ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- tendsto_diag_uniformity
theorem tendsto_diag_uniformity (f : β → α) (l : Filter β) :
    Tendsto (fun x => (f x, f x)) l (𝓤 α) 
Arguments: {α : Type ua} {β : Type ub} [UniformSpace α] (f : β → α) (l : Filter β)
Goal: Filter.Tendsto (fun x => (f x, f x)) l (𝓤 α)
-- tendsto_const_uniformity
theorem tendsto_const_uniformity {a : α} {f : Filter β} : Tendsto (fun _ => (a, a)) f (𝓤 α) 
Arguments: {α : Type ua} {β : Type ub} [UniformSpace α] {a : α} {f : Filter β}
Goal: Filter.Tendsto (fun x => (a, a)) f (𝓤 α)
-- symm_of_uniformity
theorem symm_of_uniformity {s : Set (α × α)} (hs : s ∈ 𝓤 α) :
    ∃ t ∈ 𝓤 α, (∀ a b, (a, b) ∈ t → (b, a) ∈ t) ∧ t ⊆ s 
Arguments: {α : Type ua} [UniformSpace α] {s : Set (α × α)} (hs : s ∈ 𝓤 α)
Goal: ∃ t ∈ 𝓤 α, (∀ (a b : α), (a, b) ∈ t → (b, a) ∈ t) ∧ t ⊆ s
-- comp_symm_of_uniformity
theorem comp_symm_of_uniformity {s : Set (α × α)} (hs : s ∈ 𝓤 α) :
    ∃ t ∈ 𝓤 α, (∀ {a b}, (a, b) ∈ t → (b, a) ∈ t) ∧ t ○ t ⊆ s 
Arguments: {α : Type ua} [UniformSpace α] {s : Set (α × α)} (hs : s ∈ 𝓤 α)
Goal: ∃ t ∈ 𝓤 α, (∀ {a b : α}, (a, b) ∈ t → (b, a) ∈ t) ∧ t ○ t ⊆ s
-- uniformity_le_symm
theorem uniformity_le_symm : 𝓤 α ≤ @Prod.swap α α <$> 𝓤 α 
Arguments: {α : Type ua} [UniformSpace α]
Goal: 𝓤 α ≤ Prod.swap <$> 𝓤 α
-- uniformity_eq_symm
theorem uniformity_eq_symm : 𝓤 α = @Prod.swap α α <$> 𝓤 α 
Arguments: {α : Type ua} [UniformSpace α]
Goal: 𝓤 α = Prod.swap <$> 𝓤 α
-- comap_swap_uniformity
@[simp]
theorem comap_swap_uniformity : comap (@Prod.swap α α) (𝓤 α) = 𝓤 α 
Arguments: {α : Type ua} [UniformSpace α]
Goal: Filter.comap Prod.swap (𝓤 α) = 𝓤 α
-- symmetrize_mem_uniformity
theorem symmetrize_mem_uniformity {V : Set (α × α)} (h : V ∈ 𝓤 α) : symmetrizeRel V ∈ 𝓤 α 
Arguments: {α : Type ua} [UniformSpace α] {V : Set (α × α)} (h : V ∈ 𝓤 α)
Goal: symmetrizeRel V ∈ 𝓤 α
-- UniformSpace.hasBasis_symmetric
theorem UniformSpace.hasBasis_symmetric :
    (𝓤 α).HasBasis (fun s : Set (α × α) => s ∈ 𝓤 α ∧ SymmetricRel s) id 
Arguments: {α : Type ua} [UniformSpace α]
Goal: (𝓤 α).HasBasis (fun s => s ∈ 𝓤 α ∧ SymmetricRel s) id
-- uniformity_lift_le_swap
theorem uniformity_lift_le_swap {g : Set (α × α) → Filter β} {f : Filter β} (hg : Monotone g)
    (h : ((𝓤 α).lift fun s => g (preimage Prod.swap s)) ≤ f) : (𝓤 α).lift g ≤ f 
Arguments: {α : Type ua} {β : Type ub} [UniformSpace α] {g : Set (α × α) → Filter β} {f : Filter β} (hg : Monotone g) (h : ((𝓤 α).lift fun s => g (Prod.swap ⁻¹' s)) ≤ f)
Goal: (𝓤 α).lift g ≤ f
-- uniformity_lift_le_comp
theorem uniformity_lift_le_comp {f : Set (α × α) → Filter β} (h : Monotone f) :
    ((𝓤 α).lift fun s => f (s ○ s)) ≤ (𝓤 α).lift f 
Arguments: {α : Type ua} {β : Type ub} [UniformSpace α] {f : Set (α × α) → Filter β} (h : Monotone f)
Goal: ((𝓤 α).lift fun s => f (s ○ s)) ≤ (𝓤 α).lift f
-- comp3_mem_uniformity
theorem comp3_mem_uniformity {s : Set (α × α)} (hs : s ∈ 𝓤 α) : ∃ t ∈ 𝓤 α, t ○ (t ○ t) ⊆ s 
Arguments: {α : Type ua} [UniformSpace α] {s : Set (α × α)} (hs : s ∈ 𝓤 α)
Goal: ∃ t ∈ 𝓤 α, t ○ (t ○ t) ⊆ s
-- comp_le_uniformity3
theorem comp_le_uniformity3 : ((𝓤 α).lift' fun s : Set (α × α) => s ○ (s ○ s)) ≤ 𝓤 α 
Arguments: {α : Type ua} [UniformSpace α]
Goal: ((𝓤 α).lift' fun s => s ○ (s ○ s)) ≤ 𝓤 α
-- comp_symm_mem_uniformity_sets
theorem comp_symm_mem_uniformity_sets {s : Set (α × α)} (hs : s ∈ 𝓤 α) :
    ∃ t ∈ 𝓤 α, SymmetricRel t ∧ t ○ t ⊆ s 
Arguments: {α : Type ua} [UniformSpace α] {s : Set (α × α)} (hs : s ∈ 𝓤 α)
Goal: ∃ t ∈ 𝓤 α, SymmetricRel t ∧ t ○ t ⊆ s
-- subset_comp_self_of_mem_uniformity
theorem subset_comp_self_of_mem_uniformity {s : Set (α × α)} (h : s ∈ 𝓤 α) : s ⊆ s ○ s 
Arguments: {α : Type ua} [UniformSpace α] {s : Set (α × α)} (h : s ∈ 𝓤 α)
Goal: s ⊆ s ○ s
-- comp_comp_symm_mem_uniformity_sets
theorem comp_comp_symm_mem_uniformity_sets {s : Set (α × α)} (hs : s ∈ 𝓤 α) :
    ∃ t ∈ 𝓤 α, SymmetricRel t ∧ t ○ t ○ t ⊆ s 
Arguments: {α : Type ua} [UniformSpace α] {s : Set (α × α)} (hs : s ∈ 𝓤 α)
Goal: ∃ t ∈ 𝓤 α, SymmetricRel t ∧ t ○ t ○ t ⊆ s
-- UniformSpace.mem_ball_self
theorem UniformSpace.mem_ball_self (x : α) {V : Set (α × α)} (hV : V ∈ 𝓤 α) : x ∈ ball x V 
Arguments: {α : Type ua} [UniformSpace α] (x : α) {V : Set (α × α)} (hV : V ∈ 𝓤 α)
Goal: x ∈ UniformSpace.ball x V
-- mem_ball_comp
theorem mem_ball_comp {V W : Set (β × β)} {x y z} (h : y ∈ ball x V) (h' : z ∈ ball y W) :
    z ∈ ball x (V ○ W) 
Arguments: {β : Type ub} {V W : Set (β × β)} {x y z : β} (h : y ∈ UniformSpace.ball x V) (h' : z ∈ UniformSpace.ball y W)
Goal: z ∈ UniformSpace.ball x (V ○ W)
-- ball_subset_of_comp_subset
theorem ball_subset_of_comp_subset {V W : Set (β × β)} {x y} (h : x ∈ ball y W) (h' : W ○ W ⊆ V) :
    ball x W ⊆ ball y V 
Arguments: {β : Type ub} {V W : Set (β × β)} {x y : β} (h : x ∈ UniformSpace.ball y W) (h' : W ○ W ⊆ V)
Goal: UniformSpace.ball x W ⊆ UniformSpace.ball y V
-- ball_mono
theorem ball_mono {V W : Set (β × β)} (h : V ⊆ W) (x : β) : ball x V ⊆ ball x W 
Arguments: {β : Type ub} {V W : Set (β × β)} (h : V ⊆ W) (x : β)
Goal: UniformSpace.ball x V ⊆ UniformSpace.ball x W
-- ball_inter
theorem ball_inter (x : β) (V W : Set (β × β)) : ball x (V ∩ W) = ball x V ∩ ball x W 
Arguments: {β : Type ub} (x : β) (V W : Set (β × β))
Goal: UniformSpace.ball x (V ∩ W) = UniformSpace.ball x V ∩ UniformSpace.ball x W
-- ball_inter_left
theorem ball_inter_left (x : β) (V W : Set (β × β)) : ball x (V ∩ W) ⊆ ball x V 
Arguments: {β : Type ub} (x : β) (V W : Set (β × β))
Goal: UniformSpace.ball x (V ∩ W) ⊆ UniformSpace.ball x V
-- ball_inter_right
theorem ball_inter_right (x : β) (V W : Set (β × β)) : ball x (V ∩ W) ⊆ ball x W 
Arguments: {β : Type ub} (x : β) (V W : Set (β × β))
Goal: UniformSpace.ball x (V ∩ W) ⊆ UniformSpace.ball x W
-- mem_ball_symmetry
theorem mem_ball_symmetry {V : Set (β × β)} (hV : SymmetricRel V) {x y} :
    x ∈ ball y V ↔ y ∈ ball x V 
Arguments: {β : Type ub} {V : Set (β × β)} (hV : SymmetricRel V) {x y : β}
Goal: x ∈ UniformSpace.ball y V ↔ y ∈ UniformSpace.ball x V
-- ball_eq_of_symmetry
theorem ball_eq_of_symmetry {V : Set (β × β)} (hV : SymmetricRel V) {x} :
    ball x V = { y | (y, x) ∈ V } 
Arguments: {β : Type ub} {V : Set (β × β)} (hV : SymmetricRel V) {x : β}
Goal: UniformSpace.ball x V = {y | (y, x) ∈ V}
-- mem_comp_of_mem_ball
theorem mem_comp_of_mem_ball {V W : Set (β × β)} {x y z : β} (hV : SymmetricRel V)
    (hx : x ∈ ball z V) (hy : y ∈ ball z W) : (x, y) ∈ V ○ W 
Arguments: {β : Type ub} {V W : Set (β × β)} {x y z : β} (hV : SymmetricRel V) (hx : x ∈ UniformSpace.ball z V) (hy : y ∈ UniformSpace.ball z W)
Goal: (x, y) ∈ V ○ W
-- UniformSpace.isOpen_ball
theorem UniformSpace.isOpen_ball (x : α) {V : Set (α × α)} (hV : IsOpen V) : IsOpen (ball x V) 
Arguments: {α : Type ua} [UniformSpace α] (x : α) {V : Set (α × α)} (hV : IsOpen V)
Goal: IsOpen (UniformSpace.ball x V)
-- UniformSpace.isClosed_ball
theorem UniformSpace.isClosed_ball (x : α) {V : Set (α × α)} (hV : IsClosed V) :
    IsClosed (ball x V) 
Arguments: {α : Type ua} [UniformSpace α] (x : α) {V : Set (α × α)} (hV : IsClosed V)
Goal: IsClosed (UniformSpace.ball x V)
-- mem_comp_comp
theorem mem_comp_comp {V W M : Set (β × β)} (hW' : SymmetricRel W) {p : β × β} :
    p ∈ V ○ M ○ W ↔ (ball p.1 V ×ˢ ball p.2 W ∩ M).Nonempty 
Arguments: {β : Type ub} {V W M : Set (β × β)} (hW' : SymmetricRel W) {p : β × β}
Goal: p ∈ V ○ M ○ W ↔ (UniformSpace.ball p.1 V ×ˢ UniformSpace.ball p.2 W ∩ M).Nonempty
-- mem_nhds_uniformity_iff_right
theorem mem_nhds_uniformity_iff_right {x : α} {s : Set α} :
    s ∈ 𝓝 x ↔ { p : α × α | p.1 = x → p.2 ∈ s } ∈ 𝓤 α 
Arguments: {α : Type ua} [UniformSpace α] {x : α} {s : Set α}
Goal: s ∈ 𝓝 x ↔ {p | p.1 = x → p.2 ∈ s} ∈ 𝓤 α
-- mem_nhds_uniformity_iff_left
theorem mem_nhds_uniformity_iff_left {x : α} {s : Set α} :
    s ∈ 𝓝 x ↔ { p : α × α | p.2 = x → p.1 ∈ s } ∈ 𝓤 α 
Arguments: {α : Type ua} [UniformSpace α] {x : α} {s : Set α}
Goal: s ∈ 𝓝 x ↔ {p | p.2 = x → p.1 ∈ s} ∈ 𝓤 α
-- nhdsWithin_eq_comap_uniformity_of_mem
theorem nhdsWithin_eq_comap_uniformity_of_mem {x : α} {T : Set α} (hx : x ∈ T) (S : Set α) :
    𝓝[S] x = (𝓤 α ⊓ 𝓟 (T ×ˢ S)).comap (Prod.mk x) 
Arguments: {α : Type ua} [UniformSpace α] {x : α} {T : Set α} (hx : x ∈ T) (S : Set α)
Goal: 𝓝[S] x = Filter.comap (Prod.mk x) (𝓤 α ⊓ 𝓟 (T ×ˢ S))
-- nhdsWithin_eq_comap_uniformity
theorem nhdsWithin_eq_comap_uniformity {x : α} (S : Set α) :
    𝓝[S] x = (𝓤 α ⊓ 𝓟 (univ ×ˢ S)).comap (Prod.mk x) 
Arguments: {α : Type ua} [UniformSpace α] {x : α} (S : Set α)
Goal: 𝓝[S] x = Filter.comap (Prod.mk x) (𝓤 α ⊓ 𝓟 (Set.univ ×ˢ S))
-- isOpen_iff_ball_subset
theorem isOpen_iff_ball_subset {s : Set α} : IsOpen s ↔ ∀ x ∈ s, ∃ V ∈ 𝓤 α, ball x V ⊆ s 
Arguments: {α : Type ua} [UniformSpace α] {s : Set α}
Goal: IsOpen s ↔ ∀ x ∈ s, ∃ V ∈ 𝓤 α, UniformSpace.ball x V ⊆ s
-- nhds_basis_uniformity'
theorem nhds_basis_uniformity' {p : ι → Prop} {s : ι → Set (α × α)} (h : (𝓤 α).HasBasis p s)
    {x : α} : (𝓝 x).HasBasis p fun i => ball x (s i) 
Arguments: {α : Type ua} {ι : Sort u_1} [UniformSpace α] {p : ι → Prop} {s : ι → Set (α × α)} (h : (𝓤 α).HasBasis p s) {x : α}
Goal: (𝓝 x).HasBasis p fun i => UniformSpace.ball x (s i)
-- nhds_basis_uniformity
theorem nhds_basis_uniformity {p : ι → Prop} {s : ι → Set (α × α)} (h : (𝓤 α).HasBasis p s)
    {x : α} : (𝓝 x).HasBasis p fun i => { y | (y, x) ∈ s i } 
Arguments: {α : Type ua} {ι : Sort u_1} [UniformSpace α] {p : ι → Prop} {s : ι → Set (α × α)} (h : (𝓤 α).HasBasis p s) {x : α}
Goal: (𝓝 x).HasBasis p fun i => {y | (y, x) ∈ s i}
-- nhds_eq_comap_uniformity'
theorem nhds_eq_comap_uniformity' {x : α} : 𝓝 x = (𝓤 α).comap fun y => (y, x) 
Arguments: {α : Type ua} [UniformSpace α] {x : α}
Goal: 𝓝 x = Filter.comap (fun y => (y, x)) (𝓤 α)
-- UniformSpace.mem_nhds_iff
theorem UniformSpace.mem_nhds_iff {x : α} {s : Set α} : s ∈ 𝓝 x ↔ ∃ V ∈ 𝓤 α, ball x V ⊆ s 
Arguments: {α : Type ua} [UniformSpace α] {x : α} {s : Set α}
Goal: s ∈ 𝓝 x ↔ ∃ V ∈ 𝓤 α, UniformSpace.ball x V ⊆ s
-- UniformSpace.ball_mem_nhds
theorem UniformSpace.ball_mem_nhds (x : α) ⦃V : Set (α × α)⦄ (V_in : V ∈ 𝓤 α) : ball x V ∈ 𝓝 x 
Arguments: {α : Type ua} [UniformSpace α] (x : α) ⦃V : Set (α × α)⦄ (V_in : V ∈ 𝓤 α)
Goal: UniformSpace.ball x V ∈ 𝓝 x
-- UniformSpace.ball_mem_nhdsWithin
theorem UniformSpace.ball_mem_nhdsWithin {x : α} {S : Set α} ⦃V : Set (α × α)⦄ (x_in : x ∈ S)
    (V_in : V ∈ 𝓤 α ⊓ 𝓟 (S ×ˢ S)) : ball x V ∈ 𝓝[S] x 
Arguments: {α : Type ua} [UniformSpace α] {x : α} {S : Set α} ⦃V : Set (α × α)⦄ (x_in : x ∈ S) (V_in : V ∈ 𝓤 α ⊓ 𝓟 (S ×ˢ S))
Goal: UniformSpace.ball x V ∈ 𝓝[S] x
-- UniformSpace.mem_nhds_iff_symm
theorem UniformSpace.mem_nhds_iff_symm {x : α} {s : Set α} :
    s ∈ 𝓝 x ↔ ∃ V ∈ 𝓤 α, SymmetricRel V ∧ ball x V ⊆ s 
Arguments: {α : Type ua} [UniformSpace α] {x : α} {s : Set α}
Goal: s ∈ 𝓝 x ↔ ∃ V ∈ 𝓤 α, SymmetricRel V ∧ UniformSpace.ball x V ⊆ s
-- UniformSpace.hasBasis_nhds
theorem UniformSpace.hasBasis_nhds (x : α) :
    HasBasis (𝓝 x) (fun s : Set (α × α) => s ∈ 𝓤 α ∧ SymmetricRel s) fun s => ball x s 
Arguments: {α : Type ua} [UniformSpace α] (x : α)
Goal: (𝓝 x).HasBasis (fun s => s ∈ 𝓤 α ∧ SymmetricRel s) fun s => UniformSpace.ball x s
-- UniformSpace.mem_closure_iff_symm_ball
theorem UniformSpace.mem_closure_iff_symm_ball {s : Set α} {x} :
    x ∈ closure s ↔ ∀ {V}, V ∈ 𝓤 α → SymmetricRel V → (s ∩ ball x V).Nonempty 
Arguments: {α : Type ua} [UniformSpace α] {s : Set α} {x : α}
Goal: x ∈ closure s ↔ ∀ {V : Set (α × α)}, V ∈ 𝓤 α → SymmetricRel V → (s ∩ UniformSpace.ball x V).Nonempty
-- UniformSpace.mem_closure_iff_ball
theorem UniformSpace.mem_closure_iff_ball {s : Set α} {x} :
    x ∈ closure s ↔ ∀ {V}, V ∈ 𝓤 α → (ball x V ∩ s).Nonempty 
Arguments: {α : Type ua} [UniformSpace α] {s : Set α} {x : α}
Goal: x ∈ closure s ↔ ∀ {V : Set (α × α)}, V ∈ 𝓤 α → (UniformSpace.ball x V ∩ s).Nonempty
-- UniformSpace.hasBasis_nhds_prod
theorem UniformSpace.hasBasis_nhds_prod (x y : α) :
    HasBasis (𝓝 (x, y)) (fun s => s ∈ 𝓤 α ∧ SymmetricRel s) fun s => ball x s ×ˢ ball y s 
Arguments: {α : Type ua} [UniformSpace α] (x y : α)
Goal: (𝓝 (x, y)).HasBasis (fun s => s ∈ 𝓤 α ∧ SymmetricRel s) fun s => UniformSpace.ball x s ×ˢ UniformSpace.ball y s
-- nhds_eq_uniformity
theorem nhds_eq_uniformity {x : α} : 𝓝 x = (𝓤 α).lift' (ball x) 
Arguments: {α : Type ua} [UniformSpace α] {x : α}
Goal: 𝓝 x = (𝓤 α).lift' (UniformSpace.ball x)
-- nhds_eq_uniformity'
theorem nhds_eq_uniformity' {x : α} : 𝓝 x = (𝓤 α).lift' fun s => { y | (y, x) ∈ s } 
Arguments: {α : Type ua} [UniformSpace α] {x : α}
Goal: 𝓝 x = (𝓤 α).lift' fun s => {y | (y, x) ∈ s}
-- mem_nhds_left
theorem mem_nhds_left (x : α) {s : Set (α × α)} (h : s ∈ 𝓤 α) : { y : α | (x, y) ∈ s } ∈ 𝓝 x 
Arguments: {α : Type ua} [UniformSpace α] (x : α) {s : Set (α × α)} (h : s ∈ 𝓤 α)
Goal: {y | (x, y) ∈ s} ∈ 𝓝 x
-- mem_nhds_right
theorem mem_nhds_right (y : α) {s : Set (α × α)} (h : s ∈ 𝓤 α) : { x : α | (x, y) ∈ s } ∈ 𝓝 y 
Arguments: {α : Type ua} [UniformSpace α] (y : α) {s : Set (α × α)} (h : s ∈ 𝓤 α)
Goal: {x | (x, y) ∈ s} ∈ 𝓝 y
-- exists_mem_nhds_ball_subset_of_mem_nhds
theorem exists_mem_nhds_ball_subset_of_mem_nhds {a : α} {U : Set α} (h : U ∈ 𝓝 a) :
    ∃ V ∈ 𝓝 a, ∃ t ∈ 𝓤 α, ∀ a' ∈ V, UniformSpace.ball a' t ⊆ U 
Arguments: {α : Type ua} [UniformSpace α] {a : α} {U : Set α} (h : U ∈ 𝓝 a)
Goal: ∃ V ∈ 𝓝 a, ∃ t ∈ 𝓤 α, ∀ a' ∈ V, UniformSpace.ball a' t ⊆ U
-- tendsto_right_nhds_uniformity
theorem tendsto_right_nhds_uniformity {a : α} : Tendsto (fun a' => (a', a)) (𝓝 a) (𝓤 α) 
Arguments: {α : Type ua} [UniformSpace α] {a : α}
Goal: Filter.Tendsto (fun a' => (a', a)) (𝓝 a) (𝓤 α)
-- tendsto_left_nhds_uniformity
theorem tendsto_left_nhds_uniformity {a : α} : Tendsto (fun a' => (a, a')) (𝓝 a) (𝓤 α) 
Arguments: {α : Type ua} [UniformSpace α] {a : α}
Goal: Filter.Tendsto (fun a' => (a, a')) (𝓝 a) (𝓤 α)
-- lift_nhds_left
theorem lift_nhds_left {x : α} {g : Set α → Filter β} (hg : Monotone g) :
    (𝓝 x).lift g = (𝓤 α).lift fun s : Set (α × α) => g (ball x s) 
Arguments: {α : Type ua} {β : Type ub} [UniformSpace α] {x : α} {g : Set α → Filter β} (hg : Monotone g)
Goal: (𝓝 x).lift g = (𝓤 α).lift fun s => g (UniformSpace.ball x s)
-- lift_nhds_right
theorem lift_nhds_right {x : α} {g : Set α → Filter β} (hg : Monotone g) :
    (𝓝 x).lift g = (𝓤 α).lift fun s : Set (α × α) => g { y | (y, x) ∈ s } 
Arguments: {α : Type ua} {β : Type ub} [UniformSpace α] {x : α} {g : Set α → Filter β} (hg : Monotone g)
Goal: (𝓝 x).lift g = (𝓤 α).lift fun s => g {y | (y, x) ∈ s}
-- nhds_nhds_eq_uniformity_uniformity_prod
theorem nhds_nhds_eq_uniformity_uniformity_prod {a b : α} :
    𝓝 a ×ˢ 𝓝 b = (𝓤 α).lift fun s : Set (α × α) =>
      (𝓤 α).lift' fun t => { y : α | (y, a) ∈ s } ×ˢ { y : α | (b, y) ∈ t } 
Arguments: {α : Type ua} [UniformSpace α] {a b : α}
Goal: 𝓝 a ×ˢ 𝓝 b = (𝓤 α).lift fun s => (𝓤 α).lift' fun t => {y | (y, a) ∈ s} ×ˢ {y | (b, y) ∈ t}
-- nhds_eq_uniformity_prod
theorem nhds_eq_uniformity_prod {a b : α} :
    𝓝 (a, b) =
      (𝓤 α).lift' fun s : Set (α × α) => { y : α | (y, a) ∈ s } ×ˢ { y : α | (b, y) ∈ s } 
Arguments: {α : Type ua} [UniformSpace α] {a b : α}
Goal: 𝓝 (a, b) = (𝓤 α).lift' fun s => {y | (y, a) ∈ s} ×ˢ {y | (b, y) ∈ s}
-- nhdset_of_mem_uniformity
theorem nhdset_of_mem_uniformity {d : Set (α × α)} (s : Set (α × α)) (hd : d ∈ 𝓤 α) :
    ∃ t : Set (α × α), IsOpen t ∧ s ⊆ t ∧
      t ⊆ { p | ∃ x y, (p.1, x) ∈ d ∧ (x, y) ∈ s ∧ (y, p.2) ∈ d } 
Arguments: {α : Type ua} [UniformSpace α] {d : Set (α × α)} (s : Set (α × α)) (hd : d ∈ 𝓤 α)
Goal: ∃ t, IsOpen t ∧ s ⊆ t ∧ t ⊆ {p | ∃ x y, (p.1, x) ∈ d ∧ (x, y) ∈ s ∧ (y, p.2) ∈ d}
-- nhds_le_uniformity
theorem nhds_le_uniformity (x : α) : 𝓝 (x, x) ≤ 𝓤 α 
Arguments: {α : Type ua} [UniformSpace α] (x : α)
Goal: 𝓝 (x, x) ≤ 𝓤 α
-- iSup_nhds_le_uniformity
theorem iSup_nhds_le_uniformity : ⨆ x : α, 𝓝 (x, x) ≤ 𝓤 α 
Arguments: {α : Type ua} [UniformSpace α]
Goal: ⨆ x, 𝓝 (x, x) ≤ 𝓤 α
-- nhdsSet_diagonal_le_uniformity
theorem nhdsSet_diagonal_le_uniformity : 𝓝ˢ (diagonal α) ≤ 𝓤 α 
Arguments: {α : Type ua} [UniformSpace α]
Goal: 𝓝ˢ (Set.diagonal α) ≤ 𝓤 α
-- closure_eq_uniformity
theorem closure_eq_uniformity (s : Set <| α × α) :
    closure s = ⋂ V ∈ { V | V ∈ 𝓤 α ∧ SymmetricRel V }, V ○ s ○ V 
Arguments: {α : Type ua} [UniformSpace α] (s : Set (α × α))
Goal: closure s = ⋂ V ∈ {V | V ∈ 𝓤 α ∧ SymmetricRel V}, V ○ s ○ V
-- uniformity_hasBasis_closed
theorem uniformity_hasBasis_closed :
    HasBasis (𝓤 α) (fun V : Set (α × α) => V ∈ 𝓤 α ∧ IsClosed V) id 
Arguments: {α : Type ua} [UniformSpace α]
Goal: (𝓤 α).HasBasis (fun V => V ∈ 𝓤 α ∧ IsClosed V) id
-- uniformity_eq_uniformity_closure
theorem uniformity_eq_uniformity_closure : 𝓤 α = (𝓤 α).lift' closure 
Arguments: {α : Type ua} [UniformSpace α]
Goal: 𝓤 α = (𝓤 α).lift' closure
-- Filter.HasBasis.uniformity_closure
theorem Filter.HasBasis.uniformity_closure {p : ι → Prop} {U : ι → Set (α × α)}
    (h : (𝓤 α).HasBasis p U) : (𝓤 α).HasBasis p fun i => closure (U i) 
Arguments: {α : Type ua} {ι : Sort u_1} [UniformSpace α] {p : ι → Prop} {U : ι → Set (α × α)} (h : (𝓤 α).HasBasis p U)
Goal: (𝓤 α).HasBasis p fun i => closure (U i)
-- uniformity_hasBasis_closure
theorem uniformity_hasBasis_closure : HasBasis (𝓤 α) (fun V : Set (α × α) => V ∈ 𝓤 α) closure 
Arguments: {α : Type ua} [UniformSpace α]
Goal: (𝓤 α).HasBasis (fun V => V ∈ 𝓤 α) closure
-- closure_eq_inter_uniformity
theorem closure_eq_inter_uniformity {t : Set (α × α)} : closure t = ⋂ d ∈ 𝓤 α, d ○ (t ○ d) 
Arguments: {α : Type ua} [UniformSpace α] {t : Set (α × α)}
Goal: closure t = ⋂ d ∈ 𝓤 α, d ○ (t ○ d)
-- uniformity_eq_uniformity_interior
theorem uniformity_eq_uniformity_interior : 𝓤 α = (𝓤 α).lift' interior 
Arguments: {α : Type ua} [UniformSpace α]
Goal: 𝓤 α = (𝓤 α).lift' interior
-- interior_mem_uniformity
theorem interior_mem_uniformity {s : Set (α × α)} (hs : s ∈ 𝓤 α) : interior s ∈ 𝓤 α 
Arguments: {α : Type ua} [UniformSpace α] {s : Set (α × α)} (hs : s ∈ 𝓤 α)
Goal: interior s ∈ 𝓤 α
-- mem_uniformity_isClosed
theorem mem_uniformity_isClosed {s : Set (α × α)} (h : s ∈ 𝓤 α) : ∃ t ∈ 𝓤 α, IsClosed t ∧ t ⊆ s 
Arguments: {α : Type ua} [UniformSpace α] {s : Set (α × α)} (h : s ∈ 𝓤 α)
Goal: ∃ t ∈ 𝓤 α, IsClosed t ∧ t ⊆ s
-- isOpen_iff_open_ball_subset
theorem isOpen_iff_open_ball_subset {s : Set α} :
    IsOpen s ↔ ∀ x ∈ s, ∃ V ∈ 𝓤 α, IsOpen V ∧ ball x V ⊆ s 
Arguments: {α : Type ua} [UniformSpace α] {s : Set α}
Goal: IsOpen s ↔ ∀ x ∈ s, ∃ V ∈ 𝓤 α, IsOpen V ∧ UniformSpace.ball x V ⊆ s
-- Dense.biUnion_uniformity_ball
theorem Dense.biUnion_uniformity_ball {s : Set α} {U : Set (α × α)} (hs : Dense s) (hU : U ∈ 𝓤 α) :
    ⋃ x ∈ s, ball x U = univ 
Arguments: {α : Type ua} [UniformSpace α] {s : Set α} {U : Set (α × α)} (hs : Dense s) (hU : U ∈ 𝓤 α)
Goal: ⋃ x ∈ s, UniformSpace.ball x U = Set.univ
-- DenseRange.iUnion_uniformity_ball
lemma DenseRange.iUnion_uniformity_ball {ι : Type*} {xs : ι → α}
    (xs_dense : DenseRange xs) {U : Set (α × α)} (hU : U ∈ uniformity α) :
    ⋃ i, UniformSpace.ball (xs i) U = univ 
Arguments: {α : Type ua} [UniformSpace α] {ι : Type u_2} {xs : ι → α} (xs_dense : DenseRange xs) {U : Set (α × α)} (hU : U ∈ 𝓤 α)
Goal: ⋃ i, UniformSpace.ball (xs i) U = Set.univ
-- uniformity_hasBasis_open
theorem uniformity_hasBasis_open : HasBasis (𝓤 α) (fun V : Set (α × α) => V ∈ 𝓤 α ∧ IsOpen V) id 
Arguments: {α : Type ua} [UniformSpace α]
Goal: (𝓤 α).HasBasis (fun V => V ∈ 𝓤 α ∧ IsOpen V) id
-- Filter.HasBasis.mem_uniformity_iff
theorem Filter.HasBasis.mem_uniformity_iff {p : β → Prop} {s : β → Set (α × α)}
    (h : (𝓤 α).HasBasis p s) {t : Set (α × α)} :
    t ∈ 𝓤 α ↔ ∃ i, p i ∧ ∀ a b, (a, b) ∈ s i → (a, b) ∈ t 
Arguments: {α : Type ua} {β : Type ub} [UniformSpace α] {p : β → Prop} {s : β → Set (α × α)} (h : (𝓤 α).HasBasis p s) {t : Set (α × α)}
Goal: t ∈ 𝓤 α ↔ ∃ i, p i ∧ ∀ (a b : α), (a, b) ∈ s i → (a, b) ∈ t
-- uniformity_hasBasis_open_symmetric
theorem uniformity_hasBasis_open_symmetric :
    HasBasis (𝓤 α) (fun V : Set (α × α) => V ∈ 𝓤 α ∧ IsOpen V ∧ SymmetricRel V) id 
Arguments: {α : Type ua} [UniformSpace α]
Goal: (𝓤 α).HasBasis (fun V => V ∈ 𝓤 α ∧ IsOpen V ∧ SymmetricRel V) id
-- comp_open_symm_mem_uniformity_sets
theorem comp_open_symm_mem_uniformity_sets {s : Set (α × α)} (hs : s ∈ 𝓤 α) :
    ∃ t ∈ 𝓤 α, IsOpen t ∧ SymmetricRel t ∧ t ○ t ⊆ s 
Arguments: {α : Type ua} [UniformSpace α] {s : Set (α × α)} (hs : s ∈ 𝓤 α)
Goal: ∃ t ∈ 𝓤 α, IsOpen t ∧ SymmetricRel t ∧ t ○ t ⊆ s
-- UniformSpace.has_seq_basis
theorem UniformSpace.has_seq_basis [IsCountablyGenerated <| 𝓤 α] :
    ∃ V : ℕ → Set (α × α), HasAntitoneBasis (𝓤 α) V ∧ ∀ n, SymmetricRel (V n) 
Arguments: (α : Type ua) [UniformSpace α] [(𝓤 α).IsCountablyGenerated]
Goal: ∃ V, (𝓤 α).HasAntitoneBasis V ∧ ∀ (n : ℕ), SymmetricRel (V n)
-- Filter.HasBasis.biInter_biUnion_ball
theorem Filter.HasBasis.biInter_biUnion_ball {p : ι → Prop} {U : ι → Set (α × α)}
    (h : HasBasis (𝓤 α) p U) (s : Set α) :
    (⋂ (i) (_ : p i), ⋃ x ∈ s, ball x (U i)) = closure s 
Arguments: {α : Type ua} {ι : Sort u_1} [UniformSpace α] {p : ι → Prop} {U : ι → Set (α × α)} (h : (𝓤 α).HasBasis p U) (s : Set α)
Goal: ⋂ i, ⋂ (_ : p i), ⋃ x ∈ s, UniformSpace.ball x (U i) = closure s
-- uniformContinuous_def
theorem uniformContinuous_def [UniformSpace β] {f : α → β} :
    UniformContinuous f ↔ ∀ r ∈ 𝓤 β, { x : α × α | (f x.1, f x.2) ∈ r } ∈ 𝓤 α 
Arguments: {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] {f : α → β}
Goal: UniformContinuous f ↔ ∀ r ∈ 𝓤 β, {x | (f x.1, f x.2) ∈ r} ∈ 𝓤 α
-- uniformContinuous_iff_eventually
theorem uniformContinuous_iff_eventually [UniformSpace β] {f : α → β} :
    UniformContinuous f ↔ ∀ r ∈ 𝓤 β, ∀ᶠ x : α × α in 𝓤 α, (f x.1, f x.2) ∈ r 
Arguments: {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] {f : α → β}
Goal: UniformContinuous f ↔ ∀ r ∈ 𝓤 β, ∀ᶠ (x : α × α) in 𝓤 α, (f x.1, f x.2) ∈ r
-- uniformContinuousOn_univ
theorem uniformContinuousOn_univ [UniformSpace β] {f : α → β} :
    UniformContinuousOn f univ ↔ UniformContinuous f 
Arguments: {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] {f : α → β}
Goal: UniformContinuousOn f Set.univ ↔ UniformContinuous f
-- uniformContinuous_of_const
theorem uniformContinuous_of_const [UniformSpace β] {c : α → β} (h : ∀ a b, c a = c b) :
    UniformContinuous c 
Arguments: {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] {c : α → β} (h : ∀ (a b : α), c a = c b)
Goal: UniformContinuous c
-- uniformContinuous_id
theorem uniformContinuous_id : UniformContinuous (@id α) 
Arguments: {α : Type ua} [UniformSpace α]
Goal: UniformContinuous id
-- uniformContinuous_const
theorem uniformContinuous_const [UniformSpace β] {b : β} : UniformContinuous fun _ : α => b 
Arguments: {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] {b : β}
Goal: UniformContinuous fun x => b
-- UniformContinuous.comp
nonrec theorem UniformContinuous.comp [UniformSpace β] [UniformSpace γ] {g : β → γ} {f : α → β}
    (hg : UniformContinuous g) (hf : UniformContinuous f) : UniformContinuous (g ∘ f) 
Arguments: {α : Type ua} {β : Type ub} {γ : Type uc} [UniformSpace α] [UniformSpace β] [UniformSpace γ] {g : β → γ} {f : α → β} (hg : UniformContinuous g) (hf : UniformContinuous f)
Goal: UniformContinuous (g ∘ f)
-- UniformContinuous.iterate
theorem UniformContinuous.iterate [UniformSpace β] (T : β → β) (n : ℕ) (h : UniformContinuous T) :
    UniformContinuous T^[n] 
Arguments: {β : Type ub} [UniformSpace β] (T : β → β) (n : ℕ) (h : UniformContinuous T)
Goal: UniformContinuous T^[n]
-- Filter.HasBasis.uniformContinuous_iff
theorem Filter.HasBasis.uniformContinuous_iff {ι'} [UniformSpace β] {p : ι → Prop}
    {s : ι → Set (α × α)} (ha : (𝓤 α).HasBasis p s) {q : ι' → Prop} {t : ι' → Set (β × β)}
    (hb : (𝓤 β).HasBasis q t) {f : α → β} :
    UniformContinuous f ↔ ∀ i, q i → ∃ j, p j ∧ ∀ x y, (x, y) ∈ s j → (f x, f y) ∈ t i 
Arguments: {α : Type ua} {β : Type ub} {ι : Sort u_1} [UniformSpace α] {ι' : Sort u_2} [UniformSpace β] {p : ι → Prop} {s : ι → Set (α × α)} (ha : (𝓤 α).HasBasis p s) {q : ι' → Prop} {t : ι' → Set (β × β)} (hb : (𝓤 β).HasBasis q t) {f : α → β}
Goal: UniformContinuous f ↔ ∀ (i : ι'), q i → ∃ j, p j ∧ ∀ (x y : α), (x, y) ∈ s j → (f x, f y) ∈ t i