ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Function.Bijective.cauchySeq_comp_iff
theorem Function.Bijective.cauchySeq_comp_iff {f : ℕ → ℕ} (hf : Bijective f) (u : ℕ → α) :
    CauchySeq (u ∘ f) ↔ CauchySeq u 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {f : ℕ → ℕ} (hf : Function.Bijective f) (u : ℕ → α)
Goal: CauchySeq (u ∘ f) ↔ CauchySeq u
-- CauchySeq.subseq_subseq_mem
theorem CauchySeq.subseq_subseq_mem {V : ℕ → Set (α × α)} (hV : ∀ n, V n ∈ 𝓤 α) {u : ℕ → α}
    (hu : CauchySeq u) {f g : ℕ → ℕ} (hf : Tendsto f atTop atTop) (hg : Tendsto g atTop atTop) :
    ∃ φ : ℕ → ℕ, StrictMono φ ∧ ∀ n, ((u ∘ f ∘ φ) n, (u ∘ g ∘ φ) n) ∈ V n 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {V : ℕ → Set (α × α)} (hV : ∀ (n : ℕ), V n ∈ 𝓤 α) {u : ℕ → α} (hu : CauchySeq u) {f g : ℕ → ℕ} (hf : Filter.Tendsto f Filter.atTop Filter.atTop) (hg : Filter.Tendsto g Filter.atTop Filter.atTop)
Goal: ∃ φ, StrictMono φ ∧ ∀ (n : ℕ), ((u ∘ f ∘ φ) n, (u ∘ g ∘ φ) n) ∈ V n
-- cauchySeq_iff'
theorem cauchySeq_iff' {u : ℕ → α} :
    CauchySeq u ↔ ∀ V ∈ 𝓤 α, ∀ᶠ k in atTop, k ∈ Prod.map u u ⁻¹' V 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {u : ℕ → α}
Goal: CauchySeq u ↔ ∀ V ∈ 𝓤 α, ∀ᶠ (k : ℕ × ℕ) in Filter.atTop, k ∈ Prod.map u u ⁻¹' V
-- cauchySeq_iff
theorem cauchySeq_iff {u : ℕ → α} :
    CauchySeq u ↔ ∀ V ∈ 𝓤 α, ∃ N, ∀ k ≥ N, ∀ l ≥ N, (u k, u l) ∈ V 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {u : ℕ → α}
Goal: CauchySeq u ↔ ∀ V ∈ 𝓤 α, ∃ N, ∀ k ≥ N, ∀ l ≥ N, (u k, u l) ∈ V
-- CauchySeq.prod_map
theorem CauchySeq.prod_map {γ δ} [UniformSpace β] [Preorder γ] [Preorder δ] {u : γ → α}
    {v : δ → β} (hu : CauchySeq u) (hv : CauchySeq v) : CauchySeq (Prod.map u v) 
Arguments: {α : Type u} {β : Type v} [uniformSpace : UniformSpace α] {γ : Type u_1} {δ : Type u_2} [UniformSpace β] [Preorder γ] [Preorder δ] {u : γ → α} {v : δ → β} (hu : CauchySeq u) (hv : CauchySeq v)
Goal: CauchySeq (Prod.map u v)
-- CauchySeq.prod
theorem CauchySeq.prod {γ} [UniformSpace β] [Preorder γ] {u : γ → α} {v : γ → β}
    (hu : CauchySeq u) (hv : CauchySeq v) : CauchySeq fun x => (u x, v x) 
Arguments: {α : Type u} {β : Type v} [uniformSpace : UniformSpace α] {γ : Type u_1} [UniformSpace β] [Preorder γ] {u : γ → α} {v : γ → β} (hu : CauchySeq u) (hv : CauchySeq v)
Goal: CauchySeq fun x => (u x, v x)
-- CauchySeq.eventually_eventually
theorem CauchySeq.eventually_eventually [SemilatticeSup β] {u : β → α} (hu : CauchySeq u)
    {V : Set (α × α)} (hV : V ∈ 𝓤 α) : ∀ᶠ k in atTop, ∀ᶠ l in atTop, (u k, u l) ∈ V 
Arguments: {α : Type u} {β : Type v} [uniformSpace : UniformSpace α] [SemilatticeSup β] {u : β → α} (hu : CauchySeq u) {V : Set (α × α)} (hV : V ∈ 𝓤 α)
Goal: ∀ᶠ (k : β) (l : β) in Filter.atTop, (u k, u l) ∈ V
-- UniformContinuous.comp_cauchySeq
theorem UniformContinuous.comp_cauchySeq {γ} [UniformSpace β] [Preorder γ] {f : α → β}
    (hf : UniformContinuous f) {u : γ → α} (hu : CauchySeq u) : CauchySeq (f ∘ u) 
Arguments: {α : Type u} {β : Type v} [uniformSpace : UniformSpace α] {γ : Type u_1} [UniformSpace β] [Preorder γ] {f : α → β} (hf : UniformContinuous f) {u : γ → α} (hu : CauchySeq u)
Goal: CauchySeq (f ∘ u)
-- CauchySeq.subseq_mem
theorem CauchySeq.subseq_mem {V : ℕ → Set (α × α)} (hV : ∀ n, V n ∈ 𝓤 α) {u : ℕ → α}
    (hu : CauchySeq u) : ∃ φ : ℕ → ℕ, StrictMono φ ∧ ∀ n, (u <| φ (n + 1), u <| φ n) ∈ V n 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {V : ℕ → Set (α × α)} (hV : ∀ (n : ℕ), V n ∈ 𝓤 α) {u : ℕ → α} (hu : CauchySeq u)
Goal: ∃ φ, StrictMono φ ∧ ∀ (n : ℕ), (u (φ (n + 1)), u (φ n)) ∈ V n
-- Filter.Tendsto.subseq_mem_entourage
theorem Filter.Tendsto.subseq_mem_entourage {V : ℕ → Set (α × α)} (hV : ∀ n, V n ∈ 𝓤 α) {u : ℕ → α}
    {a : α} (hu : Tendsto u atTop (𝓝 a)) : ∃ φ : ℕ → ℕ, StrictMono φ ∧ (u (φ 0), a) ∈ V 0 ∧
      ∀ n, (u <| φ (n + 1), u <| φ n) ∈ V (n + 1) 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {V : ℕ → Set (α × α)} (hV : ∀ (n : ℕ), V n ∈ 𝓤 α) {u : ℕ → α} {a : α} (hu : Filter.Tendsto u Filter.atTop (𝓝 a))
Goal: ∃ φ, StrictMono φ ∧ (u (φ 0), a) ∈ V 0 ∧ ∀ (n : ℕ), (u (φ (n + 1)), u (φ n)) ∈ V (n + 1)
-- tendsto_nhds_of_cauchySeq_of_subseq
theorem tendsto_nhds_of_cauchySeq_of_subseq [Preorder β] {u : β → α} (hu : CauchySeq u)
    {ι : Type*} {f : ι → β} {p : Filter ι} [NeBot p] (hf : Tendsto f p atTop) {a : α}
    (ha : Tendsto (u ∘ f) p (𝓝 a)) : Tendsto u atTop (𝓝 a) 
Arguments: {α : Type u} {β : Type v} [uniformSpace : UniformSpace α] [Preorder β] {u : β → α} (hu : CauchySeq u) {ι : Type u_1} {f : ι → β} {p : Filter ι} [p.NeBot] (hf : Filter.Tendsto f p Filter.atTop) {a : α} (ha : Filter.Tendsto (u ∘ f) p (𝓝 a))
Goal: Filter.Tendsto u Filter.atTop (𝓝 a)
-- cauchySeq_shift
theorem cauchySeq_shift {u : ℕ → α} (k : ℕ) : CauchySeq (fun n ↦ u (n + k)) ↔ CauchySeq u 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {u : ℕ → α} (k : ℕ)
Goal: (CauchySeq fun n => u (n + k)) ↔ CauchySeq u
-- Filter.HasBasis.cauchySeq_iff
theorem Filter.HasBasis.cauchySeq_iff {γ} [Nonempty β] [SemilatticeSup β] {u : β → α} {p : γ → Prop}
    {s : γ → Set (α × α)} (h : (𝓤 α).HasBasis p s) :
    CauchySeq u ↔ ∀ i, p i → ∃ N, ∀ m, N ≤ m → ∀ n, N ≤ n → (u m, u n) ∈ s i 
Arguments: {α : Type u} {β : Type v} [uniformSpace : UniformSpace α] {γ : Sort u_1} [Nonempty β] [SemilatticeSup β] {u : β → α} {p : γ → Prop} {s : γ → Set (α × α)} (h : (𝓤 α).HasBasis p s)
Goal: CauchySeq u ↔ ∀ (i : γ), p i → ∃ N, ∀ (m : β), N ≤ m → ∀ (n : β), N ≤ n → (u m, u n) ∈ s i
-- Filter.HasBasis.cauchySeq_iff'
theorem Filter.HasBasis.cauchySeq_iff' {γ} [Nonempty β] [SemilatticeSup β] {u : β → α}
    {p : γ → Prop} {s : γ → Set (α × α)} (H : (𝓤 α).HasBasis p s) :
    CauchySeq u ↔ ∀ i, p i → ∃ N, ∀ n ≥ N, (u n, u N) ∈ s i 
Arguments: {α : Type u} {β : Type v} [uniformSpace : UniformSpace α] {γ : Sort u_1} [Nonempty β] [SemilatticeSup β] {u : β → α} {p : γ → Prop} {s : γ → Set (α × α)} (H : (𝓤 α).HasBasis p s)
Goal: CauchySeq u ↔ ∀ (i : γ), p i → ∃ N, ∀ n ≥ N, (u n, u N) ∈ s i
-- cauchySeq_of_controlled
theorem cauchySeq_of_controlled [SemilatticeSup β] [Nonempty β] (U : β → Set (α × α))
    (hU : ∀ s ∈ 𝓤 α, ∃ n, U n ⊆ s) {f : β → α}
    (hf : ∀ ⦃N m n : β⦄, N ≤ m → N ≤ n → (f m, f n) ∈ U N) : CauchySeq f 
Arguments: {α : Type u} {β : Type v} [uniformSpace : UniformSpace α] [SemilatticeSup β] [Nonempty β] (U : β → Set (α × α)) (hU : ∀ s ∈ 𝓤 α, ∃ n, U n ⊆ s) {f : β → α} (hf : ∀ ⦃N m n : β⦄, N ≤ m → N ≤ n → (f m, f n) ∈ U N)
Goal: CauchySeq f
-- isComplete_iff_clusterPt
theorem isComplete_iff_clusterPt {s : Set α} :
    IsComplete s ↔ ∀ l, Cauchy l → l ≤ 𝓟 s → ∃ x ∈ s, ClusterPt x l 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {s : Set α}
Goal: IsComplete s ↔ ∀ (l : Filter α), Cauchy l → l ≤ 𝓟 s → ∃ x ∈ s, ClusterPt x l
-- isComplete_iff_ultrafilter
theorem isComplete_iff_ultrafilter {s : Set α} :
    IsComplete s ↔ ∀ l : Ultrafilter α, Cauchy (l : Filter α) → ↑l ≤ 𝓟 s → ∃ x ∈ s, ↑l ≤ 𝓝 x 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {s : Set α}
Goal: IsComplete s ↔ ∀ (l : Ultrafilter α), Cauchy ↑l → ↑l ≤ 𝓟 s → ∃ x ∈ s, ↑l ≤ 𝓝 x
-- isComplete_iff_ultrafilter'
theorem isComplete_iff_ultrafilter' {s : Set α} :
    IsComplete s ↔ ∀ l : Ultrafilter α, Cauchy (l : Filter α) → s ∈ l → ∃ x ∈ s, ↑l ≤ 𝓝 x 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {s : Set α}
Goal: IsComplete s ↔ ∀ (l : Ultrafilter α), Cauchy ↑l → s ∈ l → ∃ x ∈ s, ↑l ≤ 𝓝 x
-- IsComplete.union
protected theorem IsComplete.union {s t : Set α} (hs : IsComplete s) (ht : IsComplete t) :
    IsComplete (s ∪ t) 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {s t : Set α} (hs : IsComplete s) (ht : IsComplete t)
Goal: IsComplete (s ∪ t)
-- isComplete_iUnion_separated
theorem isComplete_iUnion_separated {ι : Sort*} {s : ι → Set α} (hs : ∀ i, IsComplete (s i))
    {U : Set (α × α)} (hU : U ∈ 𝓤 α) (hd : ∀ (i j : ι), ∀ x ∈ s i, ∀ y ∈ s j, (x, y) ∈ U → i = j) :
    IsComplete (⋃ i, s i) 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {ι : Sort u_1} {s : ι → Set α} (hs : ∀ (i : ι), IsComplete (s i)) {U : Set (α × α)} (hU : U ∈ 𝓤 α) (hd : ∀ (i j : ι), ∀ x ∈ s i, ∀ y ∈ s j, (x, y) ∈ U → i = j)
Goal: IsComplete (⋃ i, s i)
-- CompleteSpace.complete
class CompleteSpace (α : Type u) [UniformSpace α] : Prop where
    complete : ∀ {f : Filter α}, Cauchy f → ∃ x, f ≤ 𝓝 x
Arguments: {α : Type u} [UniformSpace α] [self : CompleteSpace α] {f : Filter α}
Goal: Cauchy f → ∃ x, f ≤ 𝓝 x
-- complete_univ
theorem complete_univ {α : Type u} [UniformSpace α] [CompleteSpace α] :
    IsComplete (univ : Set α) 
Arguments: {α : Type u} [UniformSpace α] [CompleteSpace α]
Goal: IsComplete Set.univ
-- CompleteSpace.fst_of_prod
lemma CompleteSpace.fst_of_prod [UniformSpace β] [CompleteSpace (α × β)] [h : Nonempty β] :
    CompleteSpace α 
Arguments: {α : Type u} {β : Type v} [uniformSpace : UniformSpace α] [UniformSpace β] [CompleteSpace (α × β)] [h : Nonempty β]
Goal: CompleteSpace α
-- CompleteSpace.snd_of_prod
lemma CompleteSpace.snd_of_prod [UniformSpace β] [CompleteSpace (α × β)] [h : Nonempty α] :
    CompleteSpace β 
Arguments: {α : Type u} {β : Type v} [uniformSpace : UniformSpace α] [UniformSpace β] [CompleteSpace (α × β)] [h : Nonempty α]
Goal: CompleteSpace β
-- completeSpace_prod_of_nonempty
lemma completeSpace_prod_of_nonempty [UniformSpace β] [Nonempty α] [Nonempty β] :
    CompleteSpace (α × β) ↔ CompleteSpace α ∧ CompleteSpace β 
Arguments: {α : Type u} {β : Type v} [uniformSpace : UniformSpace α] [UniformSpace β] [Nonempty α] [Nonempty β]
Goal: CompleteSpace (α × β) ↔ CompleteSpace α ∧ CompleteSpace β
-- completeSpace_of_isComplete_univ
theorem completeSpace_of_isComplete_univ (h : IsComplete (univ : Set α)) : CompleteSpace α 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] (h : IsComplete Set.univ)
Goal: CompleteSpace α
-- completeSpace_iff_isComplete_univ
theorem completeSpace_iff_isComplete_univ : CompleteSpace α ↔ IsComplete (univ : Set α) 
Arguments: {α : Type u} [uniformSpace : UniformSpace α]
Goal: CompleteSpace α ↔ IsComplete Set.univ
-- completeSpace_iff_ultrafilter
theorem completeSpace_iff_ultrafilter :
    CompleteSpace α ↔ ∀ l : Ultrafilter α, Cauchy (l : Filter α) → ∃ x : α, ↑l ≤ 𝓝 x 
Arguments: {α : Type u} [uniformSpace : UniformSpace α]
Goal: CompleteSpace α ↔ ∀ (l : Ultrafilter α), Cauchy ↑l → ∃ x, ↑l ≤ 𝓝 x
-- cauchy_iff_exists_le_nhds
theorem cauchy_iff_exists_le_nhds [CompleteSpace α] {l : Filter α} [NeBot l] :
    Cauchy l ↔ ∃ x, l ≤ 𝓝 x 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] [CompleteSpace α] {l : Filter α} [l.NeBot]
Goal: Cauchy l ↔ ∃ x, l ≤ 𝓝 x
-- cauchy_map_iff_exists_tendsto
theorem cauchy_map_iff_exists_tendsto [CompleteSpace α] {l : Filter β} {f : β → α} [NeBot l] :
    Cauchy (l.map f) ↔ ∃ x, Tendsto f l (𝓝 x) 
Arguments: {α : Type u} {β : Type v} [uniformSpace : UniformSpace α] [CompleteSpace α] {l : Filter β} {f : β → α} [l.NeBot]
Goal: Cauchy (Filter.map f l) ↔ ∃ x, Filter.Tendsto f l (𝓝 x)
-- cauchySeq_tendsto_of_complete
theorem cauchySeq_tendsto_of_complete [Preorder β] [CompleteSpace α] {u : β → α}
    (H : CauchySeq u) : ∃ x, Tendsto u atTop (𝓝 x) 
Arguments: {α : Type u} {β : Type v} [uniformSpace : UniformSpace α] [Preorder β] [CompleteSpace α] {u : β → α} (H : CauchySeq u)
Goal: ∃ x, Filter.Tendsto u Filter.atTop (𝓝 x)
-- cauchySeq_tendsto_of_isComplete
theorem cauchySeq_tendsto_of_isComplete [Preorder β] {K : Set α} (h₁ : IsComplete K)
    {u : β → α} (h₂ : ∀ n, u n ∈ K) (h₃ : CauchySeq u) : ∃ v ∈ K, Tendsto u atTop (𝓝 v) 
Arguments: {α : Type u} {β : Type v} [uniformSpace : UniformSpace α] [Preorder β] {K : Set α} (h₁ : IsComplete K) {u : β → α} (h₂ : ∀ (n : β), u n ∈ K) (h₃ : CauchySeq u)
Goal: ∃ v ∈ K, Filter.Tendsto u Filter.atTop (𝓝 v)
-- Cauchy.le_nhds_lim
theorem Cauchy.le_nhds_lim [CompleteSpace α] {f : Filter α} (hf : Cauchy f) :
    haveI := hf.1.nonempty; f ≤ 𝓝 (lim f) 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] [CompleteSpace α] {f : Filter α} (hf : Cauchy f)
Goal: f ≤ 𝓝 (lim f)
-- CauchySeq.tendsto_limUnder
theorem CauchySeq.tendsto_limUnder [Preorder β] [CompleteSpace α] {u : β → α} (h : CauchySeq u) :
    haveI := h.1.nonempty; Tendsto u atTop (𝓝 <| limUnder atTop u) 
Arguments: {α : Type u} {β : Type v} [uniformSpace : UniformSpace α] [Preorder β] [CompleteSpace α] {u : β → α} (h : CauchySeq u)
Goal: Filter.Tendsto u Filter.atTop (𝓝 (limUnder Filter.atTop u))
-- IsClosed.isComplete
theorem IsClosed.isComplete [CompleteSpace α] {s : Set α} (h : IsClosed s) : IsComplete s 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] [CompleteSpace α] {s : Set α} (h : IsClosed s)
Goal: IsComplete s
-- TotallyBounded.exists_subset
theorem TotallyBounded.exists_subset {s : Set α} (hs : TotallyBounded s) {U : Set (α × α)}
    (hU : U ∈ 𝓤 α) : ∃ t, t ⊆ s ∧ Set.Finite t ∧ s ⊆ ⋃ y ∈ t, { x | (x, y) ∈ U } 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {s : Set α} (hs : TotallyBounded s) {U : Set (α × α)} (hU : U ∈ 𝓤 α)
Goal: ∃ t ⊆ s, t.Finite ∧ s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ U}
-- totallyBounded_iff_subset
theorem totallyBounded_iff_subset {s : Set α} :
    TotallyBounded s ↔
      ∀ d ∈ 𝓤 α, ∃ t, t ⊆ s ∧ Set.Finite t ∧ s ⊆ ⋃ y ∈ t, { x | (x, y) ∈ d } 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {s : Set α}
Goal: TotallyBounded s ↔ ∀ d ∈ 𝓤 α, ∃ t ⊆ s, t.Finite ∧ s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ d}
-- Filter.HasBasis.totallyBounded_iff
theorem Filter.HasBasis.totallyBounded_iff {ι} {p : ι → Prop} {U : ι → Set (α × α)}
    (H : (𝓤 α).HasBasis p U) {s : Set α} :
    TotallyBounded s ↔ ∀ i, p i → ∃ t : Set α, Set.Finite t ∧ s ⊆ ⋃ y ∈ t, { x | (x, y) ∈ U i } 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {ι : Sort u_1} {p : ι → Prop} {U : ι → Set (α × α)} (H : (𝓤 α).HasBasis p U) {s : Set α}
Goal: TotallyBounded s ↔ ∀ (i : ι), p i → ∃ t, t.Finite ∧ s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ U i}
-- totallyBounded_of_forall_symm
theorem totallyBounded_of_forall_symm {s : Set α}
    (h : ∀ V ∈ 𝓤 α, SymmetricRel V → ∃ t : Set α, Set.Finite t ∧ s ⊆ ⋃ y ∈ t, ball y V) :
    TotallyBounded s 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {s : Set α} (h : ∀ V ∈ 𝓤 α, SymmetricRel V → ∃ t, t.Finite ∧ s ⊆ ⋃ y ∈ t, UniformSpace.ball y V)
Goal: TotallyBounded s
-- TotallyBounded.subset
theorem TotallyBounded.subset {s₁ s₂ : Set α} (hs : s₁ ⊆ s₂) (h : TotallyBounded s₂) :
    TotallyBounded s₁ 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {s₁ s₂ : Set α} (hs : s₁ ⊆ s₂) (h : TotallyBounded s₂)
Goal: TotallyBounded s₁
-- TotallyBounded.closure
theorem TotallyBounded.closure {s : Set α} (h : TotallyBounded s) : TotallyBounded (closure s) 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {s : Set α} (h : TotallyBounded s)
Goal: TotallyBounded (closure s)
-- totallyBounded_closure
@[simp]
lemma totallyBounded_closure {s : Set α} : TotallyBounded (closure s) ↔ TotallyBounded s 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {s : Set α}
Goal: TotallyBounded (closure s) ↔ TotallyBounded s
-- totallyBounded_iUnion
@[simp]
lemma totallyBounded_iUnion {ι : Sort*} [Finite ι] {s : ι → Set α} :
    TotallyBounded (⋃ i, s i) ↔ ∀ i, TotallyBounded (s i) 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {ι : Sort u_1} [Finite ι] {s : ι → Set α}
Goal: TotallyBounded (⋃ i, s i) ↔ ∀ (i : ι), TotallyBounded (s i)
-- totallyBounded_biUnion
lemma totallyBounded_biUnion {ι : Type*} {I : Set ι} (hI : I.Finite) {s : ι → Set α} :
    TotallyBounded (⋃ i ∈ I, s i) ↔ ∀ i ∈ I, TotallyBounded (s i) 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {ι : Type u_1} {I : Set ι} (hI : I.Finite) {s : ι → Set α}
Goal: TotallyBounded (⋃ i ∈ I, s i) ↔ ∀ i ∈ I, TotallyBounded (s i)
-- totallyBounded_sUnion
lemma totallyBounded_sUnion {S : Set (Set α)} (hS : S.Finite) :
    TotallyBounded (⋃₀ S) ↔ ∀ s ∈ S, TotallyBounded s 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {S : Set (Set α)} (hS : S.Finite)
Goal: TotallyBounded (⋃₀ S) ↔ ∀ s ∈ S, TotallyBounded s
-- Set.Finite.totallyBounded
lemma Set.Finite.totallyBounded {s : Set α} (hs : s.Finite) : TotallyBounded s 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {s : Set α} (hs : s.Finite)
Goal: TotallyBounded s
-- Set.Subsingleton.totallyBounded
lemma Set.Subsingleton.totallyBounded {s : Set α} (hs : s.Subsingleton) :
    TotallyBounded s 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {s : Set α} (hs : s.Subsingleton)
Goal: TotallyBounded s
-- totallyBounded_singleton
@[simp]
lemma totallyBounded_singleton (a : α) : TotallyBounded {a} 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] (a : α)
Goal: TotallyBounded {a}
-- totallyBounded_empty
@[simp]
theorem totallyBounded_empty : TotallyBounded (∅ : Set α) 
Arguments: {α : Type u} [uniformSpace : UniformSpace α]
Goal: TotallyBounded ∅
-- totallyBounded_union
@[simp]
lemma totallyBounded_union {s t : Set α} :
    TotallyBounded (s ∪ t) ↔ TotallyBounded s ∧ TotallyBounded t 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {s t : Set α}
Goal: TotallyBounded (s ∪ t) ↔ TotallyBounded s ∧ TotallyBounded t
-- TotallyBounded.union
protected lemma TotallyBounded.union {s t : Set α} (hs : TotallyBounded s) (ht : TotallyBounded t) :
    TotallyBounded (s ∪ t) 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {s t : Set α} (hs : TotallyBounded s) (ht : TotallyBounded t)
Goal: TotallyBounded (s ∪ t)
-- totallyBounded_insert
@[simp]
lemma totallyBounded_insert (a : α) {s : Set α} :
    TotallyBounded (insert a s) ↔ TotallyBounded s 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] (a : α) {s : Set α}
Goal: TotallyBounded (insert a s) ↔ TotallyBounded s
-- TotallyBounded.image
theorem TotallyBounded.image [UniformSpace β] {f : α → β} {s : Set α} (hs : TotallyBounded s)
    (hf : UniformContinuous f) : TotallyBounded (f '' s) 
Arguments: {α : Type u} {β : Type v} [uniformSpace : UniformSpace α] [UniformSpace β] {f : α → β} {s : Set α} (hs : TotallyBounded s) (hf : UniformContinuous f)
Goal: TotallyBounded (f '' s)
-- Ultrafilter.cauchy_of_totallyBounded
theorem Ultrafilter.cauchy_of_totallyBounded {s : Set α} (f : Ultrafilter α) (hs : TotallyBounded s)
    (h : ↑f ≤ 𝓟 s) : Cauchy (f : Filter α) 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {s : Set α} (f : Ultrafilter α) (hs : TotallyBounded s) (h : ↑f ≤ 𝓟 s)
Goal: Cauchy ↑f
-- totallyBounded_iff_filter
theorem totallyBounded_iff_filter {s : Set α} :
    TotallyBounded s ↔ ∀ f, NeBot f → f ≤ 𝓟 s → ∃ c ≤ f, Cauchy c 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {s : Set α}
Goal: TotallyBounded s ↔ ∀ (f : Filter α), f.NeBot → f ≤ 𝓟 s → ∃ c ≤ f, Cauchy c
-- totallyBounded_iff_ultrafilter
theorem totallyBounded_iff_ultrafilter {s : Set α} :
    TotallyBounded s ↔ ∀ f : Ultrafilter α, ↑f ≤ 𝓟 s → Cauchy (f : Filter α) 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {s : Set α}
Goal: TotallyBounded s ↔ ∀ (f : Ultrafilter α), ↑f ≤ 𝓟 s → Cauchy ↑f
-- isCompact_iff_totallyBounded_isComplete
theorem isCompact_iff_totallyBounded_isComplete {s : Set α} :
    IsCompact s ↔ TotallyBounded s ∧ IsComplete s 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {s : Set α}
Goal: IsCompact s ↔ TotallyBounded s ∧ IsComplete s
-- IsCompact.totallyBounded
protected theorem IsCompact.totallyBounded {s : Set α} (h : IsCompact s) : TotallyBounded s 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {s : Set α} (h : IsCompact s)
Goal: TotallyBounded s
-- IsCompact.isComplete
protected theorem IsCompact.isComplete {s : Set α} (h : IsCompact s) : IsComplete s 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {s : Set α} (h : IsCompact s)
Goal: IsComplete s
-- isCompact_of_totallyBounded_isClosed
theorem isCompact_of_totallyBounded_isClosed [CompleteSpace α] {s : Set α} (ht : TotallyBounded s)
    (hc : IsClosed s) : IsCompact s 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] [CompleteSpace α] {s : Set α} (ht : TotallyBounded s) (hc : IsClosed s)
Goal: IsCompact s
-- CauchySeq.totallyBounded_range
theorem CauchySeq.totallyBounded_range {s : ℕ → α} (hs : CauchySeq s) :
    TotallyBounded (range s) 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {s : ℕ → α} (hs : CauchySeq s)
Goal: TotallyBounded (Set.range s)
-- SequentiallyComplete.setSeq_mem
theorem setSeq_mem (n : ℕ) : setSeq hf U_mem n ∈ f 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {f : Filter α} (hf : Cauchy f) {U : ℕ → Set (α × α)} (U_mem : ∀ (n : ℕ), U n ∈ 𝓤 α) (n : ℕ)
Goal: SequentiallyComplete.setSeq hf U_mem n ∈ f
-- SequentiallyComplete.setSeq_mono
theorem setSeq_mono ⦃m n : ℕ⦄ (h : m ≤ n) : setSeq hf U_mem n ⊆ setSeq hf U_mem m 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {f : Filter α} (hf : Cauchy f) {U : ℕ → Set (α × α)} (U_mem : ∀ (n : ℕ), U n ∈ 𝓤 α) ⦃m n : ℕ⦄ (h : m ≤ n)
Goal: SequentiallyComplete.setSeq hf U_mem n ⊆ SequentiallyComplete.setSeq hf U_mem m
-- SequentiallyComplete.setSeq_sub_aux
theorem setSeq_sub_aux (n : ℕ) : setSeq hf U_mem n ⊆ setSeqAux hf U_mem n 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {f : Filter α} (hf : Cauchy f) {U : ℕ → Set (α × α)} (U_mem : ∀ (n : ℕ), U n ∈ 𝓤 α) (n : ℕ)
Goal: SequentiallyComplete.setSeq hf U_mem n ⊆ ↑(SequentiallyComplete.setSeqAux hf U_mem n)
-- SequentiallyComplete.setSeq_prod_subset
theorem setSeq_prod_subset {N m n} (hm : N ≤ m) (hn : N ≤ n) :
    setSeq hf U_mem m ×ˢ setSeq hf U_mem n ⊆ U N 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {f : Filter α} (hf : Cauchy f) {U : ℕ → Set (α × α)} (U_mem : ∀ (n : ℕ), U n ∈ 𝓤 α) {N m n : ℕ} (hm : N ≤ m) (hn : N ≤ n)
Goal: SequentiallyComplete.setSeq hf U_mem m ×ˢ SequentiallyComplete.setSeq hf U_mem n ⊆ U N
-- SequentiallyComplete.seq_mem
theorem seq_mem (n : ℕ) : seq hf U_mem n ∈ setSeq hf U_mem n 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {f : Filter α} (hf : Cauchy f) {U : ℕ → Set (α × α)} (U_mem : ∀ (n : ℕ), U n ∈ 𝓤 α) (n : ℕ)
Goal: SequentiallyComplete.seq hf U_mem n ∈ SequentiallyComplete.setSeq hf U_mem n
-- SequentiallyComplete.seq_pair_mem
theorem seq_pair_mem ⦃N m n : ℕ⦄ (hm : N ≤ m) (hn : N ≤ n) :
    (seq hf U_mem m, seq hf U_mem n) ∈ U N 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {f : Filter α} (hf : Cauchy f) {U : ℕ → Set (α × α)} (U_mem : ∀ (n : ℕ), U n ∈ 𝓤 α) ⦃N m n : ℕ⦄ (hm : N ≤ m) (hn : N ≤ n)
Goal: (SequentiallyComplete.seq hf U_mem m, SequentiallyComplete.seq hf U_mem n) ∈ U N
-- SequentiallyComplete.seq_is_cauchySeq
theorem seq_is_cauchySeq : CauchySeq <| seq hf U_mem 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {f : Filter α} (hf : Cauchy f) {U : ℕ → Set (α × α)} (U_mem : ∀ (n : ℕ), U n ∈ 𝓤 α) (U_le : ∀ s ∈ 𝓤 α, ∃ n, U n ⊆ s)
Goal: CauchySeq (SequentiallyComplete.seq hf U_mem)
-- SequentiallyComplete.le_nhds_of_seq_tendsto_nhds
theorem le_nhds_of_seq_tendsto_nhds ⦃a : α⦄ (ha : Tendsto (seq hf U_mem) atTop (𝓝 a)) : f ≤ 𝓝 a 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] {f : Filter α} (hf : Cauchy f) {U : ℕ → Set (α × α)} (U_mem : ∀ (n : ℕ), U n ∈ 𝓤 α) (U_le : ∀ s ∈ 𝓤 α, ∃ n, U n ⊆ s) ⦃a : α⦄ (ha : Filter.Tendsto (SequentiallyComplete.seq hf U_mem) Filter.atTop (𝓝 a))
Goal: f ≤ 𝓝 a
-- UniformSpace.complete_of_convergent_controlled_sequences
theorem complete_of_convergent_controlled_sequences (U : ℕ → Set (α × α)) (U_mem : ∀ n, U n ∈ 𝓤 α)
    (HU : ∀ u : ℕ → α, (∀ N m n, N ≤ m → N ≤ n → (u m, u n) ∈ U N) → ∃ a, Tendsto u atTop (𝓝 a)) :
    CompleteSpace α 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] [(𝓤 α).IsCountablyGenerated] (U : ℕ → Set (α × α)) (U_mem : ∀ (n : ℕ), U n ∈ 𝓤 α) (HU : ∀ (u : ℕ → α), (∀ (N m n : ℕ), N ≤ m → N ≤ n → (u m, u n) ∈ U N) → ∃ a, Filter.Tendsto u Filter.atTop (𝓝 a))
Goal: CompleteSpace α
-- UniformSpace.complete_of_cauchySeq_tendsto
theorem complete_of_cauchySeq_tendsto (H' : ∀ u : ℕ → α, CauchySeq u → ∃ a, Tendsto u atTop (𝓝 a)) :
    CompleteSpace α 
Arguments: {α : Type u} [uniformSpace : UniformSpace α] [(𝓤 α).IsCountablyGenerated] (H' : ∀ (u : ℕ → α), CauchySeq u → ∃ a, Filter.Tendsto u Filter.atTop (𝓝 a))
Goal: CompleteSpace α
-- UniformSpace.secondCountable_of_separable
theorem secondCountable_of_separable [SeparableSpace α] : SecondCountableTopology α 
Arguments: (α : Type u) [uniformSpace : UniformSpace α] [(𝓤 α).IsCountablyGenerated] [TopologicalSpace.SeparableSpace α]
Goal: SecondCountableTopology α