ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- EquicontinuousAt.closure'
theorem EquicontinuousAt.closure' {A : Set Y} {u : Y → X → α} {x₀ : X}
    (hA : EquicontinuousAt (u ∘ (↑) : A → X → α) x₀) (hu : Continuous u) :
    EquicontinuousAt (u ∘ (↑) : closure A → X → α) x₀ 
Arguments: {X : Type u_3} {Y : Type u_5} {α : Type u_7} [tX : TopologicalSpace X] [tY : TopologicalSpace Y] [uα : UniformSpace α] {A : Set Y} {u : Y → X → α} {x₀ : X} (hA : EquicontinuousAt (u ∘ Subtype.val) x₀) (hu : Continuous u)
Goal: EquicontinuousAt (u ∘ Subtype.val) x₀
-- Set.EquicontinuousAt.closure
protected theorem Set.EquicontinuousAt.closure {A : Set (X → α)} {x₀ : X}
    (hA : A.EquicontinuousAt x₀) : (closure A).EquicontinuousAt x₀ 
Arguments: {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {A : Set (X → α)} {x₀ : X} (hA : A.EquicontinuousAt x₀)
Goal: (closure A).EquicontinuousAt x₀
-- Set.EquicontinuousWithinAt.closure
protected theorem Set.EquicontinuousWithinAt.closure {A : Set (X → α)} {S : Set X} {x₀ : X}
    (hA : A.EquicontinuousWithinAt S x₀) :
    (closure A).EquicontinuousWithinAt S x₀ 
Arguments: {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {A : Set (X → α)} {S : Set X} {x₀ : X} (hA : A.EquicontinuousWithinAt S x₀)
Goal: (closure A).EquicontinuousWithinAt S x₀
-- Equicontinuous.closure'
theorem Equicontinuous.closure' {A : Set Y} {u : Y → X → α}
    (hA : Equicontinuous (u ∘ (↑) : A → X → α)) (hu : Continuous u) :
    Equicontinuous (u ∘ (↑) : closure A → X → α) 
Arguments: {X : Type u_3} {Y : Type u_5} {α : Type u_7} [tX : TopologicalSpace X] [tY : TopologicalSpace Y] [uα : UniformSpace α] {A : Set Y} {u : Y → X → α} (hA : Equicontinuous (u ∘ Subtype.val)) (hu : Continuous u)
Goal: Equicontinuous (u ∘ Subtype.val)
-- EquicontinuousOn.closure'
theorem EquicontinuousOn.closure' {A : Set Y} {u : Y → X → α} {S : Set X}
    (hA : EquicontinuousOn (u ∘ (↑) : A → X → α) S) (hu : Continuous (S.restrict ∘ u)) :
    EquicontinuousOn (u ∘ (↑) : closure A → X → α) S 
Arguments: {X : Type u_3} {Y : Type u_5} {α : Type u_7} [tX : TopologicalSpace X] [tY : TopologicalSpace Y] [uα : UniformSpace α] {A : Set Y} {u : Y → X → α} {S : Set X} (hA : EquicontinuousOn (u ∘ Subtype.val) S) (hu : Continuous (S.restrict ∘ u))
Goal: EquicontinuousOn (u ∘ Subtype.val) S
-- Set.Equicontinuous.closure
protected theorem Set.Equicontinuous.closure {A : Set <| X → α} (hA : A.Equicontinuous) :
    (closure A).Equicontinuous 
Arguments: {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {A : Set (X → α)} (hA : A.Equicontinuous)
Goal: (closure A).Equicontinuous
-- Set.EquicontinuousOn.closure
protected theorem Set.EquicontinuousOn.closure {A : Set <| X → α} {S : Set X}
    (hA : A.EquicontinuousOn S) : (closure A).EquicontinuousOn S 
Arguments: {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {A : Set (X → α)} {S : Set X} (hA : A.EquicontinuousOn S)
Goal: (closure A).EquicontinuousOn S
-- UniformEquicontinuousOn.closure'
theorem UniformEquicontinuousOn.closure' {A : Set Y} {u : Y → β → α} {S : Set β}
    (hA : UniformEquicontinuousOn (u ∘ (↑) : A → β → α) S) (hu : Continuous (S.restrict ∘ u)) :
    UniformEquicontinuousOn (u ∘ (↑) : closure A → β → α) S 
Arguments: {Y : Type u_5} {α : Type u_7} {β : Type u_9} [tY : TopologicalSpace Y] [uα : UniformSpace α] [uβ : UniformSpace β] {A : Set Y} {u : Y → β → α} {S : Set β} (hA : UniformEquicontinuousOn (u ∘ Subtype.val) S) (hu : Continuous (S.restrict ∘ u))
Goal: UniformEquicontinuousOn (u ∘ Subtype.val) S
-- UniformEquicontinuous.closure'
theorem UniformEquicontinuous.closure' {A : Set Y} {u : Y → β → α}
    (hA : UniformEquicontinuous (u ∘ (↑) : A → β → α)) (hu : Continuous u) :
    UniformEquicontinuous (u ∘ (↑) : closure A → β → α) 
Arguments: {Y : Type u_5} {α : Type u_7} {β : Type u_9} [tY : TopologicalSpace Y] [uα : UniformSpace α] [uβ : UniformSpace β] {A : Set Y} {u : Y → β → α} (hA : UniformEquicontinuous (u ∘ Subtype.val)) (hu : Continuous u)
Goal: UniformEquicontinuous (u ∘ Subtype.val)
-- Set.UniformEquicontinuous.closure
protected theorem Set.UniformEquicontinuous.closure {A : Set <| β → α}
    (hA : A.UniformEquicontinuous) : (closure A).UniformEquicontinuous 
Arguments: {α : Type u_7} {β : Type u_9} [uα : UniformSpace α] [uβ : UniformSpace β] {A : Set (β → α)} (hA : A.UniformEquicontinuous)
Goal: (closure A).UniformEquicontinuous
-- Set.UniformEquicontinuousOn.closure
protected theorem Set.UniformEquicontinuousOn.closure {A : Set <| β → α} {S : Set β}
    (hA : A.UniformEquicontinuousOn S) : (closure A).UniformEquicontinuousOn S 
Arguments: {α : Type u_7} {β : Type u_9} [uα : UniformSpace α] [uβ : UniformSpace β] {A : Set (β → α)} {S : Set β} (hA : A.UniformEquicontinuousOn S)
Goal: (closure A).UniformEquicontinuousOn S
-- Filter.Tendsto.continuousWithinAt_of_equicontinuousWithinAt
theorem Filter.Tendsto.continuousWithinAt_of_equicontinuousWithinAt {l : Filter ι} [l.NeBot]
    {F : ι → X → α} {f : X → α} {S : Set X} {x₀ : X} (h₁ : ∀ x ∈ S, Tendsto (F · x) l (𝓝 (f x)))
    (h₂ : Tendsto (F · x₀) l (𝓝 (f x₀))) (h₃ : EquicontinuousWithinAt F S x₀) :
    ContinuousWithinAt f S x₀ 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {l : Filter ι} [l.NeBot] {F : ι → X → α} {f : X → α} {S : Set X} {x₀ : X} (h₁ : ∀ x ∈ S, Filter.Tendsto (fun x_1 => F x_1 x) l (nhds (f x))) (h₂ : Filter.Tendsto (fun x => F x x₀) l (nhds (f x₀))) (h₃ : EquicontinuousWithinAt F S x₀)
Goal: ContinuousWithinAt f S x₀
-- Filter.Tendsto.continuousAt_of_equicontinuousAt
theorem Filter.Tendsto.continuousAt_of_equicontinuousAt {l : Filter ι} [l.NeBot] {F : ι → X → α}
    {f : X → α} {x₀ : X} (h₁ : Tendsto F l (𝓝 f)) (h₂ : EquicontinuousAt F x₀) :
    ContinuousAt f x₀ 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {l : Filter ι} [l.NeBot] {F : ι → X → α} {f : X → α} {x₀ : X} (h₁ : Filter.Tendsto F l (nhds f)) (h₂ : EquicontinuousAt F x₀)
Goal: ContinuousAt f x₀
-- Filter.Tendsto.continuous_of_equicontinuous
theorem Filter.Tendsto.continuous_of_equicontinuous {l : Filter ι} [l.NeBot] {F : ι → X → α}
    {f : X → α} (h₁ : Tendsto F l (𝓝 f)) (h₂ : Equicontinuous F) : Continuous f 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {l : Filter ι} [l.NeBot] {F : ι → X → α} {f : X → α} (h₁ : Filter.Tendsto F l (nhds f)) (h₂ : Equicontinuous F)
Goal: Continuous f
-- Filter.Tendsto.continuousOn_of_equicontinuousOn
theorem Filter.Tendsto.continuousOn_of_equicontinuousOn {l : Filter ι} [l.NeBot] {F : ι → X → α}
    {f : X → α} {S : Set X} (h₁ : ∀ x ∈ S, Tendsto (F · x) l (𝓝 (f x)))
    (h₂ : EquicontinuousOn F S) : ContinuousOn f S 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {l : Filter ι} [l.NeBot] {F : ι → X → α} {f : X → α} {S : Set X} (h₁ : ∀ x ∈ S, Filter.Tendsto (fun x_1 => F x_1 x) l (nhds (f x))) (h₂ : EquicontinuousOn F S)
Goal: ContinuousOn f S
-- Filter.Tendsto.uniformContinuousOn_of_uniformEquicontinuousOn
theorem Filter.Tendsto.uniformContinuousOn_of_uniformEquicontinuousOn {l : Filter ι} [l.NeBot]
    {F : ι → β → α} {f : β → α} {S : Set β} (h₁ : ∀ x ∈ S, Tendsto (F · x) l (𝓝 (f x)))
    (h₂ : UniformEquicontinuousOn F S) :
    UniformContinuousOn f S 
Arguments: {ι : Type u_1} {α : Type u_7} {β : Type u_9} [uα : UniformSpace α] [uβ : UniformSpace β] {l : Filter ι} [l.NeBot] {F : ι → β → α} {f : β → α} {S : Set β} (h₁ : ∀ x ∈ S, Filter.Tendsto (fun x_1 => F x_1 x) l (nhds (f x))) (h₂ : UniformEquicontinuousOn F S)
Goal: UniformContinuousOn f S
-- Filter.Tendsto.uniformContinuous_of_uniformEquicontinuous
theorem Filter.Tendsto.uniformContinuous_of_uniformEquicontinuous {l : Filter ι} [l.NeBot]
    {F : ι → β → α} {f : β → α} (h₁ : Tendsto F l (𝓝 f)) (h₂ : UniformEquicontinuous F) :
    UniformContinuous f 
Arguments: {ι : Type u_1} {α : Type u_7} {β : Type u_9} [uα : UniformSpace α] [uβ : UniformSpace β] {l : Filter ι} [l.NeBot] {F : ι → β → α} {f : β → α} (h₁ : Filter.Tendsto F l (nhds f)) (h₂ : UniformEquicontinuous F)
Goal: UniformContinuous f
-- EquicontinuousAt.tendsto_of_mem_closure
theorem EquicontinuousAt.tendsto_of_mem_closure {l : Filter ι} {F : ι → X → α} {f : X → α}
    {s : Set X} {x : X} {z : α} (hF : EquicontinuousAt F x) (hf : Tendsto f (𝓝[s] x) (𝓝 z))
    (hs : ∀ y ∈ s, Tendsto (F · y) l (𝓝 (f y))) (hx : x ∈ closure s) :
    Tendsto (F · x) l (𝓝 z) 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {l : Filter ι} {F : ι → X → α} {f : X → α} {s : Set X} {x : X} {z : α} (hF : EquicontinuousAt F x) (hf : Filter.Tendsto f (nhdsWithin x s) (nhds z)) (hs : ∀ y ∈ s, Filter.Tendsto (fun x => F x y) l (nhds (f y))) (hx : x ∈ closure s)
Goal: Filter.Tendsto (fun x_1 => F x_1 x) l (nhds z)
-- Equicontinuous.isClosed_setOf_tendsto
theorem Equicontinuous.isClosed_setOf_tendsto {l : Filter ι} {F : ι → X → α} {f : X → α}
    (hF : Equicontinuous F) (hf : Continuous f) :
    IsClosed {x | Tendsto (F · x) l (𝓝 (f x))} 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {l : Filter ι} {F : ι → X → α} {f : X → α} (hF : Equicontinuous F) (hf : Continuous f)
Goal: IsClosed {x | Filter.Tendsto (fun x_1 => F x_1 x) l (nhds (f x))}
-- EMetric.continuous_infEdist_hausdorffEdist
theorem continuous_infEdist_hausdorffEdist :
    Continuous fun p : α × Closeds α => infEdist p.1 p.2 
Arguments: {α : Type u} [EMetricSpace α]
Goal: Continuous fun p => EMetric.infEdist p.1 ↑p.2
-- EMetric.isClosed_subsets_of_isClosed
theorem isClosed_subsets_of_isClosed (hs : IsClosed s) :
    IsClosed { t : Closeds α | (t : Set α) ⊆ s } 
Arguments: {α : Type u} [EMetricSpace α] {s : Set α} (hs : IsClosed s)
Goal: IsClosed {t | ↑t ⊆ s}
-- EMetric.Closeds.edist_eq
theorem Closeds.edist_eq {s t : Closeds α} : edist s t = hausdorffEdist (s : Set α) t 
Arguments: {α : Type u} [EMetricSpace α] {s t : TopologicalSpace.Closeds α}
Goal: edist s t = EMetric.hausdorffEdist ↑s ↑t
-- EMetric.NonemptyCompacts.ToCloseds.uniformEmbedding
theorem NonemptyCompacts.ToCloseds.uniformEmbedding :
    UniformEmbedding (@NonemptyCompacts.toCloseds α _ _) 
Arguments: {α : Type u} [EMetricSpace α]
Goal: UniformEmbedding TopologicalSpace.NonemptyCompacts.toCloseds
-- EMetric.NonemptyCompacts.isClosed_in_closeds
theorem NonemptyCompacts.isClosed_in_closeds [CompleteSpace α] :
    IsClosed (range <| @NonemptyCompacts.toCloseds α _ _) 
Arguments: {α : Type u} [EMetricSpace α] [CompleteSpace α]
Goal: IsClosed (Set.range TopologicalSpace.NonemptyCompacts.toCloseds)
-- Metric.NonemptyCompacts.dist_eq
theorem NonemptyCompacts.dist_eq {x y : NonemptyCompacts α} :
    dist x y = hausdorffDist (x : Set α) y 
Arguments: {α : Type u} [MetricSpace α] {x y : TopologicalSpace.NonemptyCompacts α}
Goal: dist x y = Metric.hausdorffDist ↑x ↑y
-- Metric.lipschitz_infDist_set
theorem lipschitz_infDist_set (x : α) : LipschitzWith 1 fun s : NonemptyCompacts α => infDist x s 
Arguments: {α : Type u} [MetricSpace α] (x : α)
Goal: LipschitzWith 1 fun s => Metric.infDist x ↑s
-- Metric.lipschitz_infDist
theorem lipschitz_infDist : LipschitzWith 2 fun p : α × NonemptyCompacts α => infDist p.1 p.2 
Arguments: {α : Type u} [MetricSpace α]
Goal: LipschitzWith 2 fun p => Metric.infDist p.1 ↑p.2
-- Metric.uniformContinuous_infDist_Hausdorff_dist
theorem uniformContinuous_infDist_Hausdorff_dist :
    UniformContinuous fun p : α × NonemptyCompacts α => infDist p.1 p.2 
Arguments: {α : Type u} [MetricSpace α]
Goal: UniformContinuous fun p => Metric.infDist p.1 ↑p.2
-- mem_idRel
@[simp]
theorem mem_idRel {a b : α} : (a, b) ∈ @idRel α ↔ a = b 
Arguments: {α : Type ua} {a b : α}
Goal: (a, b) ∈ idRel ↔ a = b
-- idRel_subset
@[simp]
theorem idRel_subset {s : Set (α × α)} : idRel ⊆ s ↔ ∀ a, (a, a) ∈ s 
Arguments: {α : Type ua} {s : Set (α × α)}
Goal: idRel ⊆ s ↔ ∀ (a : α), (a, a) ∈ s
-- mem_compRel
@[simp]
theorem mem_compRel {α : Type u} {r₁ r₂ : Set (α × α)} {x y : α} :
    (x, y) ∈ r₁ ○ r₂ ↔ ∃ z, (x, z) ∈ r₁ ∧ (z, y) ∈ r₂ 
Arguments: {α : Type u} {r₁ r₂ : Set (α × α)} {x y : α}
Goal: (x, y) ∈ r₁ ○ r₂ ↔ ∃ z, (x, z) ∈ r₁ ∧ (z, y) ∈ r₂
-- swap_idRel
@[simp]
theorem swap_idRel : Prod.swap '' idRel = @idRel α 
Arguments: {α : Type ua}
Goal: Prod.swap '' idRel = idRel
-- Monotone.compRel
theorem Monotone.compRel [Preorder β] {f g : β → Set (α × α)} (hf : Monotone f) (hg : Monotone g) :
    Monotone fun x => f x ○ g x 
Arguments: {α : Type ua} {β : Type ub} [Preorder β] {f g : β → Set (α × α)} (hf : Monotone f) (hg : Monotone g)
Goal: Monotone fun x => f x ○ g x
-- compRel_mono
@[mono, gcongr]
theorem compRel_mono {f g h k : Set (α × α)} (h₁ : f ⊆ h) (h₂ : g ⊆ k) : f ○ g ⊆ h ○ k 
Arguments: {α : Type ua} {f g h k : Set (α × α)} (h₁ : f ⊆ h) (h₂ : g ⊆ k)
Goal: f ○ g ⊆ h ○ k
-- prod_mk_mem_compRel
theorem prod_mk_mem_compRel {a b c : α} {s t : Set (α × α)} (h₁ : (a, c) ∈ s) (h₂ : (c, b) ∈ t) :
    (a, b) ∈ s ○ t 
Arguments: {α : Type ua} {a b c : α} {s t : Set (α × α)} (h₁ : (a, c) ∈ s) (h₂ : (c, b) ∈ t)
Goal: (a, b) ∈ s ○ t
-- id_compRel
@[simp]
theorem id_compRel {r : Set (α × α)} : idRel ○ r = r 
Arguments: {α : Type ua} {r : Set (α × α)}
Goal: idRel ○ r = r
-- compRel_assoc
theorem compRel_assoc {r s t : Set (α × α)} : r ○ s ○ t = r ○ (s ○ t) 
Arguments: {α : Type ua} {r s t : Set (α × α)}
Goal: r ○ s ○ t = r ○ (s ○ t)
-- left_subset_compRel
theorem left_subset_compRel {s t : Set (α × α)} (h : idRel ⊆ t) : s ⊆ s ○ t 
Arguments: {α : Type ua} {s t : Set (α × α)} (h : idRel ⊆ t)
Goal: s ⊆ s ○ t
-- right_subset_compRel
theorem right_subset_compRel {s t : Set (α × α)} (h : idRel ⊆ s) : t ⊆ s ○ t 
Arguments: {α : Type ua} {s t : Set (α × α)} (h : idRel ⊆ s)
Goal: t ⊆ s ○ t
-- subset_comp_self
theorem subset_comp_self {s : Set (α × α)} (h : idRel ⊆ s) : s ⊆ s ○ s 
Arguments: {α : Type ua} {s : Set (α × α)} (h : idRel ⊆ s)
Goal: s ⊆ s ○ s
-- subset_iterate_compRel
theorem subset_iterate_compRel {s t : Set (α × α)} (h : idRel ⊆ s) (n : ℕ) :
    t ⊆ (s ○ ·)^[n] t 
Arguments: {α : Type ua} {s t : Set (α × α)} (h : idRel ⊆ s) (n : ℕ)
Goal: t ⊆ (fun x => s ○ x)^[n] t
-- symmetric_symmetrizeRel
theorem symmetric_symmetrizeRel (V : Set (α × α)) : SymmetricRel (symmetrizeRel V) 
Arguments: {α : Type ua} (V : Set (α × α))
Goal: SymmetricRel (symmetrizeRel V)
-- symmetrizeRel_subset_self
theorem symmetrizeRel_subset_self (V : Set (α × α)) : symmetrizeRel V ⊆ V 
Arguments: {α : Type ua} (V : Set (α × α))
Goal: symmetrizeRel V ⊆ V
-- symmetrize_mono
@[mono]
theorem symmetrize_mono {V W : Set (α × α)} (h : V ⊆ W) : symmetrizeRel V ⊆ symmetrizeRel W 
Arguments: {α : Type ua} {V W : Set (α × α)} (h : V ⊆ W)
Goal: symmetrizeRel V ⊆ symmetrizeRel W
-- SymmetricRel.mk_mem_comm
theorem SymmetricRel.mk_mem_comm {V : Set (α × α)} (hV : SymmetricRel V) {x y : α} :
    (x, y) ∈ V ↔ (y, x) ∈ V 
Arguments: {α : Type ua} {V : Set (α × α)} (hV : SymmetricRel V) {x y : α}
Goal: (x, y) ∈ V ↔ (y, x) ∈ V
-- SymmetricRel.eq
theorem SymmetricRel.eq {U : Set (α × α)} (hU : SymmetricRel U) : Prod.swap ⁻¹' U = U 
Arguments: {α : Type ua} {U : Set (α × α)} (hU : SymmetricRel U)
Goal: Prod.swap ⁻¹' U = U
-- SymmetricRel.inter
theorem SymmetricRel.inter {U V : Set (α × α)} (hU : SymmetricRel U) (hV : SymmetricRel V) :
    SymmetricRel (U ∩ V) 
Arguments: {α : Type ua} {U V : Set (α × α)} (hU : SymmetricRel U) (hV : SymmetricRel V)
Goal: SymmetricRel (U ∩ V)
-- UniformSpace.Core.comp
structure UniformSpace.Core (α : Type u) where
    uniformity : Filter (α × α)
    refl : 𝓟 idRel ≤ uniformity
    symm : Tendsto Prod.swap uniformity uniformity
    comp : (uniformity.lift' fun s => s ○ s) ≤ uniformity
Arguments: {α : Type u} (self : UniformSpace.Core α)
Goal: (self.uniformity.lift' fun s => s ○ s) ≤ self.uniformity
-- UniformSpace.Core.symm
structure UniformSpace.Core (α : Type u) where
    uniformity : Filter (α × α)
    refl : 𝓟 idRel ≤ uniformity
    symm : Tendsto Prod.swap uniformity uniformity
    comp : (uniformity.lift' fun s => s ○ s) ≤ uniformity
Arguments: {α : Type u} (self : UniformSpace.Core α)
Goal: Filter.Tendsto Prod.swap self.uniformity self.uniformity
-- UniformSpace.Core.refl
structure UniformSpace.Core (α : Type u) where
    uniformity : Filter (α × α)
    refl : 𝓟 idRel ≤ uniformity
    symm : Tendsto Prod.swap uniformity uniformity
    comp : (uniformity.lift' fun s => s ○ s) ≤ uniformity
Arguments: {α : Type u} (self : UniformSpace.Core α)
Goal: 𝓟 idRel ≤ self.uniformity
-- UniformSpace.Core.comp_mem_uniformity_sets
protected theorem UniformSpace.Core.comp_mem_uniformity_sets {c : Core α} {s : Set (α × α)}
    (hs : s ∈ c.uniformity) : ∃ t ∈ c.uniformity, t ○ t ⊆ s 
Arguments: {α : Type ua} {c : UniformSpace.Core α} {s : Set (α × α)} (hs : s ∈ c.uniformity)
Goal: ∃ t ∈ c.uniformity, t ○ t ⊆ s
-- UniformSpace.Core.ext
theorem UniformSpace.Core.ext :
    ∀ {u₁ u₂ : UniformSpace.Core α}, u₁.uniformity = u₂.uniformity → u₁ = u₂
  
Arguments: {α : Type ua} {u₁ u₂ : UniformSpace.Core α}
Goal: u₁.uniformity = u₂.uniformity → u₁ = u₂
-- UniformSpace.Core.nhds_toTopologicalSpace
theorem UniformSpace.Core.nhds_toTopologicalSpace {α : Type u} (u : Core α) (x : α) :
    @nhds α u.toTopologicalSpace x = comap (Prod.mk x) u.uniformity 
Arguments: {α : Type u} (u : UniformSpace.Core α) (x : α)
Goal: 𝓝 x = Filter.comap (Prod.mk x) u.uniformity
-- UniformSpace.nhds_eq_comap_uniformity
class UniformSpace (α : Type u) extends TopologicalSpace α where
    protected uniformity : Filter (α × α)
    protected symm : Tendsto Prod.swap uniformity uniformity
    protected comp : (uniformity.lift' fun s => s ○ s) ≤ uniformity
    protected nhds_eq_comap_uniformity (x : α) : 𝓝 x = comap (Prod.mk x) uniformity
Arguments: {α : Type u} [self : UniformSpace α] (x : α)
Goal: 𝓝 x = Filter.comap (Prod.mk x) UniformSpace.uniformity
-- UniformSpace.symm
class UniformSpace (α : Type u) extends TopologicalSpace α where
    protected uniformity : Filter (α × α)
    protected symm : Tendsto Prod.swap uniformity uniformity
    protected comp : (uniformity.lift' fun s => s ○ s) ≤ uniformity
    protected nhds_eq_comap_uniformity (x : α) : 𝓝 x = comap (Prod.mk x) uniformity
Arguments: {α : Type u} [self : UniformSpace α]
Goal: Filter.Tendsto Prod.swap UniformSpace.uniformity UniformSpace.uniformity
-- UniformSpace.comp
class UniformSpace (α : Type u) extends TopologicalSpace α where
    protected uniformity : Filter (α × α)
    protected symm : Tendsto Prod.swap uniformity uniformity
    protected comp : (uniformity.lift' fun s => s ○ s) ≤ uniformity
    protected nhds_eq_comap_uniformity (x : α) : 𝓝 x = comap (Prod.mk x) uniformity
Arguments: {α : Type u} [self : UniformSpace α]
Goal: (UniformSpace.uniformity.lift' fun s => s ○ s) ≤ UniformSpace.uniformity
-- UniformSpace.toCore_toTopologicalSpace
theorem UniformSpace.toCore_toTopologicalSpace (u : UniformSpace α) :
    u.toCore.toTopologicalSpace = u.toTopologicalSpace 
Arguments: {α : Type ua} (u : UniformSpace α)
Goal: u.toCore.toTopologicalSpace = UniformSpace.toTopologicalSpace
-- UniformSpace.ext
@[ext]
protected theorem UniformSpace.ext {u₁ u₂ : UniformSpace α} (h : 𝓤[u₁] = 𝓤[u₂]) : u₁ = u₂ 
Arguments: {α : Type ua} {u₁ u₂ : UniformSpace α} (h : 𝓤 α = 𝓤 α)
Goal: u₁ = u₂
-- UniformSpace.ext_iff
protected theorem UniformSpace.ext_iff {u₁ u₂ : UniformSpace α} :
    u₁ = u₂ ↔ ∀ s, s ∈ 𝓤[u₁] ↔ s ∈ 𝓤[u₂] 
Arguments: {α : Type ua} {u₁ u₂ : UniformSpace α}
Goal: u₁ = u₂ ↔ ∀ (s : Set (α × α)), s ∈ 𝓤 α ↔ s ∈ 𝓤 α
-- UniformSpace.ofCoreEq_toCore
theorem UniformSpace.ofCoreEq_toCore (u : UniformSpace α) (t : TopologicalSpace α)
    (h : t = u.toCore.toTopologicalSpace) : .ofCoreEq u.toCore t h = u 
Arguments: {α : Type ua} (u : UniformSpace α) (t : TopologicalSpace α) (h : t = u.toCore.toTopologicalSpace)
Goal: UniformSpace.ofCoreEq u.toCore t h = u
-- UniformSpace.replaceTopology_eq
theorem UniformSpace.replaceTopology_eq {α : Type*} [i : TopologicalSpace α] (u : UniformSpace α)
    (h : i = u.toTopologicalSpace) : u.replaceTopology h = u 
Arguments: {α : Type u_2} [i : TopologicalSpace α] (u : UniformSpace α) (h : i = UniformSpace.toTopologicalSpace)
Goal: u.replaceTopology h = u
-- UniformSpace.hasBasis_ofFun
theorem UniformSpace.hasBasis_ofFun {α : Type u} {β : Type v} [LinearOrderedAddCommMonoid β]
    (h₀ : ∃ x : β, 0 < x) (d : α → α → β) (refl : ∀ x, d x x = 0) (symm : ∀ x y, d x y = d y x)
    (triangle : ∀ x y z, d x z ≤ d x y + d y z)
    (half : ∀ ε > (0 : β), ∃ δ > (0 : β), ∀ x < δ, ∀ y < δ, x + y < ε) :
    𝓤[.ofFun d refl symm triangle half].HasBasis ((0 : β) < ·) (fun ε => { x | d x.1 x.2 < ε }) 
Arguments: {α : Type u} {β : Type v} [LinearOrderedAddCommMonoid β] (h₀ : ∃ x, 0 < x) (d : α → α → β) (refl : ∀ (x : α), d x x = 0) (symm : ∀ (x y : α), d x y = d y x) (triangle : ∀ (x y z : α), d x z ≤ d x y + d y z) (half : ∀ ε > 0, ∃ δ > 0, ∀ x < δ, ∀ y < δ, x + y < ε)
Goal: (𝓤 α).HasBasis (fun x => 0 < x) fun ε => {x | d x.1 x.2 < ε}
-- nhds_eq_comap_uniformity
theorem nhds_eq_comap_uniformity {x : α} : 𝓝 x = (𝓤 α).comap (Prod.mk x) 
Arguments: {α : Type ua} [UniformSpace α] {x : α}
Goal: 𝓝 x = Filter.comap (Prod.mk x) (𝓤 α)
-- isOpen_uniformity
theorem isOpen_uniformity {s : Set α} :
    IsOpen s ↔ ∀ x ∈ s, { p : α × α | p.1 = x → p.2 ∈ s } ∈ 𝓤 α 
Arguments: {α : Type ua} [UniformSpace α] {s : Set α}
Goal: IsOpen s ↔ ∀ x ∈ s, {p | p.1 = x → p.2 ∈ s} ∈ 𝓤 α
-- refl_le_uniformity
theorem refl_le_uniformity : 𝓟 idRel ≤ 𝓤 α 
Arguments: {α : Type ua} [UniformSpace α]
Goal: 𝓟 idRel ≤ 𝓤 α
-- refl_mem_uniformity
theorem refl_mem_uniformity {x : α} {s : Set (α × α)} (h : s ∈ 𝓤 α) : (x, x) ∈ s 
Arguments: {α : Type ua} [UniformSpace α] {x : α} {s : Set (α × α)} (h : s ∈ 𝓤 α)
Goal: (x, x) ∈ s
-- mem_uniformity_of_eq
theorem mem_uniformity_of_eq {x y : α} {s : Set (α × α)} (h : s ∈ 𝓤 α) (hx : x = y) : (x, y) ∈ s 
Arguments: {α : Type ua} [UniformSpace α] {x y : α} {s : Set (α × α)} (h : s ∈ 𝓤 α) (hx : x = y)
Goal: (x, y) ∈ s
-- symm_le_uniformity
theorem symm_le_uniformity : map (@Prod.swap α α) (𝓤 _) ≤ 𝓤 _ 
Arguments: {α : Type ua} [UniformSpace α]
Goal: Filter.map Prod.swap (𝓤 α) ≤ 𝓤 α
-- comp_le_uniformity
theorem comp_le_uniformity : ((𝓤 α).lift' fun s : Set (α × α) => s ○ s) ≤ 𝓤 α 
Arguments: {α : Type ua} [UniformSpace α]
Goal: ((𝓤 α).lift' fun s => s ○ s) ≤ 𝓤 α
-- lift'_comp_uniformity
theorem lift'_comp_uniformity : ((𝓤 α).lift' fun s : Set (α × α) => s ○ s) = 𝓤 α 
Arguments: {α : Type ua} [UniformSpace α]
Goal: ((𝓤 α).lift' fun s => s ○ s) = 𝓤 α
-- tendsto_swap_uniformity
theorem tendsto_swap_uniformity : Tendsto (@Prod.swap α α) (𝓤 α) (𝓤 α) 
Arguments: {α : Type ua} [UniformSpace α]
Goal: Filter.Tendsto Prod.swap (𝓤 α) (𝓤 α)
-- comp_mem_uniformity_sets
theorem comp_mem_uniformity_sets {s : Set (α × α)} (hs : s ∈ 𝓤 α) : ∃ t ∈ 𝓤 α, t ○ t ⊆ s 
Arguments: {α : Type ua} [UniformSpace α] {s : Set (α × α)} (hs : s ∈ 𝓤 α)
Goal: ∃ t ∈ 𝓤 α, t ○ t ⊆ s
-- eventually_uniformity_iterate_comp_subset
theorem eventually_uniformity_iterate_comp_subset {s : Set (α × α)} (hs : s ∈ 𝓤 α) (n : ℕ) :
    ∀ᶠ t in (𝓤 α).smallSets, (t ○ ·)^[n] t ⊆ s 
Arguments: {α : Type ua} [UniformSpace α] {s : Set (α × α)} (hs : s ∈ 𝓤 α) (n : ℕ)
Goal: ∀ᶠ (t : Set (α × α)) in (𝓤 α).smallSets, (fun x => t ○ x)^[n] t ⊆ s
-- eventually_uniformity_comp_subset
theorem eventually_uniformity_comp_subset {s : Set (α × α)} (hs : s ∈ 𝓤 α) :
    ∀ᶠ t in (𝓤 α).smallSets, t ○ t ⊆ s 
Arguments: {α : Type ua} [UniformSpace α] {s : Set (α × α)} (hs : s ∈ 𝓤 α)
Goal: ∀ᶠ (t : Set (α × α)) in (𝓤 α).smallSets, t ○ t ⊆ s
-- Filter.Tendsto.uniformity_trans
theorem Filter.Tendsto.uniformity_trans {l : Filter β} {f₁ f₂ f₃ : β → α}
    (h₁₂ : Tendsto (fun x => (f₁ x, f₂ x)) l (𝓤 α))
    (h₂₃ : Tendsto (fun x => (f₂ x, f₃ x)) l (𝓤 α)) : Tendsto (fun x => (f₁ x, f₃ x)) l (𝓤 α) 
Arguments: {α : Type ua} {β : Type ub} [UniformSpace α] {l : Filter β} {f₁ f₂ f₃ : β → α} (h₁₂ : Filter.Tendsto (fun x => (f₁ x, f₂ x)) l (𝓤 α)) (h₂₃ : Filter.Tendsto (fun x => (f₂ x, f₃ x)) l (𝓤 α))
Goal: Filter.Tendsto (fun x => (f₁ x, f₃ x)) l (𝓤 α)
-- Filter.Tendsto.uniformity_symm
theorem Filter.Tendsto.uniformity_symm {l : Filter β} {f : β → α × α} (h : Tendsto f l (𝓤 α)) :
    Tendsto (fun x => ((f x).2, (f x).1)) l (𝓤 α) 
Arguments: {α : Type ua} {β : Type ub} [UniformSpace α] {l : Filter β} {f : β → α × α} (h : Filter.Tendsto f l (𝓤 α))
Goal: Filter.Tendsto (fun x => ((f x).2, (f x).1)) l (𝓤 α)