ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Trivialization.Prod.continuous_inv_fun
theorem Prod.continuous_inv_fun :
    ContinuousOn (Prod.invFun' e₁ e₂) ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ) 
Arguments: {B : Type u_1} [TopologicalSpace B] {F₁ : Type u_2} [TopologicalSpace F₁] {E₁ : B → Type u_3} [TopologicalSpace (Bundle.TotalSpace F₁ E₁)] {F₂ : Type u_4} [TopologicalSpace F₂] {E₂ : B → Type u_5} [TopologicalSpace (Bundle.TotalSpace F₂ E₂)] {e₁ : Trivialization F₁ Bundle.TotalSpace.proj} {e₂ : Trivialization F₂ Bundle.TotalSpace.proj} [(x : B) → Zero (E₁ x)] [(x : B) → Zero (E₂ x)]
Goal: ContinuousOn (Trivialization.Prod.invFun' e₁ e₂) ((e₁.baseSet ∩ e₂.baseSet) ×ˢ Set.univ)
-- Trivialization.baseSet_prod
@[simp]
theorem baseSet_prod : (prod e₁ e₂).baseSet = e₁.baseSet ∩ e₂.baseSet 
Arguments: {B : Type u_1} [TopologicalSpace B] {F₁ : Type u_2} [TopologicalSpace F₁] {E₁ : B → Type u_3} [TopologicalSpace (Bundle.TotalSpace F₁ E₁)] {F₂ : Type u_4} [TopologicalSpace F₂] {E₂ : B → Type u_5} [TopologicalSpace (Bundle.TotalSpace F₂ E₂)] (e₁ : Trivialization F₁ Bundle.TotalSpace.proj) (e₂ : Trivialization F₂ Bundle.TotalSpace.proj) [(x : B) → Zero (E₁ x)] [(x : B) → Zero (E₂ x)]
Goal: (e₁.prod e₂).baseSet = e₁.baseSet ∩ e₂.baseSet
-- Trivialization.prod_symm_apply
theorem prod_symm_apply (x : B) (w₁ : F₁) (w₂ : F₂) :
    (prod e₁ e₂).toPartialEquiv.symm (x, w₁, w₂) = ⟨x, e₁.symm x w₁, e₂.symm x w₂⟩ 
Arguments: {B : Type u_1} [TopologicalSpace B] {F₁ : Type u_2} [TopologicalSpace F₁] {E₁ : B → Type u_3} [TopologicalSpace (Bundle.TotalSpace F₁ E₁)] {F₂ : Type u_4} [TopologicalSpace F₂] {E₂ : B → Type u_5} [TopologicalSpace (Bundle.TotalSpace F₂ E₂)] (e₁ : Trivialization F₁ Bundle.TotalSpace.proj) (e₂ : Trivialization F₂ Bundle.TotalSpace.proj) [(x : B) → Zero (E₁ x)] [(x : B) → Zero (E₂ x)] (x : B) (w₁ : F₁) (w₂ : F₂)
Goal: ↑(e₁.prod e₂).symm (x, w₁, w₂) = { proj := x, snd := (e₁.symm x w₁, e₂.symm x w₂) }
-- pullbackTopology_def
irreducible_def pullbackTopology : TopologicalSpace (TotalSpace F (f *ᵖ E)) 
Arguments: {B : Type u_1} (F : Type u_2) (E : B → Type u_3) {B' : Type u_4} (f : B' → B) [TopologicalSpace B'] [TopologicalSpace (Bundle.TotalSpace F E)]
Goal: pullbackTopology F E f =
  TopologicalSpace.induced Bundle.TotalSpace.proj inst✝ ⊓ TopologicalSpace.induced (Bundle.Pullback.lift f) inst✝¹
-- Pullback.continuous_proj
theorem Pullback.continuous_proj (f : B' → B) : Continuous (π F (f *ᵖ E)) 
Arguments: {B : Type u} (F : Type v) (E : B → Type w₁) {B' : Type w₂} [TopologicalSpace B'] [TopologicalSpace (Bundle.TotalSpace F E)] (f : B' → B)
Goal: Continuous Bundle.TotalSpace.proj
-- Pullback.continuous_lift
theorem Pullback.continuous_lift (f : B' → B) : Continuous (@Pullback.lift B F E B' f) 
Arguments: {B : Type u} (F : Type v) (E : B → Type w₁) {B' : Type w₂} [TopologicalSpace B'] [TopologicalSpace (Bundle.TotalSpace F E)] (f : B' → B)
Goal: Continuous (Bundle.Pullback.lift f)
-- inducing_pullbackTotalSpaceEmbedding
theorem inducing_pullbackTotalSpaceEmbedding (f : B' → B) :
    Inducing (@pullbackTotalSpaceEmbedding B F E B' f) 
Arguments: {B : Type u} (F : Type v) (E : B → Type w₁) {B' : Type w₂} [TopologicalSpace B'] [TopologicalSpace (Bundle.TotalSpace F E)] (f : B' → B)
Goal: Inducing (Bundle.pullbackTotalSpaceEmbedding f)
-- Pullback.continuous_totalSpaceMk
theorem Pullback.continuous_totalSpaceMk [∀ x, TopologicalSpace (E x)] [FiberBundle F E]
    {f : B' → B} {x : B'} : Continuous (@TotalSpace.mk _ F (f *ᵖ E) x) 
Arguments: {B : Type u} (F : Type v) (E : B → Type w₁) {B' : Type w₂} [TopologicalSpace B'] [TopologicalSpace (Bundle.TotalSpace F E)] [TopologicalSpace F] [TopologicalSpace B] [(x : B) → TopologicalSpace (E x)] [FiberBundle F E] {f : B' → B} {x : B'}
Goal: Continuous (Bundle.TotalSpace.mk x)
-- Rat.uniformSpace_eq
theorem Rat.uniformSpace_eq :
    (AbsoluteValue.abs : AbsoluteValue ℚ ℚ).uniformSpace = PseudoMetricSpace.toUniformSpace 
Goal: AbsoluteValue.abs.uniformSpace = PseudoMetricSpace.toUniformSpace
-- CompareReals.compare_uc
theorem compare_uc : UniformContinuous compareEquiv 
Goal: UniformContinuous ⇑CompareReals.compareEquiv
-- CompareReals.compare_uc_symm
theorem compare_uc_symm : UniformContinuous compareEquiv.symm 
Goal: UniformContinuous ⇑CompareReals.compareEquiv.symm
-- AbsoluteValue.hasBasis_uniformity
theorem hasBasis_uniformity :
    𝓤[abv.uniformSpace].HasBasis ((0 : 𝕜) < ·) fun ε => { p : R × R | abv (p.2 - p.1) < ε } 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {R : Type u_2} [CommRing R] (abv : AbsoluteValue R 𝕜)
Goal: (𝓤 R).HasBasis (fun x => 0 < x) fun ε => {p | abv (p.2 - p.1) < ε}
-- AbstractCompletion.uniformInducing
structure AbstractCompletion (α : Type u) [UniformSpace α] where
    space : Type u
    coe : α → space
    uniformStruct : UniformSpace space
    complete : CompleteSpace space
    separation : T0Space space
    uniformInducing : UniformInducing coe
    dense : DenseRange coe
Arguments: {α : Type u} [UniformSpace α] (self : AbstractCompletion α)
Goal: UniformInducing self.coe
-- AbstractCompletion.complete
structure AbstractCompletion (α : Type u) [UniformSpace α] where
    space : Type u
    coe : α → space
    uniformStruct : UniformSpace space
    complete : CompleteSpace space
    separation : T0Space space
    uniformInducing : UniformInducing coe
    dense : DenseRange coe
Arguments: {α : Type u} [UniformSpace α] (self : AbstractCompletion α)
Goal: CompleteSpace self.space
-- AbstractCompletion.dense
structure AbstractCompletion (α : Type u) [UniformSpace α] where
    space : Type u
    coe : α → space
    uniformStruct : UniformSpace space
    complete : CompleteSpace space
    separation : T0Space space
    uniformInducing : UniformInducing coe
    dense : DenseRange coe
Arguments: {α : Type u} [UniformSpace α] (self : AbstractCompletion α)
Goal: DenseRange self.coe
-- AbstractCompletion.separation
structure AbstractCompletion (α : Type u) [UniformSpace α] where
    space : Type u
    coe : α → space
    uniformStruct : UniformSpace space
    complete : CompleteSpace space
    separation : T0Space space
    uniformInducing : UniformInducing coe
    dense : DenseRange coe
Arguments: {α : Type u} [UniformSpace α] (self : AbstractCompletion α)
Goal: T0Space self.space
-- AbstractCompletion.closure_range
theorem closure_range : closure (range ι) = univ 
Arguments: {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α)
Goal: closure (Set.range pkg.coe) = Set.univ
-- AbstractCompletion.denseInducing
theorem denseInducing : DenseInducing ι 
Arguments: {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α)
Goal: DenseInducing pkg.coe
-- AbstractCompletion.uniformContinuous_coe
theorem uniformContinuous_coe : UniformContinuous ι 
Arguments: {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α)
Goal: UniformContinuous pkg.coe
-- AbstractCompletion.continuous_coe
theorem continuous_coe : Continuous ι 
Arguments: {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α)
Goal: Continuous pkg.coe
-- AbstractCompletion.induction_on
@[elab_as_elim]
theorem induction_on {p : hatα → Prop} (a : hatα) (hp : IsClosed { a | p a }) (ih : ∀ a, p (ι a)) :
    p a 
Arguments: {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {p : pkg.space → Prop} (a : pkg.space) (hp : IsClosed {a | p a}) (ih : ∀ (a : α), p (pkg.coe a))
Goal: p a
-- AbstractCompletion.funext
protected theorem funext [TopologicalSpace β] [T2Space β] {f g : hatα → β} (hf : Continuous f)
    (hg : Continuous g) (h : ∀ a, f (ι a) = g (ι a)) : f = g 
Arguments: {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {β : Type u_2} [TopologicalSpace β] [T2Space β] {f g : pkg.space → β} (hf : Continuous f) (hg : Continuous g) (h : ∀ (a : α), f (pkg.coe a) = g (pkg.coe a))
Goal: f = g
-- AbstractCompletion.extend_def
theorem extend_def (hf : UniformContinuous f) : pkg.extend f = pkg.denseInducing.extend f 
Arguments: {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {β : Type u_2} [UniformSpace β] {f : α → β} (hf : UniformContinuous f)
Goal: pkg.extend f = ⋯.extend f
-- AbstractCompletion.extend_coe
theorem extend_coe [T2Space β] (hf : UniformContinuous f) (a : α) : (pkg.extend f) (ι a) = f a 
Arguments: {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {β : Type u_2} [UniformSpace β] {f : α → β} [T2Space β] (hf : UniformContinuous f) (a : α)
Goal: pkg.extend f (pkg.coe a) = f a
-- AbstractCompletion.uniformContinuous_extend
theorem uniformContinuous_extend : UniformContinuous (pkg.extend f) 
Arguments: {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {β : Type u_2} [UniformSpace β] {f : α → β} [CompleteSpace β]
Goal: UniformContinuous (pkg.extend f)
-- AbstractCompletion.continuous_extend
theorem continuous_extend : Continuous (pkg.extend f) 
Arguments: {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {β : Type u_2} [UniformSpace β] {f : α → β} [CompleteSpace β]
Goal: Continuous (pkg.extend f)
-- AbstractCompletion.extend_unique
theorem extend_unique (hf : UniformContinuous f) {g : hatα → β} (hg : UniformContinuous g)
    (h : ∀ a : α, f a = g (ι a)) : pkg.extend f = g 
Arguments: {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {β : Type u_2} [UniformSpace β] {f : α → β} [CompleteSpace β] [T0Space β] (hf : UniformContinuous f) {g : pkg.space → β} (hg : UniformContinuous g) (h : ∀ (a : α), f a = g (pkg.coe a))
Goal: pkg.extend f = g
-- AbstractCompletion.extend_comp_coe
@[simp]
theorem extend_comp_coe {f : hatα → β} (hf : UniformContinuous f) : pkg.extend (f ∘ ι) = f 
Arguments: {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {β : Type u_2} [UniformSpace β] [CompleteSpace β] [T0Space β] {f : pkg.space → β} (hf : UniformContinuous f)
Goal: pkg.extend (f ∘ pkg.coe) = f
-- AbstractCompletion.uniformContinuous_map
theorem uniformContinuous_map : UniformContinuous (map f) 
Arguments: {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {β : Type u_2} [UniformSpace β] (pkg' : AbstractCompletion β) (f : α → β)
Goal: UniformContinuous (pkg.map pkg' f)
-- AbstractCompletion.continuous_map
@[continuity]
theorem continuous_map : Continuous (map f) 
Arguments: {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {β : Type u_2} [UniformSpace β] (pkg' : AbstractCompletion β) (f : α → β)
Goal: Continuous (pkg.map pkg' f)
-- AbstractCompletion.map_coe
@[simp]
theorem map_coe (hf : UniformContinuous f) (a : α) : map f (ι a) = ι' (f a) 
Arguments: {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {β : Type u_2} [UniformSpace β] (pkg' : AbstractCompletion β) {f : α → β} (hf : UniformContinuous f) (a : α)
Goal: pkg.map pkg' f (pkg.coe a) = pkg'.coe (f a)
-- AbstractCompletion.map_unique
theorem map_unique {f : α → β} {g : hatα → hatβ} (hg : UniformContinuous g)
    (h : ∀ a, ι' (f a) = g (ι a)) : map f = g 
Arguments: {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {β : Type u_2} [UniformSpace β] (pkg' : AbstractCompletion β) {f : α → β} {g : pkg.space → pkg'.space} (hg : UniformContinuous g) (h : ∀ (a : α), pkg'.coe (f a) = g (pkg.coe a))
Goal: pkg.map pkg' f = g
-- AbstractCompletion.map_id
@[simp]
theorem map_id : pkg.map pkg id = id 
Arguments: {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α)
Goal: pkg.map pkg id = id
-- AbstractCompletion.extend_map
theorem extend_map [CompleteSpace γ] [T0Space γ] {f : β → γ} {g : α → β}
    (hf : UniformContinuous f) (hg : UniformContinuous g) :
    pkg'.extend f ∘ map g = pkg.extend (f ∘ g) 
Arguments: {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {β : Type u_2} [UniformSpace β] (pkg' : AbstractCompletion β) {γ : Type u_3} [UniformSpace γ] [CompleteSpace γ] [T0Space γ] {f : β → γ} {g : α → β} (hf : UniformContinuous f) (hg : UniformContinuous g)
Goal: pkg'.extend f ∘ pkg.map pkg' g = pkg.extend (f ∘ g)
-- AbstractCompletion.map_comp
theorem map_comp {g : β → γ} {f : α → β} (hg : UniformContinuous g) (hf : UniformContinuous f) :
    pkg'.map pkg'' g ∘ pkg.map pkg' f = pkg.map pkg'' (g ∘ f) 
Arguments: {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {β : Type u_2} [UniformSpace β] (pkg' : AbstractCompletion β) {γ : Type u_3} [UniformSpace γ] (pkg'' : AbstractCompletion γ) {g : β → γ} {f : α → β} (hg : UniformContinuous g) (hf : UniformContinuous f)
Goal: pkg'.map pkg'' g ∘ pkg.map pkg' f = pkg.map pkg'' (g ∘ f)
-- AbstractCompletion.uniformContinuous_compare
theorem uniformContinuous_compare : UniformContinuous (pkg.compare pkg') 
Arguments: {α : Type u_1} [UniformSpace α] (pkg pkg' : AbstractCompletion α)
Goal: UniformContinuous (pkg.compare pkg')
-- AbstractCompletion.compare_coe
theorem compare_coe (a : α) : pkg.compare pkg' (pkg.coe a) = pkg'.coe a 
Arguments: {α : Type u_1} [UniformSpace α] (pkg pkg' : AbstractCompletion α) (a : α)
Goal: pkg.compare pkg' (pkg.coe a) = pkg'.coe a
-- AbstractCompletion.inverse_compare
theorem inverse_compare : pkg.compare pkg' ∘ pkg'.compare pkg = id 
Arguments: {α : Type u_1} [UniformSpace α] (pkg pkg' : AbstractCompletion α)
Goal: pkg.compare pkg' ∘ pkg'.compare pkg = id
-- AbstractCompletion.uniformContinuous_compareEquiv
theorem uniformContinuous_compareEquiv : UniformContinuous (pkg.compareEquiv pkg') 
Arguments: {α : Type u_1} [UniformSpace α] (pkg pkg' : AbstractCompletion α)
Goal: UniformContinuous ⇑(pkg.compareEquiv pkg')
-- AbstractCompletion.uniformContinuous_compareEquiv_symm
theorem uniformContinuous_compareEquiv_symm : UniformContinuous (pkg.compareEquiv pkg').symm 
Arguments: {α : Type u_1} [UniformSpace α] (pkg pkg' : AbstractCompletion α)
Goal: UniformContinuous ⇑(pkg.compareEquiv pkg').symm
-- AbstractCompletion.compare_comp_eq_compare
theorem compare_comp_eq_compare (γ : Type*) [TopologicalSpace γ]
    [T3Space γ] {f : α → γ} (cont_f : Continuous f) :
    letI := pkg.uniformStruct.toTopologicalSpace
    letI := pkg'.uniformStruct.toTopologicalSpace
    (∀ a : pkg.space,
      Filter.Tendsto f (Filter.comap pkg.coe (𝓝 a)) (𝓝 ((pkg.denseInducing.extend f) a))) →
      pkg.denseInducing.extend f ∘ pkg'.compare pkg = pkg'.denseInducing.extend f 
Arguments: {α : Type u_1} [UniformSpace α] (pkg pkg' : AbstractCompletion α) (γ : Type u_3) [TopologicalSpace γ] [T3Space γ] {f : α → γ} (cont_f : Continuous f)
Goal: (∀ (a : pkg.space), Filter.Tendsto f (Filter.comap pkg.coe (nhds a)) (nhds (⋯.extend f a))) →
  ⋯.extend f ∘ pkg'.compare pkg = ⋯.extend f
-- AbstractCompletion.extension₂_coe_coe
theorem extension₂_coe_coe (hf : UniformContinuous <| uncurry f) (a : α) (b : β) :
    pkg.extend₂ pkg' f (ι a) (ι' b) = f a b 
Arguments: {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {β : Type u_2} [UniformSpace β] (pkg' : AbstractCompletion β) {γ : Type u_3} [UniformSpace γ] [T0Space γ] {f : α → β → γ} (hf : UniformContinuous (Function.uncurry f)) (a : α) (b : β)
Goal: pkg.extend₂ pkg' f (pkg.coe a) (pkg'.coe b) = f a b
-- AbstractCompletion.uniformContinuous_extension₂
theorem uniformContinuous_extension₂ : UniformContinuous₂ (pkg.extend₂ pkg' f) 
Arguments: {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {β : Type u_2} [UniformSpace β] (pkg' : AbstractCompletion β) {γ : Type u_3} [UniformSpace γ] (f : α → β → γ) [CompleteSpace γ]
Goal: UniformContinuous₂ (pkg.extend₂ pkg' f)
-- AbstractCompletion.uniformContinuous_map₂
theorem uniformContinuous_map₂ (f : α → β → γ) : UniformContinuous₂ (pkg.map₂ pkg' pkg'' f) 
Arguments: {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {β : Type u_2} [UniformSpace β] (pkg' : AbstractCompletion β) {γ : Type u_3} [UniformSpace γ] (pkg'' : AbstractCompletion γ) (f : α → β → γ)
Goal: UniformContinuous₂ (pkg.map₂ pkg' pkg'' f)
-- AbstractCompletion.continuous_map₂
theorem continuous_map₂ {δ} [TopologicalSpace δ] {f : α → β → γ} {a : δ → hatα} {b : δ → hatβ}
    (ha : Continuous a) (hb : Continuous b) :
    Continuous fun d : δ => pkg.map₂ pkg' pkg'' f (a d) (b d) 
Arguments: {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {β : Type u_2} [UniformSpace β] (pkg' : AbstractCompletion β) {γ : Type u_3} [UniformSpace γ] (pkg'' : AbstractCompletion γ) {δ : Type u_4} [TopologicalSpace δ] {f : α → β → γ} {a : δ → pkg.space} {b : δ → pkg'.space} (ha : Continuous a) (hb : Continuous b)
Goal: Continuous fun d => pkg.map₂ pkg' pkg'' f (a d) (b d)
-- AbstractCompletion.map₂_coe_coe
theorem map₂_coe_coe (a : α) (b : β) (f : α → β → γ) (hf : UniformContinuous₂ f) :
    pkg.map₂ pkg' pkg'' f (ι a) (ι' b) = ι'' (f a b) 
Arguments: {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {β : Type u_2} [UniformSpace β] (pkg' : AbstractCompletion β) {γ : Type u_3} [UniformSpace γ] (pkg'' : AbstractCompletion γ) (a : α) (b : β) (f : α → β → γ) (hf : UniformContinuous₂ f)
Goal: pkg.map₂ pkg' pkg'' f (pkg.coe a) (pkg'.coe b) = pkg''.coe (f a b)
-- UniformEquiv.uniformContinuous_invFun
--@[nolint has_nonempty_instance] -- Porting note(#5171): linter not yet ported
structure UniformEquiv (α : Type*) (β : Type*) [UniformSpace α] [UniformSpace β] extends
  α ≃ β where
    uniformContinuous_toFun : UniformContinuous toFun
    uniformContinuous_invFun : UniformContinuous invFun
Arguments: {α : Type u_4} {β : Type u_5} [UniformSpace α] [UniformSpace β] (self : α ≃ᵤ β)
Goal: UniformContinuous self.invFun
-- UniformEquiv.uniformContinuous_toFun
--@[nolint has_nonempty_instance] -- Porting note(#5171): linter not yet ported
structure UniformEquiv (α : Type*) (β : Type*) [UniformSpace α] [UniformSpace β] extends
  α ≃ β where
    uniformContinuous_toFun : UniformContinuous toFun
    uniformContinuous_invFun : UniformContinuous invFun
Arguments: {α : Type u_4} {β : Type u_5} [UniformSpace α] [UniformSpace β] (self : α ≃ᵤ β)
Goal: UniformContinuous self.toFun
-- UniformEquiv.toEquiv_injective
theorem toEquiv_injective : Function.Injective (toEquiv : α ≃ᵤ β → α ≃ β)
  
Arguments: {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β]
Goal: Function.Injective UniformEquiv.toEquiv
-- UniformEquiv.uniformEquiv_mk_coe
@[simp]
theorem uniformEquiv_mk_coe (a : Equiv α β) (b c) : (UniformEquiv.mk a b c : α → β) = a 
Arguments: {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (a : α ≃ β) (b : UniformContinuous a.toFun) (c : UniformContinuous a.invFun)
Goal: ⇑{ toEquiv := a, uniformContinuous_toFun := b, uniformContinuous_invFun := c } = ⇑a
-- UniformEquiv.coe_toEquiv
@[simp]
theorem coe_toEquiv (h : α ≃ᵤ β) : ⇑h.toEquiv = h 
Arguments: {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β)
Goal: ⇑h.toEquiv = ⇑h
-- UniformEquiv.coe_symm_toEquiv
@[simp]
theorem coe_symm_toEquiv (h : α ≃ᵤ β) : ⇑h.toEquiv.symm = h.symm 
Arguments: {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β)
Goal: ⇑h.symm = ⇑h.symm
-- UniformEquiv.ext
@[ext]
theorem ext {h h' : α ≃ᵤ β} (H : ∀ x, h x = h' x) : h = h' 
Arguments: {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] {h h' : α ≃ᵤ β} (H : ∀ (x : α), h x = h' x)
Goal: h = h'
-- UniformEquiv.refl_apply
@[simps! (config := .asFn) apply]
protected def refl (α : Type*) [UniformSpace α] : α ≃ᵤ α 
Arguments: (α : Type u_4) [UniformSpace α]
Goal: ⇑(UniformEquiv.refl α) = id
-- UniformEquiv.trans_apply
@[simp]
theorem trans_apply (h₁ : α ≃ᵤ β) (h₂ : β ≃ᵤ γ) (a : α) : h₁.trans h₂ a = h₂ (h₁ a) 
Arguments: {α : Type u} {β : Type u_1} {γ : Type u_2} [UniformSpace α] [UniformSpace β] [UniformSpace γ] (h₁ : α ≃ᵤ β) (h₂ : β ≃ᵤ γ) (a : α)
Goal: (h₁.trans h₂) a = h₂ (h₁ a)
-- UniformEquiv.uniformEquiv_mk_coe_symm
@[simp]
theorem uniformEquiv_mk_coe_symm (a : Equiv α β) (b c) :
    ((UniformEquiv.mk a b c).symm : β → α) = a.symm 
Arguments: {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (a : α ≃ β) (b : UniformContinuous a.toFun) (c : UniformContinuous a.invFun)
Goal: ⇑{ toEquiv := a, uniformContinuous_toFun := b, uniformContinuous_invFun := c }.symm = ⇑a.symm
-- UniformEquiv.refl_symm
@[simp]
theorem refl_symm : (UniformEquiv.refl α).symm = UniformEquiv.refl α 
Arguments: {α : Type u} [UniformSpace α]
Goal: (UniformEquiv.refl α).symm = UniformEquiv.refl α
-- UniformEquiv.uniformContinuous
protected theorem uniformContinuous (h : α ≃ᵤ β) : UniformContinuous h 
Arguments: {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β)
Goal: UniformContinuous ⇑h
-- UniformEquiv.continuous
@[continuity]
protected theorem continuous (h : α ≃ᵤ β) : Continuous h 
Arguments: {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β)
Goal: Continuous ⇑h
-- UniformEquiv.uniformContinuous_symm
protected theorem uniformContinuous_symm (h : α ≃ᵤ β) : UniformContinuous h.symm 
Arguments: {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β)
Goal: UniformContinuous ⇑h.symm
-- UniformEquiv.continuous_symm
@[continuity]
protected theorem continuous_symm (h : α ≃ᵤ β) : Continuous h.symm 
Arguments: {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β)
Goal: Continuous ⇑h.symm
-- UniformEquiv.toHomeomorph_apply
lemma toHomeomorph_apply (e : α ≃ᵤ β) : (e.toHomeomorph : α → β) = e 
Arguments: {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (e : α ≃ᵤ β)
Goal: ⇑e.toHomeomorph = ⇑e
-- UniformEquiv.toHomeomorph_symm_apply
lemma toHomeomorph_symm_apply (e : α ≃ᵤ β) : (e.toHomeomorph.symm : β → α) = e.symm 
Arguments: {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (e : α ≃ᵤ β)
Goal: ⇑e.toHomeomorph.symm = ⇑e.symm
-- UniformEquiv.apply_symm_apply
@[simp]
theorem apply_symm_apply (h : α ≃ᵤ β) (x : β) : h (h.symm x) = x 
Arguments: {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β) (x : β)
Goal: h (h.symm x) = x
-- UniformEquiv.symm_apply_apply
@[simp]
theorem symm_apply_apply (h : α ≃ᵤ β) (x : α) : h.symm (h x) = x 
Arguments: {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β) (x : α)
Goal: h.symm (h x) = x
-- UniformEquiv.bijective
protected theorem bijective (h : α ≃ᵤ β) : Function.Bijective h 
Arguments: {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β)
Goal: Function.Bijective ⇑h
-- UniformEquiv.injective
protected theorem injective (h : α ≃ᵤ β) : Function.Injective h 
Arguments: {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β)
Goal: Function.Injective ⇑h
-- UniformEquiv.surjective
protected theorem surjective (h : α ≃ᵤ β) : Function.Surjective h 
Arguments: {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β)
Goal: Function.Surjective ⇑h
-- UniformEquiv.symm_comp_self
@[simp]
theorem symm_comp_self (h : α ≃ᵤ β) : (h.symm : β → α) ∘ h = id 
Arguments: {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β)
Goal: ⇑h.symm ∘ ⇑h = id
-- UniformEquiv.self_comp_symm
@[simp]
theorem self_comp_symm (h : α ≃ᵤ β) : (h : α → β) ∘ h.symm = id 
Arguments: {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β)
Goal: ⇑h ∘ ⇑h.symm = id
-- UniformEquiv.range_coe
theorem range_coe (h : α ≃ᵤ β) : range h = univ 
Arguments: {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β)
Goal: Set.range ⇑h = Set.univ
-- UniformEquiv.image_symm
theorem image_symm (h : α ≃ᵤ β) : image h.symm = preimage h 
Arguments: {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β)
Goal: Set.image ⇑h.symm = Set.preimage ⇑h
-- UniformEquiv.preimage_symm
theorem preimage_symm (h : α ≃ᵤ β) : preimage h.symm = image h 
Arguments: {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β)
Goal: Set.preimage ⇑h.symm = Set.image ⇑h
-- UniformEquiv.image_preimage
theorem image_preimage (h : α ≃ᵤ β) (s : Set β) : h '' (h ⁻¹' s) = s 
Arguments: {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β) (s : Set β)
Goal: ⇑h '' (⇑h ⁻¹' s) = s
-- UniformEquiv.preimage_image
theorem preimage_image (h : α ≃ᵤ β) (s : Set α) : h ⁻¹' (h '' s) = s 
Arguments: {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β) (s : Set α)
Goal: ⇑h ⁻¹' (⇑h '' s) = s
-- UniformEquiv.uniformInducing
protected theorem uniformInducing (h : α ≃ᵤ β) : UniformInducing h 
Arguments: {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β)
Goal: UniformInducing ⇑h
-- UniformEquiv.comap_eq
theorem comap_eq (h : α ≃ᵤ β) : UniformSpace.comap h ‹_› = ‹_› 
Arguments: {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β)
Goal: UniformSpace.comap (⇑h) inst✝ = inst✝¹