β => (e₁ p.1, e₂ p.2) 
Arguments: {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] {α' : Type u_1} {β' : Type u_2} [UniformSpace α'] [UniformSpace β'] {e₁ : α → α'} {e₂ : β → β'} (h₁ : UniformInducing e₁) (h₂ : UniformInducing e₂)
Goal: UniformInducing fun p => (e₁ p.1, e₂ p.2)
-- UniformInducing.denseInducing
theorem UniformInducing.denseInducing {f : α → β} (h : UniformInducing f) (hd : DenseRange f) :
    DenseInducing f 
Arguments: {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] {f : α → β} (h : UniformInducing f) (hd : DenseRange f)
Goal: DenseInducing f
-- SeparationQuotient.uniformInducing_mk
theorem SeparationQuotient.uniformInducing_mk : UniformInducing (mk : α → SeparationQuotient α) 
Arguments: {α : Type u} [UniformSpace α]
Goal: UniformInducing SeparationQuotient.mk
-- UniformInducing.injective
protected theorem UniformInducing.injective [T0Space α] {f : α → β} (h : UniformInducing f) :
    Injective f 
Arguments: {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] [T0Space α] {f : α → β} (h : UniformInducing f)
Goal: Function.Injective f
-- UniformEmbedding.inj
@[mk_iff]
structure UniformEmbedding (f : α → β) extends UniformInducing f : Prop where
    inj : Function.Injective f
Arguments: {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] {f : α → β} (self : UniformEmbedding f)
Goal: Function.Injective f
-- uniformEmbedding_iff
@[mk_iff]
structure UniformEmbedding (f : α → β) extends UniformInducing f : Prop where
    inj : Function.Injective f
Arguments: {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] (f : α → β)
Goal: UniformEmbedding f ↔ UniformInducing f ∧ Function.Injective f
-- uniformEmbedding_iff'
theorem uniformEmbedding_iff' {f : α → β} :
    UniformEmbedding f ↔ Injective f ∧ UniformContinuous f ∧ comap (Prod.map f f) (𝓤 β) ≤ 𝓤 α 
Arguments: {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] {f : α → β}
Goal: UniformEmbedding f ↔ Function.Injective f ∧ UniformContinuous f ∧ Filter.comap (Prod.map f f) (𝓤 β) ≤ 𝓤 α
-- Filter.HasBasis.uniformEmbedding_iff'
theorem Filter.HasBasis.uniformEmbedding_iff' {ι ι'} {p : ι → Prop} {p' : ι' → Prop} {s s'}
    (h : (𝓤 α).HasBasis p s) (h' : (𝓤 β).HasBasis p' s') {f : α → β} :
    UniformEmbedding f ↔ Injective f ∧
      (∀ i, p' i → ∃ j, p j ∧ ∀ x y, (x, y) ∈ s j → (f x, f y) ∈ s' i) ∧
        (∀ j, p j → ∃ i, p' i ∧ ∀ x y, (f x, f y) ∈ s' i → (x, y) ∈ s j) 
Arguments: {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] {ι : Sort u_1} {ι' : Sort u_2} {p : ι → Prop} {p' : ι' → Prop} {s : ι → Set (α × α)} {s' : ι' → Set (β × β)} (h : (𝓤 α).HasBasis p s) (h' : (𝓤 β).HasBasis p' s') {f : α → β}
Goal: UniformEmbedding f ↔
  Function.Injective f ∧
    (∀ (i : ι'), p' i → ∃ j, p j ∧ ∀ (x y : α), (x, y) ∈ s j → (f x, f y) ∈ s' i) ∧
      ∀ (j : ι), p j → ∃ i, p' i ∧ ∀ (x y : α), (f x, f y) ∈ s' i → (x, y) ∈ s j
-- Filter.HasBasis.uniformEmbedding_iff
theorem Filter.HasBasis.uniformEmbedding_iff {ι ι'} {p : ι → Prop} {p' : ι' → Prop} {s s'}
    (h : (𝓤 α).HasBasis p s) (h' : (𝓤 β).HasBasis p' s') {f : α → β} :
    UniformEmbedding f ↔ Injective f ∧ UniformContinuous f ∧
      (∀ j, p j → ∃ i, p' i ∧ ∀ x y, (f x, f y) ∈ s' i → (x, y) ∈ s j) 
Arguments: {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] {ι : Sort u_1} {ι' : Sort u_2} {p : ι → Prop} {p' : ι' → Prop} {s : ι → Set (α × α)} {s' : ι' → Set (β × β)} (h : (𝓤 α).HasBasis p s) (h' : (𝓤 β).HasBasis p' s') {f : α → β}
Goal: UniformEmbedding f ↔
  Function.Injective f ∧
    UniformContinuous f ∧ ∀ (j : ι), p j → ∃ i, p' i ∧ ∀ (x y : α), (f x, f y) ∈ s' i → (x, y) ∈ s j
-- uniformEmbedding_subtype_val
theorem uniformEmbedding_subtype_val {p : α → Prop} :
    UniformEmbedding (Subtype.val : Subtype p → α) 
Arguments: {α : Type u} [UniformSpace α] {p : α → Prop}
Goal: UniformEmbedding Subtype.val
-- uniformEmbedding_set_inclusion
theorem uniformEmbedding_set_inclusion {s t : Set α} (hst : s ⊆ t) :
    UniformEmbedding (inclusion hst) 
Arguments: {α : Type u} [UniformSpace α] {s t : Set α} (hst : s ⊆ t)
Goal: UniformEmbedding (Set.inclusion hst)
-- UniformEmbedding.comp
theorem UniformEmbedding.comp {g : β → γ} (hg : UniformEmbedding g) {f : α → β}
    (hf : UniformEmbedding f) : UniformEmbedding (g ∘ f) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [UniformSpace α] [UniformSpace β] [UniformSpace γ] {g : β → γ} (hg : UniformEmbedding g) {f : α → β} (hf : UniformEmbedding f)
Goal: UniformEmbedding (g ∘ f)
-- UniformEmbedding.of_comp_iff
theorem UniformEmbedding.of_comp_iff {g : β → γ} (hg : UniformEmbedding g) {f : α → β} :
    UniformEmbedding (g ∘ f) ↔ UniformEmbedding f 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [UniformSpace α] [UniformSpace β] [UniformSpace γ] {g : β → γ} (hg : UniformEmbedding g) {f : α → β}
Goal: UniformEmbedding (g ∘ f) ↔ UniformEmbedding f
-- Equiv.uniformEmbedding
theorem Equiv.uniformEmbedding {α β : Type*} [UniformSpace α] [UniformSpace β] (f : α ≃ β)
    (h₁ : UniformContinuous f) (h₂ : UniformContinuous f.symm) : UniformEmbedding f 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [UniformSpace β] (f : α ≃ β) (h₁ : UniformContinuous ⇑f) (h₂ : UniformContinuous ⇑f.symm)
Goal: UniformEmbedding ⇑f
-- uniformEmbedding_inl
theorem uniformEmbedding_inl : UniformEmbedding (Sum.inl : α → α ⊕ β) 
Arguments: {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β]
Goal: UniformEmbedding Sum.inl
-- uniformEmbedding_inr
theorem uniformEmbedding_inr : UniformEmbedding (Sum.inr : β → α ⊕ β) 
Arguments: {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β]
Goal: UniformEmbedding Sum.inr
-- UniformInducing.uniformEmbedding
protected theorem UniformInducing.uniformEmbedding [T0Space α] {f : α → β}
    (hf : UniformInducing f) : UniformEmbedding f 
Arguments: {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] [T0Space α] {f : α → β} (hf : UniformInducing f)
Goal: UniformEmbedding f
-- uniformEmbedding_iff_uniformInducing
theorem uniformEmbedding_iff_uniformInducing [T0Space α] {f : α → β} :
    UniformEmbedding f ↔ UniformInducing f 
Arguments: {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] [T0Space α] {f : α → β}
Goal: UniformEmbedding f ↔ UniformInducing f
-- comap_uniformity_of_spaced_out
theorem comap_uniformity_of_spaced_out {α} {f : α → β} {s : Set (β × β)} (hs : s ∈ 𝓤 β)
    (hf : Pairwise fun x y => (f x, f y) ∉ s) : comap (Prod.map f f) (𝓤 β) = 𝓟 idRel 
Arguments: {β : Type v} [UniformSpace β] {α : Type u_1} {f : α → β} {s : Set (β × β)} (hs : s ∈ 𝓤 β) (hf : Pairwise fun x y => (f x, f y) ∉ s)
Goal: Filter.comap (Prod.map f f) (𝓤 β) = 𝓟 idRel
-- uniformEmbedding_of_spaced_out
theorem uniformEmbedding_of_spaced_out {α} {f : α → β} {s : Set (β × β)} (hs : s ∈ 𝓤 β)
    (hf : Pairwise fun x y => (f x, f y) ∉ s) : @UniformEmbedding α β ⊥ ‹_› f 
Arguments: {β : Type v} [UniformSpace β] {α : Type u_1} {f : α → β} {s : Set (β × β)} (hs : s ∈ 𝓤 β) (hf : Pairwise fun x y => (f x, f y) ∉ s)
Goal: UniformEmbedding f
-- UniformEmbedding.embedding
protected theorem UniformEmbedding.embedding {f : α → β} (h : UniformEmbedding f) : Embedding f 
Arguments: {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] {f : α → β} (h : UniformEmbedding f)
Goal: Embedding f
-- UniformEmbedding.denseEmbedding
theorem UniformEmbedding.denseEmbedding {f : α → β} (h : UniformEmbedding f) (hd : DenseRange f) :
    DenseEmbedding f 
Arguments: {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] {f : α → β} (h : UniformEmbedding f) (hd : DenseRange f)
Goal: DenseEmbedding f
-- closedEmbedding_of_spaced_out
theorem closedEmbedding_of_spaced_out {α} [TopologicalSpace α] [DiscreteTopology α]
    [T0Space β] {f : α → β} {s : Set (β × β)} (hs : s ∈ 𝓤 β)
    (hf : Pairwise fun x y => (f x, f y) ∉ s) : ClosedEmbedding f 
Arguments: {β : Type v} [UniformSpace β] {α : Type u_1} [TopologicalSpace α] [DiscreteTopology α] [T0Space β] {f : α → β} {s : Set (β × β)} (hs : s ∈ 𝓤 β) (hf : Pairwise fun x y => (f x, f y) ∉ s)
Goal: ClosedEmbedding f
-- closure_image_mem_nhds_of_uniformInducing
theorem closure_image_mem_nhds_of_uniformInducing {s : Set (α × α)} {e : α → β} (b : β)
    (he₁ : UniformInducing e) (he₂ : DenseInducing e) (hs : s ∈ 𝓤 α) :
    ∃ a, closure (e '' { a' | (a, a') ∈ s }) ∈ 𝓝 b 
Arguments: {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] {s : Set (α × α)} {e : α → β} (b : β) (he₁ : UniformInducing e) (he₂ : DenseInducing e) (hs : s ∈ 𝓤 α)
Goal: ∃ a, closure (e '' {a' | (a, a') ∈ s}) ∈ 𝓝 b
-- uniformEmbedding_subtypeEmb
theorem uniformEmbedding_subtypeEmb (p : α → Prop) {e : α → β} (ue : UniformEmbedding e)
    (de : DenseEmbedding e) : UniformEmbedding (DenseEmbedding.subtypeEmb p e) 
Arguments: {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] (p : α → Prop) {e : α → β} (ue : UniformEmbedding e) (de : DenseEmbedding e)
Goal: UniformEmbedding (DenseEmbedding.subtypeEmb p e)
-- UniformEmbedding.prod
theorem UniformEmbedding.prod {α' : Type*} {β' : Type*} [UniformSpace α'] [UniformSpace β']
    {e₁ : α → α'} {e₂ : β → β'} (h₁ : UniformEmbedding e₁) (h₂ : UniformEmbedding e₂) :
    UniformEmbedding fun p : α × β => (e₁ p.1, e₂ p.2) 
Arguments: {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] {α' : Type u_1} {β' : Type u_2} [UniformSpace α'] [UniformSpace β'] {e₁ : α → α'} {e₂ : β → β'} (h₁ : UniformEmbedding e₁) (h₂ : UniformEmbedding e₂)
Goal: UniformEmbedding fun p => (e₁ p.1, e₂ p.2)