ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- uniformity_eq_comap_nhds_one
@[to_additive]
theorem uniformity_eq_comap_nhds_one : 𝓤 α = comap (fun x : α × α => x.2 / x.1) (𝓝 (1 : α)) 
Arguments: (α : Type u_1) [UniformSpace α] [Group α] [UniformGroup α]
Goal: 𝓤 α = Filter.comap (fun x => x.2 / x.1) (𝓝 1)
-- uniformity_eq_comap_nhds_one_swapped
@[to_additive]
theorem uniformity_eq_comap_nhds_one_swapped :
    𝓤 α = comap (fun x : α × α => x.1 / x.2) (𝓝 (1 : α)) 
Arguments: (α : Type u_1) [UniformSpace α] [Group α] [UniformGroup α]
Goal: 𝓤 α = Filter.comap (fun x => x.1 / x.2) (𝓝 1)
-- uniformity_eq_comap_nhds_zero_swapped
@[to_additive]
theorem uniformity_eq_comap_nhds_one_swapped :
    𝓤 α = comap (fun x : α × α => x.1 / x.2) (𝓝 (1 : α)) 
Arguments: (α : Type u_1) [UniformSpace α] [AddGroup α] [UniformAddGroup α]
Goal: 𝓤 α = Filter.comap (fun x => x.1 - x.2) (𝓝 0)
-- UniformGroup.ext
@[to_additive]
theorem UniformGroup.ext {G : Type*} [Group G] {u v : UniformSpace G} (hu : @UniformGroup G u _)
    (hv : @UniformGroup G v _)
    (h : @nhds _ u.toTopologicalSpace 1 = @nhds _ v.toTopologicalSpace 1) : u = v 
Arguments: {G : Type u_3} [Group G] {u v : UniformSpace G} (hu : UniformGroup G) (hv : UniformGroup G) (h : 𝓝 1 = 𝓝 1)
Goal: u = v
-- UniformAddGroup.ext
@[to_additive]
theorem UniformGroup.ext {G : Type*} [Group G] {u v : UniformSpace G} (hu : @UniformGroup G u _)
    (hv : @UniformGroup G v _)
    (h : @nhds _ u.toTopologicalSpace 1 = @nhds _ v.toTopologicalSpace 1) : u = v 
Arguments: {G : Type u_3} [AddGroup G] {u v : UniformSpace G} (hu : UniformAddGroup G) (hv : UniformAddGroup G) (h : 𝓝 0 = 𝓝 0)
Goal: u = v
-- UniformAddGroup.ext_iff
@[to_additive]
theorem UniformGroup.ext_iff {G : Type*} [Group G] {u v : UniformSpace G}
    (hu : @UniformGroup G u _) (hv : @UniformGroup G v _) :
    u = v ↔ @nhds _ u.toTopologicalSpace 1 = @nhds _ v.toTopologicalSpace 1 
Arguments: {G : Type u_3} [AddGroup G] {u v : UniformSpace G} (hu : UniformAddGroup G) (hv : UniformAddGroup G)
Goal: u = v ↔ 𝓝 0 = 𝓝 0
-- UniformGroup.ext_iff
@[to_additive]
theorem UniformGroup.ext_iff {G : Type*} [Group G] {u v : UniformSpace G}
    (hu : @UniformGroup G u _) (hv : @UniformGroup G v _) :
    u = v ↔ @nhds _ u.toTopologicalSpace 1 = @nhds _ v.toTopologicalSpace 1 
Arguments: {G : Type u_3} [Group G] {u v : UniformSpace G} (hu : UniformGroup G) (hv : UniformGroup G)
Goal: u = v ↔ 𝓝 1 = 𝓝 1
-- UniformAddGroup.uniformity_countably_generated
@[to_additive]
theorem UniformGroup.uniformity_countably_generated [(𝓝 (1 : α)).IsCountablyGenerated] :
    (𝓤 α).IsCountablyGenerated 
Arguments: {α : Type u_1} [UniformSpace α] [AddGroup α] [UniformAddGroup α] [(𝓝 0).IsCountablyGenerated]
Goal: (𝓤 α).IsCountablyGenerated
-- UniformGroup.uniformity_countably_generated
@[to_additive]
theorem UniformGroup.uniformity_countably_generated [(𝓝 (1 : α)).IsCountablyGenerated] :
    (𝓤 α).IsCountablyGenerated 
Arguments: {α : Type u_1} [UniformSpace α] [Group α] [UniformGroup α] [(𝓝 1).IsCountablyGenerated]
Goal: (𝓤 α).IsCountablyGenerated
-- uniformity_eq_comap_inv_mul_nhds_one
@[to_additive]
theorem uniformity_eq_comap_inv_mul_nhds_one :
    𝓤 α = comap (fun x : α × α => x.1⁻¹ * x.2) (𝓝 (1 : α)) 
Arguments: {α : Type u_1} [UniformSpace α] [Group α] [UniformGroup α]
Goal: 𝓤 α = Filter.comap (fun x => x.1⁻¹ * x.2) (𝓝 1)
-- uniformity_eq_comap_neg_add_nhds_zero
@[to_additive]
theorem uniformity_eq_comap_inv_mul_nhds_one :
    𝓤 α = comap (fun x : α × α => x.1⁻¹ * x.2) (𝓝 (1 : α)) 
Arguments: {α : Type u_1} [UniformSpace α] [AddGroup α] [UniformAddGroup α]
Goal: 𝓤 α = Filter.comap (fun x => -x.1 + x.2) (𝓝 0)
-- uniformity_eq_comap_neg_add_nhds_zero_swapped
@[to_additive]
theorem uniformity_eq_comap_inv_mul_nhds_one_swapped :
    𝓤 α = comap (fun x : α × α => x.2⁻¹ * x.1) (𝓝 (1 : α)) 
Arguments: {α : Type u_1} [UniformSpace α] [AddGroup α] [UniformAddGroup α]
Goal: 𝓤 α = Filter.comap (fun x => -x.2 + x.1) (𝓝 0)
-- uniformity_eq_comap_inv_mul_nhds_one_swapped
@[to_additive]
theorem uniformity_eq_comap_inv_mul_nhds_one_swapped :
    𝓤 α = comap (fun x : α × α => x.2⁻¹ * x.1) (𝓝 (1 : α)) 
Arguments: {α : Type u_1} [UniformSpace α] [Group α] [UniformGroup α]
Goal: 𝓤 α = Filter.comap (fun x => x.2⁻¹ * x.1) (𝓝 1)
-- Filter.HasBasis.uniformity_of_nhds_zero
@[to_additive]
theorem Filter.HasBasis.uniformity_of_nhds_one {ι} {p : ι → Prop} {U : ι → Set α}
    (h : (𝓝 (1 : α)).HasBasis p U) :
    (𝓤 α).HasBasis p fun i => { x : α × α | x.2 / x.1 ∈ U i } 
Arguments: {α : Type u_1} [UniformSpace α] [AddGroup α] [UniformAddGroup α] {ι : Sort u_3} {p : ι → Prop} {U : ι → Set α} (h : (𝓝 0).HasBasis p U)
Goal: (𝓤 α).HasBasis p fun i => {x | x.2 - x.1 ∈ U i}
-- Filter.HasBasis.uniformity_of_nhds_one
@[to_additive]
theorem Filter.HasBasis.uniformity_of_nhds_one {ι} {p : ι → Prop} {U : ι → Set α}
    (h : (𝓝 (1 : α)).HasBasis p U) :
    (𝓤 α).HasBasis p fun i => { x : α × α | x.2 / x.1 ∈ U i } 
Arguments: {α : Type u_1} [UniformSpace α] [Group α] [UniformGroup α] {ι : Sort u_3} {p : ι → Prop} {U : ι → Set α} (h : (𝓝 1).HasBasis p U)
Goal: (𝓤 α).HasBasis p fun i => {x | x.2 / x.1 ∈ U i}
-- Filter.HasBasis.uniformity_of_nhds_one_inv_mul
@[to_additive]
theorem Filter.HasBasis.uniformity_of_nhds_one_inv_mul {ι} {p : ι → Prop} {U : ι → Set α}
    (h : (𝓝 (1 : α)).HasBasis p U) :
    (𝓤 α).HasBasis p fun i => { x : α × α | x.1⁻¹ * x.2 ∈ U i } 
Arguments: {α : Type u_1} [UniformSpace α] [Group α] [UniformGroup α] {ι : Sort u_3} {p : ι → Prop} {U : ι → Set α} (h : (𝓝 1).HasBasis p U)
Goal: (𝓤 α).HasBasis p fun i => {x | x.1⁻¹ * x.2 ∈ U i}
-- Filter.HasBasis.uniformity_of_nhds_zero_neg_add
@[to_additive]
theorem Filter.HasBasis.uniformity_of_nhds_one_inv_mul {ι} {p : ι → Prop} {U : ι → Set α}
    (h : (𝓝 (1 : α)).HasBasis p U) :
    (𝓤 α).HasBasis p fun i => { x : α × α | x.1⁻¹ * x.2 ∈ U i } 
Arguments: {α : Type u_1} [UniformSpace α] [AddGroup α] [UniformAddGroup α] {ι : Sort u_3} {p : ι → Prop} {U : ι → Set α} (h : (𝓝 0).HasBasis p U)
Goal: (𝓤 α).HasBasis p fun i => {x | -x.1 + x.2 ∈ U i}
-- Filter.HasBasis.uniformity_of_nhds_one_swapped
@[to_additive]
theorem Filter.HasBasis.uniformity_of_nhds_one_swapped {ι} {p : ι → Prop} {U : ι → Set α}
    (h : (𝓝 (1 : α)).HasBasis p U) :
    (𝓤 α).HasBasis p fun i => { x : α × α | x.1 / x.2 ∈ U i } 
Arguments: {α : Type u_1} [UniformSpace α] [Group α] [UniformGroup α] {ι : Sort u_3} {p : ι → Prop} {U : ι → Set α} (h : (𝓝 1).HasBasis p U)
Goal: (𝓤 α).HasBasis p fun i => {x | x.1 / x.2 ∈ U i}
-- Filter.HasBasis.uniformity_of_nhds_zero_swapped
@[to_additive]
theorem Filter.HasBasis.uniformity_of_nhds_one_swapped {ι} {p : ι → Prop} {U : ι → Set α}
    (h : (𝓝 (1 : α)).HasBasis p U) :
    (𝓤 α).HasBasis p fun i => { x : α × α | x.1 / x.2 ∈ U i } 
Arguments: {α : Type u_1} [UniformSpace α] [AddGroup α] [UniformAddGroup α] {ι : Sort u_3} {p : ι → Prop} {U : ι → Set α} (h : (𝓝 0).HasBasis p U)
Goal: (𝓤 α).HasBasis p fun i => {x | x.1 - x.2 ∈ U i}
-- Filter.HasBasis.uniformity_of_nhds_zero_neg_add_swapped
@[to_additive]
theorem Filter.HasBasis.uniformity_of_nhds_one_inv_mul_swapped {ι} {p : ι → Prop} {U : ι → Set α}
    (h : (𝓝 (1 : α)).HasBasis p U) :
    (𝓤 α).HasBasis p fun i => { x : α × α | x.2⁻¹ * x.1 ∈ U i } 
Arguments: {α : Type u_1} [UniformSpace α] [AddGroup α] [UniformAddGroup α] {ι : Sort u_3} {p : ι → Prop} {U : ι → Set α} (h : (𝓝 0).HasBasis p U)
Goal: (𝓤 α).HasBasis p fun i => {x | -x.2 + x.1 ∈ U i}
-- Filter.HasBasis.uniformity_of_nhds_one_inv_mul_swapped
@[to_additive]
theorem Filter.HasBasis.uniformity_of_nhds_one_inv_mul_swapped {ι} {p : ι → Prop} {U : ι → Set α}
    (h : (𝓝 (1 : α)).HasBasis p U) :
    (𝓤 α).HasBasis p fun i => { x : α × α | x.2⁻¹ * x.1 ∈ U i } 
Arguments: {α : Type u_1} [UniformSpace α] [Group α] [UniformGroup α] {ι : Sort u_3} {p : ι → Prop} {U : ι → Set α} (h : (𝓝 1).HasBasis p U)
Goal: (𝓤 α).HasBasis p fun i => {x | x.2⁻¹ * x.1 ∈ U i}
-- uniformContinuous_of_tendsto_one
@[to_additive]
theorem uniformContinuous_of_tendsto_one {hom : Type*} [UniformSpace β] [Group β] [UniformGroup β]
    [FunLike hom α β] [MonoidHomClass hom α β] {f : hom} (h : Tendsto f (𝓝 1) (𝓝 1)) :
    UniformContinuous f 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [Group α] [UniformGroup α] {hom : Type u_3} [UniformSpace β] [Group β] [UniformGroup β] [FunLike hom α β] [MonoidHomClass hom α β] {f : hom} (h : Filter.Tendsto (⇑f) (𝓝 1) (𝓝 1))
Goal: UniformContinuous ⇑f
-- uniformContinuous_of_tendsto_zero
@[to_additive]
theorem uniformContinuous_of_tendsto_one {hom : Type*} [UniformSpace β] [Group β] [UniformGroup β]
    [FunLike hom α β] [MonoidHomClass hom α β] {f : hom} (h : Tendsto f (𝓝 1) (𝓝 1)) :
    UniformContinuous f 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [AddGroup α] [UniformAddGroup α] {hom : Type u_3} [UniformSpace β] [AddGroup β] [UniformAddGroup β] [FunLike hom α β] [AddMonoidHomClass hom α β] {f : hom} (h : Filter.Tendsto (⇑f) (𝓝 0) (𝓝 0))
Goal: UniformContinuous ⇑f
-- uniformContinuous_of_continuousAt_one
@[to_additive "An additive group homomorphism (a bundled morphism of a type that implements
`AddMonoidHomClass`) between two uniform additive groups is uniformly continuous provided that it
is continuous at zero. See also `continuous_of_continuousAt_zero`."]
theorem uniformContinuous_of_continuousAt_one {hom : Type*} [UniformSpace β] [Group β]
    [UniformGroup β] [FunLike hom α β] [MonoidHomClass hom α β]
    (f : hom) (hf : ContinuousAt f 1) :
    UniformContinuous f 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [Group α] [UniformGroup α] {hom : Type u_3} [UniformSpace β] [Group β] [UniformGroup β] [FunLike hom α β] [MonoidHomClass hom α β] (f : hom) (hf : ContinuousAt (⇑f) 1)
Goal: UniformContinuous ⇑f
-- uniformContinuous_of_continuousAt_zero
@[to_additive "An additive group homomorphism (a bundled morphism of a type that implements
`AddMonoidHomClass`) between two uniform additive groups is uniformly continuous provided that it
is continuous at zero. See also `continuous_of_continuousAt_zero`."]
theorem uniformContinuous_of_continuousAt_one {hom : Type*} [UniformSpace β] [Group β]
    [UniformGroup β] [FunLike hom α β] [MonoidHomClass hom α β]
    (f : hom) (hf : ContinuousAt f 1) :
    UniformContinuous f 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [AddGroup α] [UniformAddGroup α] {hom : Type u_3} [UniformSpace β] [AddGroup β] [UniformAddGroup β] [FunLike hom α β] [AddMonoidHomClass hom α β] (f : hom) (hf : ContinuousAt (⇑f) 0)
Goal: UniformContinuous ⇑f
-- MonoidHom.uniformContinuous_of_continuousAt_one
@[to_additive]
theorem MonoidHom.uniformContinuous_of_continuousAt_one [UniformSpace β] [Group β] [UniformGroup β]
    (f : α →* β) (hf : ContinuousAt f 1) : UniformContinuous f 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [Group α] [UniformGroup α] [UniformSpace β] [Group β] [UniformGroup β] (f : α →* β) (hf : ContinuousAt (⇑f) 1)
Goal: UniformContinuous ⇑f
-- AddMonoidHom.uniformContinuous_of_continuousAt_zero
@[to_additive]
theorem MonoidHom.uniformContinuous_of_continuousAt_one [UniformSpace β] [Group β] [UniformGroup β]
    (f : α →* β) (hf : ContinuousAt f 1) : UniformContinuous f 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [AddGroup α] [UniformAddGroup α] [UniformSpace β] [AddGroup β] [UniformAddGroup β] (f : α →+ β) (hf : ContinuousAt (⇑f) 0)
Goal: UniformContinuous ⇑f
-- UniformGroup.uniformContinuous_iff_open_ker
@[to_additive "A homomorphism from a uniform additive group to a discrete uniform additive group is
continuous if and only if its kernel is open."]
theorem UniformGroup.uniformContinuous_iff_open_ker {hom : Type*} [UniformSpace β]
    [DiscreteTopology β] [Group β] [UniformGroup β] [FunLike hom α β] [MonoidHomClass hom α β]
    {f : hom} :
    UniformContinuous f ↔ IsOpen ((f : α →* β).ker : Set α) 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [Group α] [UniformGroup α] {hom : Type u_3} [UniformSpace β] [DiscreteTopology β] [Group β] [UniformGroup β] [FunLike hom α β] [MonoidHomClass hom α β] {f : hom}
Goal: UniformContinuous ⇑f ↔ IsOpen ↑(↑f).ker
-- UniformAddGroup.uniformContinuous_iff_open_ker
@[to_additive "A homomorphism from a uniform additive group to a discrete uniform additive group is
continuous if and only if its kernel is open."]
theorem UniformGroup.uniformContinuous_iff_open_ker {hom : Type*} [UniformSpace β]
    [DiscreteTopology β] [Group β] [UniformGroup β] [FunLike hom α β] [MonoidHomClass hom α β]
    {f : hom} :
    UniformContinuous f ↔ IsOpen ((f : α →* β).ker : Set α) 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [AddGroup α] [UniformAddGroup α] {hom : Type u_3} [UniformSpace β] [DiscreteTopology β] [AddGroup β] [UniformAddGroup β] [FunLike hom α β] [AddMonoidHomClass hom α β] {f : hom}
Goal: UniformContinuous ⇑f ↔ IsOpen ↑(↑f).ker
-- uniformContinuous_addMonoidHom_of_continuous
@[to_additive]
theorem uniformContinuous_monoidHom_of_continuous {hom : Type*} [UniformSpace β] [Group β]
    [UniformGroup β] [FunLike hom α β] [MonoidHomClass hom α β] {f : hom} (h : Continuous f) :
    UniformContinuous f 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [AddGroup α] [UniformAddGroup α] {hom : Type u_3} [UniformSpace β] [AddGroup β] [UniformAddGroup β] [FunLike hom α β] [AddMonoidHomClass hom α β] {f : hom} (h : Continuous ⇑f)
Goal: UniformContinuous ⇑f
-- uniformContinuous_monoidHom_of_continuous
@[to_additive]
theorem uniformContinuous_monoidHom_of_continuous {hom : Type*} [UniformSpace β] [Group β]
    [UniformGroup β] [FunLike hom α β] [MonoidHomClass hom α β] {f : hom} (h : Continuous f) :
    UniformContinuous f 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [Group α] [UniformGroup α] {hom : Type u_3} [UniformSpace β] [Group β] [UniformGroup β] [FunLike hom α β] [MonoidHomClass hom α β] {f : hom} (h : Continuous ⇑f)
Goal: UniformContinuous ⇑f
-- CauchySeq.mul
@[to_additive]
theorem CauchySeq.mul {ι : Type*} [Preorder ι] {u v : ι → α} (hu : CauchySeq u)
    (hv : CauchySeq v) : CauchySeq (u * v) 
Arguments: {α : Type u_1} [UniformSpace α] [Group α] [UniformGroup α] {ι : Type u_3} [Preorder ι] {u v : ι → α} (hu : CauchySeq u) (hv : CauchySeq v)
Goal: CauchySeq (u * v)
-- CauchySeq.add
@[to_additive]
theorem CauchySeq.mul {ι : Type*} [Preorder ι] {u v : ι → α} (hu : CauchySeq u)
    (hv : CauchySeq v) : CauchySeq (u * v) 
Arguments: {α : Type u_1} [UniformSpace α] [AddGroup α] [UniformAddGroup α] {ι : Type u_3} [Preorder ι] {u v : ι → α} (hu : CauchySeq u) (hv : CauchySeq v)
Goal: CauchySeq (u + v)
-- CauchySeq.add_const
@[to_additive]
theorem CauchySeq.mul_const {ι : Type*} [Preorder ι] {u : ι → α} {x : α} (hu : CauchySeq u) :
    CauchySeq fun n => u n * x 
Arguments: {α : Type u_1} [UniformSpace α] [AddGroup α] [UniformAddGroup α] {ι : Type u_3} [Preorder ι] {u : ι → α} {x : α} (hu : CauchySeq u)
Goal: CauchySeq fun n => u n + x
-- CauchySeq.mul_const
@[to_additive]
theorem CauchySeq.mul_const {ι : Type*} [Preorder ι] {u : ι → α} {x : α} (hu : CauchySeq u) :
    CauchySeq fun n => u n * x 
Arguments: {α : Type u_1} [UniformSpace α] [Group α] [UniformGroup α] {ι : Type u_3} [Preorder ι] {u : ι → α} {x : α} (hu : CauchySeq u)
Goal: CauchySeq fun n => u n * x
-- CauchySeq.const_mul
@[to_additive]
theorem CauchySeq.const_mul {ι : Type*} [Preorder ι] {u : ι → α} {x : α} (hu : CauchySeq u) :
    CauchySeq fun n => x * u n 
Arguments: {α : Type u_1} [UniformSpace α] [Group α] [UniformGroup α] {ι : Type u_3} [Preorder ι] {u : ι → α} {x : α} (hu : CauchySeq u)
Goal: CauchySeq fun n => x * u n
-- CauchySeq.const_add
@[to_additive]
theorem CauchySeq.const_mul {ι : Type*} [Preorder ι] {u : ι → α} {x : α} (hu : CauchySeq u) :
    CauchySeq fun n => x * u n 
Arguments: {α : Type u_1} [UniformSpace α] [AddGroup α] [UniformAddGroup α] {ι : Type u_3} [Preorder ι] {u : ι → α} {x : α} (hu : CauchySeq u)
Goal: CauchySeq fun n => x + u n
-- CauchySeq.neg
@[to_additive]
theorem CauchySeq.inv {ι : Type*} [Preorder ι] {u : ι → α} (h : CauchySeq u) :
    CauchySeq u⁻¹ 
Arguments: {α : Type u_1} [UniformSpace α] [AddGroup α] [UniformAddGroup α] {ι : Type u_3} [Preorder ι] {u : ι → α} (h : CauchySeq u)
Goal: CauchySeq (-u)
-- CauchySeq.inv
@[to_additive]
theorem CauchySeq.inv {ι : Type*} [Preorder ι] {u : ι → α} (h : CauchySeq u) :
    CauchySeq u⁻¹ 
Arguments: {α : Type u_1} [UniformSpace α] [Group α] [UniformGroup α] {ι : Type u_3} [Preorder ι] {u : ι → α} (h : CauchySeq u)
Goal: CauchySeq u⁻¹
-- totallyBounded_iff_subset_finite_iUnion_nhds_one
@[to_additive]
theorem totallyBounded_iff_subset_finite_iUnion_nhds_one {s : Set α} :
    TotallyBounded s ↔ ∀ U ∈ 𝓝 (1 : α), ∃ t : Set α, t.Finite ∧ s ⊆ ⋃ y ∈ t, y • U 
Arguments: {α : Type u_1} [UniformSpace α] [Group α] [UniformGroup α] {s : Set α}
Goal: TotallyBounded s ↔ ∀ U ∈ 𝓝 1, ∃ t, t.Finite ∧ s ⊆ ⋃ y ∈ t, y • U
-- totallyBounded_iff_subset_finite_iUnion_nhds_zero
@[to_additive]
theorem totallyBounded_iff_subset_finite_iUnion_nhds_one {s : Set α} :
    TotallyBounded s ↔ ∀ U ∈ 𝓝 (1 : α), ∃ t : Set α, t.Finite ∧ s ⊆ ⋃ y ∈ t, y • U 
Arguments: {α : Type u_1} [UniformSpace α] [AddGroup α] [UniformAddGroup α] {s : Set α}
Goal: TotallyBounded s ↔ ∀ U ∈ 𝓝 0, ∃ t, t.Finite ∧ s ⊆ ⋃ y ∈ t, y +ᵥ U
-- TendstoUniformlyOnFilter.add
@[to_additive]
theorem TendstoUniformlyOnFilter.mul (hf : TendstoUniformlyOnFilter f g l l')
    (hf' : TendstoUniformlyOnFilter f' g' l l') : TendstoUniformlyOnFilter (f * f') (g * g') l l' 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [AddGroup α] [UniformAddGroup α] {ι : Type u_3} {l : Filter ι} {l' : Filter β} {f f' : ι → β → α} {g g' : β → α} (hf : TendstoUniformlyOnFilter f g l l') (hf' : TendstoUniformlyOnFilter f' g' l l')
Goal: TendstoUniformlyOnFilter (f + f') (g + g') l l'
-- TendstoUniformlyOnFilter.mul
@[to_additive]
theorem TendstoUniformlyOnFilter.mul (hf : TendstoUniformlyOnFilter f g l l')
    (hf' : TendstoUniformlyOnFilter f' g' l l') : TendstoUniformlyOnFilter (f * f') (g * g') l l' 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [Group α] [UniformGroup α] {ι : Type u_3} {l : Filter ι} {l' : Filter β} {f f' : ι → β → α} {g g' : β → α} (hf : TendstoUniformlyOnFilter f g l l') (hf' : TendstoUniformlyOnFilter f' g' l l')
Goal: TendstoUniformlyOnFilter (f * f') (g * g') l l'
-- TendstoUniformlyOnFilter.sub
@[to_additive]
theorem TendstoUniformlyOnFilter.div (hf : TendstoUniformlyOnFilter f g l l')
    (hf' : TendstoUniformlyOnFilter f' g' l l') : TendstoUniformlyOnFilter (f / f') (g / g') l l' 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [AddGroup α] [UniformAddGroup α] {ι : Type u_3} {l : Filter ι} {l' : Filter β} {f f' : ι → β → α} {g g' : β → α} (hf : TendstoUniformlyOnFilter f g l l') (hf' : TendstoUniformlyOnFilter f' g' l l')
Goal: TendstoUniformlyOnFilter (f - f') (g - g') l l'
-- TendstoUniformlyOnFilter.div
@[to_additive]
theorem TendstoUniformlyOnFilter.div (hf : TendstoUniformlyOnFilter f g l l')
    (hf' : TendstoUniformlyOnFilter f' g' l l') : TendstoUniformlyOnFilter (f / f') (g / g') l l' 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [Group α] [UniformGroup α] {ι : Type u_3} {l : Filter ι} {l' : Filter β} {f f' : ι → β → α} {g g' : β → α} (hf : TendstoUniformlyOnFilter f g l l') (hf' : TendstoUniformlyOnFilter f' g' l l')
Goal: TendstoUniformlyOnFilter (f / f') (g / g') l l'
-- TendstoUniformlyOn.mul
@[to_additive]
theorem TendstoUniformlyOn.mul (hf : TendstoUniformlyOn f g l s)
    (hf' : TendstoUniformlyOn f' g' l s) : TendstoUniformlyOn (f * f') (g * g') l s 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [Group α] [UniformGroup α] {ι : Type u_3} {l : Filter ι} {f f' : ι → β → α} {g g' : β → α} {s : Set β} (hf : TendstoUniformlyOn f g l s) (hf' : TendstoUniformlyOn f' g' l s)
Goal: TendstoUniformlyOn (f * f') (g * g') l s
-- TendstoUniformlyOn.add
@[to_additive]
theorem TendstoUniformlyOn.mul (hf : TendstoUniformlyOn f g l s)
    (hf' : TendstoUniformlyOn f' g' l s) : TendstoUniformlyOn (f * f') (g * g') l s 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [AddGroup α] [UniformAddGroup α] {ι : Type u_3} {l : Filter ι} {f f' : ι → β → α} {g g' : β → α} {s : Set β} (hf : TendstoUniformlyOn f g l s) (hf' : TendstoUniformlyOn f' g' l s)
Goal: TendstoUniformlyOn (f + f') (g + g') l s
-- TendstoUniformlyOn.sub
@[to_additive]
theorem TendstoUniformlyOn.div (hf : TendstoUniformlyOn f g l s)
    (hf' : TendstoUniformlyOn f' g' l s) : TendstoUniformlyOn (f / f') (g / g') l s 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [AddGroup α] [UniformAddGroup α] {ι : Type u_3} {l : Filter ι} {f f' : ι → β → α} {g g' : β → α} {s : Set β} (hf : TendstoUniformlyOn f g l s) (hf' : TendstoUniformlyOn f' g' l s)
Goal: TendstoUniformlyOn (f - f') (g - g') l s
-- TendstoUniformlyOn.div
@[to_additive]
theorem TendstoUniformlyOn.div (hf : TendstoUniformlyOn f g l s)
    (hf' : TendstoUniformlyOn f' g' l s) : TendstoUniformlyOn (f / f') (g / g') l s 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [Group α] [UniformGroup α] {ι : Type u_3} {l : Filter ι} {f f' : ι → β → α} {g g' : β → α} {s : Set β} (hf : TendstoUniformlyOn f g l s) (hf' : TendstoUniformlyOn f' g' l s)
Goal: TendstoUniformlyOn (f / f') (g / g') l s
-- TendstoUniformly.add
@[to_additive]
theorem TendstoUniformly.mul (hf : TendstoUniformly f g l) (hf' : TendstoUniformly f' g' l) :
    TendstoUniformly (f * f') (g * g') l 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [AddGroup α] [UniformAddGroup α] {ι : Type u_3} {l : Filter ι} {f f' : ι → β → α} {g g' : β → α} (hf : TendstoUniformly f g l) (hf' : TendstoUniformly f' g' l)
Goal: TendstoUniformly (f + f') (g + g') l
-- TendstoUniformly.mul
@[to_additive]
theorem TendstoUniformly.mul (hf : TendstoUniformly f g l) (hf' : TendstoUniformly f' g' l) :
    TendstoUniformly (f * f') (g * g') l 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [Group α] [UniformGroup α] {ι : Type u_3} {l : Filter ι} {f f' : ι → β → α} {g g' : β → α} (hf : TendstoUniformly f g l) (hf' : TendstoUniformly f' g' l)
Goal: TendstoUniformly (f * f') (g * g') l
-- TendstoUniformly.div
@[to_additive]
theorem TendstoUniformly.div (hf : TendstoUniformly f g l) (hf' : TendstoUniformly f' g' l) :
    TendstoUniformly (f / f') (g / g') l 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [Group α] [UniformGroup α] {ι : Type u_3} {l : Filter ι} {f f' : ι → β → α} {g g' : β → α} (hf : TendstoUniformly f g l) (hf' : TendstoUniformly f' g' l)
Goal: TendstoUniformly (f / f') (g / g') l
-- TendstoUniformly.sub
@[to_additive]
theorem TendstoUniformly.div (hf : TendstoUniformly f g l) (hf' : TendstoUniformly f' g' l) :
    TendstoUniformly (f / f') (g / g') l 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [AddGroup α] [UniformAddGroup α] {ι : Type u_3} {l : Filter ι} {f f' : ι → β → α} {g g' : β → α} (hf : TendstoUniformly f g l) (hf' : TendstoUniformly f' g' l)
Goal: TendstoUniformly (f - f') (g - g') l
-- UniformCauchySeqOn.add
@[to_additive]
theorem UniformCauchySeqOn.mul (hf : UniformCauchySeqOn f l s) (hf' : UniformCauchySeqOn f' l s) :
    UniformCauchySeqOn (f * f') l s 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [AddGroup α] [UniformAddGroup α] {ι : Type u_3} {l : Filter ι} {f f' : ι → β → α} {s : Set β} (hf : UniformCauchySeqOn f l s) (hf' : UniformCauchySeqOn f' l s)
Goal: UniformCauchySeqOn (f + f') l s
-- UniformCauchySeqOn.mul
@[to_additive]
theorem UniformCauchySeqOn.mul (hf : UniformCauchySeqOn f l s) (hf' : UniformCauchySeqOn f' l s) :
    UniformCauchySeqOn (f * f') l s 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [Group α] [UniformGroup α] {ι : Type u_3} {l : Filter ι} {f f' : ι → β → α} {s : Set β} (hf : UniformCauchySeqOn f l s) (hf' : UniformCauchySeqOn f' l s)
Goal: UniformCauchySeqOn (f * f') l s
-- UniformCauchySeqOn.sub
@[to_additive]
theorem UniformCauchySeqOn.div (hf : UniformCauchySeqOn f l s) (hf' : UniformCauchySeqOn f' l s) :
    UniformCauchySeqOn (f / f') l s 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [AddGroup α] [UniformAddGroup α] {ι : Type u_3} {l : Filter ι} {f f' : ι → β → α} {s : Set β} (hf : UniformCauchySeqOn f l s) (hf' : UniformCauchySeqOn f' l s)
Goal: UniformCauchySeqOn (f - f') l s