ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Metric.tendstoUniformlyOnFilter_iff
theorem tendstoUniformlyOnFilter_iff {F : ι → β → α} {f : β → α} {p : Filter ι} {p' : Filter β} :
    TendstoUniformlyOnFilter F f p p' ↔
      ∀ ε > 0, ∀ᶠ n : ι × β in p ×ˢ p', dist (f n.snd) (F n.fst n.snd) < ε 
Arguments: {α : Type u} {β : Type v} {ι : Type u_2} [PseudoMetricSpace α] {F : ι → β → α} {f : β → α} {p : Filter ι} {p' : Filter β}
Goal: TendstoUniformlyOnFilter F f p p' ↔ ∀ ε > 0, ∀ᶠ (n : ι × β) in p ×ˢ p', dist (f n.2) (F n.1 n.2) < ε
-- Metric.tendstoLocallyUniformlyOn_iff
theorem tendstoLocallyUniformlyOn_iff [TopologicalSpace β] {F : ι → β → α} {f : β → α}
    {p : Filter ι} {s : Set β} :
    TendstoLocallyUniformlyOn F f p s ↔
      ∀ ε > 0, ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, ∀ᶠ n in p, ∀ y ∈ t, dist (f y) (F n y) < ε 
Arguments: {α : Type u} {β : Type v} {ι : Type u_2} [PseudoMetricSpace α] [TopologicalSpace β] {F : ι → β → α} {f : β → α} {p : Filter ι} {s : Set β}
Goal: TendstoLocallyUniformlyOn F f p s ↔ ∀ ε > 0, ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, ∀ᶠ (n : ι) in p, ∀ y ∈ t, dist (f y) (F n y) < ε
-- Metric.tendstoUniformlyOn_iff
theorem tendstoUniformlyOn_iff {F : ι → β → α} {f : β → α} {p : Filter ι} {s : Set β} :
    TendstoUniformlyOn F f p s ↔ ∀ ε > 0, ∀ᶠ n in p, ∀ x ∈ s, dist (f x) (F n x) < ε 
Arguments: {α : Type u} {β : Type v} {ι : Type u_2} [PseudoMetricSpace α] {F : ι → β → α} {f : β → α} {p : Filter ι} {s : Set β}
Goal: TendstoUniformlyOn F f p s ↔ ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, dist (f x) (F n x) < ε
-- Metric.tendstoLocallyUniformly_iff
theorem tendstoLocallyUniformly_iff [TopologicalSpace β] {F : ι → β → α} {f : β → α}
    {p : Filter ι} :
    TendstoLocallyUniformly F f p ↔
      ∀ ε > 0, ∀ x : β, ∃ t ∈ 𝓝 x, ∀ᶠ n in p, ∀ y ∈ t, dist (f y) (F n y) < ε 
Arguments: {α : Type u} {β : Type v} {ι : Type u_2} [PseudoMetricSpace α] [TopologicalSpace β] {F : ι → β → α} {f : β → α} {p : Filter ι}
Goal: TendstoLocallyUniformly F f p ↔ ∀ ε > 0, ∀ (x : β), ∃ t ∈ 𝓝 x, ∀ᶠ (n : ι) in p, ∀ y ∈ t, dist (f y) (F n y) < ε
-- Metric.tendstoUniformly_iff
theorem tendstoUniformly_iff {F : ι → β → α} {f : β → α} {p : Filter ι} :
    TendstoUniformly F f p ↔ ∀ ε > 0, ∀ᶠ n in p, ∀ x, dist (f x) (F n x) < ε 
Arguments: {α : Type u} {β : Type v} {ι : Type u_2} [PseudoMetricSpace α] {F : ι → β → α} {f : β → α} {p : Filter ι}
Goal: TendstoUniformly F f p ↔ ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ (x : β), dist (f x) (F n x) < ε
-- Metric.cauchy_iff
protected theorem cauchy_iff {f : Filter α} :
    Cauchy f ↔ NeBot f ∧ ∀ ε > 0, ∃ t ∈ f, ∀ x ∈ t, ∀ y ∈ t, dist x y < ε 
Arguments: {α : Type u} [PseudoMetricSpace α] {f : Filter α}
Goal: Cauchy f ↔ f.NeBot ∧ ∀ ε > 0, ∃ t ∈ f, ∀ x ∈ t, ∀ y ∈ t, dist x y < ε
-- Metric.nhds_basis_ball
theorem nhds_basis_ball : (𝓝 x).HasBasis (0 < ·) (ball x) 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α}
Goal: (𝓝 x).HasBasis (fun x => 0 < x) (Metric.ball x)
-- Metric.mem_nhds_iff
theorem mem_nhds_iff : s ∈ 𝓝 x ↔ ∃ ε > 0, ball x ε ⊆ s 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α} {s : Set α}
Goal: s ∈ 𝓝 x ↔ ∃ ε > 0, Metric.ball x ε ⊆ s
-- Metric.eventually_nhds_iff
theorem eventually_nhds_iff {p : α → Prop} :
    (∀ᶠ y in 𝓝 x, p y) ↔ ∃ ε > 0, ∀ ⦃y⦄, dist y x < ε → p y 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α} {p : α → Prop}
Goal: (∀ᶠ (y : α) in 𝓝 x, p y) ↔ ∃ ε > 0, ∀ ⦃y : α⦄, dist y x < ε → p y
-- Metric.eventually_nhds_iff_ball
theorem eventually_nhds_iff_ball {p : α → Prop} :
    (∀ᶠ y in 𝓝 x, p y) ↔ ∃ ε > 0, ∀ y ∈ ball x ε, p y 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α} {p : α → Prop}
Goal: (∀ᶠ (y : α) in 𝓝 x, p y) ↔ ∃ ε > 0, ∀ y ∈ Metric.ball x ε, p y
-- Metric.eventually_nhds_prod_iff
theorem eventually_nhds_prod_iff {f : Filter ι} {x₀ : α} {p : α × ι → Prop} :
    (∀ᶠ x in 𝓝 x₀ ×ˢ f, p x) ↔ ∃ ε > (0 : ℝ), ∃ pa : ι → Prop, (∀ᶠ i in f, pa i) ∧
      ∀ {x}, dist x x₀ < ε → ∀ {i}, pa i → p (x, i) 
Arguments: {α : Type u} {ι : Type u_2} [PseudoMetricSpace α] {f : Filter ι} {x₀ : α} {p : α × ι → Prop}
Goal: (∀ᶠ (x : α × ι) in 𝓝 x₀ ×ˢ f, p x) ↔
  ∃ ε > 0, ∃ pa, (∀ᶠ (i : ι) in f, pa i) ∧ ∀ {x : α}, dist x x₀ < ε → ∀ {i : ι}, pa i → p (x, i)
-- Metric.eventually_prod_nhds_iff
theorem eventually_prod_nhds_iff {f : Filter ι} {x₀ : α} {p : ι × α → Prop} :
    (∀ᶠ x in f ×ˢ 𝓝 x₀, p x) ↔ ∃ pa : ι → Prop, (∀ᶠ i in f, pa i) ∧
      ∃ ε > 0, ∀ {i}, pa i → ∀ {x}, dist x x₀ < ε → p (i, x) 
Arguments: {α : Type u} {ι : Type u_2} [PseudoMetricSpace α] {f : Filter ι} {x₀ : α} {p : ι × α → Prop}
Goal: (∀ᶠ (x : ι × α) in f ×ˢ 𝓝 x₀, p x) ↔
  ∃ pa, (∀ᶠ (i : ι) in f, pa i) ∧ ∃ ε > 0, ∀ {i : ι}, pa i → ∀ {x : α}, dist x x₀ < ε → p (i, x)
-- Metric.nhds_basis_closedBall
theorem nhds_basis_closedBall : (𝓝 x).HasBasis (fun ε : ℝ => 0 < ε) (closedBall x) 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α}
Goal: (𝓝 x).HasBasis (fun ε => 0 < ε) (Metric.closedBall x)
-- Metric.nhds_basis_ball_inv_nat_succ
theorem nhds_basis_ball_inv_nat_succ :
    (𝓝 x).HasBasis (fun _ => True) fun n : ℕ => ball x (1 / (↑n + 1)) 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α}
Goal: (𝓝 x).HasBasis (fun x => True) fun n => Metric.ball x (1 / (↑n + 1))
-- Metric.nhds_basis_ball_inv_nat_pos
theorem nhds_basis_ball_inv_nat_pos :
    (𝓝 x).HasBasis (fun n => 0 < n) fun n : ℕ => ball x (1 / ↑n) 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α}
Goal: (𝓝 x).HasBasis (fun n => 0 < n) fun n => Metric.ball x (1 / ↑n)
-- Metric.nhds_basis_ball_pow
theorem nhds_basis_ball_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) :
    (𝓝 x).HasBasis (fun _ => True) fun n : ℕ => ball x (r ^ n) 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α} {r : ℝ} (h0 : 0 < r) (h1 : r < 1)
Goal: (𝓝 x).HasBasis (fun x => True) fun n => Metric.ball x (r ^ n)
-- Metric.nhds_basis_closedBall_pow
theorem nhds_basis_closedBall_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) :
    (𝓝 x).HasBasis (fun _ => True) fun n : ℕ => closedBall x (r ^ n) 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α} {r : ℝ} (h0 : 0 < r) (h1 : r < 1)
Goal: (𝓝 x).HasBasis (fun x => True) fun n => Metric.closedBall x (r ^ n)
-- Metric.isOpen_iff
theorem isOpen_iff : IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, ball x ε ⊆ s 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α}
Goal: IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, Metric.ball x ε ⊆ s
-- Metric.isOpen_ball
theorem isOpen_ball : IsOpen (ball x ε) 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ}
Goal: IsOpen (Metric.ball x ε)
-- Metric.ball_mem_nhds
theorem ball_mem_nhds (x : α) {ε : ℝ} (ε0 : 0 < ε) : ball x ε ∈ 𝓝 x 
Arguments: {α : Type u} [PseudoMetricSpace α] (x : α) {ε : ℝ} (ε0 : 0 < ε)
Goal: Metric.ball x ε ∈ 𝓝 x
-- Metric.closedBall_mem_nhds
theorem closedBall_mem_nhds (x : α) {ε : ℝ} (ε0 : 0 < ε) : closedBall x ε ∈ 𝓝 x 
Arguments: {α : Type u} [PseudoMetricSpace α] (x : α) {ε : ℝ} (ε0 : 0 < ε)
Goal: Metric.closedBall x ε ∈ 𝓝 x
-- Metric.closedBall_mem_nhds_of_mem
theorem closedBall_mem_nhds_of_mem {x c : α} {ε : ℝ} (h : x ∈ ball c ε) : closedBall c ε ∈ 𝓝 x 
Arguments: {α : Type u} [PseudoMetricSpace α] {x c : α} {ε : ℝ} (h : x ∈ Metric.ball c ε)
Goal: Metric.closedBall c ε ∈ 𝓝 x
-- Metric.nhdsWithin_basis_ball
theorem nhdsWithin_basis_ball {s : Set α} :
    (𝓝[s] x).HasBasis (fun ε : ℝ => 0 < ε) fun ε => ball x ε ∩ s 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α} {s : Set α}
Goal: (𝓝[s] x).HasBasis (fun ε => 0 < ε) fun ε => Metric.ball x ε ∩ s
-- Metric.mem_nhdsWithin_iff
theorem mem_nhdsWithin_iff {t : Set α} : s ∈ 𝓝[t] x ↔ ∃ ε > 0, ball x ε ∩ t ⊆ s 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α} {s t : Set α}
Goal: s ∈ 𝓝[t] x ↔ ∃ ε > 0, Metric.ball x ε ∩ t ⊆ s
-- Metric.tendsto_nhdsWithin_nhdsWithin
theorem tendsto_nhdsWithin_nhdsWithin [PseudoMetricSpace β] {t : Set β} {f : α → β} {a b} :
    Tendsto f (𝓝[s] a) (𝓝[t] b) ↔
      ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, x ∈ s → dist x a < δ → f x ∈ t ∧ dist (f x) b < ε 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] {s : Set α} [PseudoMetricSpace β] {t : Set β} {f : α → β} {a : α} {b : β}
Goal: Filter.Tendsto f (𝓝[s] a) (𝓝[t] b) ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, x ∈ s → dist x a < δ → f x ∈ t ∧ dist (f x) b < ε
-- Metric.tendsto_nhdsWithin_nhds
theorem tendsto_nhdsWithin_nhds [PseudoMetricSpace β] {f : α → β} {a b} :
    Tendsto f (𝓝[s] a) (𝓝 b) ↔
      ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, x ∈ s → dist x a < δ → dist (f x) b < ε 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] {s : Set α} [PseudoMetricSpace β] {f : α → β} {a : α} {b : β}
Goal: Filter.Tendsto f (𝓝[s] a) (𝓝 b) ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, x ∈ s → dist x a < δ → dist (f x) b < ε
-- Metric.tendsto_nhds_nhds
theorem tendsto_nhds_nhds [PseudoMetricSpace β] {f : α → β} {a b} :
    Tendsto f (𝓝 a) (𝓝 b) ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, dist x a < δ → dist (f x) b < ε 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} {a : α} {b : β}
Goal: Filter.Tendsto f (𝓝 a) (𝓝 b) ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, dist x a < δ → dist (f x) b < ε
-- Metric.continuousAt_iff
theorem continuousAt_iff [PseudoMetricSpace β] {f : α → β} {a : α} :
    ContinuousAt f a ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, dist x a < δ → dist (f x) (f a) < ε 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} {a : α}
Goal: ContinuousAt f a ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, dist x a < δ → dist (f x) (f a) < ε
-- Metric.continuousWithinAt_iff
theorem continuousWithinAt_iff [PseudoMetricSpace β] {f : α → β} {a : α} {s : Set α} :
    ContinuousWithinAt f s a ↔
      ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, x ∈ s → dist x a < δ → dist (f x) (f a) < ε 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} {a : α} {s : Set α}
Goal: ContinuousWithinAt f s a ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, x ∈ s → dist x a < δ → dist (f x) (f a) < ε
-- Metric.continuousOn_iff
theorem continuousOn_iff [PseudoMetricSpace β] {f : α → β} {s : Set α} :
    ContinuousOn f s ↔ ∀ b ∈ s, ∀ ε > 0, ∃ δ > 0, ∀ a ∈ s, dist a b < δ → dist (f a) (f b) < ε 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} {s : Set α}
Goal: ContinuousOn f s ↔ ∀ b ∈ s, ∀ ε > 0, ∃ δ > 0, ∀ a ∈ s, dist a b < δ → dist (f a) (f b) < ε
-- Metric.continuous_iff
theorem continuous_iff [PseudoMetricSpace β] {f : α → β} :
    Continuous f ↔ ∀ b, ∀ ε > 0, ∃ δ > 0, ∀ a, dist a b < δ → dist (f a) (f b) < ε 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β}
Goal: Continuous f ↔ ∀ (b : α), ∀ ε > 0, ∃ δ > 0, ∀ (a : α), dist a b < δ → dist (f a) (f b) < ε
-- Metric.tendsto_nhds
theorem tendsto_nhds {f : Filter β} {u : β → α} {a : α} :
    Tendsto u f (𝓝 a) ↔ ∀ ε > 0, ∀ᶠ x in f, dist (u x) a < ε 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] {f : Filter β} {u : β → α} {a : α}
Goal: Filter.Tendsto u f (𝓝 a) ↔ ∀ ε > 0, ∀ᶠ (x : β) in f, dist (u x) a < ε
-- Metric.continuousAt_iff'
theorem continuousAt_iff' [TopologicalSpace β] {f : β → α} {b : β} :
    ContinuousAt f b ↔ ∀ ε > 0, ∀ᶠ x in 𝓝 b, dist (f x) (f b) < ε 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [TopologicalSpace β] {f : β → α} {b : β}
Goal: ContinuousAt f b ↔ ∀ ε > 0, ∀ᶠ (x : β) in 𝓝 b, dist (f x) (f b) < ε
-- Metric.continuousWithinAt_iff'
theorem continuousWithinAt_iff' [TopologicalSpace β] {f : β → α} {b : β} {s : Set β} :
    ContinuousWithinAt f s b ↔ ∀ ε > 0, ∀ᶠ x in 𝓝[s] b, dist (f x) (f b) < ε 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [TopologicalSpace β] {f : β → α} {b : β} {s : Set β}
Goal: ContinuousWithinAt f s b ↔ ∀ ε > 0, ∀ᶠ (x : β) in 𝓝[s] b, dist (f x) (f b) < ε
-- Metric.continuousOn_iff'
theorem continuousOn_iff' [TopologicalSpace β] {f : β → α} {s : Set β} :
    ContinuousOn f s ↔ ∀ b ∈ s, ∀ ε > 0, ∀ᶠ x in 𝓝[s] b, dist (f x) (f b) < ε 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [TopologicalSpace β] {f : β → α} {s : Set β}
Goal: ContinuousOn f s ↔ ∀ b ∈ s, ∀ ε > 0, ∀ᶠ (x : β) in 𝓝[s] b, dist (f x) (f b) < ε
-- Metric.continuous_iff'
theorem continuous_iff' [TopologicalSpace β] {f : β → α} :
    Continuous f ↔ ∀ (a), ∀ ε > 0, ∀ᶠ x in 𝓝 a, dist (f x) (f a) < ε 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [TopologicalSpace β] {f : β → α}
Goal: Continuous f ↔ ∀ (a : β), ∀ ε > 0, ∀ᶠ (x : β) in 𝓝 a, dist (f x) (f a) < ε
-- Metric.tendsto_atTop
theorem tendsto_atTop [Nonempty β] [SemilatticeSup β] {u : β → α} {a : α} :
    Tendsto u atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) a < ε 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [Nonempty β] [SemilatticeSup β] {u : β → α} {a : α}
Goal: Filter.Tendsto u Filter.atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) a < ε
-- Metric.tendsto_atTop'
theorem tendsto_atTop' [Nonempty β] [SemilatticeSup β] [NoMaxOrder β] {u : β → α} {a : α} :
    Tendsto u atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n > N, dist (u n) a < ε 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [Nonempty β] [SemilatticeSup β] [NoMaxOrder β] {u : β → α} {a : α}
Goal: Filter.Tendsto u Filter.atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n > N, dist (u n) a < ε
-- Metric.isOpen_singleton_iff
theorem isOpen_singleton_iff {α : Type*} [PseudoMetricSpace α] {x : α} :
    IsOpen ({x} : Set α) ↔ ∃ ε > 0, ∀ y, dist y x < ε → y = x 
Arguments: {α : Type u_3} [PseudoMetricSpace α] {x : α}
Goal: IsOpen {x} ↔ ∃ ε > 0, ∀ (y : α), dist y x < ε → y = x
-- Metric.exists_ball_inter_eq_singleton_of_mem_discrete
theorem exists_ball_inter_eq_singleton_of_mem_discrete [DiscreteTopology s] {x : α} (hx : x ∈ s) :
    ∃ ε > 0, Metric.ball x ε ∩ s = {x} 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} [DiscreteTopology ↑s] {x : α} (hx : x ∈ s)
Goal: ∃ ε > 0, Metric.ball x ε ∩ s = {x}
-- Metric.exists_closedBall_inter_eq_singleton_of_discrete
theorem exists_closedBall_inter_eq_singleton_of_discrete [DiscreteTopology s] {x : α} (hx : x ∈ s) :
    ∃ ε > 0, Metric.closedBall x ε ∩ s = {x} 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} [DiscreteTopology ↑s] {x : α} (hx : x ∈ s)
Goal: ∃ ε > 0, Metric.closedBall x ε ∩ s = {x}
-- Dense.exists_dist_lt
theorem _root_.Dense.exists_dist_lt {s : Set α} (hs : Dense s) (x : α) {ε : ℝ} (hε : 0 < ε) :
    ∃ y ∈ s, dist x y < ε 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} (hs : Dense s) (x : α) {ε : ℝ} (hε : 0 < ε)
Goal: ∃ y ∈ s, dist x y < ε
-- DenseRange.exists_dist_lt
nonrec theorem _root_.DenseRange.exists_dist_lt {β : Type*} {f : β → α} (hf : DenseRange f) (x : α)
    {ε : ℝ} (hε : 0 < ε) : ∃ y, dist x (f y) < ε 
Arguments: {α : Type u} [PseudoMetricSpace α] {β : Type u_3} {f : β → α} (hf : DenseRange f) (x : α) {ε : ℝ} (hε : 0 < ε)
Goal: ∃ y, dist x (f y) < ε
-- Metric.uniformSpace_eq_bot
protected lemma uniformSpace_eq_bot :
    ‹PseudoMetricSpace α›.toUniformSpace = ⊥ ↔
      ∃ r : ℝ, 0 < r ∧ Pairwise (r ≤ dist · · : α → α → Prop) 
Arguments: {α : Type u} [PseudoMetricSpace α]
Goal: PseudoMetricSpace.toUniformSpace = ⊥ ↔ ∃ r, 0 < r ∧ Pairwise fun x x_1 => r ≤ dist x x_1
-- DiscreteTopology.of_forall_le_dist
lemma DiscreteTopology.of_forall_le_dist {α} [PseudoMetricSpace α] {r : ℝ} (hpos : 0 < r)
    (hr : Pairwise (r ≤ dist · · : α → α → Prop)) : DiscreteTopology α 
Arguments: {α : Type u_3} [PseudoMetricSpace α] {r : ℝ} (hpos : 0 < r) (hr : Pairwise fun x x_1 => r ≤ dist x x_1)
Goal: DiscreteTopology α
-- Metric.uniformity_edist_aux
theorem Metric.uniformity_edist_aux {α} (d : α → α → ℝ≥0) :
    ⨅ ε > (0 : ℝ), 𝓟 { p : α × α | ↑(d p.1 p.2) < ε } =
      ⨅ ε > (0 : ℝ≥0∞), 𝓟 { p : α × α | ↑(d p.1 p.2) < ε } 
Arguments: {α : Type u_3} (d : α → α → ℝ≥0)
Goal: ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | ↑(d p.1 p.2) < ε} = ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | ↑(d p.1 p.2) < ε}
-- Metric.uniformity_edist
theorem Metric.uniformity_edist : 𝓤 α = ⨅ ε > 0, 𝓟 { p : α × α | edist p.1 p.2 < ε } 
Arguments: {α : Type u} [PseudoMetricSpace α]
Goal: 𝓤 α = ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | edist p.1 p.2 < ε}
-- Metric.eball_top_eq_univ
theorem Metric.eball_top_eq_univ (x : α) : EMetric.ball x ∞ = Set.univ 
Arguments: {α : Type u} [PseudoMetricSpace α] (x : α)
Goal: EMetric.ball x ⊤ = Set.univ
-- Metric.emetric_ball
@[simp]
theorem Metric.emetric_ball {x : α} {ε : ℝ} : EMetric.ball x (ENNReal.ofReal ε) = ball x ε 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ}
Goal: EMetric.ball x (ENNReal.ofReal ε) = Metric.ball x ε
-- Metric.emetric_ball_nnreal
@[simp]
theorem Metric.emetric_ball_nnreal {x : α} {ε : ℝ≥0} : EMetric.ball x ε = ball x ε 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ≥0}
Goal: EMetric.ball x ↑ε = Metric.ball x ↑ε
-- Metric.emetric_closedBall
theorem Metric.emetric_closedBall {x : α} {ε : ℝ} (h : 0 ≤ ε) :
    EMetric.closedBall x (ENNReal.ofReal ε) = closedBall x ε 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ} (h : 0 ≤ ε)
Goal: EMetric.closedBall x (ENNReal.ofReal ε) = Metric.closedBall x ε
-- Metric.emetric_closedBall_nnreal
@[simp]
theorem Metric.emetric_closedBall_nnreal {x : α} {ε : ℝ≥0} :
    EMetric.closedBall x ε = closedBall x ε 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ≥0}
Goal: EMetric.closedBall x ↑ε = Metric.closedBall x ↑ε
-- Metric.emetric_ball_top
@[simp]
theorem Metric.emetric_ball_top (x : α) : EMetric.ball x ⊤ = univ 
Arguments: {α : Type u} [PseudoMetricSpace α] (x : α)
Goal: EMetric.ball x ⊤ = Set.univ
-- Metric.inseparable_iff
theorem Metric.inseparable_iff {x y : α} : Inseparable x y ↔ dist x y = 0 
Arguments: {α : Type u} [PseudoMetricSpace α] {x y : α}
Goal: Inseparable x y ↔ dist x y = 0
-- PseudoMetricSpace.replaceUniformity_eq
theorem PseudoMetricSpace.replaceUniformity_eq {α} [U : UniformSpace α] (m : PseudoMetricSpace α)
    (H : 𝓤[U] = 𝓤[PseudoEMetricSpace.toUniformSpace]) : m.replaceUniformity H = m 
Arguments: {α : Type u_3} [U : UniformSpace α] (m : PseudoMetricSpace α) (H : 𝓤 α = 𝓤 α)
Goal: m.replaceUniformity H = m
-- PseudoMetricSpace.replaceTopology_eq
theorem PseudoMetricSpace.replaceTopology_eq {γ} [U : TopologicalSpace γ] (m : PseudoMetricSpace γ)
    (H : U = m.toUniformSpace.toTopologicalSpace) : m.replaceTopology H = m 
Arguments: {γ : Type u_3} [U : TopologicalSpace γ] (m : PseudoMetricSpace γ) (H : U = UniformSpace.toTopologicalSpace)
Goal: m.replaceTopology H = m
-- PseudoMetricSpace.replaceBornology_eq
theorem PseudoMetricSpace.replaceBornology_eq {α} [m : PseudoMetricSpace α] [B : Bornology α]
    (H : ∀ s, @IsBounded _ B s ↔ @IsBounded _ PseudoMetricSpace.toBornology s) :
    PseudoMetricSpace.replaceBornology _ H = m 
Arguments: {α : Type u_3} [m : PseudoMetricSpace α] [B : Bornology α] (H : ∀ (s : Set α), Bornology.IsBounded s ↔ Bornology.IsBounded s)
Goal: m.replaceBornology H = m
-- Real.dist_eq
theorem Real.dist_eq (x y : ℝ) : dist x y = |x - y| 
Arguments: (x y : ℝ)
Goal: dist x y = |x - y|
-- Real.nndist_eq
theorem Real.nndist_eq (x y : ℝ) : nndist x y = Real.nnabs (x - y) 
Arguments: (x y : ℝ)
Goal: nndist x y = Real.nnabs (x - y)
-- Real.nndist_eq'
theorem Real.nndist_eq' (x y : ℝ) : nndist x y = Real.nnabs (y - x) 
Arguments: (x y : ℝ)
Goal: nndist x y = Real.nnabs (y - x)
-- Real.dist_0_eq_abs
theorem Real.dist_0_eq_abs (x : ℝ) : dist x 0 = |x| 
Arguments: (x : ℝ)
Goal: dist x 0 = |x|
-- Real.sub_le_dist
theorem Real.sub_le_dist (x y : ℝ) : x - y ≤ dist x y 
Arguments: (x y : ℝ)
Goal: x - y ≤ dist x y
-- Real.ball_eq_Ioo
theorem Real.ball_eq_Ioo (x r : ℝ) : ball x r = Ioo (x - r) (x + r) 
Arguments: (x r : ℝ)
Goal: Metric.ball x r = Set.Ioo (x - r) (x + r)
-- Real.closedBall_eq_Icc
theorem Real.closedBall_eq_Icc {x r : ℝ} : closedBall x r = Icc (x - r) (x + r) 
Arguments: {x r : ℝ}
Goal: Metric.closedBall x r = Set.Icc (x - r) (x + r)
-- Real.Ioo_eq_ball
theorem Real.Ioo_eq_ball (x y : ℝ) : Ioo x y = ball ((x + y) / 2) ((y - x) / 2) 
Arguments: (x y : ℝ)
Goal: Set.Ioo x y = Metric.ball ((x + y) / 2) ((y - x) / 2)
-- Real.Icc_eq_closedBall
theorem Real.Icc_eq_closedBall (x y : ℝ) : Icc x y = closedBall ((x + y) / 2) ((y - x) / 2) 
Arguments: (x y : ℝ)
Goal: Set.Icc x y = Metric.closedBall ((x + y) / 2) ((y - x) / 2)
-- Metric.uniformity_eq_comap_nhds_zero
theorem Metric.uniformity_eq_comap_nhds_zero :
    𝓤 α = comap (fun p : α × α => dist p.1 p.2) (𝓝 (0 : ℝ)) 
Arguments: {α : Type u} [PseudoMetricSpace α]
Goal: 𝓤 α = Filter.comap (fun p => dist p.1 p.2) (𝓝 0)
-- cauchySeq_iff_tendsto_dist_atTop_0
theorem cauchySeq_iff_tendsto_dist_atTop_0 [Nonempty β] [SemilatticeSup β] {u : β → α} :
    CauchySeq u ↔ Tendsto (fun n : β × β => dist (u n.1) (u n.2)) atTop (𝓝 0) 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [Nonempty β] [SemilatticeSup β] {u : β → α}
Goal: CauchySeq u ↔ Filter.Tendsto (fun n => dist (u n.1) (u n.2)) Filter.atTop (𝓝 0)
-- tendsto_uniformity_iff_dist_tendsto_zero
theorem tendsto_uniformity_iff_dist_tendsto_zero {f : ι → α × α} {p : Filter ι} :
    Tendsto f p (𝓤 α) ↔ Tendsto (fun x => dist (f x).1 (f x).2) p (𝓝 0) 
Arguments: {α : Type u} {ι : Type u_2} [PseudoMetricSpace α] {f : ι → α × α} {p : Filter ι}
Goal: Filter.Tendsto f p (𝓤 α) ↔ Filter.Tendsto (fun x => dist (f x).1 (f x).2) p (𝓝 0)
-- Filter.Tendsto.congr_dist
theorem Filter.Tendsto.congr_dist {f₁ f₂ : ι → α} {p : Filter ι} {a : α}
    (h₁ : Tendsto f₁ p (𝓝 a)) (h : Tendsto (fun x => dist (f₁ x) (f₂ x)) p (𝓝 0)) :
    Tendsto f₂ p (𝓝 a) 
Arguments: {α : Type u} {ι : Type u_2} [PseudoMetricSpace α] {f₁ f₂ : ι → α} {p : Filter ι} {a : α} (h₁ : Filter.Tendsto f₁ p (𝓝 a)) (h : Filter.Tendsto (fun x => dist (f₁ x) (f₂ x)) p (𝓝 0))
Goal: Filter.Tendsto f₂ p (𝓝 a)
-- tendsto_iff_of_dist
theorem tendsto_iff_of_dist {f₁ f₂ : ι → α} {p : Filter ι} {a : α}
    (h : Tendsto (fun x => dist (f₁ x) (f₂ x)) p (𝓝 0)) : Tendsto f₁ p (𝓝 a) ↔ Tendsto f₂ p (𝓝 a) 
Arguments: {α : Type u} {ι : Type u_2} [PseudoMetricSpace α] {f₁ f₂ : ι → α} {p : Filter ι} {a : α} (h : Filter.Tendsto (fun x => dist (f₁ x) (f₂ x)) p (𝓝 0))
Goal: Filter.Tendsto f₁ p (𝓝 a) ↔ Filter.Tendsto f₂ p (𝓝 a)
-- dist_dist_dist_le_left
theorem dist_dist_dist_le_left (x y z : α) : dist (dist x z) (dist y z) ≤ dist x y 
Arguments: {α : Type u} [PseudoMetricSpace α] (x y z : α)
Goal: dist (dist x z) (dist y z) ≤ dist x y
-- dist_dist_dist_le_right
theorem dist_dist_dist_le_right (x y z : α) : dist (dist x y) (dist x z) ≤ dist y z 
Arguments: {α : Type u} [PseudoMetricSpace α] (x y z : α)
Goal: dist (dist x y) (dist x z) ≤ dist y z
-- dist_dist_dist_le
theorem dist_dist_dist_le (x y x' y' : α) : dist (dist x y) (dist x' y') ≤ dist x x' + dist y y' 
Arguments: {α : Type u} [PseudoMetricSpace α] (x y x' y' : α)
Goal: dist (dist x y) (dist x' y') ≤ dist x x' + dist y y'
-- nhds_comap_dist
theorem nhds_comap_dist (a : α) : ((𝓝 (0 : ℝ)).comap (dist · a)) = 𝓝 a 
Arguments: {α : Type u} [PseudoMetricSpace α] (a : α)
Goal: Filter.comap (fun x => dist x a) (𝓝 0) = 𝓝 a
-- tendsto_iff_dist_tendsto_zero
theorem tendsto_iff_dist_tendsto_zero {f : β → α} {x : Filter β} {a : α} :
    Tendsto f x (𝓝 a) ↔ Tendsto (fun b => dist (f b) a) x (𝓝 0) 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] {f : β → α} {x : Filter β} {a : α}
Goal: Filter.Tendsto f x (𝓝 a) ↔ Filter.Tendsto (fun b => dist (f b) a) x (𝓝 0)
-- Metric.ball_subset_interior_closedBall
theorem ball_subset_interior_closedBall : ball x ε ⊆ interior (closedBall x ε) 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ}
Goal: Metric.ball x ε ⊆ interior (Metric.closedBall x ε)
-- Metric.mem_closure_iff
theorem mem_closure_iff {s : Set α} {a : α} : a ∈ closure s ↔ ∀ ε > 0, ∃ b ∈ s, dist a b < ε 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} {a : α}
Goal: a ∈ closure s ↔ ∀ ε > 0, ∃ b ∈ s, dist a b < ε
-- Metric.mem_closure_range_iff
theorem mem_closure_range_iff {e : β → α} {a : α} :
    a ∈ closure (range e) ↔ ∀ ε > 0, ∃ k : β, dist a (e k) < ε 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] {e : β → α} {a : α}
Goal: a ∈ closure (Set.range e) ↔ ∀ ε > 0, ∃ k, dist a (e k) < ε
-- Metric.mem_closure_range_iff_nat
theorem mem_closure_range_iff_nat {e : β → α} {a : α} :
    a ∈ closure (range e) ↔ ∀ n : ℕ, ∃ k : β, dist a (e k) < 1 / ((n : ℝ) + 1) 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] {e : β → α} {a : α}
Goal: a ∈ closure (Set.range e) ↔ ∀ (n : ℕ), ∃ k, dist a (e k) < 1 / (↑n + 1)
-- Metric.mem_of_closed'
theorem mem_of_closed' {s : Set α} (hs : IsClosed s) {a : α} :
    a ∈ s ↔ ∀ ε > 0, ∃ b ∈ s, dist a b < ε 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} (hs : IsClosed s) {a : α}
Goal: a ∈ s ↔ ∀ ε > 0, ∃ b ∈ s, dist a b < ε