ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- cauchySeq_of_le_tendsto_0
theorem cauchySeq_of_le_tendsto_0 {s : β → α} (b : β → ℝ)
    (h : ∀ n m N : β, N ≤ n → N ≤ m → dist (s n) (s m) ≤ b N) (h₀ : Tendsto b atTop (𝓝 0)) :
    CauchySeq s 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [Nonempty β] [SemilatticeSup β] {s : β → α} (b : β → ℝ) (h : ∀ (n m N : β), N ≤ n → N ≤ m → dist (s n) (s m) ≤ b N) (h₀ : Filter.Tendsto b Filter.atTop (𝓝 0))
Goal: CauchySeq s
-- cauchySeq_bdd
theorem cauchySeq_bdd {u : ℕ → α} (hu : CauchySeq u) : ∃ R > 0, ∀ m n, dist (u m) (u n) < R 
Arguments: {α : Type u} [PseudoMetricSpace α] {u : ℕ → α} (hu : CauchySeq u)
Goal: ∃ R > 0, ∀ (m n : ℕ), dist (u m) (u n) < R
-- cauchySeq_iff_le_tendsto_0
theorem cauchySeq_iff_le_tendsto_0 {s : ℕ → α} :
    CauchySeq s ↔
      ∃ b : ℕ → ℝ,
        (∀ n, 0 ≤ b n) ∧
          (∀ n m N : ℕ, N ≤ n → N ≤ m → dist (s n) (s m) ≤ b N) ∧ Tendsto b atTop (𝓝 0) 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : ℕ → α}
Goal: CauchySeq s ↔
  ∃ b,
    (∀ (n : ℕ), 0 ≤ b n) ∧ (∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (s n) (s m) ≤ b N) ∧ Filter.Tendsto b Filter.atTop (𝓝 0)
-- Metric.exists_subseq_bounded_of_cauchySeq
lemma Metric.exists_subseq_bounded_of_cauchySeq (u : ℕ → α) (hu : CauchySeq u) (b : ℕ → ℝ)
    (hb : ∀ n, 0 < b n) :
    ∃ f : ℕ → ℕ, StrictMono f ∧ ∀ n, ∀ m ≥ f n, dist (u m) (u (f n)) < b n 
Arguments: {α : Type u} [PseudoMetricSpace α] (u : ℕ → α) (hu : CauchySeq u) (b : ℕ → ℝ) (hb : ∀ (n : ℕ), 0 < b n)
Goal: ∃ f, StrictMono f ∧ ∀ (n m : ℕ), m ≥ f n → dist (u m) (u (f n)) < b n
-- CantorScheme.map_mem
theorem map_mem (x : (inducedMap A).1) (n : ℕ) : (inducedMap A).2 x ∈ A (res x n) 
Arguments: {β : Type u_1} {α : Type u_2} {A : List β → Set α} (x : ↑(CantorScheme.inducedMap A).fst) (n : ℕ)
Goal: (CantorScheme.inducedMap A).snd x ∈ A (PiNat.res (↑x) n)
-- CantorScheme.ClosureAntitone.antitone
protected theorem ClosureAntitone.antitone [TopologicalSpace α] (hA : ClosureAntitone A) :
    CantorScheme.Antitone A 
Arguments: {β : Type u_1} {α : Type u_2} {A : List β → Set α} [TopologicalSpace α] (hA : CantorScheme.ClosureAntitone A)
Goal: CantorScheme.Antitone A
-- CantorScheme.Antitone.closureAntitone
protected theorem Antitone.closureAntitone [TopologicalSpace α] (hanti : CantorScheme.Antitone A)
    (hclosed : ∀ l, IsClosed (A l)) : ClosureAntitone A 
Arguments: {β : Type u_1} {α : Type u_2} {A : List β → Set α} [TopologicalSpace α] (hanti : CantorScheme.Antitone A) (hclosed : ∀ (l : List β), IsClosed (A l))
Goal: CantorScheme.ClosureAntitone A
-- CantorScheme.Disjoint.map_injective
theorem Disjoint.map_injective (hA : CantorScheme.Disjoint A) : Injective (inducedMap A).2 
Arguments: {β : Type u_1} {α : Type u_2} {A : List β → Set α} (hA : CantorScheme.Disjoint A)
Goal: Function.Injective (CantorScheme.inducedMap A).snd
-- CantorScheme.VanishingDiam.dist_lt
theorem VanishingDiam.dist_lt (hA : VanishingDiam A) (ε : ℝ) (ε_pos : 0 < ε) (x : ℕ → β) :
    ∃ n : ℕ, ∀ (y) (_ : y ∈ A (res x n)) (z) (_ : z ∈ A (res x n)), dist y z < ε 
Arguments: {β : Type u_1} {α : Type u_2} {A : List β → Set α} [PseudoMetricSpace α] (hA : CantorScheme.VanishingDiam A) (ε : ℝ) (ε_pos : 0 < ε) (x : ℕ → β)
Goal: ∃ n, ∀ y ∈ A (PiNat.res x n), ∀ z ∈ A (PiNat.res x n), dist y z < ε
-- CantorScheme.VanishingDiam.map_continuous
theorem VanishingDiam.map_continuous [TopologicalSpace β] [DiscreteTopology β]
    (hA : VanishingDiam A) : Continuous (inducedMap A).2 
Arguments: {β : Type u_1} {α : Type u_2} {A : List β → Set α} [PseudoMetricSpace α] [TopologicalSpace β] [DiscreteTopology β] (hA : CantorScheme.VanishingDiam A)
Goal: Continuous (CantorScheme.inducedMap A).snd
-- CantorScheme.ClosureAntitone.map_of_vanishingDiam
theorem ClosureAntitone.map_of_vanishingDiam [CompleteSpace α] (hdiam : VanishingDiam A)
    (hanti : ClosureAntitone A) (hnonempty : ∀ l, (A l).Nonempty) : (inducedMap A).1 = univ 
Arguments: {β : Type u_1} {α : Type u_2} {A : List β → Set α} [PseudoMetricSpace α] [CompleteSpace α] (hdiam : CantorScheme.VanishingDiam A) (hanti : CantorScheme.ClosureAntitone A) (hnonempty : ∀ (l : List β), (A l).Nonempty)
Goal: (CantorScheme.inducedMap A).fst = Set.univ
-- LipschitzAdd.lipschitz_add
class LipschitzAdd [AddMonoid β] : Prop where
  lipschitz_add : ∃ C, LipschitzWith C fun p : β × β => p.1 + p.2
Arguments: {β : Type u_2} [PseudoMetricSpace β] [AddMonoid β] [self : LipschitzAdd β]
Goal: ∃ C, LipschitzWith C fun p => p.1 + p.2
-- LipschitzMul.lipschitz_mul
@[to_additive]
class LipschitzMul [Monoid β] : Prop where
  lipschitz_mul : ∃ C, LipschitzWith C fun p : β × β => p.1 * p.2
Arguments: {β : Type u_2} [PseudoMetricSpace β] [Monoid β] [self : LipschitzMul β]
Goal: ∃ C, LipschitzWith C fun p => p.1 * p.2
-- lipschitzWith_lipschitz_const_mul_edist
@[to_additive]
theorem lipschitzWith_lipschitz_const_mul_edist [_i : LipschitzMul β] :
    LipschitzWith (LipschitzMul.C β) fun p : β × β => p.1 * p.2 
Arguments: {β : Type u_2} [PseudoMetricSpace β] [Monoid β] [_i : LipschitzMul β]
Goal: LipschitzWith (LipschitzMul.C β) fun p => p.1 * p.2
-- lipschitzWith_lipschitz_const_add_edist
@[to_additive]
theorem lipschitzWith_lipschitz_const_mul_edist [_i : LipschitzMul β] :
    LipschitzWith (LipschitzMul.C β) fun p : β × β => p.1 * p.2 
Arguments: {β : Type u_2} [PseudoMetricSpace β] [AddMonoid β] [_i : LipschitzAdd β]
Goal: LipschitzWith (LipschitzAdd.C β) fun p => p.1 + p.2
-- lipschitz_with_lipschitz_const_mul
@[to_additive]
theorem lipschitz_with_lipschitz_const_mul :
    ∀ p q : β × β, dist (p.1 * p.2) (q.1 * q.2) ≤ LipschitzMul.C β * dist p q 
Arguments: {β : Type u_2} [PseudoMetricSpace β] [Monoid β] [LipschitzMul β] (p q : β × β)
Goal: dist (p.1 * p.2) (q.1 * q.2) ≤ ↑(LipschitzMul.C β) * dist p q
-- lipschitz_with_lipschitz_const_add
@[to_additive]
theorem lipschitz_with_lipschitz_const_mul :
    ∀ p q : β × β, dist (p.1 * p.2) (q.1 * q.2) ≤ LipschitzMul.C β * dist p q 
Arguments: {β : Type u_2} [PseudoMetricSpace β] [AddMonoid β] [LipschitzAdd β] (p q : β × β)
Goal: dist (p.1 + p.2) (q.1 + q.2) ≤ ↑(LipschitzAdd.C β) * dist p q
-- BoundedSMul.dist_smul_pair'
class BoundedSMul : Prop where
  dist_smul_pair' : ∀ x : α, ∀ y₁ y₂ : β, dist (x • y₁) (x • y₂) ≤ dist x 0 * dist y₁ y₂
  dist_pair_smul' : ∀ x₁ x₂ : α, ∀ y : β, dist (x₁ • y) (x₂ • y) ≤ dist x₁ x₂ * dist y 0
Arguments: {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] [Zero α] [Zero β] [SMul α β] [self : BoundedSMul α β] (x : α) (y₁ y₂ : β)
Goal: dist (x • y₁) (x • y₂) ≤ dist x 0 * dist y₁ y₂
-- BoundedSMul.dist_pair_smul'
class BoundedSMul : Prop where
  dist_smul_pair' : ∀ x : α, ∀ y₁ y₂ : β, dist (x • y₁) (x • y₂) ≤ dist x 0 * dist y₁ y₂
  dist_pair_smul' : ∀ x₁ x₂ : α, ∀ y : β, dist (x₁ • y) (x₂ • y) ≤ dist x₁ x₂ * dist y 0
Arguments: {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] [Zero α] [Zero β] [SMul α β] [self : BoundedSMul α β] (x₁ x₂ : α) (y : β)
Goal: dist (x₁ • y) (x₂ • y) ≤ dist x₁ x₂ * dist y 0
-- dist_smul_pair
theorem dist_smul_pair (x : α) (y₁ y₂ : β) : dist (x • y₁) (x • y₂) ≤ dist x 0 * dist y₁ y₂ 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] [Zero α] [Zero β] [SMul α β] [BoundedSMul α β] (x : α) (y₁ y₂ : β)
Goal: dist (x • y₁) (x • y₂) ≤ dist x 0 * dist y₁ y₂
-- dist_pair_smul
theorem dist_pair_smul (x₁ x₂ : α) (y : β) : dist (x₁ • y) (x₂ • y) ≤ dist x₁ x₂ * dist y 0 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] [Zero α] [Zero β] [SMul α β] [BoundedSMul α β] (x₁ x₂ : α) (y : β)
Goal: dist (x₁ • y) (x₂ • y) ≤ dist x₁ x₂ * dist y 0
-- KuratowskiEmbedding.embeddingOfSubset_coe
theorem embeddingOfSubset_coe : embeddingOfSubset x a n = dist a (x n) - dist (x 0) (x n) 
Arguments: {α : Type u} {n : ℕ} [MetricSpace α] (x : ℕ → α) (a : α)
Goal: ↑(KuratowskiEmbedding.embeddingOfSubset x a) n = dist a (x n) - dist (x 0) (x n)
-- KuratowskiEmbedding.embeddingOfSubset_dist_le
theorem embeddingOfSubset_dist_le (a b : α) :
    dist (embeddingOfSubset x a) (embeddingOfSubset x b) ≤ dist a b 
Arguments: {α : Type u} [MetricSpace α] (x : ℕ → α) (a b : α)
Goal: dist (KuratowskiEmbedding.embeddingOfSubset x a) (KuratowskiEmbedding.embeddingOfSubset x b) ≤ dist a b
-- KuratowskiEmbedding.embeddingOfSubset_isometry
theorem embeddingOfSubset_isometry (H : DenseRange x) : Isometry (embeddingOfSubset x) 
Arguments: {α : Type u} [MetricSpace α] (x : ℕ → α) (H : DenseRange x)
Goal: Isometry (KuratowskiEmbedding.embeddingOfSubset x)
-- KuratowskiEmbedding.exists_isometric_embedding
theorem exists_isometric_embedding (α : Type u) [MetricSpace α] [SeparableSpace α] :
    ∃ f : α → ℓ^∞(ℕ), Isometry f 
Arguments: (α : Type u) [MetricSpace α] [TopologicalSpace.SeparableSpace α]
Goal: ∃ f, Isometry f
-- kuratowskiEmbedding.isometry
protected theorem kuratowskiEmbedding.isometry (α : Type u) [MetricSpace α] [SeparableSpace α] :
    Isometry (kuratowskiEmbedding α) 
Arguments: (α : Type u) [MetricSpace α] [TopologicalSpace.SeparableSpace α]
Goal: Isometry (kuratowskiEmbedding α)
-- LipschitzOnWith.extend_lp_infty
theorem LipschitzOnWith.extend_lp_infty [PseudoMetricSpace α] {s : Set α} {ι : Type*}
    {f : α → ℓ^∞(ι)} {K : ℝ≥0} (hfl : LipschitzOnWith K f s) :
    ∃ g : α → ℓ^∞(ι), LipschitzWith K g ∧ EqOn f g s 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} {ι : Type u_1} {f : α → ↥(lp (fun i => ℝ) ⊤)} {K : ℝ≥0} (hfl : LipschitzOnWith K f s)
Goal: ∃ g, LipschitzWith K g ∧ Set.EqOn f g s
-- UniformSpace.Completion.uniformContinuous_dist
protected theorem uniformContinuous_dist :
    UniformContinuous fun p : Completion α × Completion α ↦ dist p.1 p.2 
Arguments: {α : Type u} [PseudoMetricSpace α]
Goal: UniformContinuous fun p => dist p.1 p.2
-- UniformSpace.Completion.continuous_dist
protected theorem continuous_dist [TopologicalSpace β] {f g : β → Completion α} (hf : Continuous f)
    (hg : Continuous g) : Continuous fun x ↦ dist (f x) (g x) 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [TopologicalSpace β] {f g : β → UniformSpace.Completion α} (hf : Continuous f) (hg : Continuous g)
Goal: Continuous fun x => dist (f x) (g x)
-- UniformSpace.Completion.dist_eq
@[simp]
protected theorem dist_eq (x y : α) : dist (x : Completion α) y = dist x y 
Arguments: {α : Type u} [PseudoMetricSpace α] (x y : α)
Goal: dist (↑α x) (↑α y) = dist x y
-- UniformSpace.Completion.dist_self
protected theorem dist_self (x : Completion α) : dist x x = 0 
Arguments: {α : Type u} [PseudoMetricSpace α] (x : UniformSpace.Completion α)
Goal: dist x x = 0
-- UniformSpace.Completion.dist_comm
protected theorem dist_comm (x y : Completion α) : dist x y = dist y x 
Arguments: {α : Type u} [PseudoMetricSpace α] (x y : UniformSpace.Completion α)
Goal: dist x y = dist y x
-- UniformSpace.Completion.dist_triangle
protected theorem dist_triangle (x y z : Completion α) : dist x z ≤ dist x y + dist y z 
Arguments: {α : Type u} [PseudoMetricSpace α] (x y z : UniformSpace.Completion α)
Goal: dist x z ≤ dist x y + dist y z
-- UniformSpace.Completion.mem_uniformity_dist
protected theorem mem_uniformity_dist (s : Set (Completion α × Completion α)) :
    s ∈ 𝓤 (Completion α) ↔ ∃ ε > 0, ∀ {a b}, dist a b < ε → (a, b) ∈ s 
Arguments: {α : Type u} [PseudoMetricSpace α] (s : Set (UniformSpace.Completion α × UniformSpace.Completion α))
Goal: s ∈ 𝓤 (UniformSpace.Completion α) ↔ ∃ ε > 0, ∀ {a b : UniformSpace.Completion α}, dist a b < ε → (a, b) ∈ s
-- UniformSpace.Completion.uniformity_dist'
protected theorem uniformity_dist' :
    𝓤 (Completion α) = ⨅ ε : { ε : ℝ // 0 < ε }, 𝓟 { p | dist p.1 p.2 < ε.val } 
Arguments: {α : Type u} [PseudoMetricSpace α]
Goal: 𝓤 (UniformSpace.Completion α) = ⨅ ε, 𝓟 {p | dist p.1 p.2 < ↑ε}
-- UniformSpace.Completion.uniformity_dist
protected theorem uniformity_dist : 𝓤 (Completion α) = ⨅ ε > 0, 𝓟 { p | dist p.1 p.2 < ε } 
Arguments: {α : Type u} [PseudoMetricSpace α]
Goal: 𝓤 (UniformSpace.Completion α) = ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | dist p.1 p.2 < ε}
-- UniformSpace.Completion.eq_of_dist_eq_zero
@[deprecated eq_of_dist_eq_zero (since := "2024-03-10")]
protected theorem eq_of_dist_eq_zero (x y : Completion α) (h : dist x y = 0) : x = y 
Arguments: {α : Type u} [PseudoMetricSpace α] (x y : UniformSpace.Completion α) (h : dist x y = 0)
Goal: x = y
-- UniformSpace.Completion.coe_isometry
theorem coe_isometry : Isometry ((↑) : α → Completion α) 
Arguments: {α : Type u} [PseudoMetricSpace α]
Goal: Isometry ↑α
-- UniformSpace.Completion.edist_eq
@[simp]
protected theorem edist_eq (x y : α) : edist (x : Completion α) y = edist x y 
Arguments: {α : Type u} [PseudoMetricSpace α] (x y : α)
Goal: edist (↑α x) (↑α y) = edist x y
-- LipschitzWith.completion_extension
theorem LipschitzWith.completion_extension [MetricSpace β] [CompleteSpace β] {f : α → β}
    {K : ℝ≥0} (h : LipschitzWith K f) : LipschitzWith K (Completion.extension f) 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [MetricSpace β] [CompleteSpace β] {f : α → β} {K : ℝ≥0} (h : LipschitzWith K f)
Goal: LipschitzWith K (UniformSpace.Completion.extension f)
-- LipschitzWith.completion_map
theorem LipschitzWith.completion_map [PseudoMetricSpace β] {f : α → β} {K : ℝ≥0}
    (h : LipschitzWith K f) : LipschitzWith K (Completion.map f) 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} {K : ℝ≥0} (h : LipschitzWith K f)
Goal: LipschitzWith K (UniformSpace.Completion.map f)
-- Isometry.completion_extension
theorem Isometry.completion_extension [MetricSpace β] [CompleteSpace β] {f : α → β}
    (h : Isometry f) : Isometry (Completion.extension f) 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [MetricSpace β] [CompleteSpace β] {f : α → β} (h : Isometry f)
Goal: Isometry (UniformSpace.Completion.extension f)
-- Isometry.completion_map
theorem Isometry.completion_map [PseudoMetricSpace β] {f : α → β}
    (h : Isometry f) : Isometry (Completion.map f) 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} (h : Isometry f)
Goal: Isometry (UniformSpace.Completion.map f)
-- Filter.HasBasis.cauchy_iff
theorem Filter.HasBasis.cauchy_iff {ι} {p : ι → Prop} {s : ι → Set (α × α)} (h : (𝓤 α).HasBasis p s)
    {f : Filter α} :
    Cauchy f ↔ NeBot f ∧ ∀ i, p i → ∃ t ∈ f, ∀ x ∈ t, ∀ y ∈ t, (x, y) ∈ s i 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {ι : Sort u_1} {p : ι → Prop} {s : ι → Set (α × α)} (h : (𝓤 α).HasBasis p s) {f : Filter α}
Goal: Cauchy f ↔ f.NeBot ∧ ∀ (i : ι), p i → ∃ t ∈ f, ∀ x ∈ t, ∀ y ∈ t, (x, y) ∈ s i
-- cauchy_iff'
theorem cauchy_iff' {f : Filter α} :
    Cauchy f ↔ NeBot f ∧ ∀ s ∈ 𝓤 α, ∃ t ∈ f, ∀ x ∈ t, ∀ y ∈ t, (x, y) ∈ s 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {f : Filter α}
Goal: Cauchy f ↔ f.NeBot ∧ ∀ s ∈ 𝓤 α, ∃ t ∈ f, ∀ x ∈ t, ∀ y ∈ t, (x, y) ∈ s
-- cauchy_iff
theorem cauchy_iff {f : Filter α} : Cauchy f ↔ NeBot f ∧ ∀ s ∈ 𝓤 α, ∃ t ∈ f, t ×ˢ t ⊆ s 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {f : Filter α}
Goal: Cauchy f ↔ f.NeBot ∧ ∀ s ∈ 𝓤 α, ∃ t ∈ f, t ×ˢ t ⊆ s
-- cauchy_iff_le
lemma cauchy_iff_le {l : Filter α} [hl : l.NeBot] :
    Cauchy l ↔ l ×ˢ l ≤ 𝓤 α 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {l : Filter α} [hl : l.NeBot]
Goal: Cauchy l ↔ l ×ˢ l ≤ 𝓤 α
-- Cauchy.ultrafilter_of
theorem Cauchy.ultrafilter_of {l : Filter α} (h : Cauchy l) :
    Cauchy (@Ultrafilter.of _ l h.1 : Filter α) 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {l : Filter α} (h : Cauchy l)
Goal: Cauchy ↑(Ultrafilter.of l)
-- cauchy_map_iff
theorem cauchy_map_iff {l : Filter β} {f : β → α} :
    Cauchy (l.map f) ↔ NeBot l ∧ Tendsto (fun p : β × β => (f p.1, f p.2)) (l ×ˢ l) (𝓤 α) 
Arguments: {α : Type u} {β : Type v} [uniformSpace : UniformSpace α] {l : Filter β} {f : β → α}
Goal: Cauchy (Filter.map f l) ↔ l.NeBot ∧ Filter.Tendsto (fun p => (f p.1, f p.2)) (l ×ˢ l) (𝓤 α)
-- cauchy_map_iff'
theorem cauchy_map_iff' {l : Filter β} [hl : NeBot l] {f : β → α} :
    Cauchy (l.map f) ↔ Tendsto (fun p : β × β => (f p.1, f p.2)) (l ×ˢ l) (𝓤 α) 
Arguments: {α : Type u} {β : Type v} [uniformSpace : UniformSpace α] {l : Filter β} [hl : l.NeBot] {f : β → α}
Goal: Cauchy (Filter.map f l) ↔ Filter.Tendsto (fun p => (f p.1, f p.2)) (l ×ˢ l) (𝓤 α)
-- Cauchy.mono
theorem Cauchy.mono {f g : Filter α} [hg : NeBot g] (h_c : Cauchy f) (h_le : g ≤ f) : Cauchy g 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {f g : Filter α} [hg : g.NeBot] (h_c : Cauchy f) (h_le : g ≤ f)
Goal: Cauchy g
-- Cauchy.mono'
theorem Cauchy.mono' {f g : Filter α} (h_c : Cauchy f) (_ : NeBot g) (h_le : g ≤ f) : Cauchy g 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {f g : Filter α} (h_c : Cauchy f)
Goal: g.NeBot → g ≤ f → Cauchy g
-- cauchy_nhds
theorem cauchy_nhds {a : α} : Cauchy (𝓝 a) 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {a : α}
Goal: Cauchy (𝓝 a)
-- cauchy_pure
theorem cauchy_pure {a : α} : Cauchy (pure a) 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {a : α}
Goal: Cauchy (pure a)
-- Filter.Tendsto.cauchy_map
theorem Filter.Tendsto.cauchy_map {l : Filter β} [NeBot l] {f : β → α} {a : α}
    (h : Tendsto f l (𝓝 a)) : Cauchy (map f l) 
Arguments: {α : Type u} {β : Type v} [uniformSpace : UniformSpace α] {l : Filter β} [l.NeBot] {f : β → α} {a : α} (h : Filter.Tendsto f l (𝓝 a))
Goal: Cauchy (Filter.map f l)
-- Cauchy.mono_uniformSpace
lemma Cauchy.mono_uniformSpace {u v : UniformSpace β} {F : Filter β} (huv : u ≤ v)
    (hF : Cauchy (uniformSpace := u) F) : Cauchy (uniformSpace := v) F 
Arguments: {β : Type v} {u v : UniformSpace β} {F : Filter β} (huv : u ≤ v) (hF : Cauchy F)
Goal: Cauchy F
-- cauchy_inf_uniformSpace
lemma cauchy_inf_uniformSpace {u v : UniformSpace β} {F : Filter β} :
    Cauchy (uniformSpace := u ⊓ v) F ↔
    Cauchy (uniformSpace := u) F ∧ Cauchy (uniformSpace := v) F 
Arguments: {β : Type v} {u v : UniformSpace β} {F : Filter β}
Goal: Cauchy F ↔ Cauchy F ∧ Cauchy F
-- cauchy_iInf_uniformSpace
lemma cauchy_iInf_uniformSpace {ι : Sort*} [Nonempty ι] {u : ι → UniformSpace β}
    {l : Filter β} :
    Cauchy (uniformSpace := ⨅ i, u i) l ↔ ∀ i, Cauchy (uniformSpace := u i) l 
Arguments: {β : Type v} {ι : Sort u_1} [Nonempty ι] {u : ι → UniformSpace β} {l : Filter β}
Goal: Cauchy l ↔ ∀ (i : ι), Cauchy l
-- cauchy_iInf_uniformSpace'
lemma cauchy_iInf_uniformSpace' {ι : Sort*} {u : ι → UniformSpace β}
    {l : Filter β} [l.NeBot] :
    Cauchy (uniformSpace := ⨅ i, u i) l ↔ ∀ i, Cauchy (uniformSpace := u i) l 
Arguments: {β : Type v} {ι : Sort u_1} {u : ι → UniformSpace β} {l : Filter β} [l.NeBot]
Goal: Cauchy l ↔ ∀ (i : ι), Cauchy l
-- cauchy_comap_uniformSpace
lemma cauchy_comap_uniformSpace {u : UniformSpace β} {f : α → β} {l : Filter α} :
    Cauchy (uniformSpace := comap f u) l ↔ Cauchy (map f l) 
Arguments: {α : Type u} {β : Type v} {u : UniformSpace β} {f : α → β} {l : Filter α}
Goal: Cauchy l ↔ Cauchy (Filter.map f l)
-- cauchy_prod_iff
lemma cauchy_prod_iff [UniformSpace β] {F : Filter (α × β)} :
    Cauchy F ↔ Cauchy (map Prod.fst F) ∧ Cauchy (map Prod.snd F) 
Arguments: {α : Type u} {β : Type v} [uniformSpace : UniformSpace α] [UniformSpace β] {F : Filter (α × β)}
Goal: Cauchy F ↔ Cauchy (Filter.map Prod.fst F) ∧ Cauchy (Filter.map Prod.snd F)
-- Cauchy.prod
theorem Cauchy.prod [UniformSpace β] {f : Filter α} {g : Filter β} (hf : Cauchy f) (hg : Cauchy g) :
    Cauchy (f ×ˢ g) 
Arguments: {α : Type u} {β : Type v} [uniformSpace : UniformSpace α] [UniformSpace β] {f : Filter α} {g : Filter β} (hf : Cauchy f) (hg : Cauchy g)
Goal: Cauchy (f ×ˢ g)
-- le_nhds_of_cauchy_adhp_aux
theorem le_nhds_of_cauchy_adhp_aux {f : Filter α} {x : α}
    (adhs : ∀ s ∈ 𝓤 α, ∃ t ∈ f, t ×ˢ t ⊆ s ∧ ∃ y, (x, y) ∈ s ∧ y ∈ t) : f ≤ 𝓝 x 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {f : Filter α} {x : α} (adhs : ∀ s ∈ 𝓤 α, ∃ t ∈ f, t ×ˢ t ⊆ s ∧ ∃ y, (x, y) ∈ s ∧ y ∈ t)
Goal: f ≤ 𝓝 x
-- le_nhds_of_cauchy_adhp
theorem le_nhds_of_cauchy_adhp {f : Filter α} {x : α} (hf : Cauchy f) (adhs : ClusterPt x f) :
    f ≤ 𝓝 x 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {f : Filter α} {x : α} (hf : Cauchy f) (adhs : ClusterPt x f)
Goal: f ≤ 𝓝 x
-- le_nhds_iff_adhp_of_cauchy
theorem le_nhds_iff_adhp_of_cauchy {f : Filter α} {x : α} (hf : Cauchy f) :
    f ≤ 𝓝 x ↔ ClusterPt x f 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {f : Filter α} {x : α} (hf : Cauchy f)
Goal: f ≤ 𝓝 x ↔ ClusterPt x f
-- Cauchy.map
nonrec theorem Cauchy.map [UniformSpace β] {f : Filter α} {m : α → β} (hf : Cauchy f)
    (hm : UniformContinuous m) : Cauchy (map m f) 
Arguments: {α : Type u} {β : Type v} [uniformSpace : UniformSpace α] [UniformSpace β] {f : Filter α} {m : α → β} (hf : Cauchy f) (hm : UniformContinuous m)
Goal: Cauchy (Filter.map m f)
-- Cauchy.comap
nonrec theorem Cauchy.comap [UniformSpace β] {f : Filter β} {m : α → β} (hf : Cauchy f)
    (hm : comap (fun p : α × α => (m p.1, m p.2)) (𝓤 β) ≤ 𝓤 α) [NeBot (comap m f)] :
    Cauchy (comap m f) 
Arguments: {α : Type u} {β : Type v} [uniformSpace : UniformSpace α] [UniformSpace β] {f : Filter β} {m : α → β} (hf : Cauchy f) (hm : Filter.comap (fun p => (m p.1, m p.2)) (𝓤 β) ≤ 𝓤 α) [(Filter.comap m f).NeBot]
Goal: Cauchy (Filter.comap m f)
-- Cauchy.comap'
theorem Cauchy.comap' [UniformSpace β] {f : Filter β} {m : α → β} (hf : Cauchy f)
    (hm : Filter.comap (fun p : α × α => (m p.1, m p.2)) (𝓤 β) ≤ 𝓤 α)
    (_ : NeBot (Filter.comap m f)) : Cauchy (Filter.comap m f) 
Arguments: {α : Type u} {β : Type v} [uniformSpace : UniformSpace α] [UniformSpace β] {f : Filter β} {m : α → β} (hf : Cauchy f) (hm : Filter.comap (fun p => (m p.1, m p.2)) (𝓤 β) ≤ 𝓤 α)
Goal: (Filter.comap m f).NeBot → Cauchy (Filter.comap m f)
-- CauchySeq.tendsto_uniformity
theorem CauchySeq.tendsto_uniformity [Preorder β] {u : β → α} (h : CauchySeq u) :
    Tendsto (Prod.map u u) atTop (𝓤 α) 
Arguments: {α : Type u} {β : Type v} [uniformSpace : UniformSpace α] [Preorder β] {u : β → α} (h : CauchySeq u)
Goal: Filter.Tendsto (Prod.map u u) Filter.atTop (𝓤 α)
-- CauchySeq.nonempty
theorem CauchySeq.nonempty [Preorder β] {u : β → α} (hu : CauchySeq u) : Nonempty β 
Arguments: {α : Type u} {β : Type v} [uniformSpace : UniformSpace α] [Preorder β] {u : β → α} (hu : CauchySeq u)
Goal: Nonempty β
-- CauchySeq.mem_entourage
theorem CauchySeq.mem_entourage {β : Type*} [SemilatticeSup β] {u : β → α} (h : CauchySeq u)
    {V : Set (α × α)} (hV : V ∈ 𝓤 α) : ∃ k₀, ∀ i j, k₀ ≤ i → k₀ ≤ j → (u i, u j) ∈ V 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {β : Type u_1} [SemilatticeSup β] {u : β → α} (h : CauchySeq u) {V : Set (α × α)} (hV : V ∈ 𝓤 α)
Goal: ∃ k₀, ∀ (i j : β), k₀ ≤ i → k₀ ≤ j → (u i, u j) ∈ V
-- Filter.Tendsto.cauchySeq
theorem Filter.Tendsto.cauchySeq [SemilatticeSup β] [Nonempty β] {f : β → α} {x}
    (hx : Tendsto f atTop (𝓝 x)) : CauchySeq f 
Arguments: {α : Type u} {β : Type v} [uniformSpace : UniformSpace α] [SemilatticeSup β] [Nonempty β] {f : β → α} {x : α} (hx : Filter.Tendsto f Filter.atTop (𝓝 x))
Goal: CauchySeq f
-- cauchySeq_const
theorem cauchySeq_const [SemilatticeSup β] [Nonempty β] (x : α) : CauchySeq fun _ : β => x 
Arguments: {α : Type u} {β : Type v} [uniformSpace : UniformSpace α] [SemilatticeSup β] [Nonempty β] (x : α)
Goal: CauchySeq fun x_1 => x
-- cauchySeq_iff_tendsto
theorem cauchySeq_iff_tendsto [Nonempty β] [SemilatticeSup β] {u : β → α} :
    CauchySeq u ↔ Tendsto (Prod.map u u) atTop (𝓤 α) 
Arguments: {α : Type u} {β : Type v} [uniformSpace : UniformSpace α] [Nonempty β] [SemilatticeSup β] {u : β → α}
Goal: CauchySeq u ↔ Filter.Tendsto (Prod.map u u) Filter.atTop (𝓤 α)
-- CauchySeq.comp_tendsto
theorem CauchySeq.comp_tendsto {γ} [Preorder β] [SemilatticeSup γ] [Nonempty γ] {f : β → α}
    (hf : CauchySeq f) {g : γ → β} (hg : Tendsto g atTop atTop) : CauchySeq (f ∘ g) 
Arguments: {α : Type u} {β : Type v} [uniformSpace : UniformSpace α] {γ : Type u_1} [Preorder β] [SemilatticeSup γ] [Nonempty γ] {f : β → α} (hf : CauchySeq f) {g : γ → β} (hg : Filter.Tendsto g Filter.atTop Filter.atTop)
Goal: CauchySeq (f ∘ g)
-- CauchySeq.comp_injective
theorem CauchySeq.comp_injective [SemilatticeSup β] [NoMaxOrder β] [Nonempty β] {u : ℕ → α}
    (hu : CauchySeq u) {f : β → ℕ} (hf : Injective f) : CauchySeq (u ∘ f) 
Arguments: {α : Type u} {β : Type v} [uniformSpace : UniformSpace α] [SemilatticeSup β] [NoMaxOrder β] [Nonempty β] {u : ℕ → α} (hu : CauchySeq u) {f : β → ℕ} (hf : Function.Injective f)
Goal: CauchySeq (u ∘ f)